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diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..d7b82bc --- /dev/null +++ b/.gitattributes @@ -0,0 +1,4 @@ +*.txt text eol=lf +*.htm text eol=lf +*.html text eol=lf +*.md text eol=lf diff --git a/LICENSE.txt b/LICENSE.txt new file mode 100644 index 0000000..6312041 --- /dev/null +++ b/LICENSE.txt @@ -0,0 +1,11 @@ +This eBook, including all associated images, markup, improvements, +metadata, and any other content or labor, has been confirmed to be +in the PUBLIC DOMAIN IN THE UNITED STATES. + +Procedures for determining public domain status are described in +the "Copyright How-To" at https://www.gutenberg.org. + +No investigation has been made concerning possible copyrights in +jurisdictions other than the United States. Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..16ec76f --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #66078 (https://www.gutenberg.org/ebooks/66078) diff --git a/old/66078-0.txt b/old/66078-0.txt deleted file mode 100644 index fc52592..0000000 --- a/old/66078-0.txt +++ /dev/null @@ -1,13322 +0,0 @@ -The Project Gutenberg eBook of A Treatise on Mechanics, by Henry Kater - -This eBook is for the use of anyone anywhere in the United States and -most other parts of the world at no cost and with almost no restrictions -whatsoever. You may copy it, give it away or re-use it under the terms -of the Project Gutenberg License included with this eBook or online at -www.gutenberg.org. If you are not located in the United States, you -will have to check the laws of the country where you are located before -using this eBook. - -Title: A Treatise on Mechanics - -Author: Henry Kater - Dionysius Lardner - -Release Date: August 17, 2021 [eBook #66078] - -Language: English - -Character set encoding: UTF-8 - -Produced by: Thiers Halliwell, deaurider and the Online Distributed - Proofreading Team at https://www.pgdp.net (This file was - produced from images generously made available by The Internet - Archive) - -*** START OF THE PROJECT GUTENBERG EBOOK A TREATISE ON MECHANICS *** -Transcriber’s notes: - -The text of this e-book has mostly been preserved in its original form -including some inconsistency of hyphenation and use of diacritics -(aeriform/aëriform). Three spelling typos have been corrected -(arrangment → arrangement, pully → pulley, dye → die) as have typos -in equations on pages 40 and 43. Some missing punctuation has been -corrected silently (periods, commas, incorrect quotes). Footnotes have -been positioned below the relevant paragraphs. - -In this plain text version, italic text is denoted by _underscores_ -and bold text by *asterisks*. A caret (^) indicates that the following -character is superscripted, e.g. D^2 (D squared). - - - - A - - TREATISE ON MECHANICS, - - BY - - CAPTAIN HENRY KATER, V. PRES: R.S. - - ----and---- - - DIONYSIUS LARDNER, D.C.L. F.R.S. &c. &c. - - A NEW EDITION REVISED & CORRECTED. - 1852. - - [Illustration: _H. Corbould del._ _E. Finder fc._] - - London: - PRINTED FOR LONGMAN, BROWN, GREEN & LONGMANS. PATERNOSTER ROW: - - - - -ADVERTISEMENT. - - -This Treatise on Mechanics, which was originally published in 1830, -is the work of Dr. Lardner, with the exception of the twenty-first -chapter, which was written by the late Captain Kater. The present -edition has been revised and corrected by Dr. Lardner. - - _London, January, 1852._ - - - - -CONTENTS. - - - CHAP. I. - - PROPERTIES OF MATTER. - - Organs of Sense.--Sensations.--Properties or Qualities.--Observation. - --Comparison and Generalisation.--Particular and general Qualities.-- - Magnitude.--Size.--Volume.--Lines.--Surfaces.--Edges.--Area.--Length. - --Impenetrability.--Apparent Penetration.--Figure.--Different from - Volume.--Atoms.--Molecules.--Matter separable.--Particles.--Force.-- - Cohesion of Atoms.--Hypothetical Phrases unnecessary.--Attraction. 1 - - - CHAP. II. - - PROPERTIES OF MATTER, CONTINUED. - - Divisibility.--Unlimited Divisibility.--Wollaston’s micrometric - Wire. --Method of making it.--Thickness of a Soap Bubble.--Wings of - Insects. --Gilding of Wire for Embroidery.--Globules of the Blood.-- - Animalcules.--Their minute Organisation.--Ultimate Atoms.--Crystals.-- - Porosity.--Volume.--Density.--Quicksilver passing through Pores of - Wood.--Filtration.--Porosity of Hydrophane.--Compressibility.-- - Elasticity.--Dilatability.--Heat.--Contraction of Metal used to - restore the Perpendicular to Walls of a Building.--Impenetrability of - Air.--Compressibility of it.--Elasticity of it.--Liquids not absolutely - incompressible.--Experiments.--Elasticity of Fluids.-- Aeriform - Fluids.--Domestic Fire Box.--Evolution of Heat by compressed Air. 9 - - - CHAP. III. - - INERTIA. - - Inertia.--Matter Incapable of spontaneous Change.--Impediments to - Motion.--Motion of the Solar System.--Law of Nature.--Language used - to express Inertia sometimes faulty.--Familiar Examples of Inertia. 27 - - - CHAP. IV. - - ACTION AND REACTION. - - Inertia in a single Body.--Consequences of Inertia in two or more - Bodies.--Examples.--Effects of Impact.--Motion not estimated by - Speed or Velocity alone.--Examples.--Rule for estimating the - Quantity of Motion.--Action and Reaction.--Examples of.--Velocity - of two Bodies after Impact.--Rule for finding the common Velocity - after Impact.--Magnet and Iron.--Feather and Cannon Ball impinging. - --Newton’s Laws of Motion.--Inutility of.--Familiar Effects - resulting from Consequences of Inertia. 34 - - - CHAP. V. - - COMPOSITION AND RESOLUTION OF FORCE. - - Motion and Pressure.--Force.--Attraction.--Parallelogram of Forces. - --Resultant.--Components.--Composition of Force.--Resolution of - Force.--Illustrative Experiments.--Composition of Pressures.-- - Theorems regulating Pressures also regulate Motion.--Examples.-- - Resolution of Motion.--Forces in Equilibrium.--Composition of Motion - and Pressure.--Illustrations.--Boat in a Current.--Motions of Fishes. - --Flight of Birds.--Sails of a Vessel.--Tacking.--Equestrian Feats. - --Absolute and relative Motion. 48 - - - CHAP. VI. - - ATTRACTION. - - Impulse.--Mechanical State of Bodies.--Absolute Rest.--Uniform and - rectilinear Motion.--Attractions.--Molecular or atomic.--Interstitial - Spaces in Bodies.--Repulsion and Attraction.--Cohesion.--In Solids - and Fluids.--Manufacture of Shot.--Capillary Attractions.--Shortening - of Rope by Moisture.--Suspension of Liquids in capillary Tubes.-- - Capillary Siphon.--Affinity between Quicksilver and Gold.--Examples - of Affinity.--Sulphuric Acid and Water.--Oxygen and Hydrogen.--Oxygen - and Quicksilver.--Magnetism.--Electricity and Electro-Magnetism.-- - Gravitation.--Its Law.--Examples of.--Depends on the Mass.-- - Attraction between the Earth and detached Bodies on its Surface.-- - Weight.--Gravitation of the Earth.--Illustrated by Projectiles.-- - Plumb-Line.--Cavendish’s Experiments. 63 - - - CHAP. VII. - - TERRESTRIAL GRAVITY. - - Phenomena of falling Bodies.--Gravity greater at the Poles than - Equator.--Heavy and light Bodies fall with equal Speed to the Earth. - --Experiment.--Increased Velocity of falling Bodies.--Principles of - uniformly accelerated Motion.--Relations between the Height, Time, - and Velocity.--Attwood’s Machine.--Retarded Motion. 84 - - - CHAP. VIII. - - OF THE MOTION OF BODIES ON INCLINED PLANES AND CURVES. - - Force perpendicular to a Plane.--Oblique Force.--Inclined Plane.-- - Weight produces Pressure and Motion.--Motion uniformly accelerated.-- - Space moved through in a given Time.--Increased Elevation produces - increased Force.--Perpendicular and horizontal Plane.--Final - Velocity.--Motion down a Curve.--Depends upon Velocity and Curvature. - --Centrifugal Force.--Circle of Curvature.--Radius of Curvature.-- - Whirling Table.--Experiments.--Solar System.--Examples of centrifugal - Force. 85 - - - CHAP. IX. - - THE CENTRE OF GRAVITY. - - Terrestrial Attraction the combined Action of parallel Forces.-- - Single equivalent Force.--Examples.--Method of finding the Centre of - Gravity.--Line of Direction.--Globe.--Oblate Spheroid.--Prolate - Spheroid.--Cube.--Straight Wand.--Flat Plate.--Triangular Plate.-- - Centre of Gravity not always within the Body.--A Ring.--Experiments. - --Stable, instable, and neutral Equilibrium.--Motion and Position of - the Arms and Feet.--Effect of the Knee-Joint.--Positions of a Dancer. - --Porter under a Load.--Motion of a Quadruped.--Rope Dancing.-- - Centre of Gravity of two Bodies separated from each other.-- - Mathematical and experimental Examples.--The Conservation of the - Motion of the Centre of Gravity.--Solar System.--Centre of Gravity - sometimes called Centre of Inertia. 107 - - - CHAP. X. - - THE MECHANICAL PROPERTIES OF AN AXIS. - - An Axis.--Planets and common spinning Top.--Oscillation or Vibration. - --Instantaneous and continued Forces.--Percussion.--Continued Force. - --Rotation.--Impressed Forces.--Properties of a fixed Axis.--Movement - of the Force round the Axis.--Leverage of the Force.--Impulse - perpendicular to, but not crossing, the Axis.--Radius of Gyration.-- - Centre of Gyration.--Moment of Inertia.--Principal Axes.--Centre of - Percussion. 128 - - - CHAP. XI. - - OF THE PENDULUM. - - Isochronism.--Experiments.--Simple Pendulum.--Examples illustrative - of.--Length of.--Experiments of Kater, Biot, Sabine, and others.-- - Huygens’ Cycloidal Pendulum. 145 - - - CHAP. XII. - - OF SIMPLE MACHINES. - - Statics.--Dynamics.--Force.--Power.--Weight.--Lever.--Cord.-- - Inclined Plane. 160 - - - CHAP. XIII. - - OF THE LEVER. - - Arms.--Fulcrum.--Three Kinds of Levers.--Crow Bar.--Handspike.--Oar. - --Nutcrackers.--Turning Lathe.--Steelyard.--Rectangular Lever.-- - Hammer.--Load between two Bearers.--Combination of Levers.-- - Equivalent Lever. 167 - - - CHAP. XIV. - - OF WHEEL-WORK. - - Wheel and Axle.--Thickness of the Rope.--Ways of applying the Power. - --Projecting Pins.--Windlass.--Winch.--Axle.--Horizontal Wheel.-- - Tread-Mill.--Cranes.--Water-Wheels.--Paddle-Wheel.--Rachet-Wheel.-- - Rack.--Spring of a Watch.--Fusee.--Straps or Cords.--Examples of.-- - Turning Lathe.--Revolving Shafts.--Spinning Machinery.--Saw-Mill.-- - Pinion.--Leaves.--Crane.--Spur-Wheels.--Crown-Wheels.--Bevelled - Wheels.--Hunting-Cog.--Chronometers.--Hair-Spring.--Balance-Wheel. 178 - - - CHAP. XV. - - OF THE PULLEY. - - Cord.--Sheave.--Fixed Pulley.--Fire Escapes.--Single moveable - Pulley.--Systems of Pulleys.--Smeaton’s Tackle.--White’s Pulley.-- - Advantage of.--Runner.--Spanish Bartons. 199 - - - CHAP. XVI. - - ON THE INCLINED PLANE, WEDGE, AND SCREW. - - Inclined Plane.--Effect of a Weight on.--Power of.--Roads.--Power - Oblique to the Plane.--Plane sometimes moves under the Weight.-- - Wedge.--Sometimes formed of two inclined Planes.--More powerful - as its Angle is acute.--Where used.--Limits to the Angle.--Screw. - --Hunter’s Screw.--Examples.--Micrometer Screw. 209 - - - CHAP. XVII. - - ON THE REGULATION AND ACCUMULATION OF FORCE. - - Uniformity of Operation.--Irregularity of prime Mover.--Water-Mill. - --Wind-Mill.--Steam Pressure.--Animal Power.--Spring.--Regulators. - --Steam-Engine.--Governor.--Self-acting Damper.--Tachometer.-- - Accumulation of Power.--Examples.--Hammer.--Flail.--Bow-string.-- - Fire Arms.--Air-Gun.--Steam-Gun.--Inert Matter a Magazine for - Force.--Fly-Wheel.--Condensed Air.--Rolling Metal.--Coining-Press. 224 - - - CHAP. XVIII. - - MECHANICAL CONTRIVANCES FOR MODIFYING MOTION. - - Division of Motion into rectilinear and rotatory.--Continued and - reciprocating.--Examples.--Flowing Water.--Wind.--Animal Motion.-- - Falling of a Body.--Syringe-Pump.--Hammer.--Steam-Engine.--Fulling - Mill.--Rose-Engine.--Apparatus of Zureda.--Leupold’s Application - of it.--Hooke’s universal Joint.--Circular and alternate Motion.-- - Examples.--Watt’s Methods of connecting the Motion of the Piston - with that of the Beam.--Parallel Motion. 245 - - - CHAP. XIX. - - OF FRICTION AND THE RIGIDITY OF CORDAGE. - - Friction and Rigidity.--Laws of Friction.--Rigidity of Cordage.-- - Strength of Materials.--Resistance from Friction.--Independent of - the Magnitude of Surfaces.--Examples.--Vince’s Experiments.--Effect - of Velocity.--Means for diminishing Friction.--Friction Wheels.-- - Angle of Repose.--Best Angle of Draught.--Rail-Roads.--Stiffness - of Ropes. 260 - - - CHAP. XX. - - ON THE STRENGTH OF MATERIALS. - - Difficulty of determining the Laws which govern the Strength of - Materials.--Forces tending to separate the Parts of a Solid.--Laws - by which Solids resist Compression.--Euler’s theory.--Transverse - Strength of Solids.--Strength diminished by the Increase of Height. - --Lateral or Transverse Strain.--Limits of Magnitude.--Relative - Strength of small Animals greater than large ones. 272 - - - CHAP. XXI. - - ON BALANCES AND PENDULUMS. - - Weight.--Time.--The Balance.--Fulcrum.--Centre of Gravity of.-- - Sensibility of.--Positions of the Fulcrum.--Beam variously - constructed.--Troughton’s Balance.--Robinson’s Balance.--Kater’s - Balance.--Method of adjusting a Balance.--Use of it.--Precautions - necessary.--Of Weights.--Adjustment of.--Dr. Black’s Balance.-- - Steelyard.--Roman Statera or Steelyard.--Convenience of.--C. Paul’s - Steelyard.--Chinese Steelyard.--Danish Balance.--Bent Lever Balance. - --Brady’s Balance.--Weighing Machine for Turnpike Roads.--Instruments - for Weighing by means of a Spring.--Spring Steelyard.--Salter’s - Spring Balance.--Marriott’s Dial Weighing Machine.--Dynamometer.-- - Compensation Pendulums.--Barton’s Gridiron Pendulum.--Table of linear - Expansion.--Second Table.--Harrison’s Pendulum.--Troughton’s - Pendulum.--Benzenberg’s Pendulum.--Ward’s Compensation Pendulum.-- - Compensation Tube of Julien le Roy.--Deparcieux’s Compensation.-- - Kater’s Pendulum.--Reed’s Pendulum.--Ellicott’s Pendulum.--Mercurial - Pendulum.--Graham’s Pendulum.--Compensation Pendulum of Wood and - Lead.--Smeaton’s Pendulum.--Brown’s Mode of Adjustment. 278 - - - - -THE - -ELEMENTS OF MECHANICS. - - - - -CHAP. I. - -PROPERTIES OF MATTER--MAGNITUDE--IMPENETRABILITY--FIGURE--FORCE. - - -(1.) Placed in the material world, Man is continually exposed to the -action of an infinite variety of objects by which he is surrounded. The -body, to which the thinking and living principles have been united, -is an apparatus exquisitely contrived to receive and to transmit -impressions. Its various parts are organised with obvious reference to -the several external agents by which it is to be effected. Each organ -is designed to convey to the mind immediate notice of some peculiar -action, and is accordingly endued with a corresponding susceptibility. -This adaptation of such organs to the particular influences of -material agents, is rendered still more conspicuous when we consider -that, however delicate its structure, each organ is wholly insensible -to every influence except that to which it appears to be specially -appropriated. The eye, so intensely susceptible of impressions from -light, is not at all affected by those of sound; while the fine -mechanism of the ear, so sensitively alive to every effect of the -latter class, is altogether insensible to the former. The splendour of -excessive light may occasion blindness, and deafness may result from -the roar of a cannonade; but neither the sight nor the hearing can be -injured by the most extreme action of that principle which is designed -to affect the other. - -Thus the organs of sense are instruments by which the mind is enabled -to determine the existence and the qualities of external things. The -effects which these objects produce upon the mind through the organs, -are called _sensations_, and these sensations are the immediate -elements of all human knowledge. MATTER is the general name which -has been given to that substance, which, under forms infinitely -various, affects the senses. Metaphysicians have differed in defining -this principle. Some have even doubted of its existence. But these -discussions are beyond the sphere of mechanical philosophy, the -conclusions of which are in nowise affected by them. Our investigations -here relate, not to matter as an abstract existence, but to those -qualities which we discover in it by the senses, and of the existence -of which we are sure, however the question as to matter itself may be -decided. When we speak of “bodies,” we mean those things, whatever they -be, which excite in our minds certain sensations; and the powers to -excite those sensations are called “properties,” or “qualities.” - -(2.) To ascertain by observation the properties of bodies, is the -first step towards obtaining a knowledge of nature. Hence man becomes -a natural philosopher the moment he begins to feel and to perceive. -The first stage of life is a state of constant and curious excitement. -Observation and attention, ever awake, are engaged upon a succession -of objects new and wonderful. The large repository of the memory is -opened, and every hour pours into it unbounded stores of natural facts -and appearances, the rich materials of future knowledge. The keen -appetite for discovery implanted in the mind for the highest ends, -continually stimulated by the presence of what is novel, renders torpid -every other faculty, and the powers of reflection and comparison are -lost in the incessant activity and unexhausted vigour of observation. -After a season, however, the more ordinary classes of phenomena cease -to excite by their novelty. Attention is drawn from the discovery of -what is new, to the examination of what is familiar. From the external -world the mind turns in upon itself, and the feverish astonishment -of childhood gives place to the more calm contemplation of incipient -maturity. The vast and heterogeneous mass of phenomena collected by -past experience is brought under review. The great work of comparison -begins. Memory produces her stores, and reason arranges them. Then -succeed those first attempts at generalisation which mark the dawn of -science in the mind. - -To compare, to classify, to generalise, seem to be instinctive -propensities peculiar to man. They separate him from inferior animals -by a wide chasm. It is to these powers that all the higher mental -attributes may be traced, and it is from their right application that -all progress in science must arise. Without these powers, the phenomena -of nature would continue a confused heap of crude facts, with which the -memory might be loaded, but from which the intellect would derive no -advantage. Comparison and generalisation are the great digestive organs -of the mind, by which only nutrition can be extracted from this mass of -intellectual food, and without which, observation the most extensive, -and attention the most unremitting, can be productive of no real or -useful advancement in knowledge. - -(3.) Upon reviewing those properties of bodies which the senses -most frequently present to us, we observe that very few of them are -essential to, and inseparable from, matter. The greater number may be -called _particular_ or _peculiar qualities_, being found in some bodies -but not in others. Thus the property of attracting iron is peculiar to -the loadstone, and not observable in other substances. One body excites -the sensation of green, another of red, and a third is deprived of all -colour. A few characteristic and essential qualities are, however, -inseparable from matter in whatever state, or under whatever form it -exist. Such properties alone can be considered as tests of materiality. -Where their presence is neither manifest to sense, nor demonstrable by -reason, _there_ matter is not. The principal of these qualities are -_magnitude_ and _impenetrability_. - -(4.) _Magnitude._--Every body occupies space, that is, it has -magnitude. This is a property observable by the senses in all bodies -which are not so minute as to elude them, and which the understanding -can trace to the smallest particle of matter. It is impossible, by any -stretch of imagination, even to conceive a portion of matter so minute -as to have no magnitude. - -The _quantity_ of space which a body occupies is sometimes called its -_magnitude_. In colloquial phraseology, the word _size_ is used to -express this notion; but the most correct term, and that which we shall -generally adopt is _volume_. Thus we say, the volume of the earth is so -many cubic miles, the volume of this room is so many cubic feet. - -The external limits of the magnitude of a body are _lines_ and -_surfaces_, lines being the limits which separate the several surfaces -of the same body. The linear limits of a body are also called _edges_. -Thus the line which separates the top of a chest from one of its sides -is called an edge. - -The _quantity_ of a surface is called its _area_, and the _quantity_ -of a line is called its _length_. Thus we say, the _area_ of a field -is so many acres, the _length_ of a rope is so many yards. The word -“magnitude” is, however, often used indifferently for volume, area, -and length. If the objects of investigation were of a more complex -and subtle character, as in metaphysics, this unsteady application of -terms might be productive of confusion, and even of error; but in this -science the meaning of the term is evident, from the way in which it is -applied, and no inconvenience is found to arise. - -(5.) _Impenetrability._--This property will be most clearly explained -by defining the positive quality from which it takes its name, and -of which it merely signifies the absence. A substance would be -_penetrable_ if it were such as to allow another to pass through the -space which it occupies, without disturbing its component parts. Thus, -if a comet striking the earth could enter it at one side, and, passing -through it, emerge from the other without separating or deranging any -bodies on or within the earth, then the earth would be penetrable by -the comet. When bodies are said to be impenetrable, it is therefore -meant that one cannot pass through another without displacing some or -all of the component parts of that other. There are many instances of -apparent penetration; but in all these, the parts of the body which -seem to be penetrated are displaced. Thus, if the point of a needle be -plunged in a vessel of water, all the water which previously filled the -space into which the needle enters will be displaced, and the level of -the water will rise in the vessel to the same height as it would by -pouring in so much more water as would fill the space occupied by the -needle. - -(6.) _Figure._--If the hand be placed upon a solid body, we become -sensible of its impenetrability, by the obstruction which it opposes to -the entrance of the hand within its dimensions. We are also sensible -that this obstruction commences at certain places; that it has certain -determinate limits; that these limitations are placed in certain -directions relatively to each other. The mutual relation which is found -to subsist between these boundaries of a body, gives us the notion of -its _figure_. The _figure_ and _volume_ of a body should be carefully -distinguished. Each is entirely independent of the other. Bodies having -very different _volumes_ may have the same _figure_; and in like manner -bodies differing in _figure_ may have the same _volume_. The figure of -a body is what in popular language is called its _shape_ or _form_. The -volume of a body is that which is commonly called its _size_. It will -hence be easily understood, that one body (a globe, for example) may -have ten times the volume of another (globe), and yet have the same -figure; and that two bodies (as a die and a globe) may have _figures_ -altogether different, and yet have equal _volumes_. What we have here -observed of volumes will also be applicable to lengths and areas. The -arc of a circle and a straight line may have the same length, although -they have different figures; and, on the other hand, two arcs of -different circles may have the same figure, but very unequal lengths. -The surface of a ball is curved, that of the table plane; and yet the -_area_ of the surface of the ball may be equal to that of the table. - -(7.) _Atoms--Molecules._--Impenetrability must not be confounded -with inseparability. Every body which has been brought under human -observation is separable into parts; and these parts, however small, -are separable into others, still more minute. To this process of -division no practical limit has ever been found. Nevertheless, many -of the phenomena which the researches of those who have successfully -examined the laws of nature have developed, render it highly probable -that all bodies are composed of elementary parts which are indivisible -and unalterable. The component parts, which may be called _atoms_, are -so minute, as altogether to elude the senses, even when aided by the -most powerful scientific instruments. The word _molecule_ is often used -to signify component parts of a body so small as to escape sensible -observation, but not ultimate atoms, each molecule being supposed to -be formed of several atoms, arranged according to some determinate -figure. _Particle_ is used also to express small component parts, but -more generally is applied to those which are not too minute to be -discoverable by observation. - -(8.) _Force._--If the particles of matter were endued with no property -in relation to one another, except their mutual impenetrability, the -universe would be like a mass of sand, without variety of state or -form. Atoms, when placed in juxtaposition, would neither cohere, as in -solid bodies, nor repel each other, as in aeriform substances. On the -contrary, we find that in some cases the atoms which compose bodies -are not simply placed together, but a certain effect is manifested in -their strong coherence. If they were merely placed in juxtaposition, -their separation would be effected as easily as any one of them could -be removed from one place to another. Take a piece of iron, and -attempt to separate its parts: the effort will be strongly resisted, -and it will be a matter of much greater facility to move the whole -mass. It appears, therefore, that in such cases the parts which are in -juxtaposition _cohere_ and resist their mutual separation. This effect -is denominated _force_; and the constituent atoms are said to cohere -with a greater or less degree of force, according as they oppose a -greater or less resistance to their mutual separation. - -The coherence of particles in juxtaposition is an effect of the same -class as the mutual approach of particles placed at a distance from -each other. It is not difficult to perceive that the same influence -which causes the bodies A and B to approach each other, when placed at -some distance asunder, will, when they unite, retain them together, -and oppose a resistance to their separation. Hence this effect of -the mutual approximation of bodies towards each other is also called -_force_. - -Force is generally defined to be “whatever produces or opposes the -production of motion in matter.” In this sense, it is a name for -the unknown cause of a known effect. It would, however, be more -philosophical to give the name, not to the _cause_, of which we are -ignorant, but to the _effect_, of which we have sensible evidence. -To observe and to classify is the whole business of the natural -philosopher. When _causes_ are referred to, it is implied, that effects -of the same class arise from the agency of the same cause. However -probable this assumption may be, it is altogether unnecessary. All -the objects of science, the enlargement of mind, the extension and -improvement of knowledge, the facility of its acquisition, are obtained -by generalisation alone, and no good can arise from tainting our -conclusions with the possible errors of hypotheses. - -It may be here, once for all, observed, that the phraseology of -causation and hypotheses has become so interwoven with the language -of science, that it is impossible to avoid the frequent use of it. -Thus, we say, “the magnet _attracts_ iron;” the expression _attract_ -intimating the cause of the observed effect. In such cases, however, -we must be understood to mean the _effect itself_, finding it less -inconvenient to continue the use of the received phrases, modifying -their signification, than to introduce new ones. - -Force, when manifested by the mutual approach or cohesion of bodies, is -also called _attraction_, and it is variously denominated, according to -the circumstances under which it is observed to act. Thus, the force -which holds together the atoms of solid bodies is called _cohesive -attraction_. The force which draws bodies to the surface of the earth, -when placed above it, is called the _attraction of gravitation_. The -force which is exhibited by the mutual approach, or adhesion, of the -loadstone and iron, is called _magnetic attraction_, and so on. - -When force is manifested by the motion of bodies from each other, it -is called _repulsion_. Thus, if a piece of glass, having been briskly -rubbed with a silk handkerchief, touch successively two feathers, these -feathers, if brought near each other, will move asunder. This effect is -called _repulsion_, and the feathers are said to _repel_ each other. - -(9.) The influence which forces have upon the form, state, arrangement, -and motions of material substances is the principal object of physical -science. In its strict sense, MECHANICS is a term of very extensive -signification. According to the more popular usage, however, it has -been generally applied to that part of physical science which includes -the investigation of the phenomena of motion and rest, pressure and -other effects developed by the mutual action of solid masses. The -consideration of similar phenomena, exhibited in bodies of the liquid -form, is consigned to HYDROSTATICS, and that of aeriform fluids to -PNEUMATICS. - - - - -CHAP. II. - -DIVISIBILITY--POROSITY--DENSITY--COMPRESSIBILITY--ELASTICITY--DILATABILITY. - - -(10.) Besides the qualities of magnitude and impenetrability, there are -several other general properties of bodies contemplated in mechanical -philosophy, and to which we shall have frequent occasion to refer. -Those which we shall notice in the present chapter are, - - 1. Divisibility. - 2. Porosity--Density. - 3. Compressibility--Elasticity. - 4. Dilatability. - -(11.) _Divisibility._--Observation and experience prove that all bodies -of sensible magnitude, even the most solid, consist of parts which are -separable. To the practical subdivision of matter there seems to be no -assignable limit. Numerous examples of the division of matter, to a -degree almost exceeding belief, may be found in experimental enquiries -instituted in physical science; the useful arts furnish many instances -not less striking; but, perhaps, the most conspicuous proofs which can -be produced, of the extreme minuteness of which the parts of matter -are susceptible, arise from the consideration of certain parts of the -organised world. - -(12.) The relative places of stars in the heavens, as seen in the -field of view of a telescope, are marked by fine lines of wire placed -before the eye-glass, and which cross each other at right angles. The -stars appearing in the telescope as mere lucid points without sensible -magnitude, it is necessary that the wires which mark their places -should have a corresponding tenuity. But these wires being magnified -by the eye-glass would have an apparent thickness, which would render -them inapplicable to this purpose, unless their real dimensions were of -a most uncommon degree of minuteness. To obtain wire for this purpose, -Dr. Wollaston invented the following process:--A piece of fine -platinum wire, _a b_, is extended along the axis of a cylindrical -mould, A B, _fig. 1._ Into this mould, at A, molten silver -is poured. Since the heat necessary for the fusion of platinum is much -greater than that which retains silver in the liquid form, the wire -_a b_ remains solid, while the mould A B is filled with -the silver. When the metal has become solid by being cooled, and has -been removed from the mould, a cylindrical bar of silver is obtained, -having a platinum wire in its axis. This bar is then wire-drawn, by -forcing it successively through holes C, D, E, F, G, H, diminishing -in magnitude, the first hole being a little less than the wire at -the beginning of the process. By these means the platinum _a b_ -is wire-drawn at the same time and in the same proportion with the -silver, so that whatever be the original proportion of the thickness -of the wire _a b_ to that of the mould A B, the same will -be the proportion of the platinum wire to the whole at the several -thicknesses C, D, &c. If we suppose the mould A B to be ten times -the thickness of the wire _a b_, then the silver wire, throughout -the whole process, will be ten times the thickness of the platinum -wire which it includes within it. The silver wire may be drawn to a -thickness not exceeding the 300th of an inch. The platinum will thus -not exceed the 3000th of an inch. The wire is then dipped in nitric -acid, which dissolves the silver, but leaves the platinum solid. By -this method Dr. Wollaston succeeded in obtaining wire, the diameter of -which did not exceed the 18000th of an inch. A quantity of this wire, -equal in bulk to a common die used in games of chance, would extend -from Paris to Rome. - -(13.) Newton succeeded in determining the thickness of very thin laminæ -of transparent substances by observing the colours which they reflect. -A soap bubble is a thin shell of water, and is observed to reflect -different colours from different parts of its surface. Immediately -before the bubble bursts, a black spot may be observed near the top. At -this part the thickness has been proved not to exceed the 2,500,000th -of an inch. - -The transparent wings of certain insects are so attenuated in their -structure that 50,000 of them placed over each other would not form a -pile a quarter of an inch in height. - -(14.) In the manufacture of embroidery it is necessary to obtain very -fine gilt silver threads. To accomplish this, a cylindrical bar of -silver, weighing 360 ounces, is covered with about two ounces of gold. -This gilt bar is then wire-drawn, as in the first example, until it -is reduced to a thread so fine that 3400 feet of it weigh less than -an ounce. The wire is then flattened by passing it between rollers -under a severe pressure, a process which increases its length, so that -about 4000 feet shall weigh one ounce. Hence, one foot will weigh the -4000th part of an ounce. The proportion of the gold to the silver in -the original bar was that of 2 to 360, or 1 to 180. Since the same -proportion is preserved after the bar has been wire-drawn, it follows -that the quantity of gold which covers one foot of the fine wire is the -180th part of the 4000th of an ounce; that is the 720,000th part of an -ounce. - -The quantity of gold which covers one inch of this wire will be twelve -times less than that which covers one foot. Hence, this quantity will -be the 8,640,000th part of an ounce. If this inch be again divided -into 100 equal parts, every part will be distinctly visible without -the aid of microscopes. The gold which covers this small but visible -portion is the 864,000,000th part of an ounce. But we may proceed even -further; this portion of the wire may be viewed by a microscope which -magnifies 500 times, so that the 500th part of it will thus become -visible. In this manner, therefore, an ounce of gold may be divided -into 432,000,000,000 visible parts, each of which will possess all the -characters and qualities found in the largest masses of the metal. -It will retain its solidity, texture, and colour; it will resist the -same agents, and enter into combination with the same substances. If -the gilt wire be dipped in nitric acid, the silver within the coating -will be dissolved, but the hollow tube of gold which surrounded it will -still cohere and remain suspended. - -(15.) The organised world offers still more remarkable examples of the -inconceivable subtilty of matter. - -The blood which flows in the veins of animals is not, as it seems, -an uniformly red liquid. It consists of flat discs of a red colour, -floating in a transparent fluid called _serum_. In different species -these discs differ both in figure and in magnitude. In man and all -animals which suckle their young, they are perfectly circular or nearly -so. In birds, reptiles, and fishes, they are of oval form. In the human -species, the diameter of these discs is about the 3500th of an inch. -Hence it follows, that in a drop of blood which would remain suspended -from the point of a fine needle, there must be about 3,000,000 of such -discs. - -Small as these discs are, the animal kingdom presents beings whose -whole bodies are still more minute. Animalcules have been discovered, -whose magnitude is such, that a million of them do not exceed the -bulk of a grain of sand; and yet each of these creatures is composed -of members as curiously organised as those of the largest species; -they have life and spontaneous motion, and are endued with sense and -instinct. In the liquids in which they live, they are observed to -move with astonishing speed and activity; nor are their motions blind -and fortuitous, but evidently governed by choice, and directed to an -end. They use food and drink, from which they derive nutrition, and -are therefore furnished with a digestive apparatus. They have great -muscular power, and are furnished with limbs and muscles of strength -and flexibility. They are susceptible of the same appetites, and -obnoxious to the same passions, the gratification of which is attended -with the same results as in our own species. Spallanzani observes, that -certain animalcules devour others so voraciously, that they fatten and -become indolent and sluggish by over-feeding. After a meal of this -kind, if they be confined in distilled water, so as to be deprived of -all food, their condition becomes reduced; they regain their spirit -and activity, and amuse themselves in the pursuit of the more minute -animals, which are supplied to them; they swallow these without -depriving them of life, for, by the aid of the microscope, the one -has been observed moving within the body of the other. These singular -appearances are not matters of idle and curious observation. They lead -us to enquire what parts are necessary to produce such results. Must we -not conclude that these creatures have heart, arteries, veins, muscles, -sinews, tendons, nerves, circulating fluids, and all the concomitant -apparatus of a living organised body? And if so, how inconceivably -minute must those parts be! If a globule of their blood bears the same -proportion to their whole bulk as a globule of our blood bears to our -magnitude, what powers of calculation can give an adequate notion of -its minuteness? - -(16.) These and many other phenomena observed in the immediate -productions of nature, or developed by mechanical and chemical -processes, prove that the materials of which bodies are formed are -susceptible of minuteness which infinitely exceeds the powers of -sensible observation, even when those powers have been extended by all -the aids of science. Shall we then conclude that matter is infinitely -divisible, and that there are no original constituent atoms of -determinate magnitude and figure at which all subdivision must cease? -Such an inference would be unwarranted, even had we no other means of -judging the question, except those of direct observation; for it would -be imposing that limit on the works of nature which she has placed -upon our powers of observing them. Aided by reason, however, and a due -consideration of certain phenomena which come within our immediate -powers of observation, we are frequently able to determine other -phenomena which are beyond those powers. The diurnal motion of the -earth is not perceived by us, because all things around us participate -in it, preserve their relative position, and appear to be at rest. But -reason tells us that such a motion must produce the alternations of -day and night, and the rising and setting of all the heavenly bodies; -appearances which are plainly observable, and which betray the cause -from which they arise. Again, we cannot place ourselves at a distance -from the earth, and behold the axis on which it revolves, and observe -its peculiar obliquity to the orbit in which the earth moves; but we -see and feel the vicissitudes of the seasons, an effect which is the -immediate consequence of that inclination, and by which we are able to -detect it. - -(17.) So it is in the present case. Although we are unable by direct -observation to prove the existence of constituent material atoms of -determinate figure, yet there are many observable phenomena which -render their existence in the highest degree probable, if not morally -certain. The most remarkable of this class of effects is observed in -the crystallisation of salts. When salt is dissolved in a sufficient -quantity of pure water, it mixes with the water in such a manner as -wholly to disappear to the sight and touch, the mixture being one -uniform transparent liquid like the water itself, before its union -with the salt. The presence of the salt in the water may, however, be -ascertained by weighing the mixture, which will be found to exceed -the original weight of the water by the exact amount of the weight -of the salt. It is a well-known fact, that a certain degree of heat -will convert water into vapour, and that the same degree of heat does -not produce the same effect upon salt. The mixture of salt and water -being exposed to this temperature, the water will gradually evaporate, -disengaging itself from the salt with which it has been combined. When -so much of the water has evaporated, that what remains is insufficient -to keep in solution the whole of the salt, a part of the latter thus -disengaged from the water will return to the solid state. The saline -constituent will not in this case collect in irregular solid molecules; -but will exhibit itself in particles of regular figure, terminated by -plane surfaces, the figure being always the same for the same species -of salt, but different for different species. These particles are -called _crystals_. There are several circumstances in the formation of -these _crystals_ which merit attention. - -If one of them be detached from the others, and the progress of its -formation observed, it will be found gradually to increase, always -preserving its original figure. Since its increase must be caused -by the continued accession of saline molecules, disengaged by the -evaporation of the water, it follows that these molecules must be so -formed, that by attaching themselves successively to the crystal, they -maintain the regularity of its bounding planes, and preserve their -mutual inclinations unvaried. - -Suppose a crystal to be taken from the liquid during the process of -crystallisation, and a piece broken from it so as to destroy the -regularity of its form: if the crystal thus broken be restored to the -liquid, it will be observed gradually to resume its regular form, the -atoms of salt successively dismissed by the vaporising water filling -up the irregular cavities produced by the fracture. Hence it follows, -that the saline particles which compose the surface of the crystal, -and those which form the interior of its mass, are similar, and exert -similar attractions on the atoms disengaged by the water. - -All these details of the process of crystallisation are very evident -indications of a determinate figure in the ultimate atoms of the -substances which are crystallised. But besides the substances which are -thus reduced by art to the form of crystals, there are larger classes -which naturally exist in that state. There are certain planes, called -_planes of cleavage_, in the directions of which natural crystals are -easily divided. These planes, in substances of the same kind, always -have the same relative position, but differ in different substances. -The surfaces of the planes of cleavage are quite invisible before the -crystal is divided; but when the parts are separated, these surfaces -exhibit a most intense polish, which no effort of art can equal. - -We may conceive crystallised substances to be regular mechanical -structures formed of atoms of a certain figure, on which the figure of -the whole structure must depend. The planes of cleavage are parallel to -the sides of the constituent atoms; and their directions, therefore, -form so many conditions for the determination of their figure. The -shape of the atoms being thus determined, it is not difficult to assign -all the various ways in which they may be arranged, so as to produce -figures which are accordingly found to correspond with the various -forms of crystals of the same substance. - -(18.) When these phenomena are duly considered and compared, little -doubt can remain that all substances susceptible of crystallisation, -consist of atoms of determinate figure. This is the case with all solid -bodies whatever, which have come under scientific observation, for -they have been severally found in or reduced to a crystallised form. -Liquids crystallise in freezing, and if aëriform fluids could by any -means be reduced to the solid form, they would probably also manifest -the same effect. Hence it appears reasonable to presume, that all -bodies are composed of atoms; that the different qualities with which -we find different substances endued, depend on the magnitude and figure -of these atoms; that these atoms are indestructible and immutable by -any natural process, for we find the qualities which depend on them -unchangeably the same under all the influences to which they have been -submitted since their creation; that these atoms are so minute in their -magnitude, that they cannot be observed by any means which human art -has yet contrived; but still that magnitudes can be assigned which they -do not exceed. - -It is proper, however, to observe here, that the various theorems -of mechanical science do not rest upon any hypothesis concerning -these atoms as a basis. These theorems are not inferred from this -or any other supposition, and therefore their truth would not be in -anywise disturbed, even though it should be established that matter is -physically divisible _in infinitum_. The basis of mechanical science -is _observed facts_, and, since the reasoning is demonstrative, the -conclusions have the same degree of certainty as the facts from which -they are deduced. - -(19.) _Porosity._--The _volume_ of a body is the quantity of space -included within its external surfaces. The _mass_ of a body, is the -collection of atoms or material particles of which it consists. Two -atoms or particles are said to be in contact, when they have approached -each other until arrested by their mutual impenetrability. If the -component particles of a body were in contact, the _volume_ would be -completely occupied by the _mass_. But this is not the case. We shall -presently prove, that the component particles of no known substance are -in absolute contact. Hence it follows that the volume consists partly -of material particles, and partly of interstitial spaces, which spaces -are either absolutely void and empty, or filled by some substance of a -different kind from the body in question. These interstitial spaces are -called _pores_. - -In bodies which are constituted uniformly throughout their entire -dimensions, the component particles and the pores are uniformly -distributed through the volume; that is, a given space in one part -of the volume will contain the same quantity of matter and the same -quantity of pores as an equal space in another part. - -(20.) The proportion of the quantity of matter to the magnitude is -called the _density_. Thus if of two substances, one contain in a -given space twice as much matter as the other, it is said to be “twice -as dense.” The density of bodies is, therefore, proportionate to the -closeness or proximity of their particles; and it is evident, that the -greater the density, the less will be the porosity. - -The pores of a body are frequently filled with another body of a more -subtle nature. If the pores of a body on the surface of the earth, and -exposed to the atmosphere, be greater than the atoms of air, then the -air may pervade the pores. This is found to be the case with many sorts -of wood which have an open grain. If a piece of such wood, or of chalk, -or of sugar, be pressed to the bottom of a vessel of water, the air -which fills the pores will be observed to escape in bubbles and to rise -to the surface, the water entering the pores, and taking its place. - -If a tall vessel or tube, having a wooden bottom, be filled with -quicksilver, the liquid metal will be forced by its own weight through -the pores of the wood, and will be seen escaping in a silver shower -from the bottom. - -(21.) The process of filtration, in the arts, depends on the presence -of pores of such a magnitude as to allow a passage to the liquid, but -to refuse it to those impurities from which it is to be disengaged. -Various substances are used as filtres; but, whatever be used, this -circumstance should always be remembered, that no substance can be -separated from a liquid by filtration, except one whose particles -are larger than those of the liquid. In general, filtres are used to -separate _solid_ impurities from a liquid. The most ordinary filtres -are soft stone, paper, and charcoal. - -(22.) All organised substances in the animal and vegetable kingdoms -are, from their very natures, porous in a high degree. Minerals are -porous in various degrees. Among the silicious stones is one called -_hydrophane_, which manifests its porosity in a very remarkable manner. -The stone, in its ordinary state, is semi-transparent. If, however, -it be plunged in water, when it is withdrawn it is as translucent -as glass. The pores, in this case, previously filled with air, are -pervaded by the water, between which and the stone there subsists -a physical relation, by which the one renders the other perfectly -transparent. - -Larger mineral masses exhibit degrees of porosity not less striking. -Water percolates through the sides and roofs of caverns and grottoes, -and being impregnated with calcareous and other earths, forms -stalactites, or pendant protuberances, which present a curious -appearance. - -(23.) _Compressibility._--That quality, in virtue of which a body -allows its volume to be diminished without diminishing its mass, is -called _compressibility_. This effect is produced by bringing the -constituent particles more close together, and thereby increasing the -density and diminishing the pores. This effect may be produced in -several ways; but the name “compressibility” is only applied to it -when it is caused by the agency of mechanical force, as by pressure or -percussion. - -All known bodies, whatever be their nature, are capable of having their -dimensions reduced without diminishing their mass; and this is one of -the most conclusive proofs that all bodies are porous, or that the -constituent atoms are not in contact; for the space by which the volume -may be diminished must, before the diminution, consist of pores. - -(24.) _Elasticity._--Some bodies, when compressed by mechanical agency, -will resume their former dimensions with a certain energy when relieved -from the operation of the force which has compressed them. This -property is called _elasticity_; and it follows, from this definition, -that all elastic bodies must be compressible, although the converse is -not true, compressibility not necessarily implying elasticity. - -(25.) _Dilatability._--This quality is the opposite of compressibility. -It is the capability observed in bodies to have their volume enlarged -without increasing their mass. This effect may be produced in several -ways. In ordinary circumstances, a body may exist under the constant -action of a pressure by which its volume and density are determined. It -may happen, that on the occasional removal of that pressure, the body -will _dilate_ by a quality inherent in its constitution. This is the -case with common air. Dilatation may also be the effect of heat, as -will presently appear. - -The several qualities of bodies which we have noticed in this chapter, -when viewed in relation to each other, present many circumstances -worthy of attention. - -(26.) It is a physical law, of high generality, that an increase -in the temperature, or degree of heat by which a body is affected, -is accompanied by an increase of volume; and that a diminution of -temperature is accompanied by a diminution of volume. The exceptions -to this law will be noticed and explained in our treatise on HEAT. -Hence it appears that the reduction of temperature is an effect which, -considered mechanically, is equivalent to compression or condensation, -since it diminishes the volume without altering the mass; and since -this is an effect of which all bodies whatever are susceptible, it -follows that all bodies whatever have _pores_. (23.) - -The fact, that the elevation of temperature produces an increase of -volume, is manifested by numerous experiments. - -(27.) If a flaccid bladder be tied at the mouth, so as to stop the -escape of air, and be then held before a fire, it will gradually swell, -and assume the appearance of being fully inflated. The small quantity -of air contained in the bladder is, in this case, so much dilated by -the heat, that it occupies a considerably increased space, and fills -the bladder, of which it before only occupied a small part. When the -bladder is removed from the fire, and allowed to resume its former -temperature, the air returns to its former dimensions, and the bladder -becomes again flaccid. - -(28.) Let A B, _fig. 2._ be a glass tube, with a bulb at the -end A; and let the bulb A, and a part of the tube, be filled with any -liquid, coloured so as to be visible. Let C be the level of the liquid -in the tube. If the bulb be now exposed to heat, by being plunged in -hot water, the level of the liquid C will rapidly rise towards B. This -effect is produced by the dilatation of the liquid in the bulb, which -filling a greater space, a part of it is forced into the tube. This -experiment may easily be made with a common glass tube and a little -port wine. - -Thermometers are constructed on this principle, the rise of the liquid -in the tube being used as an indication of the degree of heat which -causes it. A particular account of these useful instruments will be -found in our treatise on HEAT. - -(29.) The change of dimension of solids produced by changes of -temperature being much less than that of bodies in the liquid or -aeriform state, is not so easily observable. A remarkable instance -occurs in the process of shoeing the wheels of carriages. The rim of -iron with which the wheel is to be bound, is made in the first instance -of a diameter somewhat less than that of the wheel; but being raised by -the application of fire to a very high temperature, its volume receives -such an increase, that it will be sufficient to embrace and surround -the wheel. When placed upon the wheel it is cooled, and suddenly -contracting its dimensions, binds the parts of the wheel firmly -together, and becomes securely seated in its place upon the fellies. - -(30.) It frequently happens that the stopper of a glass bottle or -decanter becomes fixed in its place so firmly, that the exertion of -force sufficient to withdraw it would endanger the vessel. In this -case, if a cloth wetted with hot-water be applied to the neck of the -bottle, the glass will expand, and the neck will be enlarged, so as to -allow the stopper to be easily withdrawn. - -(31.) The contraction of metal consequent upon change of temperature -was applied some time ago in Paris to restore the walls of a tottering -building to their proper position. In the _Conservatoire des Arts -et Métiers_, the walls of a part of the building were forced out of -the perpendicular by the weight of the roof, so that each wall was -leaning outwards. M. Molard conceived the notion of applying the -irresistible force with which metals contract in cooling, to draw the -walls together. Bars of iron were placed in parallel directions across -the building, and at right-angles to the direction of the walls. Being -passed through the walls, nuts were screwed on their ends, outside the -building. Every alternate bar was then heated by lamps, and the nuts -screwed close to the walls. The bars were then cooled, and the lengths -being diminished by contraction, the nuts on their extremities were -drawn together, and with them the walls were drawn through an equal -space. The same process was repeated with the intermediate bars, and -so on alternately until the walls were brought into a perpendicular -position. - -(32.) Since there is a continual change of temperature in all bodies on -the surface of the globe, it follows, that there is also a continual -change of magnitude. The substances which surround us are constantly -swelling and contracting, under the vicissitudes of heat and cold. They -grow smaller in winter, and dilate in summer. They swell their bulk -on a warm day, and contract it on a cold one. These curious phenomena -are not noticed, only because our ordinary means of observation are -not sufficiently accurate to appreciate them. Nevertheless, in some -familiar instances the effect is very obvious. In warm weather the -flesh swells, the vessels appear filled, the hand is plump, and the -skin distended. In cold weather, when the body has been exposed to the -open air, the flesh appears to contract, the vessels shrink, and the -skin shrivels. - -(33.) The phenomena attending change of temperature are conclusive -proofs of the universal porosity of material substances, but they are -not the only proofs. Many substances admit of compression by the mere -agency of mechanical force. - -Let a small piece of cork be placed floating on the surface of water -in a basin or other vessel, and an empty glass goblet be inverted over -the cork, so that its edge just meets the water. A portion of air will -then be confined in the goblet, and detached from the remainder of -the atmosphere. If the goblet be now pressed downwards, so as to be -entirely immersed, it will be observed, that the water will not fill -it, being excluded by the _impenetrability_ of the air inclosed in it. -This experiment, therefore, is decisive of the fact, that air, one of -the most subtle and attenuated substances we know of, possesses the -quality of impenetrability. It absolutely excludes any other body from -the space which it occupies at any given moment. - -But although the water does not fill the goblet, yet if the position -of the cork which floats upon its surface be noticed, it will be -found that the level of the water within has risen above its edge or -rim. In fact, the water has partially filled the goblet, and the air -has been forced to contract its dimensions. This effect is produced by -the pressure of the incumbent water forcing the surface in the goblet -against the air, which yields until it is so far compressed that it -acquires a force able to withstand this pressure. Thus it appears -that air is capable of being reduced in its dimensions by mechanical -pressure, independently of the agency of heat. It is _compressible_. - -That this effect is the consequence of the pressure of the liquid will -be easily made manifest by showing that, as the pressure is increased, -the air is proportionally contracted in its dimensions; and as it is -diminished, the dimensions are on the other hand enlarged. If the -depth of the goblet in the water be increased, the cork will be seen -to rise in it, showing that the increased pressure, at the greater -depth, causes the air in the goblet to be more condensed. If, on the -other hand, the goblet be raised toward the surface, the cork will be -observed to descend toward the edge, showing that as it is relieved -from the pressure of the liquid, the air gradually approaches to its -primitive dimensions. - -(34.) These phenomena also prove, that air has the property of -_elasticity_. If it were simply compressible, and not elastic, it would -retain the dimensions to which it was reduced by the pressure of the -liquid; but this is not found to be the result. As the compressing -force is diminished, so in the same proportion does the air, by its -elastic virtue, exert a force by which it resumes its former dimensions. - -That it is the air alone which excludes the water from the goblet, -in the preceding experiments, can easily be proved. When the goblet -is sunk deep in the vessel of water, let it be inclined a little to -one side until its mouth is presented towards the side of the vessel; -let this inclination be so regulated, that the surface of the water -in the goblet shall just reach its edge. Upon a slight increase of -inclination, air will be observed to escape from the goblet, and to -rise in bubbles to the surface of the water. If the goblet be then -restored to its position, it will be found that the cork will rise -higher in it than before the escape of the air. The water in this case -rises and fills the space which the air allowed to escape has deserted. -The same process may be repeated until all the air has escaped, and -then the goblet will be completely filled by the water. - -(35.) Liquids are compressible by mechanical force in so slight a -degree, that they are considered in all hydrostatical treatises -as incompressible fluids. They are, however, not absolutely -incompressible, but yield slightly to very intense pressure. The -question of the compressibility of liquids was raised at a remote -period in the history of science. Nearly two centuries ago, an -experiment was instituted at the Academy _del Cimento_ in Florence, -to ascertain whether water be compressible. With this view, a hollow -ball of gold was filled with the liquid, and the aperture exactly -and firmly closed. The globe was then submitted to a very severe -pressure, by which its figure was slightly changed. Now it is proved -in geometry, that a globe has this peculiar property, that any -change whatever in its figure must necessarily diminish its volume -or contents. Hence it was inferred, that if the water did not issue -through the pores of the gold, or burst the globe, its compressibility -would be established. The result of the experiment was, that the water -_did_ ooze through the pores, and covered the surface of the globe, -presenting the appearance of dew, or of steam cooled by the metal. -But this experiment was inconclusive. It is quite true, that if the -water _had not_ escaped upon the change of figure of the globe, the -_compressibility_ of the liquid would have been established. The escape -of the water does not, however, prove its _incompressibility_. To -accomplish this, it would be necessary first to measure accurately the -volume of water which transuded by compression, and next to measure -the diminution of volume which the vessel suffered by its change of -figure. If this diminution were greater than the volume of water which -escaped, it would follow that the water remaining in the globe had -been compressed, notwithstanding the escape of the remainder. But this -could never be accomplished with the delicacy and exactitude necessary -in such an experiment; and, consequently, as far as the question of -the compressibility of water was concerned, nothing was proved. It -forms, however, a very striking illustration of the porosity of so -dense a substance as gold, and proves that its pores are larger than -the elementary particles of water, since these are capable of passing -through them. - -(36.) It has since been proved, that water, and other liquids, are -compressible. In the year 1761, Canton communicated to the Royal -Society the results of some experiments which proved this fact. He -provided a glass tube with a bulb, such as that described in (28), and -filled the bulb and a part of the tube with water well purified from -air. He then placed this in an apparatus called a condenser, by which -he was enabled to submit the surface of the liquid in the tube to very -intense pressure of condensed air. He found that the level of the -liquid in the tube fell in a perceptible degree upon the application of -the pressure. The same experiment established the fact, that liquids -are _elastic_; for upon removing the pressure, the liquid rose to its -original level, and therefore resumed its former dimensions. - -(37.) Elasticity does not always accompany compressibility. If lead or -iron be submitted to the hammer, it may be hardened and diminished in -its volume; but it will not resume its former volume after each stroke -of the hammer. - -(38.) There are some bodies which maintain the state of density in -which they are commonly found by the continual agency of mechanical -pressure; and such bodies are endued with a quality, in virtue of which -they would enlarge their dimensions without limit, if the pressure -which confines them were removed. Such bodies are called _elastic -fluids_ or _gases_, and always exist in the form of common air, in -whose mechanical properties they participate. They are hence often -called _aeriform fluids_. - -Those who are provided with an air-pump can easily establish this -property experimentally. Take a flaccid bladder, such as that already -described in (27.), and place it under the glass receiver of an -air-pump: by this instrument we shall be able to remove the air which -surrounds the bladder under the receiver, so as to relieve the small -quantity of air which is inclosed in the bladder from the pressure -of the external air: when this is accomplished, the bladder will be -observed to swell, as if it were inflated, and will be perfectly -distended. The air contained in it, therefore, has a tendency to -dilate, which takes effect when it ceases to be resisted by the -pressure of surrounding air. - -(39.) It has been stated that the increase or diminution of temperature -is accompanied by an increase or diminution of volume. Related to this, -there is another phenomenon too remarkable to pass unnoticed, although -this is not the proper place to dwell upon it: it is the converse of -the former; viz. that an increase or diminution of bulk is accompanied -by a diminution or increase of temperature. As the application of heat -from some foreign source produces an increase of dimensions, so if the -dimensions be increased from any other cause, a corresponding portion -of the heat which the body had before the enlargement, will be absorbed -in the process, and the temperature will be thereby diminished. In the -same way, since the abstraction of heat causes a diminution of volume, -so if that diminution be caused by any other means, the body will _give -out_ the heat which in the other case was abstracted, and will rise in -its temperature. - -Numerous and well-known facts illustrate these observations. A smith by -hammering a piece of bar iron, and thereby compressing it, will render -it _red hot_. When air is violently compressed, it becomes so hot as -to ignite cotton and other substances. An ingenious instrument for -producing a light for domestic uses has been constructed, consisting -of a small cylinder, in which a solid piston moves air-tight: a little -tinder, or dry sponge, is attached to the bottom of the piston, which -is then violently forced into the cylinder: the air between the bottom -of the cylinder and the piston becomes intensely compressed, and -evolves so much heat as to light the tinder. - -In all the cases where friction or percussion produces heat or fire, -it is because they are means of compression. The effects of flints, of -pieces of wood rubbed together, the warmth produced by friction on the -flesh, are all to be attributed to the same cause. - - - - -CHAP. III. - -INERTIA. - - -(40.) The quality of matter which is of all others the most important -in mechanical investigations, is that which has been called _Inertia_. - -Matter is incapable of spontaneous change. This is one of the earliest -and most universal results of human observation: it is equivalent to -stating that mere matter is deprived of life; for spontaneous action -is the only test of the presence of the living principle. If we see a -mass of matter undergo any change, we never seek for the cause of that -change in the body itself; we look for some external cause producing -it. This inability for voluntary change of state or qualities is a -more general principle than inertia. At any given moment of time a -body must be in one or other of two states, rest or motion. _Inertia_, -or _inactivity_, signifies the total absence of power to change this -state. A body endued with inertia cannot of itself, and independent of -all external influence, commence to move from a state of rest; neither -can it when moving arrest its progress and become quiescent. - -(41.) The same property by which a body is unable by any power of its -own to pass from a state of rest to one of motion, or _vice versâ_, -also renders it incapable of increasing or diminishing any motion which -it may have received from an external cause. If a body be moving in a -certain direction at the rate of ten miles per hour, it cannot by any -energy of its own change its rate of motion to eleven or nine miles an -hour. This is a direct consequence of that manifestation of inertia -which has just been explained. For the same power which would cause a -body moving at ten miles an hour to increase its rate to eleven miles, -would also cause the same body at rest to commence moving at the rate -of one mile an hour; and the same power which would cause a body moving -at the rate of ten miles an hour to move at the rate of nine miles in -the hour, would cause the same body moving at the rate of one mile -an hour to become quiescent. It therefore appears, that to increase -or diminish the motion of a body is an effect of the same kind as to -change the state of rest into that of motion, or _vice versâ_. - -(42.) The effects and phenomena which hourly fall under our observation -afford unnumbered examples of the inability of lifeless matter to put -itself into motion, or to increase any motion which may have been -communicated to it. But it does not happen that we have the same -direct and frequent evidence of its inability to destroy or diminish -any motion which it may have received. And hence it arises, that while -no one will deny to matter the former effect of inertia, few will at -first acknowledge the latter. Indeed, even so late as the time of -KEPLER, philosophers themselves held it as a maxim, that “matter is -more inclined to rest than to motion;” we ought not, therefore, to be -surprised if in the present day those who have not been conversant with -physical science are slow to believe that a body once put in motion -would continue for ever to move with the same velocity, if it were not -stopped by some external cause. - -Reason, assisted by observation, will, however, soon dispel this -illusion. Experience shows us in various ways, that the same causes -which destroy motion in one direction are capable of producing as -much motion in the opposite direction. Thus, if a wheel, spinning on -its axis with a certain velocity, be stopped by a hand seizing one of -the spokes, the effort which accomplishes this is exactly the same as, -had the wheel been previously at rest, would have put it in motion in -the opposite direction with the same velocity. If a carriage drawn -by horses be in motion, the same exertion of power in the horses is -necessary to stop it, as would be necessary to _back_ it, if it were -at rest. Now, if this be admitted as a general principle, it must be -evident that a body which can destroy or diminish its own motion must -also be capable of putting itself into motion from a state of rest, -or of increasing any motion which it has received. But this latter is -contrary to all experience, and therefore we are compelled to admit -that a body cannot diminish or destroy any motion which it has received. - -Let us enquire why we are more disposed to admit the inability of -matter to produce than to destroy motion in itself. We see most of -those motions which take place around us on the surface of the earth -subject to gradual decay, and if not renewed from time to time, at -length cease. A stone rolled along the ground, a wheel revolving on -its axis, the heaving of the deep after a storm, and all other motions -produced in bodies by external causes, decay, when the exciting -cause is suspended; and if that cause do not renew its action, they -ultimately cease. - -But is there no exciting cause, on the other hand, which thus gradually -deprives those bodies of their motion?--and if that cause were -removed, or its intensity diminished, would not the motion continue, -or be more slowly retarded? When a stone is rolled along the ground, -the inequalities of its shape as well as those of the ground are -impediments, which retard and soon destroy its motion. Render the -stone round, and the ground level, and the motion will be considerably -prolonged. But still small asperities will remain on the stone, and on -the surface over which it rolls: substitute for the stone a ball of -highly-polished steel, moving on a highly-polished steel plane, truly -level, and the motion will continue without sensible diminution for -a very long period; but even here, and in every instance of motions -produced by art, minute asperities must exist on the surfaces which -move in contact with each other, which must resist, gradually diminish, -and ultimately destroy the motion. - -Independently of the obstructions to the continuation of motion arising -from friction, there is another impediment to which all motions on the -surface of the earth are liable--the resistance of the air. How much -this may affect the continuation of motion appears by many familiar -effects. On a calm day carry an open umbrella with its concave side -presented in the direction in which you are moving, and a powerful -resistance will be opposed to your progress, which will increase with -every increase of the speed with which you move. - -(43.) We are not, however, without direct experience to prove, that -motions when unresisted will for ever continue. In the heavens we find -an apparatus, which furnishes a sublime verification of this principle. -There, removed from all casual obstructions and resistances, the vast -bodies of the universe roll on in their appointed paths with unerring -regularity, preserving without diminution all that motion which they -received at their creation from the hand which launched them into -space. This alone, unsupported by other reasons, would be sufficient -to establish the quality of inertia; but viewed in connection with the -other circumstances previously mentioned, no doubt can remain that this -is an universal law of nature. - -(44.) It has been proved, that inability to change the _quantity_ of -motion is a consequence of _inertia_. The inability to change the -_direction_ of motion is another consequence of this quality. The same -cause which increases or diminishes motion, would also give motion -to a body at rest; and therefore we infer that the same inability -which prevents a body from moving itself, will also prevent it from -increasing or diminishing any motion which it has received. In the -same manner we can show, that any cause which changes the direction of -motion would also give motion to a body at rest; and therefore if a -body change the direction of its own motion, the same body might move -itself from a state of rest; and therefore the power of changing the -direction of any motion which it may have received is inconsistent with -the quality of inertia. - -(45.) If a body, moving from A, _fig. 3._ to B, receive at -B a blow in the direction C B E, it will immediately -change its direction to that of another line B D. The cause which -produces this change of direction would have put the body in motion in -the direction B E, had it been quiescent at B when it sustained -the blow. - -(46.) Again, suppose G H to be a hard plane surface; and let the -body be supposed to be perfectly inelastic. When it strikes the surface -at B, it will commence to move along it in the direction B H. This -change of direction is produced by the resistance of the surface. If -the body, instead of meeting the surface in the direction A B, had -moved in the direction E B, perpendicular to it, all motion would -have been destroyed, and the body reduced to a state of rest. - -(47.) By the former example it appears that the deflecting cause would -have put a quiescent body in motion, and by the latter it would have -reduced a moving body to a state of rest. Hence the phenomenon of a -change of direction is to be referred to the same class as the change -from rest to motion, or from motion to rest. The quality of inertia -is, therefore, inconsistent with any change in the direction of motion -which does not arise from an external cause. - -(48.) From all that has been here stated, we may infer generally, that -an inanimate parcel of matter is incapable of changing its state of -rest or motion; that, in whatever state it be, in that state it must -for ever continue, unless disturbed by some external cause; that -if it be in motion, that motion must always be _uniform_, or must -proceed at the same rate, equal spaces being moved over in the same -time: any increase of its rate must betray some impelling cause; any -diminution must proceed from an impeding cause, and neither of these -causes can exist in the body itself; that such motion must not only be -constantly at the same uniform rate, but also must be always in the -same direction, any deflection from one uniform direction necessarily -arising from some external influence. - -The language sometimes used to explain the property of inertia in -popular works, is eminently calculated to mislead the student. The -terms resistance and stubbornness to move are faulty in this respect. -Inertia implies absolute passiveness, a perfect indifference to rest -or motion. It implies as strongly the absence of all resistance to -the reception of motion, as it does the absence of all power to move -itself. The term _vis inertiæ_ or _force of inactivity_, so frequently -used even by authors pretending to scientific accuracy, is still more -reprehensible. It is a contradiction in terms; the term _inactivity_ -implying the absence of all force. - - * * * * * - -(49.) Before we close this chapter, it may be advantageous to point out -some practical and familiar examples of the general law of inertia. -The student must, however, recollect, that the great object of -science is generalisation, and that his mind is to be elevated to the -contemplation of the _laws_ of nature, and to receive a habit the very -reverse of that which disposes us to enjoy the descent from generals to -particulars. Instances, taken from the occurrences of ordinary life, -may, however, be useful in verifying the general law, and in impressing -it upon the memory; and for this reason, we shall occasionally in the -present treatise refer to such examples; always, however, keeping -them in subservience to the general principles of which they are -manifestations, and on which the attention of the student should never -cease to be fixed. - -(50.) If a carriage, a horse, or a boat, moving with speed, be suddenly -retarded or stopped, by any cause which does not at the same time -affect passengers, riders, or any loose bodies which are carried, they -will be precipitated in the direction of the motion; because by reason -of their inertia, they persevere in the motion which they shared in -common with that which transported them, and are not deprived of that -motion by the same cause. - -(51.) If a passenger leap from a carriage in rapid motion, he will fall -in the direction in which the carriage is moving at the moment his feet -meet the ground; because his body, on quitting the vehicle, retains, by -its inertia, the motion which it had in common with it. When he reaches -the ground, this motion is destroyed by the resistance of the ground to -the feet, but is retained in the upper and heavier part of the body; so -that the same effect is produced as if the feet had been tripped. - -(52.) When a carriage is once put in motion with a determinate speed -on a level road, the only force necessary to sustain the motion is -that which is sufficient to overcome the friction of the road; but -at starting a greater expenditure of force is necessary, inasmuch as -not only the friction is to be overcome, but the force with which the -vehicle is intended to move must be communicated to it. Hence we see -that horses make a much greater exertion at starting than subsequently, -when the carriage is in motion; and we may also infer the inexpediency -of attempting to start at full speed, especially with heavy carriages. - -(53.) _Coursing_ owes all its interest to the instinctive consciousness -of the nature of inertia which seems to govern the measures of the -hare. The greyhound is a comparatively heavy body moving at the same -or greater speed in pursuit. The hare _doubles_, that is, suddenly -changes the direction of her course, and turns back at an oblique angle -with the direction in which she had been running. The greyhound, unable -to resist the tendency of its body to persevere in the rapid motion it -had acquired, is urged forward many yards before it is able to check -its speed and return to the pursuit. Meanwhile the hare is gaining -ground in the other direction, so that the animals are at a very -considerable distance asunder when the pursuit is recommenced. In this -way a hare, though much less fleet than a greyhound, will often escape -it. - -In racing, the horses shoot far beyond the winning-post before their -course can be arrested. - - - - -CHAP. IV. - -ACTION AND REACTION. - - -(54.) The effects of inertia or inactivity, considered in the last -chapter, are such as may be manifested by a single insulated body, -without reference to, or connection with, any other body whatever. They -might all be recognised if there were but one body existing in the -universe. There are, however, other important results of this law, to -the development of which two bodies at least are necessary. - -(55.) If a mass A, _fig._ 4., moving towards C, impinge upon an equal -mass, which is quiescent at B, the two masses will move together -towards C after the impact. But it will be observed, that their speed -after the impact will be only half that of A before it. Thus, after the -impact, A loses half its velocity; and B, which was before quiescent, -receives exactly this amount of motion. It appears, therefore, in this -case, that B receives exactly as much motion as A loses: so that the -real quantity of motion from B to C is the same as the quantity of -motion from A to B. - -Now, suppose that B consisted of two masses, each equal to A, it would -be found that in this case the velocity of the triple mass after impact -would be one-third of the velocity from A to B. Thus, after impact, A -loses two-thirds of its velocity and, B consisting of two masses each -equal to A, each of these two receives one-third of A’s motion; so that -the whole motion received by B is two-thirds of the motion of A before -impact. By the impact, therefore, exactly as much motion is received by -B as is lost by A. - -A similar result will be obtained, whatever proportion may subsist -between the masses A and B. Suppose B to be ten times A; then the whole -motion of A must, after the impact, be distributed among the parts of -the united masses of A and B: but these united masses are, in this -case, eleven times the mass of A. Now, as they all move with a common -motion, it follows that A’s former motion must be equally distributed -among them; so that each part shall have an eleventh part of it. -Therefore the velocity after impact will be the eleventh part of the -velocity of A before it. Thus A loses by the impact ten-eleventh parts -of its motion, which are precisely what B receives. - -Again, if the masses of A and B be 5 and 7, then the united mass after -impact will be 12. The motion of A before impact will be equally -distributed between these twelve parts, so that each part will have -a twelfth of it; but five of these parts belong to the mass A, and -seven to B. Hence B will receive seven-twelfths, while A retains -five-twelfths. - -(56.) In general, therefore, when a mass A in motion impinges on a mass -B at rest, to find the motion of the united mass after impact, “divide -the whole motion of A into as many equal parts as there are equal -component masses in A and B together, and then B will receive by the -impact as many parts of this motion as it has equal component masses.” - -This is an immediate consequence of the property of inertia, explained -in the last chapter. If we were to suppose that by their mutual impact -A were to give to B either more or less motion than that which it (A) -loses, it would necessarily follow, that either A or B must have a -power of producing or of resisting motion, which would be inconsistent -with the quality of inertia already defined. For if A give to B _more_ -motion than it loses, all the overplus or excess must be excited in B -by the _action_ of A; and, therefore, A is not inactive, but is capable -of exciting motion which it does not possess. On the other hand, B -cannot receive from A _less_ motion than A loses, because then B must -be admitted to have the power by its resistance of destroying all the -deficiency; a power essentially active, and inconsistent with the -quality of inertia. - -(57.) If we contemplate the effects of impact, which we have now -described, as facts ascertained by experiment (which they may be), -we may take them as further verification of the universality of the -quality of inertia. But, on the other hand, we may view them as -phenomena which may certainly be predicted from the previous knowledge -of that quality; and this is one of many instances of the advantage -which science possesses over knowledge _merely_ practical. Having -obtained by observation or experience a certain number of simple facts, -and thence deduced the general qualities of bodies, we are enabled, -by demonstrative reasoning, to discover _other facts_ which have -never fallen under our observation, or, if so, may have never excited -attention. In this way philosophers have discovered certain small -motions and slight changes which have taken place among the heavenly -bodies, and have directed the attention of astronomical observers to -them, instructing them with the greatest precision as to the exact -moment of time and the point of the firmament to which they should -direct the telescope, in order to witness the predicted event. - -(58.) Since by the quality of inertia a body can neither generate -nor destroy motion, it follows that when two bodies act upon each -other in any way whatever, the total quantity of motion in a given -direction, after the action takes place, must be the same as before -it, for otherwise some motion would be produced by the action of the -bodies, which would contradict the principle that they are inert. The -word “action” is here applied, perhaps improperly, but according to the -usage of mechanical writers, to express a certain phenomenon or effect. -It is, therefore, not to be understood as implying any active principle -in the bodies to which it is attributed. - -(59.) In the cases of collision of which we have spoken, one of the -masses B was supposed to be quiescent before the impact. We shall now -suppose it to be moving in the same direction as A, that is, towards C, -but with a less velocity, so that A shall overtake it, and impinge upon -it. After the impact, the two masses will move towards C with a common -velocity, the amount of which we now propose to determine. - -If the masses A and B be equal, then their motions or velocities added -together must be the motion of the united mass after impact, since no -motion can either be created or destroyed by that event. But as A and B -move with a common motion, this sum must be equally distributed between -them, and therefore each will move with a velocity equal to half the -sum of their velocities before the impact. Thus, if A have the velocity -7, and B have 5, the velocity of the united mass after impact is 6, -being the half of 12, the sum of 7 and 5. - -If A and B be not equal, suppose them divided into equal component -parts, and let A consist of 8, and B of 6, equal masses: let the -velocity of A be 17, so that the motion of each of the 8 parts being -17, the motion of the whole will be 136. In the same manner, let the -velocity of B be 10, the motion of each part being 10, the whole motion -of the 6 parts will be 60. The sum of the two motions, therefore, -towards C is 196; and since none of this can be lost by the impact, -nor any motion added to it, this must also be the whole motion of the -united masses after impact. Being equally distributed among the 14 -component parts of which these united masses consist, each part will -have a fourteenth of the whole motion. Hence, 196 being divided by 14, -we obtain the quotient 14, which is the velocity with which the whole -moves. - -(60.) In general, therefore, when two masses moving in the same -direction impinge one upon the other, and after impact move together, -their common velocity may be determined by the following rule: “Express -the masses and velocities by numbers in the usual way, and multiply -the numbers expressing the masses by the numbers which express the -velocities; the two products thus obtained being added together, and -their sum divided by the sum of the numbers expressing the masses, the -quotient will be the number expressing the required velocity.” - -(61.) From the preceding details, it appears that _motion_ is not -adequately estimated by _speed_ or _velocity_. For example, a certain -mass A, moving at a determinate rate, has a certain quantity of motion. -If another equal mass B be added to A, and a similar velocity be given -to it, as much more motion will evidently be called into existence. In -other words, the _two_ equal masses A and B united have _twice_ as much -motion as the single mass A had when moving alone, and with the same -speed. The same reasoning will show that _three_ equal masses will with -the same speed have _three times_ the motion of any one of them. In -general, therefore, the velocity being the same, the quantity of motion -will always be increased or diminished in the same proportion as the -mass moved is increased or diminished. - -(62.) On the other hand, the quantity of motion does not depend on the -mass _only_, but also on the speed. If a certain determinate mass move -with a certain determinate speed, another equal mass which moves with -twice the speed, that is, which moves over twice the space in the same -time, will have twice the quantity of motion. In this manner, the mass -being the same, the quantity of motion will increase or diminish in the -same proportion as the velocity. - -(63.) The true estimate, then, of the quantity of motion is found -by multiplying together the numbers which express the mass and the -velocity. Thus, in the example which has been last given of the impact -of masses, the quantities of motion before and after impact appear to -be as follow: - - Before Impact. | After Impact. - | - Mass of A 8 | Mass of A 8 - Velocity of A 17 | Common velocity 14 - -----------------+ -------------- - Quantity of } 8 × 17[1] or 136 | Quantity of } 8 × 14 or 112 - motion of A } | motion of A } - -----------------+ -------------- - Mass of B 6 | Mass of B 6 - Velocity of B 10 | Common velocity 14 - -----------------+ -------------- - Quantity of } 6 × 10 or 60 | Quantity of } 6 × 14 = 84 - motion of B } | motion of B } - -----------------+ -------------- - - * The sign × placed between two numbers meant that they are to be - multiplied together. - -By this calculation it appears that in the impact A has lost a quantity -of motion expressed by 24, and that B has received exactly that amount. -The effect, therefore, of the impact is a _transfer_ of motion from A -to B; but no new motion is produced in the direction A C which did -not exist before. This is obviously consistent with the property of -inertia, and indeed an inevitable result of it. - -These results may be generalised and more clearly and concisely -expressed by the aid of the symbols of arithmetic. - -Let _a_ express the velocity of A. - -Let _b_ express the velocity of B. - -Let _x_ express the velocity of the united masses of A and B after -impact, each of these velocities being expressed in feet per second, -and the masses of A and B being expressed by the weight in pounds. - -We shall then have the momenta or moving forces of A and B before -impact, expressed by A × _a_ and B × _b_, and the moving force of the -united mass after impact will be expressed by (A + B) × _x_. - -The moving force of A after impact is A × _x_, and therefore the force -it loses by the collision will be (A × _a_ - A × _x_). The force of B -after impact will be B × _x_, and therefore the force it gains will be -B × _x_ - B × _b_. But since the force lost by A must be equal to the -force gained by B, we shall have - - A × _a_ - A × _x_ = B × _x_ - B × _b_ - -from which it is easy to infer - - (A + B) × _x_ = A × _a_ + B × _b_ - -and if it be required to express the velocity of the united masses -after impact, we have - - _x_ = (A × _a_ + B × _b_)/(A + B) - -When it is said that A × _a_ and B × _b_ express the moving forces of -A and B, it must be understood that the _unit_ of momentum or moving -force is in the case here supposed, the force with which a mass of -matter weighing 1 lb. would move if its velocity were 1 foot per -second, and accordingly the forces with which A and B move before -impact are as many times this as there are units respectively in the -numbers signified by the general symbols A × _a_ and B × _b_. - -In like manner, the force of the united masses after impact is as many -times greater than that of 1 lb. moving through 1 foot per second -as there are units in the numbers expressed by (A + B) × _x_. - -(64.) These phenomena present an example of a law deduced from the -property of inertia, and generally expressed thus--“action and reaction -are equal, and in contrary directions.” The student must, however, be -cautious not to receive these terms in their ordinary acceptation. -After the full explanation of inertia given in the last chapter, it -is, perhaps, scarcely necessary here to repeat, that in the phenomena -manifested by the motion of two bodies, there can be neither “action” -nor “reaction,” properly so called. The bodies are absolutely incapable -either of action or resistance. The sense in which these words must -be received, as used in the _law_, is merely an expression of the -_transfer_ of a certain quantity of motion from one body to another, -which is called an _action_ in the body which loses the motion, and a -_reaction_ in the body which receives it. The _accession_ of motion to -the latter is said to proceed from the _action_ of the former; and the -_loss_ of the same motion in the former is ascribed to the _reaction_ -of the latter. The whole phraseology is, however, most objectionable -and unphilosophical, and is calculated to create wrong notions. - -(65.) The bodies impinging were, in the last case, supposed to move in -the same direction. We shall now consider the case in which they move -in opposite directions. - -First, let the masses A and B be supposed to be equal, and moving in -opposite directions, with the same velocity. Let C, _fig. 5._, be -the point at which they meet. The equal motions in opposite directions -will, in this case, destroy each other, and both masses will be -reduced to a state of rest. Thus, the mass A loses all its motion in -the direction A C, which it may be supposed to transfer to B at -the moment of impact. But B having previously had an equal quantity -of motion in the direction B C, will now have two equal motions -impressed upon it, in directions immediately opposite; and these -motions neutralising each other, the mass becomes quiescent. In this -case, therefore, as in all the former examples, each body transfers -to the other all the motion which it loses, consistently with the -principle of “action and reaction.” - -The masses A and B being still supposed equal, let them move towards -C with different velocities. Let A move with the velocity 10, and B -with the velocity 6. Of the 10 parts of motion with which A is endued, -6 being transferred to B, will destroy the equal velocity 6, which B -has in the direction B C. The bodies will then move together in -the direction C B, the four remaining parts of A’s motion being -equally distributed between them. Each body will, therefore, have two -parts of A’s original motion, and 2 therefore will be their common -velocity after impact. In this case, A loses 8 of the 10 parts of its -motion in the direction A C. On the other hand, B loses the entire -of its 6 parts of motion in the direction B C, and receives 2 -parts in the direction A C. This is equivalent to receiving 8 -parts of A’s motion in the direction A C. Thus, according to the -law of “action and reaction,” B receives exactly what A loses. - -Finally, suppose that both the masses and velocities of A and B are -unequal. Let the mass of A be 8, and its velocity 9: and let the mass -of B be 6, and its velocity 5. The quantity of motion of A will be 72, -and that of B, in the opposite direction, will be 30. Of the 72 parts -of motion, which A has in the direction A C, 30 being transferred -to B, will destroy all its 30 parts of motion in the direction -B C, and the two masses will move in the direction C B, with -the remaining 42 parts of motion, which will be equally distributed -among their 14 component masses. Each component part will, therefore, -receive 3 parts of motion; and accordingly 3 will be the common -velocity of the united mass after impact. - -(66.) When two masses moving in opposite directions impinge and move -together, their common velocity after impact may be found by the -following rule:--“Multiply the numbers expressing the masses by those -which express the velocities respectively, and subtract the lesser -product from the greater; divide the remainder by the sum of the -numbers expressing the masses, and the quotient will be the common -velocity; the direction will be that of the mass which has the greater -quantity of motion.” - -It may be shown without difficulty, that the example which we have -just given obeys the law of “action and reaction.” - - Before impact. | After impact. - | - Mass of A 8 | Mass of A 8 - Velocity of A 9 | Common velocity 3 - ------------+ ----------- - Quantity of motion } 8 × 9 or 72 | Quantity of motion } 8 × 3 or 24 - in direction A C } | in direction A C } - ------------+ ----------- - Mass of B 6 | Mass of B 6 - Velocity of B 5 | Common velocity 3 - ------------+ ----------- - Quantity of motion } 6 × 5 or 30 | Quantity of motion } 6 × 3 or 18 - in direction B C } | in direction A C } - ------------+ ----------- - -Hence it appears that the quantity of motion in the direction A C -of which A has been deprived by the impact is 48, the difference -between 72 and 24. On the other hand, B loses by the impact the -quantity 30 in the direction B C, which is equivalent to receiving -30 in the direction A C. But it also acquires a quantity 18 in -the direction A C, which, added to the former 30, gives a total -of 48 received by B in the direction A C. Thus the same quantity -of motion which A loses in the direction A C, is received by B in -the same direction. The law of “action and reaction” is, therefore, -fulfilled. - -This result may in like manner be generalised. Retaining the former -symbols, the moving forces of A and B before impact will be A × _a_ and -B × _b_ and their forces after impact will be A × _x_ and B × _x_. The -force lost by A will therefore be A × _a_ - A × _x_. The mass B will -have lost all the force B × _b_ which it had in its former direction, -and will have received the force B × _x_ in the opposite direction. -Therefore the actual force imparted to B by the collision will be B -× _b_ + B × _x_. But since the force lost by A must be equal to that -imparted to B, we shall have - - A × _a_ - A × _x_ = B × _b_ + B × _x_ - -and therefore - - (A + B) × _x_ = A × _a_ - B × _b_ - -and if the common velocity after impact be required, we have - - _x_ = (A × _a_ - B × _b_)/(A + B) - -As a general rule, therefore, to find the common velocity after impact. -Multiply the weights by the previous velocities and take their sum if -the bodies move in the same direction, and their difference if they -move in opposite directions, and divide the one or the other by the sum -of their weights. The greatest will be the velocity after impact. - -(67.) The examples of the equality of action and reaction in the -collision of bodies may be exhibited experimentally by a very simple -apparatus. Let A, _fig. 6._, and B be two balls of soft clay, or -any other substance which is inelastic, or nearly so, and let these -be suspended from C by equal strings, so that they may be in contact; -and let a graduated arc, of which the centre is C, be placed so that -the balls may oscillate over it. One of the balls being moved from its -place of rest along the arc, and allowed to descend upon the other -through a certain number of degrees, will strike the other with a -velocity corresponding to that number of degrees, and both balls will -then move together with a velocity which may be estimated by the number -of degrees of the arc through which they rise. - -(68.) In all these cases in which we have explained the law of “action -and reaction,” the transfer of motion from one body to the other has -been made by impact or collision. The phenomenon has been selected only -because it is the most ordinary way in which bodies are seen to affect -each other. The law is, however, universal, and will be fulfilled -in whatever manner the bodies may affect each other. Thus A may be -connected with B by a flexible string, which, at the commencement of -A’s motion, is slack. Until the string becomes stretched, that is, -until A’s distance from B becomes equal to the length of the string, -A will continue to have all the motion first impressed upon it. But -when the string is stretched, a part of that motion is transferred to -B, which is then drawn after A; and whatever motion B in this way -receives, A must lose. All that has been observed of the effect of -motion transferred by impact will be equally applicable in this case. - -Again, if B, _fig. 4._, be a magnet moving in the direction -B C with a certain quantity of motion, and while it is so moving a -mass of iron be placed at rest at A, the attraction of the magnet will -draw the iron after it towards C, and will thus communicate to the iron -a certain quantity of motion in the direction of C. All the motion thus -communicated to the iron A must be lost by the magnet B. - -If the magnet and the iron were both placed quiescent at B and A, the -attraction of the magnet would cause the iron to move from A towards B; -but the magnet in this case not having any motion, cannot be literally -said to _transfer_ a motion to the iron. At the moment, however, when -the iron begins to move from A towards B, the magnet will be observed -to begin also to move from B towards A; and if the velocities of the -two bodies be expressed by numbers, and respectively multiplied by the -numbers expressing their masses, the quantities of motion thus obtained -will be found to be exactly equal. We have already explained why a -quantity of motion received in the direction B A, is equivalent -to the same quantity lost in the direction A B. Hence it appears, -that the magnet in receiving as much motion in the direction B A, -as it gives in the direction A B, suffers an effect which is -equivalent to losing as much motion directed towards C as it has -communicated to the iron in the same direction. - -In the same manner, if the body B had any property in virtue of which -it might _repel_ A, it would itself be repelled with the same quantity -of motion. In a word, whatever be the manner in which the bodies may -affect each other, whether by collision, traction, attraction, or -repulsion, or by whatever other name the phenomenon may be designated, -still it is an inevitable consequence, that any motion, in a given -direction, which one of the bodies may receive, must be accompanied by -a loss of motion in the same direction, and to the same amount, by -the other body, or the acquisition of as much motion in the contrary -direction; or, finally, by a loss in the same direction, and an -acquisition of motion in the contrary direction, the combined amount of -which is equal to the motion received by the former. - -(69.) From the principle, that the force of a body in motion depends on -the mass and the velocity, it follows, that any body, however small, -may be made to move with the same force as any other body, however -great, by giving to the smaller body a velocity which bears to that of -the greater the same proportion as the mass of the greater bears to the -mass of the smaller. Thus a feather, ten thousand of which would have -the same weight as a cannon-ball, would move with the same force if -it had ten thousand times the velocity; and in such a case, these two -bodies encountering in opposite directions, would mutually destroy each -other’s motion. - -(70.) The consequences of the property of inertia, which have been -explained in the present and preceding chapters, have been given by -Newton, in his PRINCIPIA, and, after him, in most English treatises on -mechanics, under the form of three propositions, which are called the -“laws of motion.” They are as follow:-- - - -I. - -“Every body must persevere in its state of rest, or of uniform motion -in a straight line, unless it be compelled to change that state by -forces impressed upon it.” - - -II. - -“Every change of motion must be proportional to the impressed force, -and must be in the direction of that straight line in which the force -is impressed.” - - -III. - -“Action must always be equal and contrary to reaction; or the actions -of two bodies upon each other must be equal, and directed towards -contrary sides.” - -When _inertia_ and _force_ are defined, the first law becomes an -identical proposition. The second law cannot be rendered perfectly -intelligible until the student has read the chapter on the composition -and resolution of forces, for, in fact, it is intended as an expression -of the whole body of results in that chapter. The third law has -been explained in the present chapter, as far as it can be rendered -intelligible in the present stage of our progress. - -We have noticed these formularies more from a respect for the -authorities by which they have been proposed and adopted, than from any -persuasion of their utility. Their full import cannot be comprehended -until nearly the whole of elementary mechanics has been acquired, and -then all such summaries become useless. - - * * * * * - -(71.) The consequences deduced from the consideration of the quality -of inertia in this chapter, will account for many effects which fall -under our notice daily, and with which we have become so familiar, that -they have almost ceased to excite curiosity. One of the facts of which -we have most frequent practical illustration is, that the quantity of -motion or _moving force_, as it is sometimes called, is estimated by -the velocity of the motion, and the weight or mass of the thing moved -conjointly. - -If the same force impel two balls, one of one pound weight, and the -other of two pounds, it follows, since the balls can neither give force -to themselves, nor resist that which is impressed upon them, that they -will move with the same force. But the lighter ball will move with -twice the speed of the heavier. The impressed force which is manifested -by giving velocity to a double mass in the one, is engaged in giving a -double velocity to the other. - -If a cannon-ball were forty times the weight of a musket-ball, but the -musket-ball moved with forty times the velocity of the cannon-ball, -both would strike any obstacle with the same force, and would overcome -the same resistance; for the one would acquire from its velocity as -much force as the other derives from its weight. - -A very small velocity may be accompanied by enormous force, if the mass -which is moved with that velocity be proportionally great. A large -ship, floating near the pier wall, may approach it with so small a -velocity as to be scarcely perceptible, and yet the force will be so -great as to crush a small boat. - -A grain of shot flung from the hand, and striking the person, will -occasion no pain, and indeed will scarcely be felt, while a block of -stone having the same velocity would occasion death. - -If a body in motion strike a body at rest, the striking body must -sustain as great a shock from the collision as if it had been at rest, -and struck by the other body with the same force. For the loss of force -which it sustains in the one direction, is an effect of the same kind -as if, being at rest, it had received as much force in the opposite -direction. If a man, walking rapidly or running, encounters another -standing still, he suffers as much from the collision as the man -against whom he strikes. - -If a leaden bullet be discharged against a plank of hard wood, it will -be found that the round shape of the ball is destroyed, and that it -has itself suffered a force by the impact, which is equivalent to the -effect which it produces upon the plank. - -When two bodies moving in opposite directions meet, each body sustains -as great a shock as if, being at rest, it had been struck by the other -body with the united forces of the two. Thus, if two equal balls, -moving at the rate of ten feet in a second, meet, each will be struck -with the same force as if, being at rest, the other had moved against -it at the rate of twenty feet in a second. In this case one part of the -shock sustained arises from the loss of force in one direction, and -another from the reception of force in the opposite direction. - -For this reason, two persons walking in opposite directions receive -from their encounter a more violent shock than might be expected. If -they be of nearly equal weight, and one be walking at the rate of three -and the other four miles an hour, each sustains the same shock as if he -had been at rest, and struck by the other running at the rate of seven -miles an hour. - -This principle accounts for the destructive effects arising from ships -running foul of each other at sea. If two ships of 500 tons burden -encounter each other, sailing at ten knots an hour, each sustains the -shock which, being at rest, it would receive from a vessel of 1000 tons -burden sailing ten knots an hour. - -It is a mistake to suppose, that when a large and small body encounter, -the small body suffers a greater shock than the large one. The shock -which they sustain must be the same; but the large body may be better -able to bear it. - -When the fist of a pugilist strikes the body of his antagonist, it -sustains as great a shock as it gives; but the fist being more fitted -to endure the blow, the injury and pain are inflicted on his opponent. -This is not the case, however, when fist meets fist. Then the parts -in collision are equally sensitive and vulnerable, and the effect is -aggravated by both having approached each other with great force. The -effect of the blow is the same as if one fist, being held at rest, were -struck by the other with the combined force of both. - - - - -CHAP. V. - -THE COMPOSITION AND RESOLUTION OF FORCE. - - -(72.) Motion and pressure are terms too familiar to need explanation. -It may be observed, generally, that definitions in the first rudiments -of a science are seldom, if ever, comprehended. The force of words is -learned by their application; and it is not until a definition becomes -useless, that we are taught the meaning of the terms in which it is -expressed. Moreover, we are perhaps justified in saying, that in the -mathematical sciences the fundamental notions are of so uncompounded a -character, that definitions, when developed and enlarged upon, often -draw us into metaphysical subtleties and distinctions, which, whatever -be their merit or importance, would be here altogether misplaced. We -shall, therefore, at once take it for granted, that the words _motion_ -and _pressure_ express phenomena or effects which are the subjects -of constant experience and hourly observation; and if the scientific -use of these words be more precise than their general and popular -application, that precision will soon be learned by their frequent use -in the present treatise. - -(73.) FORCE is the name given in mechanics to whatever produces motion -or pressure. This word is also often used to express the motion or -pressure itself; and when the cause of the motion or pressure is not -known, this is the only correct use of the word. Thus, when a piece of -iron moves toward a magnet, it is usual to say that the cause of the -motion is “the attraction of the magnet;” but in effect we are ignorant -of the _cause_ of this phenomenon; and the name _attraction_ would -be better applied to the effect of which we have experience. In like -manner the _attraction_ and _repulsion_ of electrified bodies should be -understood, not as names for unknown causes, but as words expressing -observed appearances or effects. - -When a certain phraseology has, however, gotten into general use, it -is neither easy nor convenient to supersede it. We shall, therefore, -be compelled, in speaking of motion and pressure, to use the language -of causation; but must advise the student that it is effects and not -causes which will be expressed. - -(74.) If two forces act upon the same point of a body in different -directions, a single force may be assigned, which, acting on that -point, will produce the same result as the united effects of the other -two. - -Let P, _fig. 7._, be the point on which the two forces act, and -let their directions be P A and P B. From the point P, upon -the line P A, take a length P _a_, consisting of as many inches -as there are ounces in the force P A; and, in like manner, take P -_b_, in the direction P B, consisting of as many inches as there -are ounces in the force P B. Through _a_ draw a line parallel to -P B, and through _b_ draw a line parallel to P A, and suppose -that these lines meet at _c_. Then draw P C. A single force, -acting in the direction P C, and consisting of as many ounces as -the line P c consists of inches, will produce upon the point P -the same effect as the two forces P A and P B produce acting -together. - -(75.) The figure P _a c b_ is called in GEOMETRY a -_parallelogram_; the lines P _a_, P _b_, are called its _sides_, and -the line P _c_ is called its _diagonal_. Thus the method of finding an -equivalent for two forces, which we have just explained, is generally -called “the parallelogram of forces,” and is usually expressed thus: -“If two forces be represented in quantity and direction by the sides of -a parallelogram, an equivalent force will be represented in quantity -and direction by its diagonal.” - -(76.) A single force, which is thus mechanically equivalent to two or -more other forces, is called their _resultant_, and relatively to it -they are called its _components_. In any mechanical investigation, -when the resultant is used for the components, which it always may -be, the process is called “the composition of force.” It is, however, -frequently expedient to substitute for a single force two or more -forces, to which it is mechanically equivalent, or of which it is the -resultant. This process is called “the resolution of force.” - -(77.) To verify experimentally the theorem of the parallelogram -of forces is not difficult. Let two small wheels, M N, -_fig. 8._, with grooves in their edges to receive a thread, be -attached to an upright board, or to a wall. Let a thread be passed over -them, having weights A and B, hooked upon loops at its extremities. -From any part P of the thread between the wheels let a weight C be -suspended: it will draw the thread downwards, so as to form an angle -M P N, and the apparatus will settle itself at rest in some -determinate position. In this state it is evident that since the weight -C, acting in the direction P C, balances the weights A and B, -acting in the directions P M and P N, these two forces must -be mechanically equivalent to a force equal to the weight C, and acting -directly upwards from P. The weight C is therefore the quantity of the -resultant of the forces P M and P N; and the direction of the -resultant is that of a line drawn directly upwards from P. - -To ascertain how far this is consistent with the theorem of “the -parallelogram of forces,” let a line P O be drawn upon the upright -board to which the wheels are attached, from the point P upward, in the -direction of the thread C P. Also, let lines be drawn upon the -board immediately under the threads P M and P N. From the -point P, on the line P O, take as many inches as there are ounces -in the weight C. Let the part of P O thus measured be P _c_, and -from _c_ draw _c a_ parallel to P N, and _c b_ parallel -to P M. If the sides P _a_ and P _b_ of the parallelogram thus -formed be measured, it will be found that P _a_ will consist of as many -inches as there are ounces in the weight A, and P _b_ of as many inches -as there are ounces in the weight B. - -In this illustration, _ounces_ and _inches_ have been used as the -subdivisions of _weight_ and _length_. It is scarcely necessary to -state, that any other measures of these quantities would serve as well, -only observing that the same denominations must be preserved in all -parts of the same investigation. - -(78.) Among the philosophical apparatus of the University of London, -is a very simple and convenient instrument which I constructed for -the experimental illustration of this important theorem. The wheels -M N are attached to the tops of two tall stands, the heights -of which may be varied at pleasure by an adjusting screw. A jointed -parallelogram, A B C D, _fig. 9._, is formed, whose -sides are divided into inches, and the joints at A and B are moveable, -so as to vary the lengths of the sides at pleasure. The joint C is -fixed at the extremity of a ruler, also divided into inches, while -the opposite joint A is attached to a brass loop, which surrounds the -diagonal ruler loosely, so as to slide freely along it. An adjusting -screw is provided in this loop so as to clamp it in any required -position. - -In making the experiment, the sides A B and A D, C B -and C D are adjusted by the joints B and A to the same number -of inches respectively as there are ounces in the weights A and B, -_fig. 8._ Then the diagonal A C is adjusted by the loop and -screw at A, to as many inches as there are ounces in the weight C. -This done, the point A is placed behind P, _fig. 8._, and the -parallelogram is held upright, so that the diagonal A C shall be -in the direction of the vertical thread P C. The sides A B -and A D will then be found to take the direction of the threads -P M and P N. By changing the weights and the lengths of the -diagonal and sides of the parallelogram, the experiment may be easily -varied at pleasure. - -(79.) In the examples of the composition of forces which we have here -given, the effects of the forces are the production of pressures, or, -to speak more correctly, the theorem which we have illustrated, is “the -composition of pressures.” For the point P is supposed to be at rest, -and to be drawn or pressed in the directions P M and P N. -In the definition which has been given of the word force, it is -declared to include motions as well as pressures. In fact, if motion be -resisted, the effect is converted into pressure. The same cause acting -upon a body, will either produce motion or pressure, according as the -body is free or restrained. If the body be free, motion ensues; if -restrained, pressure, or both these effects together. It is therefore -consistent with analogy to expect that the same theorems which regulate -pressures, will also be applicable to motions; and we find accordingly -a most exact correspondence. - -(80.) If a body have a motion in the direction A B, and at the -point P it receive another motion, such as would carry it in the -direction P C, _fig. 10._, were it previously quiescent at -P, it is required to determine the direction which the body will take, -and the speed with which it will move, under these circumstances. - -Let the velocity with which the body is moving from A to B be such, -that it would move through a certain space, suppose P N, in one -second of time, and let the velocity of the motion impressed upon it -at P be such, that if it had no previous motion it would move from P -to M in one second. From the point M draw a line parallel to P B, -and from N draw a line parallel to P C, and suppose these lines to -meet at some point, as O. Then draw the line P O. In consequence -of the two motions, which are at the same time impressed upon the body -at P, it will move in the straight line from P to O. - -Thus the two motions, which are expressed in quantity and direction -by the sides of a parallelogram, will, when given to the same body, -produce a single motion, expressed in quantity and direction by its -diagonal; a theorem which is to motions exactly what the former was to -pressures. - -There are various methods of illustrating experimentally the -composition of motion. An ivory ball, being placed upon a perfectly -level square table, at one of the corners, and receiving two equal -impulses, in the directions of the sides of the table, will move along -the diagonal. Apparatus for this experiment differ from each other only -in the way of communicating the impulses to the ball. - -(81.) As two motions simultaneously communicated to a body are -equivalent to a single motion in an intermediate direction, so -also a single motion may be mechanically replaced, by two motions -in directions expressed by the sides of any parallelogram, whose -diagonal represents the single motion. This process is “the resolution -of motion,” and gives considerable clearness and facility to many -mechanical investigations. - -(82.) It is frequently necessary to express the portion of a given -force, which acts in some given direction different from the -immediate direction of the force itself. Thus, if a force act from -A, _fig. 11._, in the direction A C, we may require to -estimate what part of that force acts in the direction A B. If the -force be a pressure, take as many inches A P from A, on the line -A C, as there are ounces in the force, and from P draw P M -perpendicular to A B; then the part of the force which acts along -A B will be as many ounces as there are inches in A M. The -force A B is mechanically equivalent to two forces, expressed by -the sides A M and A N of the parallelogram; but A N, -being perpendicular to A B, can have no effect on a body at A, -in the direction of A B, and therefore the effective part of the -force A P in the direction A B is expressed by A M. - -(83.) Any number of forces acting on the same point of a body may -be replaced by a single force, which is mechanically equivalent to -them, and which is, therefore, their resultant. This composition may -be effected by the successive application of the parallelogram of -forces. Let the several forces be called A, B, C, D, E, &c. Draw the -parallelogram whose sides express the forces A and B, and let its -diagonal be A′. The force expressed by A′ will be equivalent to A and -B. Then draw the parallelogram whose sides express the forces A′ and -C, and let its diagonal be B′. This diagonal will express a force -mechanically equivalent to A′ and C. But A′ is mechanically equivalent -to A and B, and therefore B′ is mechanically equivalent to A, B, and -C. Next construct a parallelogram, whose sides express the forces B′ -and D, and let its diagonal be C′. The force expressed by C′ will be -mechanically equivalent to the forces B′ and D; but the force B′ is -equivalent to A, B, C, and therefore C′ is equivalent to A, B, C, and -D. By continuing this process it is evident, that a single force may be -found, which will be equivalent to, and may be always substituted for, -any number of forces which act upon the same point. - -If the forces which act upon the point neutralise each other, so that -no motion can ensue, they are said to be in equilibrium. - -(84.) Examples of the composition of motion and pressure are -continually presenting themselves. They occur in almost every instance -of motion or force which falls under our observation. The difficulty is -to find an example which, strictly speaking, is a simple motion. - -When a boat is rowed across a river, in which there is a current, it -will not move in the direction in which it is impelled by the oars. -Neither will it take the direction of the stream, but will proceed -exactly in that intermediate direction which is determined by the -composition of force. - -Let A, _fig. 12._, be the place of the boat at starting; and -suppose that the oars are so worked as to impel the boat towards B -with a force which would carry it to B in one hour, if there were no -current in the river. But, on the other hand, suppose the rapidity of -the current is such, that without any exertion of the rowers the boat -would float down the stream in one hour to C. From C draw C D -parallel to A B, and draw the straight line A D diagonally. -The combined effect of the oars and the current will be, that the boat -will be carried along A D, and will arrive at the opposite bank in -one hour, at the point D. - -If the object be, therefore, to reach the point B, starting from A, -the rowers must calculate, as nearly as possible, the velocity of the -current. They must imagine a certain point E at such a distance above -B that the boat would be floated by the stream from E to B in the time -taken in crossing the river in the direction A E, if there were no -current. If they row towards the point E, the boat will arrive at the -point B, moving in the line A B. - -In this case the boat is impelled by two forces, that of the oars -in the direction A E, and that of the current in the direction -A C. The result will be, according to the parallelogram of forces, -a motion in the diagonal A B. - -The wind and tide acting upon a vessel is a case of a similar kind. -Suppose that the wind is made to impel the vessel in the direction of -the keel; while the tide may be acting in any direction oblique to that -of the keel. The course of the vessel is determined exactly in the same -manner as that of the boat in the last example. - -The action of the oars themselves, in impelling the boat, is an example -of the composition of force. Let A, _fig. 13._, be the head, -and B the stern of the boat. The boatman presents his face towards -B, and places the oars so that their blades press against the water -in the directions C E, D F. The resistance of the water -produces forces on the side of the boat, in the directions G L and -H L, which, by the composition of force, are equivalent to die -diagonal force K L, in the direction of the keel. - -Similar observations will apply to almost every body impelled by -instruments projecting from its sides, and acting against a fluid. The -motions of fishes, the act of swimming, the flight of birds, are all -instances of the same kind. - -(85.) The action of wind upon the sails of a vessel, and the force -thereby transmitted to the keel, modified by the rudder, is a problem -which is solved by the principles of the composition and resolution -of force; but it is of too complicated and difficult a nature to be -introduced with all its necessary conditions and limitations in this -place. The question may, however, be simplified, if we consider the -canvass of the sails to be stretched so completely as to form a plane -surface. Let A B, _fig. 14._, be the position of the sail, -and let the wind blow in the direction C D. If the line C D -be taken to express the force of the wind, let D E C F -be a parallelogram, of which it is the diagonal. The force C D is -equivalent to two forces, one in the direction F D of the plane -of the canvass, and the other E D perpendicular to the sail. -The effect, therefore, is the same as if there were _two winds_, one -blowing in the direction of F D or B A, that is against the -edge of the sail, and the other, E D, blowing full against its -face. It is evident that the former will produce no effect whatever -upon the sail, and that the latter will urge the vessel in the -direction D G. - -Let us now consider this force D G as acting in the diagonal of -the parallelogram D H G I. It will be equivalent to two -forces, D H and D I, acting along the sides. One of these -forces, D H, is in the direction of the keel, and the other, -D I, at right angles to the length of the vessel, so as to urge -it _sideways_. The form of the vessel is evidently such as to offer a -great resistance to the latter force, and very little to the former. -It consequently proceeds with considerable velocity in the direction -D H of its keel, and makes way very slowly in the sideward -direction D I. The latter effect is called _lee-way_. - -From this explanation it will be easily understood, how a wind which -is nearly opposed to the course of a vessel may, nevertheless, be made -to impel it by the effect of sails. The angle B D V, formed -by the sail and the direction of the keel, may be very oblique, as may -also be the angle C D B formed by the direction of the wind -and that of the sail. Therefore the angle C D V, made up of -these two, and which is that formed by the direction of the wind and -that of the keel, may be very oblique. In _fig. 15._ the wind -is nearly contrary to the direction of the keel, and yet there is an -impelling force expressed by the line D H, the line C D -expressing, as before, the whole force of the wind. - -In this example there are two successive decompositions of force. -First, the original force of the wind C D is resolved into two, -E D and F D; and next the element E D, or its equal -D G, is resolved into D I and D H; so that the original -force is resolved into three, viz. F D, D I, D H, -which, taken together, are mechanically equivalent to it. The part -F D is entirely ineffectual; it glides off on the surface of the -canvass without producing any effect upon the vessel. The part D I -produces _lee-way_, and the part D H impels. - -[Illustration: _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -(86.) If the wind, however, be directly contrary to the course which -it is required that the vessel should take, there is no position which -can be given to the sails which will impel the vessel. In this case -the required course itself is resolved into two, in which the vessel -sails alternately, a process which is called _tacking_. Thus, suppose -the vessel is required to move from A to E, _fig. 16._, the wind -setting from E to A. The motion A B being resolved into two, by -being assumed as the diagonal of a parallelogram, the sides A _a_, _a_ -B of the parallelogram are successively sailed over, and the vessel by -this means arrives at B, instead of moving along the diagonal A B. -In the same manner she moves along B _b_, _b_ C, C _c_, _c_ D, D _d_, -_d_ E, and arrives at E. She thus sails continually at a sufficient -angle with the wind to obtain an impelling force, yet at a sufficiently -small angle to make way in her proposed course. - -The consideration of the effect of the rudder, which we have omitted in -the preceding illustration, affords another instance of the resolution -of force. We shall not, however, pursue this example further. - -(87.) A body falling from the top of the mast when the vessel is in -full sail, is an example of the composition of motion. It might be -expected, that during the descent of the body, the vessel having sailed -forward, would leave it behind, and that, therefore, it would fall -in the water behind the stern, or at least on the deck, considerably -behind the mast. On the other hand, it is found to fall at the foot -of the mast, exactly as it would if the vessel were not in motion. To -account for this, let A B, _fig. 17._, be the position of -the mast when the body at the top is disengaged. The mast is moving -onwards with the vessel in the direction A C, so that in the time -which the body would take to fall to the deck, the top of the mast -would move from A to C. But the body being on the mast at the moment it -is disengaged, has this motion A C in common with the mast; and -therefore in its descent it is affected by two motions, viz. that of -the vessel expressed by A C, and its descending motion expressed -by A B. Hence, by the composition of motion, it will be found -at the opposite angle D of the parallelogram, at the end of the fall. -During the fall, however, the mast has moved with the vessel, and has -advanced to C D, so that the body falls at the foot of the mast. - -(88.) An instance of the composition of motion, which is worthy of -some attention, as it affords a proof of the diurnal motion of the -earth, is derived from observing the descent of a body from a very high -tower. To render the explanation of this more simple, we shall suppose -the tower to be on the equator of the earth. Let E P Q, -_fig. 18._, be a section of the earth through the equator, and -let P T be the tower. Let us suppose that the earth moves on its -axis in the direction E P Q. The foot P of the tower will, -therefore, in one day move over the circle E P Q, while the -top T moves over the greater circle T T′ R. Hence it is -evident, that the top of the tower moves with greater speed than the -foot, and therefore in the same time moves through a greater space. Now -suppose a body placed at the top; it participates in the motion which -the top of the tower has in common with the earth. If it be disengaged, -it also receives the descending motion T P. Let us suppose that -the body would take five seconds to fall from T to P, and that in the -same time the top T is moved by the rotation of the earth from T to -T′, the foot being moved from P to P′. The falling body is therefore -endued with two motions, one expressed by T T′, and the other by -T P. The combined effect of these will be found in the usual way -by the parallelogram. Take T _p_ equal to T T′; the body will -move from T to _p_ in the time of the fall, and will meet the ground -at _p_. But since T T′ is greater than P P′, it follows -that the point _p_ must be at a distance from P′ equal to the excess -of T T′ above P P′. Hence the body will not fall exactly -at the foot of the tower, but at a certain distance from it, in the -direction of the earth’s motion, that is, eastward. This is found, by -experiment, to be actually the case; and the distance from the foot of -the tower, at which the body is observed to fall, agrees with that -which is computed from the motion of the earth, to as great a degree of -exactness as could be expected from the nature of the experiment. - -(89.) The properties of compounded motions cause some of the equestrian -feats exhibited at public spectacles to be performed by a kind of -exertion very different from that which the spectators generally -attribute to the performer. For example, the horseman standing on the -saddle leaps over a garter extended over the horse at right angles -to his motion; the horse passing under the garter, the rider lights -upon the saddle at the opposite side. The exertion of the performer, -in this case, is not that which he would use were he to leap from the -ground over a garter at the same height. In the latter case, he would -make an exertion to rise, and, at the same time, to project his body -forward. In the case, however, of the horseman, he merely makes that -exertion which is necessary to rise directly upwards to a sufficient -height to clear the garter. The motion which he has in common with the -horse, compounded with the elevation acquired by his muscular power, -accomplishes the leap. - -To explain this more fully, let A B C, _fig. 19._, be -the direction in which the horse moves, A being the point at which -the rider quits the saddle, and C the point at which he returns to -it. Let D be the highest point which is to be cleared in the leap. At -A the rider makes a leap towards the point E, and this must be done -at such a distance from B, that he would rise from B to E in the time -in which the horse moves from A to B. On departing from A, the rider -has, therefore, two motions, represented by the lines A E and -A B, by which he will move from the point A to the opposite angle -D of the parallelogram. At D, the exertion of the leap being overcome -by the weight of his body, he begins to return downward, and would fall -from D to B in the time in which the horse moves from B to C. But at -D he still retains the motion which he had in common with the horse; -and therefore, in leaving the point D, he has two motions, expressed -by the lines D F and D B. The compounded effects of these -motions carry him from D to C. Strictly speaking, his motion from A to -D, and from D to C, is not in straight lines, but in a curve. It is not -necessary here, however, to attend to this circumstance. - -(90.) If a billiard-ball strike the cushion of the table obliquely, -it will be reflected from it in a certain direction, forming an angle -with the direction in which it struck it. This affords an example of -the resolution and composition of motion. We shall first consider the -effect which would ensue if the ball struck the cushion perpendicularly. - -Let A B, _fig. 20._, be the cushion, and C D the -direction in which the ball moves towards it. If the ball and the -cushion were perfectly inelastic, the resistance of the cushion would -destroy the motion of the ball, and it would be reduced to a state of -rest at D. If, on the other hand, the ball were perfectly elastic, it -would be reflected from the cushion, and would receive as much motion -from D to C after the impact, as it had from C to D before it. Perfect -elasticity, however, is a quality which is never found in these bodies. -They are always elastic, but imperfectly so. Consequently the ball -after the impact will be reflected from D towards C, but with a less -motion than that with which it approached from C to D. - -Now let us suppose that the ball, instead of moving from C to D, moves -from E to D. The force with which it strikes D being expressed by -D E′, equal to E D, may be resolved into two, D F and -D C′. The resistance of the cushion destroys D C′, and the -elasticity produces a contrary force in the direction D C, but -less than D C or D C′, because that elasticity is imperfect. -The line D C expressing the force in the direction C D, let -D G (less than D C) express the reflective force in the -direction D C. The other element D F, into which the force -D E′ is resolved by the impact, is not destroyed or modified by -the cushion, and therefore, on leaving the cushion at D, the ball is -influenced by two forces, D F (which is equal to C E) and -D G. Consequently it will move in the diagonal D H. - -(91.) The angle E D C is in this case called the “angle of -incidence,” and C D H is called “the angle of reflection.” -It is evident, from what has been just inferred, that the ball, being -imperfectly elastic, the angle of incidence must always be less than -the angle of reflection, and with the same obliquity of incidence, -the more imperfect the elasticity is, the less will be the angle of -reflection. - -In the impact of a perfectly elastic body, the angle of reflection -would be equal to the angle of incidence. For then the line D G, -expressing the reflective force, would be taken equal to C D, -and the angle C D H would be equal to C D E. This -is found by experiment to be the case when light is reflected from a -polished surface of glass or metal. - -Motion is sometimes distinguished into _absolute_ and _relative_. What -“relative motion” means is easily explained. If a man walk upon the -deck of a ship from stem to stern, he has a relative motion which is -measured by the space upon the deck over which he walks in a given -time. But while he is thus walking from stem to stern, the ship and -its contents, including himself, are impelled through the deep in -the opposite direction. If it so happen that the motion of the man, -from stem to stern, be exactly equal to the motion of the ship in the -contrary way, the man will be, relatively to the surface of the sea -and that of the earth, at rest. Thus, relatively to the ship, he is in -motion, while, relatively to the surface of the earth, he is at rest. -But still this is not absolute rest. The surface itself is moving by -the diurnal rotation of the earth upon its axis, as well as by the -annual motion in its orbit round the sun. These motions, and others to -which the earth is subject, must be all compounded by the theorem of -the parallelogram of forces before we can obtain the _absolute state_ -of the body with respect to motion or rest. - - - - -CHAP. VI. - -ATTRACTION. - - -(92.) Whatever produces, or tends to produce, a change in the state -of a particle or mass of matter with respect to motion or rest, is a -force. Rest, or uniform rectilinear motion, are therefore the only -states in which any body can exist which is not subject to the present -action of some force. We are not, however, entitled to conclude, that -because a body is observed in one or other of these states, it is -therefore uninfluenced by any forces. It may be under the immediate -action of forces which neutralise each other: thus two forces may be -acting upon it which are equal, and in opposite directions. In such -a case, its state of rest, or of uniform rectilinear motion, will be -undisturbed. The state of uniform rectilinear motion declares more with -respect to the body than the state of rest; for the former betrays the -action of a force upon the body at some antecedent period; this action -having been suspended, while its effect continues to be observed in the -motion which it has produced. - -(93.) When the state of a body is changed from rest to uniform -rectilinear motion, the action of the force is only momentary, in which -case it is called an _impulse_. If a body in uniform rectilinear motion -receive an impulse in the direction in which it is moving, the effect -will be, that it will continue to move uniformly in the same direction, -but its velocity will be increased by the amount of speed which the -impulse would have given it had it been previously quiescent. Thus, if -the previous motion be at the rate of ten feet in a second, and the -impulse be such as would move it from a state of rest at five feet in -a second, the velocity, after the impulse, will be fifteen feet in a -second. - -But if the impulse be received in a direction immediately opposed to -the previous motion, then it will diminish the speed by that amount of -velocity which it would give to the body had it been previously at -rest. In the example already given, if the impulse were opposed to the -previous motion, the velocity of the body after the impulse would be -five feet in a second. If the impulse received in the direction opposed -to the motion be such as would give to the body at rest a velocity -equal to that with which it is moving, then the effect will be, that -after the impulse no motion will exist; and if the impulse would give -it a still greater velocity, the body will be moved in the opposite -direction with an uniform velocity equal to the excess of that due to -the impulse over that which the body previously had. - -When a body in a state of uniform motion receives an impulse in -a direction not coinciding with that of its motion, it will move -uniformly after the impulse in an intermediate direction, which may be -determined by the principles established for the composition of motion -in the last chapter. - -Thus it appears, that whenever the state of a body is changed either -from rest to uniform rectilinear motion or _vice versa_, or from one -state of uniform rectilinear motion to another, differing from that -either in velocity or direction, or in both, the phenomenon is produced -by that peculiar modification of force whose action continues but for a -single instant, and which has been called _an impulse_. - -(94.) In most cases, however, the mechanical state of a body is -observed to be subject to a continual change or tendency to change. We -are surrounded by innumerable examples of this. A body is placed on the -table. A continual pressure is excited on the surface of the table. -This pressure is only the consequence of the continual tendency of the -body to move downwards. If the body were excited by a force of the -nature of an impulse, the effect upon the table would be instantaneous, -and would immediately cease. It would, in fact, be _a blow_. But the -continuation of the pressure proves the continuation of the action of -the force. - -If the table be removed from beneath the body, the force which excites -it being no longer resisted, will produce motion; it is manifested, -not as before, by a tendency to produce motion, but by the actual -exhibition of that phenomenon. Now if the exciting force were an -impulse, the body would descend to the ground with an uniform velocity. -On the other hand, as will hereafter appear, every moment of its fall -increases its speed, and that speed is greatest at the instant it meets -the ground. - -A piece of iron placed at a distance from a magnet approaches it, but -not with an uniform velocity. The force of the magnet continues to act -during the approach of the iron, and each moment gives it increased -motion. - -(95.) The forces which are thus in constant operation, proceed from -secret agencies which the human mind has never been able to detect. All -the analogies of nature prove that they are not the immediate results -of the divine will, but are secondary causes, that is, effects of -some more remote principles. To ascend to these secondary causes, and -thus as it were approach one step nearer to the Creator, is the great -business of philosophy; and the most certain means for accomplishing -this, is diligently to observe, to compare, and to classify the -phenomena, and to avoid assuming the existence of any thing which -has not either been directly observed, or which cannot be inferred -demonstratively from natural phenomena. Philosophy should follow -nature, and not lead her. - -While the law of inertia, established by observation and reason, -declares the inability of matter, from any principle resident in it, to -change its state, all the phenomena of the universe prove that state -to be in constant but regular fluctuation. There is not in existence -a single instance of the phenomenon of absolute rest, or of motion -which is absolutely uniform and rectilinear. In bodies, or the parts -of bodies, there is no known instance of simple passive juxtaposition -unaccompanied by pressure or tension, or some other “tendency to -motion.” Innumerable secret powers are ever at work, compensating, -as it were, for inertia, and supplying the material world with a -substitute for the principles of action and will, which give such -immeasurable superiority to the character of life. - -(96.) The forces which are thus in continual operation, whose existence -is demonstrated by their observed effects, but whose nature, seat, and -mode of operation are unknown to us, are called by the general name -_attractions_. These forces are classified according to the analogies -which prevail among their effects, in the same manner, and according -to the same principles, as organised beings are grouped in natural -history. In that department of natural science, when individuals are -distributed in classes, the object is merely to generalise, and thereby -promote the enlargement of knowledge; but nothing is or ought to be -thus assumed respecting the essence, or real internal constitution of -the individuals. According to their external and observable characters -and qualities they are classed; and this classification should never be -adduced as an evidence of any thing except that similitude of qualities -to which it owed its origin. - -Phenomena are to the natural philosopher what organised beings are to -the naturalist. He groups and classifies them on the same principles, -and with a like object. And as the naturalist gives to each species a -name applicable to the individual beings which exhibit corresponding -qualities, so the philosopher gives to each force or attraction a name -corresponding to the phenomena of which it is the cause. The naturalist -is ignorant of the real essence or internal constitution of the thing -which he nominates, and of the manner in which it comes to possess or -exhibit those qualities which form the basis of his classification; -and the natural philosopher is equally ignorant of the nature, seat, -and mode of operation of the force which he assigns as the cause of an -observed class of effects. - -These observations respecting the true import of the term “attraction” -seem the more necessary to be premised, because the general phraseology -of physical science, taken as language is commonly received, will seem -to convey something more. The names of the several attractions which -we shall have to notice, frequently refer the seat of the cause to -specific objects, and seem to imply something respecting its mode of -operation. Thus, when we say “the magnet attracts a piece of iron,” the -true philosophical import of the words is, “that a piece of iron placed -in the vicinity of the magnet, will move towards it, or placed in -contact, will adhere to it, so that some force is necessary to separate -them.” In the ordinary sense, however, something more than this simple -fact is implied. It is insinuated that the magnet is the seat of the -force which gives motion to the iron; that in the production of the -phenomenon, the magnet is an _agent_ exerting a certain influence, of -which the iron is the _subject_. Of all this, however, there is no -proof; on the contrary, since the magnet must move towards the iron -with just as much force as the iron moves towards the magnet, there is -as much reason to place the seat of the force in the iron, and consider -it as an agent affecting the magnet. But, in fact, the influence -which produces this phenomenon may not be resident in either the one -body or the other. It may be imagined to be a property of a medium in -which both are placed, or to arise from some third body, the presence -of which is not immediately observed. However attractive these and -like speculations may be, they cannot be allowed a place in physical -investigations, nor should consequences drawn from such hypotheses be -allowed to taint our conclusions with their uncertainty. - -The student ought, therefore, to be aware, that whatever may seem -to be implied by the language used in this science in relation to -attractions, nothing is permitted to form the basis of reasoning -respecting them except _their effects_; and whatever be the common -signification of the terms used, it is to these effects, and to these -alone, they should be referred. - -(97.) Attractions may be primarily distributed into two classes; one -consisting of those which exist between the molecules or constituent -parts of bodies, and the other between bodies themselves. The former -are sometimes called, for distinction, _molecular_ or _atomic_ -attractions. - -Without the agency of molecular forces, the whole face of nature would -be deprived of variety and beauty; the universe would be a confused -heap of material atoms dispersed through space, without form, shape, -coherence, or motion. Bodies would neither have the forms of solid, -liquid, or air; heat and light would no longer produce their wonted -effects; organised beings could not exist; life itself, as connected -with body, would be extinct. Atoms of matter, whether distant or in -juxtaposition, would have no tendency to change their places, and all -would be eternal stillness and rest. If, then, we are asked for a proof -of the existence of molecular forces, we may point to the earth and -to the heavens; we may name every object which can be seen or felt. -The whole material world is one great result of the influence of these -powerful agents. - -(98.) It has been proved (11. _et seq._) that the constituent particles -of bodies are of inconceivable minuteness, and that they are not in -immediate contact (23), but separated from each other by interstitial -spaces, which, like the atoms themselves, although too small to be -directly observed, yet are incontestably proved to exist, by observable -phenomena, from which their existence demonstratively follows. The -resistance which every body opposes to compression, proves that a -repulsive influence prevails between the particles, and that this -repulsion is the cause which keeps the atoms separate, and maintains -the interstitial spaces just mentioned. Although this repulsion is -found to exist between the molecules of all substances whatever, yet -it has different degrees of energy in different bodies. This is proved -by the fact, that some substances admit of easy compression, while in -others, the exertion of considerable force is necessary to produce the -smallest diminution in bulk. - -The space around each atom of a body, through which this repulsive -influence extends, is generally limited, and immediately beyond it, a -force of the opposite kind is manifested, viz. attraction. Thus, in -solid bodies, the particles resist separation as well as compression, -and the application of force is as necessary to break the body, or -divide it into separate parts, as to force its particles into closer -aggregation. It is by virtue of this attraction that solid bodies -maintain their figure, and that their parts are not separated and -scattered like those of fluids, merely by their own weight. This force -is called the _attraction of cohesion_. - -The cohesive force acts in different substances with different degrees -of energy: in some its intensity is very great; but the sphere of its -influence apparently very limited. This is the case with all bodies -which are hard, strong, and brittle, which no force can extend or -stretch in any perceptible degree, and which require a great force to -break or tear them asunder. Such, for example, is cast iron, certain -stones, and various other substances. In some bodies the cohesive force -is weak, but the sphere of its action considerable. Bodies which are -easily extended, without being broken or torn asunder, furnish examples -of this. Such are Indian-rubber, or caoutchouc, several animal and -vegetable products, and, in general, all solids of a soft and viscid -kind. - -Between these extremes, the cohesive force may be observed in various -degrees. In lead and other soft metals, its sphere of action is -greater, and its energy less, than in the former examples; but its -sphere less, and energy greater, than in the latter ones. It is from -the influence of this force, and that of the repulsion, whose sphere of -action is still closer to the component atoms, that all the varieties -of texture which we denominate hard, soft, tough, brittle, ductile, -pliant, &c. arise. - -After having been broken, or otherwise separated, the parts of a solid -may be again united by their cohesion, provided any considerable -number of points be brought into sufficiently close contact. When this -is done by mechanical means, however, the cohesion is not so strong -as before their separation, and a comparatively small force will be -sufficient again to disunite them. Two pieces of lead freshly cut, with -smooth surfaces, will adhere when pressed together, and will require a -considerable force to separate them. In the same manner if a piece of -Indian-rubber be torn, the parts separated will again cohere, by being -brought together with a slight pressure. The union of the parts in -such instances is easy, because the sphere through which the influence -of cohesion extends is considerable; but even in bodies in which -this influence extends through a more limited space, the cohesion of -separate pieces will be manifested, provided their surfaces be highly -polished, so as to insure the near approach of a great number of their -particles. Thus, two polished surfaces of glass, metal, or stone, will -adhere when brought into contact. - -In all these cases, if the bodies be disunited by mechanical force, -they will separate at exactly the parts at which they had been united, -so that after their separation no part of the one will adhere to the -other; proving that the force of cohesion of the surfaces brought into -contact is less than that which naturally held the particles of each -together. - -(99.) When a body is in the liquid form, the weight of its particles -greatly predominates over their mutual cohesion, and consequently if -such a body be unconfined it will be scattered by its own weight; if -it be placed in any vessel, it will settle itself, by the force of its -weight, into the lowest parts, so that no space in the vessel below -the upper surface of the liquid will be unoccupied. The particles of -a solid body placed in the vessel have exactly the same tendency, by -reason of their weight; but this tendency is resisted and prevented -from taking effect by their strong cohesion. - -Although this cohesion in solids is much greater than in liquids, and -productive of more obvious effects, yet the principle is not altogether -unobserved in liquids. Water converted into vapour by heat, is divided -into inconceivably minute particles, which ascend in the atmosphere. -When it is there deprived of a part of that heat which gave it the -vaporous form, the particles, in virtue of their cohesive force, -collect into round drops, in which form they descend to the earth. - -In the same manner, if a liquid be allowed to fall gradually from the -lip of a vessel, it will not be dismissed in particles indefinitely -small, as if its mass were incoherent, like sand or powder, but will -fall in drops of considerable magnitude. In proportion as the cohesive -force is greater, these drops affect a greater size. Thus, oil and -viscid liquids fall in large drops; ether, alcohol, and others in small -ones. - -Two drops of rain trickling down a window pane will coalesce when they -approach each other; and the same phenomenon is still more remarkable, -if a few drops of quicksilver be scattered on an horizontal plate of -glass. - -It is the cohesive principle which gives rotundity to grains of shot: -the liquid metal is allowed to fall like rain from a great elevation. -In its descent the drops become truly globular, and before they reach -the end of their fall they are hardened by cooling, so that they retain -their shape. - -It is also, probably, to the cohesive attraction that we should assign -the globular forms of all the great bodies of the universe; the sun, -planets, satellites, &c., which originally may have been in the liquid -state. - -(100.) Molecular attraction is also exhibited between the particles of -liquids and solids. A drop of water will not descend freely when it -is in contact with a perpendicular glass plane: it will adhere to the -glass; its descent will be retarded; and if its weight be insufficient -to overcome the adhesive force, it will remain suspended. - -If a plate of glass be placed upon the surface of water without being -permitted to sink, it will require more force to raise it from the -water than is sufficient merely to balance the weight of the glass. -This shows the adhesion of the water and glass, and also the cohesive -force with which the particles of the water resist separation. - -If a needle be dipped in certain liquids, a drop will remain suspended -at its point when withdrawn from them: and, in general, when a solid -body has been immersed in a liquid and withdrawn, it is _wet_; that -is, some of the liquid has adhered to its surfaces. If no attraction -existed between the solid and liquid, the solid would be in the same -state after immersion as before. This is proved by liquids and solids -between which no attraction exists. If a piece of glass be immersed in -mercury, it will be in the same state when withdrawn as before it was -immersed. No mercury will adhere to it; it will not be _wet_. - -When it rains, the person and vesture are affected only because this -attraction exists between them and water. If it rained mercury, none -would adhere to them. - -(101.) When molecular attraction is exhibited by liquids pervading the -interstices of porous bodies, ascending in crevices or in the bores of -small tubes, it is called _capillary attraction_. Instances of this -are innumerable. Liquids are thus drawn into the pores of sponge, -sugar, lamp-wick, &c. The animal and vegetable kingdom furnish numerous -examples of this class of effects. - -A weight being suspended by a dry rope, will be drawn upwards through -a considerable height, if the rope be moistened with a wet sponge. The -attraction of the particles composing the rope for the water is in this -case so powerful, that the tension produced by several hundred weight -cannot expel them. - -A glass tube, of small bore, being dipped in water tinged by mixture -with a little ink, will retain a quantity of the liquid suspended when -withdrawn. The height of the liquid in the tube will be seen by looking -through it. It is found that the less the bore of the tube is, the -greater will be the height of the column sustained. A series of such -tubes fixed in the same frame, with their lower orifices at the same -level, and with bores gradually decreasing, being dipped in the liquid, -will exhibit columns gradually increasing. - -A _capillary syphon_ is formed of a hank of cotton threads, one end of -which is immersed in the vessel containing the liquid, and the other is -carried into the vessel into which the liquid is to be transferred. The -liquid may be thus drawn from the one vessel into the other. The same -effect may be produced by a glass syphon with a small bore. - -(102.) It frequently happens that a _molecular repulsion_ is exhibited -between a solid and a liquid. If a piece of wood be immersed in -quicksilver, the liquid will be depressed at that part of the surface -which is near the wood; and in like manner, if it be contained in a -glass vessel, it will be depressed at the edges. In a barometer tube, -the surface of the mercury is convex, owing partly to the repulsion -between the glass and mercury. - -All solids, however, do not repel mercury. If any golden trinket be -dipped in that liquid, or even be exposed for a moment to contact -with it, the gold will be instantly intermingled with particles of -quicksilver, the metal changes its colour, and becomes white like -silver, and the mercury can only be extricated by a difficult process. -Chains, seals, rings, &c. should always be laid aside by those engaged -in experiments or other processes in which mercury is used. - -(103.) Of all the forms under which molecular force is exhibited, -that in which it takes the name of _affinity_ is attended with the -most conspicuous effects. Affinity is in chemistry what inertia is -in mechanics, the basis of the science. The present treatise is not -the proper place for any detailed account of this important class -of natural phenomena. Those who seek such knowledge are referred -to our treatise on CHEMISTRY. Since, however, affinity sometimes -influences the mechanical state of bodies, and affects their mechanical -properties, it will be necessary here to state so much respecting it as -to render intelligible those references which we may have occasion to -make to such effects. - -When the particles of different bodies are brought into close contact, -and more especially when, being in a fluid state, they are mixed -together, their union is frequently observed to produce a compound -body, differing in its qualities from either of the component bodies. -Thus the bulk of the compound is often greater or less than the united -volumes of the component bodies. The component bodies may be of the -ordinary temperature of the atmosphere, and yet the compound may be of -a much higher or lower temperature. The components may be liquid, and -the compound solid. The colour of the compound may bear no resemblance -whatever to that of the components. The species of molecular action -between the components, which produce these and similar, effects, is -called _affinity_. - -(104.) We shall limit ourselves here to the statement of a few examples -of these phenomena. - -If a pint of water and a pint of sulphuric acid be mixed, the compound -will be considerably less than a quart. The density of the mixture is, -therefore, greater than that which would result from the mere diffusion -of the particles of the one fluid through those of the other. The -particles have assumed a greater proximity, and therefore exhibit a -mutual attraction. - -In this experiment, although the liquids before being mixed be of the -temperature of the surrounding air, the mixture will be so intensely -hot, that the vessel which contains it cannot be touched without pain. - -If the two aeriform fluids, called oxygen and hydrogen, be mixed -together in a certain proportion, the compound will be water. In this -case, the components are different from the compound, not merely in -the one being _air_ and the other _liquid_, but in other respects -not less striking. The compound water extinguishes fire, and yet of -the components, hydrogen is one of the most inflammable substances -in nature, and the presence of oxygen is indispensably necessary to -sustain the phenomenon of combustion. - -Oxygen gas, united with quicksilver, produces a compound of a black -colour, the quicksilver being white and the gas colourless. When -these substances are combined in another proportion, they give a red -compound. - -(105.) Having noticed the principal molecular forces, we shall now -proceed to the consideration of those attractions which are exhibited -between bodies existing in masses. The influence of molecular -attractions is limited to insensible distances. On the contrary, the -forces which are now to be noticed act at considerable distances, -and to the influence of some there is no limit, the effect, however, -decreasing as the distance increases. - -The effect of the loadstone on iron is well known, and is one of this -class of forces. For a detailed account of this force, and the various -phenomena of which it is the cause, the reader is referred to our -treatise on MAGNETISM. - -When glass, wax, amber, and other substances are submitted to friction -with silken or woollen cloth, they are observed to attract feathers, -and other light bodies placed near them. A like effect is produced -in several other ways, and is attended with other phenomena, the -discussion of which forms a principal part of physical science. The -force thus exhibited is called electricity. For details respecting it, -and for its connection with magnetism, the reader is referred to our -treatises on ELECTRICITY and ELECTRO-MAGNETISM. - -(106.) These attractions exist either between bodies of particular -kinds, or are developed by reducing the bodies which manifest them to -a certain state by friction, or some other means. There is, however, -an attraction, which is manifested between bodies of all species, and -under all circumstances whatever; an attraction, the intensity of which -is wholly independent of the nature of the bodies, and only depends on -their masses and mutual distances. Thus, if a mass of metal and a mass -of clay be placed in the vast abyss of space, at a mile asunder, they -will instantly commence to approach each other with certain velocities. -Again, if a mass of stone and of wood respectively equal to the former, -be placed at a like distance, they will also commence to approach -each other with the same velocities as the former. This universal -attraction, which only depends on the quantity of the masses and their -mutual distances, is called the “attraction of gravitation.” We shall -first explain the “law” of this attraction, and shall then point out -some of the principal phenomena by which its existence and its laws are -known. - -(107.) The “law of gravitation” sometimes from its universality called -the “law of nature,” may be explained as follows: - -Let us suppose two masses, A and B, placed beyond the influence or -attraction of any other bodies, in a state of rest, and at any proposed -distance from each other. By their mutual attraction they will approach -each other, but not with the same velocity. The velocity of A will be -greater than that of B, in the same proportion as its mass is less -than that of B. Thus, if the mass of B be twice that of A, while A -approaches B through a space of two feet, B will approach A through a -space of one foot. Hence it follows, that the force with which A moves -towards B is equal to the force with which B moves towards A (68). This -is only a consequence of the property of inertia, and is an example of -the equality of action and reaction, as explained in Chapter IV. The -velocity with which A and B approach each other is estimated by the -diminution of their distance, A B, by their mutual approach in a -given time. Thus, if in one second A move towards B through a space of -two feet, and in the same time B moves towards A through the space of -one foot, they will approach each other through a space of three feet -in a second, which will be their relative velocity (91). - -If the mass of B be doubled, it will attract A with double the former -force, or, what is the same, will cause A to approach B with double the -former velocity. If the mass of B be trebled, it will attract A with -treble the first force, and, in general, while the distance A B -remains the same, the attractive force of B upon A will increase or -diminish in exactly the same proportion as the mass of B is increased -or diminished. - -In the same manner, if the mass A be doubled, it will be attracted by -B with a double force, because B exerts the same degree of attraction -on every part of the mass A, and any addition which it may receive will -not diminish or otherwise affect the influence of B on its former mass. - -To express this in general arithmetical symbols let _a_ and _b_ express -the space through which A and B respectively would be moved towards -each other by their mutual attraction. We would then have - - A × _a_ = B × _b_. - -Thus, it is a general law of gravitation, that so long as the distance -between two bodies remains the same, each will attract and be attracted -by the other, in proportion to its mass; and any increase or decrease -of the mass will cause a corresponding increase or decrease in the -amount of the attraction. - -(108.) We shall now explain the law, according to which the attraction -is changed, by changing the distance between the bodies. At the -distance of one mile the body B attracts A with a certain force. At the -distance of two miles, the masses not being changed, the attraction of -B upon A will be one-fourth of its amount at the distance of one mile. -At the distance of three miles, it will be one-ninth of its original -amount; at four miles, it is reduced to a sixteenth, and so on. The -following table exhibits the diminution of the attraction corresponding -to the successive increase of distance: - - +-----------+---+----+----+----+----+----+----+----+----+ - |Distance | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | &c.| - +-----------+---+----+----+----+----+----+----+----+----+ - |Attraction | 1 | 1/4| 1/9|1/16|1/25|1/36|1/49|1/64| &c.| - +-----------+---+----+----+----+----+----+----+----+----+ - -In ARITHMETIC, that number which is found by multiplying any proposed -number by itself, is called its _square_. Thus 4, that is, 2 multiplied -by 2, is the square of 2; 9 that is, 3 times 3, is the square of 3, and -so on. On inspecting the above table, it will be apparent, therefore, -that the attraction of gravitation decreases in the same proportion as -the square of the distance from the attracting body increases, the mass -of both bodies in this case being supposed to remain the same; but if -the mass of either be increased or diminished, the attraction will be -increased or diminished in the same proportion. - -(109.) Hence the _law of gravitation_ may be thus expressed: “The -mutual attraction of two bodies increases in the same proportion -as their masses are increased, and as the square of their distance -is decreased; and it decreases in proportion as their masses are -decreased, and as the square of their distance is increased.” - -This law may be more clearly expressed by means of general symbols. -Let _f_ express the force with which a mass weighing 1 lb. will -attract another mass weighing 1 lb., at the distance of 1 foot. -The force with which they will mutually attract, when removed to the -distance expressed in feet by D, will be - - _f_/D^2 - -that is, the force _f_ divided by the square of the number D. - -If one of the bodies, instead of weighing 1 lb., weigh the number -of pounds expressed by A, their mutual attraction will be increased A -times, and will therefore be expressed by - - (A × _f_)/D^2 - -In fine, if the other be also the number of pounds expressed by B, -their mutual attraction will be - - (A × B × _f_)/D^2 - -(110.) Having explained the law of gravitation, we shall now proceed to -show how the existence of this force is proved, and its law discovered. - -The earth is known to be a globular mass of matter, incomparably -greater than any of the detached bodies which are found upon its -surface. If one of these bodies suspended at any proposed height -above the surface of the earth be disengaged, it will be observed to -descend perpendicularly to the earth, that is, in the direction of the -earth’s centre. The force with which it descends will also be found -to be in proportion to the mass, without any regard to the species of -the body. These circumstances are consistent with the account which -we have given of gravitation. But by that account we should expect, -that as the falling body is attracted with a certain force towards the -earth, the earth itself should be attracted towards it by the same -force; and instead of the falling body moving towards the earth, which -is the phenomenon observed, the earth and it should move towards each -other, and meet at some intermediate point. This, in fact, is the case, -although it is impossible to render the motion of the earth observable, -for reasons which will easily be understood. - -Since all the bodies around us participate in this motion, it would -not be directly observable, even though its quantity were sufficiently -great to be perceived under other circumstances. But setting aside -this consideration, the space through which the earth moves in such a -case is too minute to be the subject of sensible observation. It has -been stated (107), that when two bodies attract each other, the space -through which the greater approaches the lesser, bears to that through -which the lesser approaches the greater, the same proportion as the -mass of the lesser bears to the mass of the greater. Now the mass of -the earth is more than 1000,000,000,000,000 times the mass of any body -which is observed to fall on its surface; and therefore if even the -largest body which can come under observation were to fall through an -height of 500 feet, the corresponding motion of the earth would be -through a space less than the 1000,000,000,000,000th part of 500 feet, -which is less than the 100,000,000,000th part of an inch. - -The attraction between the earth and detached bodies on its surface is -not only exhibited by the descent of these bodies when unsupported, -but by their pressure when supported. This pressure is what is called -_weight_. The phenomena of weight, and the descent of heavy bodies, -will be fully investigated in the next chapter. - -(111.) It is not alone by the direct fall of bodies that the -gravitation of the earth is manifested. The curvilinear motion of -bodies projected in directions different from the perpendicular, is -a combination of the effects of the uniform velocity which has been -given to the projectile by the impulse which it has received, and the -accelerated velocity which it receives from the earth’s attraction. -Suppose a body placed at any point P, _fig. 21._, above the -surface of the earth, and let P C be the direction of the earth’s -centre. If the body were allowed to move without receiving any impulse, -it would descend to the earth in the direction P A, with an -accelerated motion. But suppose that at the moment of its departure -from P, it receives an impulse in the direction P B, which would -carry it to B in the time the body would fall from P to A, then, by -the composition of motion, the body must at the end of that time be -found in the line B D, parallel to P A. If the motion in -the direction of P A were uniform, the body P would in this case -move in the straight line from P to D. But this is not the case. The -velocity of the body in the direction P A is at first so small as -to produce very little deflection of its motion from the line P B. -As the velocity, however, increases, this deflection increases, so that -it moves from P to D in a curve, which is convex, towards P B. - -The greater the velocity of the projectile in the direction P A, -the greater sweep the curve will take. Thus it will successively take -the forms P D, P E, P F, &c., and that velocity can be -computed, which (setting aside the resistance of the air) would cause -the projectile to go completely round the earth, and return to the -point P from which it departed. In this case, the body P would continue -to revolve round the earth like the moon. Hence it is obvious, that the -phenomenon of the revolution of the moon round the earth, is nothing -more than the combined effects of the earth’s attraction, and the -impulse which it received when launched into space by the hand of its -Creator. - -(112.) This is a great step in the analysis of the phenomenon of -gravitation. We have thus reduced to the same class two effects -apparently very dissimilar, the rectilinear descent of a heavy body, -and the nearly circular revolution of the moon round the earth. Hence -we are conducted to a generalisation still more extensive. - -As the moon’s revolution round the earth, in an orbit nearly circular, -is caused by the combination of the earth’s attraction, and an original -projectile impulse, so also the singular phenomena of the planets’ -revolution round the sun in orbits nearly circular, must be considered -an effect of the same class, as well as the revolution of the -satellites of those planets which are attended by such bodies. Although -the orbits in which the comets move deviate very much from circles, yet -this does not hinder the application of the same principle to them, -their deviation from circles not depending on the sun’s attraction, but -only on the direction and force of the original impulse which put them -in motion. - -(113.) We therefore conclude that gravitation is the principle -which, as it were, animates the universe. All the great changes and -revolutions of the bodies which compose our system, can be traced -to or derived from this principle. It still remains to show how -that remarkable law, by which this force is declared to increase or -decrease in the same proportion as the square of the distance from -the attracting body is decreased or increased, may be verified and -established. - -It has been shown, that the curvilinear path of a projectile -depends on, and can be derived, by mathematical reasoning, from the -consideration of the intensity of the earth’s attraction, and the -force of the original impulse, or the velocity of projection. In the -same manner, by a reverse process, when we know the curve in which a -projectile moves, we can infer the amount of the attracting force which -gives the curvature to its path. In this way, from our knowledge of the -curvature of the moon’s orbit, and the velocity with which she moves, -the intensity of the attraction which the earth exerts upon her can be -exactly ascertained. Upon comparing this with the force of gravitation -at the earth’s surface, it is found that the latter is as many times -greater than the former, as the square of the moon’s distance is -greater than the square of the distance of a body on the surface of the -earth from its centre. - -(114.) If this were the only fact which could be brought to establish -the law of gravitation, it might be thought to be an accidental -relation, not necessarily characterising the attraction of gravitation. -Upon examining the orbits and velocities of the several planets, the -same result is, however, obtained. It is found that the forces with -which they are severally attracted by the sun are great, in exactly the -same proportion as the squares of the several numbers expressing their -distances are small. The mutual gravitation of bodies on the surface of -the earth towards each other is lost in the predominating force exerted -by the earth upon all of them. Nevertheless, in some cases, this effect -has not only been observed, but actually measured. - -A plumb-line, under ordinary circumstances, hangs in a direction truly -vertical; but if it be near a large mass of matter, as a mountain, -it has been observed to be deflected from the true vertical, towards -the mountain. This effect was observed by Dr. Maskeline near the -mountain called Skehallien, in Scotland, and by French astronomers near -Chimboraco. For particulars of these observations, see our treatise on -GEODÆSY. - -Cavendish succeeded in exhibiting the effects of the mutual gravitation -of metallic spheres. Two globes of lead A, B, each about a foot in -diameter, were placed at a certain distance asunder. A light rod, -to the ends of which were attached small metallic balls C, D, was -suspended at its centre E from a fine wire, and the rod was placed -as in _fig. 22._, so that the attractions of each of the leaden -globes had a tendency to turn the rod round the centre E in the same -direction. A manifest effect was produced upon the balls C, D, by the -gravitation of the spheres. In this experiment, care must be taken that -no magnetic substance is intermixed with the materials of the balls. - -Having so far stated the principles on which the law of gravitation is -established, we shall dismiss this subject without further details, -since it more properly belongs to the subject of PHYSICAL ASTRONOMY; to -which we refer the reader for a complete demonstration of the law, and -for the detailed development of its various and important consequences. - - - - -CHAP. VII. - -TERRESTRIAL GRAVITY. - - -(115.) GRAVITATION is the general name given to this attraction, by -whatever masses of matter it may be manifested. As exhibited in the -effects produced by the earth upon surrounding bodies, it is called -“terrestrial gravity.” - -As the attraction of the earth is directed towards its centre, it might -be expected that two plumb-lines should appear not to be parallel, but -so inclined to each other as to converge to a point under the surface -of the earth. Thus, if A B and C D, _fig. 23._, be two -plumb-lines, each will be directed to the centre O, where, if their -directions were continued, they would meet. In like manner, if two -bodies were allowed to fall from A and C, they would descend in the -directions A B and C D, which converge to O. Observation, -on the contrary, shows, that plumb-lines suspended in places not far -distant from each other are truly parallel; and that bodies allowed -to fall descend in parallel lines. This apparent parallelism of the -direction of terrestrial gravity is accounted for by the enormous -proportion which the magnitude of the earth bears to the distance -between the two plumb-lines or the two falling bodies which are -compared. If the distance between the places B, D, were 1200 feet, the -inclination of the lines A B and C D would not amount to a -quarter of a minute, or the 240th part of a degree. But the distance, -in cases where the parallelism is assumed, is never greater than, and -seldom so great as, a few yards; and hence the inclination of the -directions A B and C D is too small to be appreciated by any -practical measure. In the investigation of the phenomena of falling -bodies, we shall, therefore, assume, that all the particles of the -same body are attracted in parallel directions, perpendicular to an -horizontal plane. - -(116.) Since the intensity of terrestrial gravity increases as the -square of the distance decreases, it might be expected that, as a -falling body approaches the earth, the force which accelerates it -should be continually increasing, and, strictly speaking, it is so. But -any height through which we observe falling bodies to descend bears so -very small a proportion to the whole distance from the centre, that -the change of intensity of the force of gravity is quite beyond any -practical means of estimating it. The radius, or the distance from -the surface of the earth to its centre, is 4000 miles. Now, suppose -a body descended through the height of half a mile, a distance very -much beyond those used in experimental enquiries, the distances from -the centre, at the beginning and end of the fall, are then in the -proportion of 8000 to 8001, and therefore the proportion of the force -of attraction at the commencement to the force at the end, being that -of the squares of these numbers, is 64,000,000 to 64,016,001, which, in -the whole descent, is an increase of about one part in 4000; a quantity -practically insignificant. We shall, therefore, in explaining the laws -of falling bodies, assume that, in the entire descent, the body is -urged by a force of uniform intensity. - -Although the force which attracts all parts of the same body during -its descent in a given place is the same, yet the force of gravity, -at different parts of the earth’s surface, has different intensities. -The intensity diminishes with the latitude, so that it is greater -towards the poles, and lesser towards the equator. The causes of -this variation, its law, and the experimental proofs of it, will be -explained, when we shall treat of centrifugal force, and the motion of -pendulums. It is sufficient merely to advert to it in this place. - -(117.) Since the earth’s attraction acts separately and equally on -every particle of matter, without regard to the nature or species of -the body, it follows that all bodies, of whatever kind, or whatever -be their masses, must be moved with the same velocity. If two equal -particles of matter be placed at a certain distance above the surface -of the earth, they will fall in parallel lines, and with exactly the -same speed, because the earth attracts them equally. In the same -manner, a thousand particles would fall with equal velocities. Now, -these circumstances will in no wise be changed if those 1000 particles, -instead of existing separately, be aggregated into two solid masses, -one consisting of 990 particles, and the other of 10. We shall thus -have a heavy body and a light one, and, according to our reasoning, -they must fall to the earth with the same speed. - -Common experience, however, is not always consistent with this -doctrine. What are called light substances, as feathers, gold-leaf, -paper, &c., are observed to fall slowly and irregularly, while -heavier masses, as solid pieces of metal, stones, &c., fall rapidly. -Nay, there are not a few instances in which the earth, instead of -attracting bodies, seems to repel them, as in the case of smoke, -vapours, balloons, and other substances which actually ascend. We are -to consider that the mass of the earth is not the only agent engaged in -these phenomena. The earth is surrounded by an atmosphere composed of -an elastic or aeriform fluid. This atmosphere has certain properties, -which will be explained in our treatise on PNEUMATICS, and which are -the causes of the anomalous circumstances alluded to. Light bodies -rise in the atmosphere, for the same reason that a piece of cork rises -from the bottom of a vessel of water; and other light bodies fall more -slowly than heavy ones, for the same reason that an egg in water falls -to the bottom more slowly than a leaden bullet. This treatise is not -the place to give a direct explanation of these phenomena. It will -be sufficient for our present purpose to show, that if there were no -atmosphere, all bodies, heavy and light, would fall at the same rate. -This may easily be accomplished by the aid of an air-pump. Having -by that instrument abstracted the air from a tall glass vessel, we -are enabled, by means of a wire passing air-tight through a hole in -the top, to let fall several bodies from the top of the vessel to the -bottom. These, whether they be feathers, paper, gold-leaf, pieces of -money, &c. all descend with the same speed, and strike the bottom at -the same moment. - -(118.) Every one who has seen a heavy body fall from a height, has -witnessed the fact, that its velocity increases as it approaches -the ground. But if this were not observable by the eye, it would be -betrayed by the effects. It is well known, that the force with which a -body strikes the ground increases with the height from whence it has -fallen. This force, however, is proportional to the velocity which it -has at the moment it meets the ground, and therefore this velocity -increases with the height. - -When the observations on attraction in the last chapter are well -understood, it will be evident that the velocity which a body has -acquired in falling from any height, is the accumulated effects of the -attraction of terrestrial gravity during the whole time of the fall. -Each instant of the fall a new impulse is given to the body, from which -it receives additional velocity; and its final velocity is composed -of the aggregation of all the small increments of velocity which are -thus communicated. As we are at present to suppose the intensity of the -attraction invariable, it will follow that the velocity communicated to -the body in each instant of time will be the same, and therefore that -the whole quantity of velocity produced or accumulated at the end of -any time is proportional to the length of that time. Thus, if a certain -velocity be produced in a body having fallen for one second, twice that -velocity will be produced when it has fallen for two seconds, thrice -that velocity in three seconds, and so on. Such is the fundamental -principle or characteristic of _uniformly accelerated motion_. - -(119.) In examining the circumstances of the descent of a body, the -time of the fall and the velocity at each instant of that time are not -the only things to be attended to. The spaces through which it falls in -given intervals of time, counted either from the commencement of its -fall, or from any proposed epoch of the descent, are equally important -objects of enquiry. To estimate the space in reference to the time and -the final velocity, we must consider that this space has been moved -through with varying speed. From a state of rest at the beginning of -the fall, the speed gradually increases with the time, and the final -velocity is greater still than that which the body had at any preceding -instant during its descent. We cannot, therefore, _directly_ appreciate -the space moved through in this case by the time and final velocity. -But as the velocity increases uniformly with the time, we shall obtain -the average speed, by finding that which the body had in the middle of -the interval which elapsed between the beginning and end of the fall, -and thus the space through which the body has actually fallen is that -through which it would move in the same time with this average velocity -uniformly continued. - -But since the velocity which the body receives in any time, counted -from the beginning of its descent, is in the proportion of that time, -it follows that the velocity of the body after half the whole time of -descent is half the final velocity. From whence it appears, that the -height from which a body falls in any proposed time is equal to the -space through which a body would move in the same time with half the -final velocity, and it is therefore equal to half the space which would -be moved through in the same time with the final velocity. - -(120.) It follows from this reasoning, that between the three -quantities, the height, the time, and the final velocity, which enter -into the investigation of the phenomena of falling bodies, there are -two fixed relations: _First_, the time, counted from the beginning of -the fall and the final velocity, are proportional the one to the other; -so that as one increases, the other increases in the same proportion. -_Secondly_, the height being equal to half the space which would be -moved through in the _time_ of the fall, with the _final velocity_, -must have a fixed proportion to these two quantities, viz. the _time_ -and the _final velocity_, or must be proportional to the product of the -two numbers which express them. - -But since the time is always proportional to the final velocity, they -may be expressed by equal numbers, and the product of equal numbers -is the square of either of them. Hence, the product of the numbers -expressing the time and final velocity is equivalent to the square -of the number expressing the time, or to the square of the number -expressing the final velocity. Hence we infer, that the height is -always proportional to the square of the time of the fall, or to the -square of the final velocity. - -(121.) The use of a few mathematical characters will render these -results more distinct, even to students not conversant with -mathematical science. - -Let S = the height from which the body falls, expressed in feet. - - V = the velocity at the end of the fall in feet per second. - - T = the number of seconds in the time of the fall. - - _g_ = the number of feet through which a body would fall in one - second. - -It will therefore follow that the velocity acquired in one second will -be 2_g_, and the velocity acquired in T seconds will therefore be 2_g_ -× T; so that - - V = 2_g_ × T [1] - -Since the space which a body falls through in T seconds is found by -multiplying the space it falls through in one second by T^2, we shall -have - - S = _g_ × T^2 [2] - -from which, combined with [1] we deduce - - S = V^2/(4_g_) [3] - - S = (1/2)V × T [4] - -By these formularies, if the height through which a body falls freely -in one second be known, the height through which it will fall in any -proposed time may be computed. For since the height is proportional -to the square of the time, the height through which it will fall in -_two_ seconds will be _four_ times that which it falls through in -_one_ second. In _three_ seconds it will fall through _nine_ times -that space; in _four_ seconds, _sixteen_ times; in _five_ seconds, -_twenty-five_ times, and so on. The following, therefore, is a general -rule to find the height through which a body will fall in any given -time: “Reduce the given time to seconds, take the square of the number -of seconds in it, and multiply the height through which a body falls in -one second by that number; the result will be the height sought.” - -The following table exhibits the heights and corresponding times as far -as 10 seconds: - - +-------+---+---+---+----+----+----+----+----+----+-----+ - |Time | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | - +-------+---+---+---+----+----+----+----+----+----+-----+ - |Height | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | - +-------+---+---+---+----+----+----+----+----+----+-----+ - -Each unit in the numbers of the first row expresses a second of time, -and each unit in those of the second row expresses the height through -which a body falls freely in a second. - -(122.) If a body fall continually for several successive seconds, -the spaces which it falls through in each succeeding second have a -remarkable relation among each other, which may be easily deduced from -the preceding table. Taking the space moved through in the first second -still as our unit, four times that space will be moved through in the -first two seconds. Subtract from this 1, the space moved through in the -first second, and the remainder 3 is the space through which the body -falls in the _second_ second. In like manner if 4, the height fallen -through in the first two seconds, be subtracted from 9, the height -fallen through in the first three seconds, the remainder 5 will be the -space fallen through in the third second. To find the space fallen -through in the fourth second, subtract 9, the space fallen through in -the first three seconds, from 16, the space fallen through in the first -four seconds, and the result is 7, and so on. It thus appears that if -the space fallen through in the first second be called 1, the spaces -described in the second, third, fourth, fifth, &c. seconds, will be -expressed by the odd numbers respectively, 3, 5, 7, 9, &c. This places -in a striking point of view the accelerated motion of a falling body, -the spaces moved through in each succeeding second being continually -increased. - -(123.) If velocity be estimated by the space through which the body -would move uniformly in one second, then the final velocity of a body -falling for one second will be 2; for with that final velocity the body -would in one second move through twice the height through which it has -fallen. - -(124.) Since the final velocity increases in the same proportion as -the time, it follows that after two seconds it is twice its amount -after one, and after three seconds thrice that, and so on. Thus, the -following table exhibits the final velocities corresponding to the -times of descent: - - +---------------+---+---+---+---+----+----+----+----+----+----+ - |Time | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | - +---------------+---+---+---+---+----+----+----+----+----+----+ - |Final velocity | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | - +---------------+---+---+---+---+----+----+----+----+----+----+ - -The numbers in the second row express the spaces through which a body -with the final velocity would move in one second, the unit being, as -usual, the space through which a body falls freely in one second. - -(125.) Having thus developed theoretically the laws which characterise -the descent of bodies, falling freely by the force of gravity, or by -any other uniform force of the same kind, it is necessary that we -should show how these laws can be exhibited by actual experiment. -There are some circumstances attending the fall of heavy bodies which -would render it difficult, if not impossible, to illustrate, by the -direct observation of this phenomenon, the properties which have -been explained in this chapter. A body falling freely by the force -of gravity, as we shall hereafter prove, descends in one second of -time through a height of about 16 feet[1]; in two seconds, it would, -therefore, fall through four times that space, or 64 feet; in three -seconds, through 9 times the height, or 144 feet; and in four seconds, -through 256 feet. In order, therefore, to be enabled to observe the -phenomena for only four seconds, we should command an height of at -least 256 feet. But further; the velocity at the end of the first -second would be at the rate of 32 feet per second; at the end of the -second second, it would be 64 feet per second; and towards the end of -the fall it would be about 120 feet per second. It is evident that this -great degree of rapidity would be a serious impediment to accurate -observation, even though we should be able to command the requisite -height. It appears therefore that the number expressed by _g_ in the -preceding formulæ is 16·083. - -[1] More exactly through 16-1/12 feet, or 193 inches. - -It occurred to Mr. George Attwood, a mathematician and natural -philosopher of the last century, that all the phenomena of falling -bodies might be experimentally exhibited and accurately observed, if -a force of the same kind as gravity, viz. an uniformly accelerating -force, be used, but of a much less intensity; so that while the motion -continues to be governed by the same laws, its quantity may be so much -diminished, that the final velocity, even after a descent of many -seconds, shall be so moderated as to admit of most deliberate and exact -observation. This being once accomplished, nothing more would remain -but to find the height through which a body would fall in one second, -or, what is the same, the proportion of the force of gravity to the -mitigated but uniform accelerating force thus substituted for it. - -(126.) To realise this notion, Attwood constructed a wheel turning on -its axle with very little friction, and having a groove on its edge -to receive a string. Over this wheel, and in the groove, he placed a -fine silken cord, to the ends of which were attached equal cylindrical -weights. Thus placed, the weights perfectly balance each other, and no -motion ensues. To one of the weights he then added a small quantity, so -as to give it a slight preponderance. The loaded weight now began to -descend, drawing up on the other side the unloaded weight. The descent -of the loaded weight, under these circumstances, is a motion exactly -of the _same kind_ as the descent of a heavy body falling freely by -the force of gravity; that is, it increases according to the same -laws, though at a very diminished rate. To explain this, suppose that -the loaded weight descends from a state of rest through one inch in -a second, it will descend through 4 inches in two seconds, through 9 -in three, through 16 in four, and so on. Thus in 20 seconds, it would -descend through 400 inches, or 33 feet 4 inches, a height which, if it -were necessary, could easily be commanded. - -It might, perhaps, be thought, that since the weights suspended at the -ends of the thread are in equilibrium, and therefore have no tendency -either to move or to resist motion, the additional weight placed upon -one of them ought to descend as rapidly as it would if it were allowed -to fall freely and unconnected with them. It is very true that this -weight will receive from the attraction of the earth the same force -when placed upon one of the suspended weights, as it would if it were -disengaged from them; but in the consequences which ensue, there is -this difference. If it were unconnected with the suspended weights, -the whole force impressed upon it would be expended in accelerating -its descent; but being connected with the equal weights which sustain -each other in equilibrium, by the silken cord passing over the wheel, -the force which is impressed upon the added weight is expended, not -as before, in giving velocity to the added weight alone, but to it -together with the two equal weights appended to the string, one of -which descends with the added weight, and the other rises on the -opposite side of the wheel. Hence, setting aside any effect which the -wheel itself produces, the velocity of the descent must be lessened -just in proportion as the mass among which the impressed force is to -be distributed is increased; and therefore the _rate_ of the fall -bears to that of a body falling freely the same proportion as the added -weight bears to the sum of the masses of the equal suspended weights -and the added weight. Thus the smaller the added weight is, and the -greater the equal suspended weights are, the slower will the rate of -descent be. - -To render the circumstances of the fall conveniently observable, a -vertical shaft (see _fig. 24._) is usually provided, which is -placed behind the descending weight. This pillar is divided to inches -and halves, and of course may be still more minutely graduated, if -necessary. A stage to receive the falling weight is moveable on this -pillar, and capable of being fixed in any proposed position by an -adjusting screw. A pendulum vibrating seconds, the beat of which ought -to be very audible, is placed near the observer. The loaded weight -being thus allowed to descend for any proposed time, or from any -required height, all the circumstances of the descent may be accurately -observed, and the several laws already explained in this chapter may be -experimentally verified. - -(127.) The laws which govern the descent of bodies by gravity, being -reversed, will be applicable to the ascent of bodies projected upwards. -If a body be projected directly upwards with any given velocity, it -will rise to the height from which it should have fallen to acquire -that velocity. The earth’s attraction will, in this case, gradually -deprive the body of the velocity which is communicated to it at the -moment at which it is projected. Consequently, the phenomenon will be -that of _retarded motion_. At each part of its ascent it will have the -same velocity which it would have if it descended to the same place -from the highest point to which it rises. Hence it is clear, that all -the particulars relative to the ascent of bodies may be immediately -inferred from those of their descent, and therefore this subject -demands no further notice. - -To complete the investigation of the phenomena of falling bodies, it -would now only remain to explain the method of ascertaining the exact -height through which a body would descend in one second, if unresisted -by the atmosphere, or any other disturbing cause. As the solution -of this problem, however, requires the aid of principles not yet -explained, it must for the present be postponed. - - - - -CHAP. VIII. - -OF THE MOTION OF BODIES ON INCLINED PLANES AND CURVES. - - -(128.) In the last chapter, we investigated the phenomena of bodies -descending freely in the vertical direction, and determined the laws -which govern, not their motion alone, but that of bodies urged by any -uniformly accelerating force whatever. We shall now consider some of -the most ordinary cases in which the free descent of bodies is impeded, -and the effects of their gravitation modified. - -(129.) If a body, urged by any forces whatever, be placed upon a -hard unyielding surface, it will evidently remain at rest, if the -resultant (76) of all the forces which are applied to it be directed -perpendicularly against the surface. In this case, the effect produced -is pressure, but no motion ensues. If only one force act upon the -body, it will remain at rest, provided the direction of that force be -perpendicular to the surface. - -But the effect will be different, if the resultant of the forces which -are applied to the body be oblique to the surface. In that case this -resultant, which, for simplicity, may be taken as a single force, may -be considered as mechanically equivalent to two forces (76), one in the -direction of the surface, and the other perpendicular to it. The latter -element will be resisted, and will produce a pressure; the former will -cause the body to move. This will perhaps be more clearly apprehended -by the aid of a diagram. - -Let A B, _fig. 25._, be the surface, and let P be a particle -of matter placed upon it, and urged by a force in the direction -P D, perpendicular to A B. It is manifest, that this force -can only press the particle P against A B, but cannot give it any -motion. - -But let us suppose, that the force which urges P is in a direction -P F, oblique to A B. Taking P F as the diagonal of a -parallelogram, whose sides are P D and P C (74), the force -P F is mechanically equivalent to two forces, expressed by the -lines P D and P C. But P D, being perpendicular to -A B, produces pressure without motion, and P C, being in -the direction of A B, produces motion without pressure. Thus -the effect of the force P F is distributed between motion and -pressure in a certain proportion, which depends on the obliquity of -its direction to that of the surface. The two extreme cases are, 1. -When it is in the direction of the surface; it then produces motion -without pressure: and, 2. When it is perpendicular to the surface; it -then produces pressure without motion. In all intermediate directions, -however, it will produce both these effects. - -(130.) It will be very apparent, that the more oblique the direction -of the force P F is to A B, the greater will be that part -of it which produces motion, and the less will be that which produces -pressure. This will be evident by inspecting _fig. 26._ In this -figure the line P F, which represents the force, is equal to -P F in _fig. 25._ But P D, which expresses the pressure, -is less in _fig. 26._ than in _fig. 25._, while P C, -which expresses the motion, is greater. So long, then, as the obliquity -of the directions of the surface and the force remain unchanged, so -long will the distribution of the force between motion and pressure -remain the same; and therefore, if the force itself remain the same, -the parts of it which produce motion and pressure will be respectively -equal. - -(131.) These general principles being understood, no difficulty can -arise in applying them to the motion of bodies urged on inclined planes -or curves by the force of gravity. If a body be placed on an unyielding -horizontal plane, it will remain at rest, producing a pressure on the -plane equal to the total amount of its weight. For in this case the -force which urges the body, being that of terrestrial gravity, its -direction is vertical, and therefore perpendicular to the horizontal -plane. - -But if the body P, _fig. 25._, be placed upon a plane A B, -oblique to the direction of the force of gravity, then, according to -what has been proved (129), the weight of the body will be distributed -into two parts, P C and P D; one, P D, producing a -pressure on the plane A B, and the other, P C, producing -motion down the plane. Since the obliquity of the perpendicular -direction P F of the weight to that of the plane A B must be -the same on whatever part of the plane the weight may be placed, it -follows (130), that the proportion P C of the weight which urges -the body down the plane must be the same throughout its whole descent. - -(132.) Hence it may easily be inferred, that the force down the -plane is uniform; for since the weight of the body P is always the -same, and since its proportion to that part which urges it down the -plane is the same, it follows that the quantity of this part cannot -vary. The motion of a heavy body down an inclined plane is therefore -an uniformly-accelerated motion, and is characterised by all the -properties of uniformly-accelerated motion, explained in the last -chapter. - -Since P F represents the force of gravity, that is, the force -with which the body would descend freely in the vertical direction, -and P C the force with which it moves down the plane, it follows -that a body would fall freely in the vertical direction from P to F -in the same time as on the plane it would move from P to C. In this -manner, therefore, when the height through which a body would fall -vertically is known, the space through which it would descend in the -same time down any given inclined plane may be immediately determined. -For let A B, _fig. 25._, be the given inclined plane, and let -P F be the space through which the body would fall in one second. -From F draw F C perpendicular to the plane, and the space P C -is that through which the body P will fall in one second on the plane. - -(133.) As the angle B A H, which measures the elevation -of the plane, is increased, the obliquity of the vertical direction -P F with the plane is also increased. Consequently, according to -what has been proved (130), it follows, that as the elevation of the -plane is increased, the force which urges the body down the plane is -also increased, and as the elevation is diminished, the force suffers a -corresponding diminution. The two extreme cases are, 1. When the plane -is raised until it becomes perpendicular, in which case the weight is -permitted to fall freely, without exerting any pressure upon the plane; -and, 2. When the plane is depressed until it becomes horizontal, in -which case the whole weight is supported, and there is no motion. - -From these circumstances it follows, that by means of an inclined plane -we can obtain an uniformly-accelerating force of any magnitude less -than that of gravity. - -We have here omitted, and shall for the present in every instance -omit, the effects of _friction_, by which the motion down the plane -is retarded. Having first investigated the mechanical properties of -bodies supposed to be free from friction, we shall consider friction -separately, and show how the present results are modified by it. - -(134.) The accelerating forces on different inclined planes may be -compared by the principle explained in (131). Let _figs. 25._ -and _26._ be two inclined planes, and take the lines P F in each -figure equal, both expressing the force of gravity, then P C will -be the force which in each case urges the body down the plane. - -As the force down an inclined plane is less than that which urges a -body falling freely in the vertical direction, the space through which -the body must fall to attain a certain final velocity must be just so -much greater as the accelerating force is less. On this principle we -shall be able to determine the final velocity in descending through any -space on a plane, compared with the final velocity attained in falling -freely in the vertical direction. Suppose the body P, _fig. 27._, -placed at the top of the plane, and from H draw the perpendicular -H C. If B H represent the force of gravity, B C will -represent the force down the plane (131). In order that the body -moving down the plane shall have a final velocity equal to that of -one which has fallen freely from B to H, it will be necessary that it -should move from B down the plane, through a space which bears the same -proportion to B H as B H does to B C. But since the -triangle A B H is in all respects similar to H B C, -only made upon a larger scale, the line A B bears the same -proportion to B H as B H bears to B C. Hence, in falling -on the inclined plane from B to A, the final velocity is the same as in -falling freely from B to H. - -It is evident that the same will be true at whatever level an -horizontal line be drawn. Thus, if I K be horizontal, the final -velocity in falling on the plane from B to I will be the same as the -final velocity in falling freely from B to K. - -(135.) The motion of a heavy body down a curve differs in an important -respect from the motion down an inclined plane. Every part of the -plane being equally inclined to the vertical direction, the effect of -gravity in the direction of the plane is uniform; and, consequently, -the phenomena obey all the established laws of uniformly-accelerated -motion. If, however, we suppose the line B A, on which the -body P descends, to be curved as in _fig. 28._, the obliquity -of its direction at different parts, to the direction P F of -gravity, will evidently vary. In the present instance, this obliquity -is greater towards B and less towards A, and hence the part of the -force of gravity which gives motion to the body is greater towards -B than towards A (130). The force, therefore, which urges the body, -instead of being uniform as in the inclined plane, is here gradually -diminished. The rate of this diminution depends entirely on the nature -of the curve, and can be deduced from the properties of the curve by -mathematical reasoning. The details of such an investigation are not, -however, of a sufficiently elementary character to allow of being -introduced with advantage into this treatise. We must therefore limit -ourselves to explain such of the results as may be necessary for the -development of the other parts of the science. - -(136.) When a heavy body is moved down an inclined plane by the force -of gravity, the plane has been proved to sustain a pressure, arising -from a certain part of the weight P D, _fig. 25._, which -acts perpendicularly to the plane. This is also the case in moving -down a curve such as B A, _fig. 28._ In this case, also, the -whole weight is distributed between that part which is directed down -the curve, and that which, being perpendicular to the curve, produces -a pressure upon it. There is, however, another cause which produces -pressure upon the curve, and which has no operation in the case of -the inclined plane. By the property of inertia, when a body is put in -motion in any direction, it must persevere in that direction, unless -it be deflected from it by an efficient force. In the motion down an -inclined plane the direction is never changed, and therefore by its -inertia the falling body retains all the motion impressed upon it -continually in the same direction; but when it descends upon a curve, -its direction is constantly varying, and the resistance of the curve -being the deflecting cause, the curve must sustain a pressure equal to -that force, which would thus be capable of continually deflecting the -body from the rectilinear path in which it would move in virtue of its -inertia. This pressure entirely depends on the curvature of the path -in which the body is constrained to move, and on its inertia, and is -therefore altogether independent of the weight, and would, in fact, -exist if the weight were without effect. - -(137.) This pressure has been denominated _centrifugal force_, because -it evinces a tendency of the moving body to _fly from_ the centre of -the curve in which it is moved. Its quantity depends conjointly on the -velocity of the motion and the curvature of the path through which -the body is moved. As circles may be described with every degree of -curvature, according to the length of the radius, or the distance from -their circumference to their centre, it follows that, whatever be the -curve in which the body moves, a circle can always be assigned which -has the same curvature as is found at any proposed point of the given -curve. Such a circle is called “the circle of curvature” at that point -of the curve; and as all curves, except the circle, vary their degrees -of curvature at different points, it follows that different parts of -the same curve will have different circles of curvature. It is evident -that the greater the radius of a circle is, the less is its curvature: -thus the circle with the radius A B, _fig. 29._, is more -curved than that whose radius is C D, and that in the exact -proportion of the radius C D to the radius A B. The radius of -the circle of curvature for any part of a curve is called “the radius -of curvature” of that part. - -(138.) The centrifugal pressure increases as the radius of curvature -increases; but it also has a dependence on the velocity with which the -moving body swings round the centre of the circle of curvature. This -velocity is estimated either by the actual space through which the body -moves, or by the _angular velocity_ of a line drawn from the centre of -the circle to the moving body. That body carries one end of this line -with it, while the other remains fixed at the centre. As this angular -swing round the centre increases, the centrifugal pressure increases. -To estimate the rate at which this pressure in general varies, it is -necessary to multiply the square of the number expressing the angular -velocity by that which expresses the radius of curvature, and the force -increases in the same proportion as the product thus obtained. - -(139.) We have observed that the same causes which produce pressure -on a body restrained, will produce motion if the body be free. -Accordingly, if a body be moved by any efficient cause in a curve, it -will, by reason of the centrifugal force, _fly off_, and the moving -force with which it will thus retreat from the centre round which -it is whirled will be a measure of the centrifugal force. Upon this -principle an apparatus called a _whirling table_ has been constructed, -for the purpose of exhibiting experimental illustrations of the laws -of centrifugal force. By this machine we are enabled to place any -proposed weights at any given distances from centres round which they -are whirled, either with the same angular velocity, or with velocities -having a certain proportion. Threads attached to the whirling weights -are carried to the centres round which they respectively revolve, and -there, passing over pulleys, are connected with weights which may be -varied at pleasure. When the whirling weights fly from their respective -centres, by reason of the centrifugal force, they draw up the weights -attached to the other ends of the threads, and the amount of the -centrifugal force is estimated by the weight which it is capable of -raising. - -With this instrument the following experiments may be exhibited:-- - -Exp. 1. Equal weights whirled with the same velocity at equal distances -from the centre raise the same weight, and therefore have the same -centrifugal force. - -Exp. 2. Equal weights whirled with the same angular velocity at -distances from the centre in the proportion of one to two, will raise -weights in the same proportion. Therefore the centrifugal forces are in -that proportion. - -Exp. 3. Equal weights whirled at equal distances with angular -velocities which are as one to two, will raise weights as one to four, -that is, as the squares of the angular velocities. Therefore the -centrifugal forces are in that proportion. - -Exp. 4. Equal weights whirled at distances which are as two to three, -with angular velocities which are as one to two, will raise weights -which are as two to twelve; that is, as the products of the distances -two and three, and the squares one and four, of the angular velocities. -Hence, the centrifugal forces are in this proportion. - -The centrifugal force must also increase as the mass of the body moved -increases; for, like attraction, each particle of the moving body is -separately and equally affected by it. Hence a double mass, moving -at the same distance, and with the same velocity, will have a double -force. The following experiment verifies this:-- - -Exp. 5. If weights, which are as one to two, be whirled at equal -distances with the same velocity, they will raise weights which are as -one to two. - -The law which governs centrifugal force may then be expressed in -general symbols briefly thus:-- - -Let _c_ = the centrifugal force with which a weight of one lb. -revolving in a circle in one second, the radius of which is one foot, -would act on a string connecting it with the centre. The force with -which it would act on a string, the length of which is R feet, would -be _c_ × R; and if instead of revolving in one second it revolved in T -seconds, the force would be - - (_c_ × R)/T^2; - -and if the revolving mass were W lbs. the force would be - - C = (_c_ × W × R)/T^2. - -This formula includes the entire theory of centrifugal force. - -But it can be shown that the number expressed by _c_ is 1·226, and -consequently - - C = (1·226 × W × R)/T^2. - -It is often more convenient to use the number of revolutions made in -a given time than the time of one revolution. Let N then express the -number of revolutions, or fraction of a revolution, made in one second, -and we shall have - - T = 1/N. - -Therefore - - C = 1·226 × W × R × N^2. - -(140.) The consideration of centrifugal force proves, that if a body -be observed to move in a curvilinear path, some efficient cause must -exist which prevents it from flying off, and which compels it to -revolve round the centre. If the body be connected with the centre by -a thread, cord, or rod, then the effect of the centrifugal force is -to give tension to the thread, cord, or rod. If an unyielding curved -surface be placed on the convex side of the path, then the force will -produce pressure on this surface. But if a body is observed to move -in a curve without any visible material connection with its centre, -and without any obstruction on the convex side of its path to resist -its retreat, as is the case with the motions of the planets round -the sun, and the satellites round the planets, it is usual to assign -the cause to the attraction of the body which occupies the centre: -in the present instance the sun is that body, and it is customary to -say that the _attraction_ of the sun, neutralising the effects of the -centrifugal force of the planets, _retains them_ in their orbits. We -have elsewhere animadverted on the inaccurate and unphilosophical style -of this phraseology, in which terms are admitted which intimate not -only an unknown cause, but assign its seat, and intimate something of -its nature. All that we are entitled to declare in this case is, that -a motion is continually impressed upon the planet; that this motion is -directed towards the sun; that it counteracts the centrifugal force; -but from whence this motion proceeds, whether it be a virtue resident -in the sun, or a property of the medium or space in which both sun and -planets are placed, or whatever other influence may be its proximate -cause, we are altogether ignorant. - - * * * * * - -(141.) Numerous examples of the effects of centrifugal force may be -produced. - -If a stone or other weight be placed in a sling, which is whirled round -by the hand in a direction perpendicular to the ground, the stone will -not fall out of the sling, even when it is at the top of its circuit, -and, consequently, has no support beneath it. The centrifugal force, in -this case, acting from the hand, which is the centre of rotation, is -greater than the weight of the body, and therefore prevents its fall. - -In like manner, a glass of water may be whirled so rapidly that even -when the mouth of the glass is presented downwards, the water will -still be retained in it by the centrifugal force. - -If a bucket of water be suspended by a number of threads, and these -threads be twisted by turning round the bucket many times in the same -direction, on allowing the cords to untwist, the bucket will be whirled -rapidly round, and the water will be observed to rise on its sides and -sink at its centre, owing to the centrifugal force with which it is -driven from the centre. This effect might be carried so far, that all -the water would flow over and leave the bucket nearly empty. - -(142.) A carriage, or horseman, or pedestrian, passing a corner moves -in a curve, and suffers a centrifugal force, which increases with -the velocity, and which impresses on the body a force directed from -the corner. An animal causes its weight to resist this force, by -voluntarily inclining its body towards the corner. In this case, let -A B, _fig. 30._, be the body; C D is the direction of -the weight perpendicular to the ground, and C F is the direction -of the centrifugal force parallel to the ground and _from_ the corner. -The body A B is inclined to the corner, so that the diagonal force -(74), which is mechanically equivalent to the weight and centrifugal -force, shall be in the direction C A, and shall therefore produce -the pressure of the feet upon the ground. - -As the velocity is increased, the centrifugal force is also increased, -and therefore a greater inclination of the body is necessary to resist -it. We accordingly find that the more rapidly a corner is turned, the -more the animal inclines his body towards it. - -A carriage, however, not having voluntary motion, cannot make this -compensation for the disturbing force which is called into existence -by the gradual change of direction of the motion; consequently it -will, under certain circumstances, be overturned, falling of course -outwards, or _from_ the corner. If A B be the carriage, and -C, _fig. 31._, the place at which the weight is principally -collected, this point C will be under the influence of two forces: the -weight, which may be represented by the perpendicular C D, and the -centrifugal force, which will be represented by a line C F, which -shall have the same proportion to C D as the centrifugal force -has to the weight. Now the combined effect of these two forces will be -the same as the effect of a single force, represented by C G. -Thus, the pressure of the carriage on the road is brought nearer to the -outer wheel B. If the centrifugal force bear the same proportion to the -weight as C F (or D B), _fig. 32._, bears to C D, -the whole pressure is thrown upon the wheel B. - -If the centrifugal force bear to the weight a greater proportion than -D B has to C D, then the line C F, which represents it, -_fig. 33._, will be greater than D B. The diagonal C G, -which represents the combined effects of the weight and centrifugal -force, will in this case pass outside the wheel B, and therefore this -resultant will be unresisted. To perceive how far it will tend to -overturn the carriage, let the force C G be resolved into two, one -in the direction of C B, and the other C K, perpendicular to -C B. The former C B will be resisted by the road, but the -latter C K will tend to lift the carriage over the external wheel. -If the velocity and the curvature of the course be continued for a -sufficient time to enable this force C K to elevate the weight, -so that the line of direction shall fall on B, the carriage will be -overthrown. - -It is evident from what has been now stated, that the chances of -overthrow under these circumstances depend on the proportion of -B D to C D, or what is to the same purpose, of the distance -between the wheels to the height of the principal seat of the load. -It will be shown in the next chapter, that there is a certain point, -called the centre of gravity, at which the entire weight of the vehicle -and its load may be conceived to be concentrated. This is the point -which in the present investigation we have marked C. The security of -the carriage, therefore, depends on the greatness of the distance -between the wheels and the smallness of the elevation of the centre of -gravity above the road; for either or both of these circumstances will -increase the proportion of B D to C D. - -(143.) In the equestrian feat exhibited in the ring at the -amphitheatre, when the horse moves round with the performer standing on -the saddle, both the horse and rider incline continually towards the -centre of the ring, and the inclination increases with the velocity of -the motion: by this inclination their weights counteract the effect -of the centrifugal force, exactly as in the case already mentioned -(142.) - -[Illustration: _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -(144.) If a body be allowed to fall by its weight down a convex -surface, such as A B, _fig. 34._, it would continue upon the -surface until it arrive at B but for the effect of the centrifugal -force: this, giving it a motion from the centre of the curve, will -cause it to quit the curve at a certain point C, which can be easily -found by mathematical computation. - -(145.) The most remarkable and important manifestation of centrifugal -force is observed in the effects produced by the rotation of the earth -upon its axis. Let the circle in _fig. 35._ represent a section -of the earth, A B being the axis on which it revolves. This -rotation causes the matter which composes the mass of the earth to -revolve in circles round the different points of the axis as centres -at the various distances at which the component parts of this mass -are placed. As they all revolve with the same angular velocity, they -will be affected by centrifugal forces, which will be greater or less -in proportion as their distances from the centre are greater or less. -Consequently the parts of the earth which are situated about the -equator, D, will be more strongly affected by centrifugal force than -those about the poles, A B. The effect of this difference has been -that the component matter about the equator has actually been driven -farther from the centre than that about the poles, so that the figure -of the earth has swelled out at the sides, and appears proportionally -depressed at the top and bottom, resembling the shape of an orange. An -exaggerated representation of this figure is given in _fig. 36._; -the real difference between the distances of the poles and equator -from the centre being too small to be perceptible in a diagram. The -exact proportion of C A to C D has never yet been certainly -ascertained. Some observations make C D exceed C A by 1/277, -and others by only 1/333. The latter, however, seems the more probable. -It may be considered to be included between these limits. - -The same cause operates more powerfully in other planets which revolve -more rapidly on their axes. Jupiter and Saturn have forms which are -considerably more elliptical. - -(146.) The centrifugal force of the earth’s rotation also affects -detached bodies on its surface. If such bodies were not held upon the -surface by the earth’s attraction, they would be immediately flung -off by the whirling motion in which they participate. The centrifugal -force, however, really diminishes the effects of the earth’s attraction -on those bodies, or, what is the same, diminishes their weights. If -the earth did not revolve on its axis, the weight of bodies in all -places equally distant from the centre would be the same; but this is -not so when the bodies, as they do, move round with the earth. They -acquire from the centrifugal force a tendency to fly from the axis, -which increases with their distance from that axis, and is therefore -greater the nearer they are to the equator, and less as they approach -the pole. But there is another reason why the centrifugal force is more -efficient, in the opposition which it gives to gravity near the equator -than near the poles. This force does not act from the centre of the -earth, but is directed from the earth’s axis. It is, therefore, not -directly opposed to gravity, except on the equator itself. On leaving -the equator, and proceeding towards the poles, it is less and less -opposed to gravity, as will be plain on inspecting _fig. 35._, -where the lines P C all represent the direction of gravity, and -the lines P F represent the direction of the centrifugal force. - -Since, then, as we proceed from the equator towards the poles, not only -the amount of the centrifugal force is continually diminished, but also -it acts less and less in opposition to gravity, it follows that the -weights of bodies are most diminished by it at the equator, and less so -towards the poles. - -Since bodies are commonly weighed by balancing them against other -bodies of known weight, it may be asked, how the phenomena we have been -just describing can be ascertained as a matter of fact? for whatever -be the body against which it may be balanced, that body must suffer -just as much diminution of weight as every other, and consequently, all -being diminished in the same proportion, the balance will be preserved -though the weights be changed. - -To render this effect observable, it will be necessary to compare the -effects of gravity with some phenomenon which is not affected by the -centrifugal force of the earth’s rotation, and which will be the same -at every part of the earth. The means of accomplishing this will be -explained in a subsequent chapter. - - - - -CHAP. IX. - -THE CENTRE OF GRAVITY. - - -(147.) By the earth’s attraction, all the particles which compose -the mass of a body are solicited by equal forces in parallel -directions downwards. If these component particles were placed in mere -juxtaposition, without any mechanical connection, the force impressed -on any one of them could in nowise affect the others, and the mass -would, in such a case, be contemplated as an aggregation of small -particles of matter, each urged by an independent force. But the bodies -which are the subjects of investigation in mechanical science are not -found in this state. Solid bodies are coherent masses, the particles -of which are firmly bound together, so that any force which affects -one, being modified according to circumstances, will be transmitted -through the whole body. Liquids accommodate themselves to the shape of -the surfaces on which they rest, and forces affecting any one part are -transmitted to others, in a manner depending on the peculiar properties -of this class of bodies. - -As all bodies, which are subjects of mechanical enquiry, on the surface -of the earth, must be continually influenced by terrestrial gravity, -it is desirable to obtain some easy and summary method of estimating -the effect of this force. To consider it, as is unavoidable in the -first instance, the combined action of an infinite number of equal and -parallel forces soliciting the elementary molecules downwards, would -be attended with manifest inconvenience. An infinite number of forces, -and an infinite subdivision of the mass, would form parts of every -mechanical problem. - -To overcome this difficulty, and to obtain all the ease and simplicity -which can be desired in elementary investigations, it is only necessary -to determine some force, whose single effect shall be equivalent to the -combined effects of the gravitation of all the molecules of the body. -If this can be accomplished, that single force might be introduced into -all problems to represent the whole effect of the earth’s attraction, -and no regard need be had to any particles of the body, except that on -which this force acts. - -(148.) To discover such a force, if it exist, we shall first enquire -what properties must necessarily characterise it. Let A B, -_fig. 37._, be a solid body placed near the surface of the -earth. Its particles are all solicited downwards, in the directions -represented by the arrows. Now, if there be any single force equivalent -to these combined effects, two properties may be at once assigned to -it: 1. It must be presented downwards, in the common direction of -those forces to which it is mechanically equivalent; and, 2. it must -be equal in intensity to their sum, or, what is the same, to the force -with which the whole mass would descend. We shall then suppose it to -have this intensity, and to have the direction of the arrow D E. -Now, if the single force, in the direction D E, be equivalent -to all the separate attractions which affect the particles, we may -suppose all these attractions removed, and the body A B influenced -only by a single attraction, acting in the direction D E. This -being admitted, it follows that if the body be placed upon a prop, -immediately under the direction of the line D E, or be suspended -from a fixed point immediately above its direction, it will remain -motionless. For the whole attracting force in the direction D E -will, in the one case, press the body on the prop, and, in the other -case, will give tension to the cord, rod, or whatever other means of -suspension be used. - -(149.) But suppose the body were suspended from some point P, not in -the direction of the line D E. Let P C be the direction of -the thread by which the body is suspended. Its whole weight, according -to the supposition which we have adopted, must then act in the -direction C E. Taking C F to represent the weight; it may be -considered as mechanically equivalent to two forces (74), C I and -C H. Of these C H, acting directly from the point P, merely -produces pressure upon it, and gives tension to the cord P C; but -C I, acting at right angles to C P, produces motion round P -as a centre, and in the direction C I, towards a vertical line -P G, drawn through the point P. If the body A B had been on -the other side of the line P G, it would have moved in like manner -towards it, and therefore in the direction contrary to its present -motion. - -Hence we must infer, that when the body is suspended from a fixed -point, it cannot remain at rest, if that fixed point be not placed in -the direction of the line D E; and, on the other hand, that if -the fixed point _be_ in the direction of that line, it cannot move. A -practical test is thus suggested, by which the line D E may be at -once discovered. Let a thread be attached to any point of the body, and -let it be suspended by this thread from a hook or other fixed point. -The direction of the thread, when the body becomes quiescent, will -be that of a single force equivalent to the gravitation of all the -component parts of the mass. - -(150.) An enquiry is here suggested: does the direction of the -equivalent force thus determined depend on the position of the body -with respect to the surface of the earth, and how is the direction -of the equivalent force affected by a change in that position? This -question may be at once solved if the body be suspended by different -points, and the directions which the suspending thread takes in each -case relatively to the figure and dimensions of the body examined. - -The body being suspended in this manner from any point, let a small -hole be bored through it, in the exact direction of the thread, so that -if the thread were continued below the point where it is attached to -the body, it would pass through this hole. The body being successively -suspended by several different points on its surface, let as many small -holes be bored through it in the same manner. If the body be then cut -through, so as to discover the directions which the several holes have -taken, they will be all found to cross each other at one point within -the body; or the same fact may be discovered thus: a thin wire, which -nearly fills the holes being passed through any one of them, it will be -found to intercept the passage of a similar wire through any other. - -This singular fact teaches us, what indeed can be proved by -mathematical reasoning without experiment, that there is _one_ point in -every body through which the single force, which is equivalent to the -gravitation of all its particles, must pass, in whatever position the -body be placed. This point is called _the centre of gravity_. - -(151.) In whatever situation a body may be placed, the centre of -gravity will have a tendency to descend in the direction of a -line perpendicular to the horizon, and which is called the _line -of direction_ of the weight. If the body be altogether free and -unrestricted by any resistance or impediment, the centre of gravity -will actually descend in this direction, and all the other points of -the body will move with the same velocity in parallel directions, -so that during its fall the position of the parts of the body, with -respect to the ground, will be unaltered. But if the body, as is most -usual, be subject to some resistance or restraint, it will either -remain unmoved, its weight being expended in exciting pressure on the -restraining points or surfaces, or it will move in a direction and -with a velocity depending on the circumstances which restrain it. - -In order to determine these effects, to predict the pressure produced -by the weight if the body be quiescent, or the mixed effects of motion -and pressure, if it be not so, it is necessary in all cases to be -able to assign the place of the centre of gravity. When the magnitude -and figure of the body, and the density of the matter which occupies -its dimensions, are known, the place of the centre of gravity can be -determined with the greatest precision by mathematical calculation. -The process by which this is accomplished, however, is not of a -sufficiently elementary nature to be properly introduced into this -treatise. To render it intelligible would require the aid of some -of the most advanced analytical principles; and even to express the -position of the point in question, except in very particular instances, -would be impossible, without the aid of peculiar symbols. - -(152.) There are certain particular forms of body in which, when they -are uniformly dense, the place of the centre of gravity can be easily -assigned, and proved by reasoning, which is generally intelligible; -but in all cases whatever, this point may be easily determined by -experiment. - -(153.) If a body uniformly dense have such a shape that a point may be -found on either side of which in all directions around it the materials -of the body are similarly distributed, that point will obviously be -the centre of gravity. For if it be supported, the gravitation of the -particles on one side drawing them downwards, is resisted by an effect -of exactly the same kind and of equal amount on the opposite side, and -so the body remains balanced on the point. - -The most remarkable body of this kind is a globe, the centre of which -is evidently its centre of gravity. - -A figure, such as _fig. 38._, called an _oblate spheroid_, has its -centre of gravity at its centre, C. Such is the figure of the earth. -The same may be observed of the elliptical solid, _fig. 39._, -which is called a prolate spheroid. - -A cube, and some other regular solids, bounded by plane surfaces, have -a point within them, such as above described, and which is, therefore, -their centre of gravity. Such are _fig. 40._ - -A straight wand of uniform thickness has its centre of gravity at the -centre of its length; and a cylindrical body has its centre of gravity -in its centre, at the middle of its length or axis. Such is the point -C, _fig. 41._ - -A flat plate of any uniform substance, and which has in every part -an equal thickness, has its centre of gravity at the middle of its -thickness, and under a point of its surface, which is to be determined -by its shape. If it be circular or elliptical, this point is its -centre. If it have any regular form, bounded by straight edges, it is -that point which is equally distant from its several angles, as C in -_fig. 42._ - -(154.) There are some cases in which, although the place of the centre -of gravity is not so obvious as in the examples just given, still -it may be discovered without any mathematical process, which is not -easily understood. Suppose A B C, _fig. 43._, to be -a flat triangular plate of uniform thickness and density. Let it be -imagined to be divided into narrow bars, by lines parallel to the side -A C, as represented in the figure. Draw B D from the angle -B to the middle point D of the side A C. It is not difficult to -perceive, that B D will divide equally all the bars into which the -triangle is conceived to be divided. Now if the flat triangular plate -A B C be placed in a horizontal position on a straight edge -coinciding with the line B D, it will be balanced: for the bars -parallel to A C will be severally balanced by the edge immediately -under their middle point; since that middle point is the centre of -gravity of each bar. Since, then, the triangle is balanced on the edge, -the centre of gravity must be somewhere immediately over it, and must, -therefore, be within the plate at some point under the line B D. - -The same reasoning will prove that the centre of gravity of the plate -is under the line A E, drawn from the angle A to the middle -point E of the side B C. To perceive this, it is only necessary -to consider the triangle divided into bars parallel to B C, -and thence to show that it will be balanced on an edge placed under -A E. Since then the centre of gravity of the plate is under the -line B D, and also under A E, it must be under the point G, -at which these lines cross each other; and it is accordingly at a depth -beneath G, equal to half the thickness of the plate. - -This may be experimentally verified by taking a piece of tin or card, -and cutting it into a triangular form. The point G being found by -drawing B D and A E, which divide two sides equally, it will -be balanced if placed upon the point of a pin at G. - -The centre of gravity of a triangle being thus determined, we shall -be able to find the position of the centre of gravity of any plate of -uniform thickness and density which is bounded by straight edges, as -will be shown hereafter. (173.) - -(155.) The centre of gravity is not always included within the volume -of the body, that is, it is not enclosed by its surfaces. Numerous -examples of this can be produced. If a piece of wire be bent into any -form, the centre of gravity will rarely be in the wire. Suppose it be -brought to the form of a ring. In that case, the centre of gravity of -the wire will be the centre of the circle, a point not forming any -part of the wire itself: nevertheless this point may be proved to have -the characteristic property of the centre of gravity; for if the ring -be suspended by any point, the centre of the ring must always settle -itself under the point of suspension. If this centre could be supposed -to be connected with the ring by very fine threads, whose weight would -be insignificant, and which might be united by a knot or otherwise at -the centre, the ring would be balanced upon a point placed under the -knot. - -In like manner, if the wire be formed into an ellipse, or any other -curve similarly arranged round a centre point, that point will be its -centre of gravity. - -(156.) To find the centre of gravity experimentally, the method -described in (149, 150) may be used. In this case two points of -suspension will be sufficient to determine it; for the directions of -the suspending cord being continued through the body, will cross each -other at the centre of gravity. These directions may also be found -by placing the body on a sharp point, and adjusting it so as to be -balanced upon it. In this case a line drawn through the body directly -upwards from the point will pass through the centre of gravity, and -therefore two such lines must cross at that point. - -(157.) If the body have two flat parallel surfaces like sheet metal, -stiff paper, card, board, &c., the centre of gravity may be found by -balancing the body in two positions on an horizontal straight edge. -The point where the lines marked by the edge cross each other will -be immediately under the centre of gravity. This may be verified by -showing that the body will be balanced on a point thus placed, or that -if it be suspended, the point thus determined will always come under -the point of suspension. - -The position of the centre of gravity of such bodies may also be found -by placing the body on an horizontal table having a straight edge. -The body being moved beyond the edge until it is in that position in -which the slightest disturbance will cause it to fall, the centre of -gravity will then be immediately over the edge. This being done in two -positions, the centre of gravity will be determined as before. - -(158.) It has been already stated, that when the body is perfectly -free, the centre of gravity must necessarily move downwards, in a -direction perpendicular to an horizontal plane. When the body is not -free, the circumstances which restrain it generally permit the centre -of gravity to move in certain directions, but obstruct its motion in -others. Thus if a body be suspended from a fixed point by a flexible -cord, the centre of gravity is free to move in every direction except -those which would carry it farther from the point of suspension than -the length of the cord. Hence if we conceive a globe or sphere to -surround the point of suspension on every side to a distance equal to -that of the centre of gravity from the point of suspension, when the -cord is fully stretched, the centre of gravity will be at liberty to -move in every direction within this sphere. - -There are an infinite variety of circumstances under which the motion -of a body may be restrained, and in which a most important and useful -class of mechanical problems originate. Before we notice others, -we shall, however, examine that which has just been described more -particularly. - -Let P, _fig. 44._, be the point of suspension, and C the centre -of gravity, and suppose the body so placed that C shall be within the -sphere already described. The cord will therefore be slackened, and in -this state the body will be free. The centre of gravity will therefore -descend in the perpendicular direction until the cord becomes fully -extended; the tension will then prevent its further motion in the -perpendicular direction. The downward force must now be considered as -the diagonal of a parallelogram, and equivalent to two forces C D -and C E, in the directions of the sides, as already explained in -(149). The force C D will bring the centre of gravity into the -direction P F, perpendicularly under the point of suspension. -Since the force of gravity acts continually on C in its approach to -P F, it will move towards that line with accelerated speed, and -when it has arrived there it will have acquired a force to which no -obstruction is immediately opposed, and consequently by its inertia -it retains this force, and moves beyond P F on the other side. -But when the point C gets into the line P F, it is in the lowest -possible position; for it is at the lowest point of the sphere which -limits its motion. When it passes to the other side of P F, it -must therefore begin to ascend, and the force of gravity, which, in the -former case, accelerated its descent, will now for the same reason, and -with equal energy, oppose its ascent. This will be easily understood. -Let C′ be any point which it may have attained in ascending; -C′ G′, the force of gravity, is now equivalent to C′ D′ and -C′ E′. The latter as before produces tension; but the former -C′ D′ is in a direction immediately opposed to the motion, and -therefore retards it. This retardation will continue until all the -motion acquired by the body in its descent from the first position -has been destroyed, and then it will begin to return to P F, and -so it will continue to vibrate from the one side to the other until -the friction on the point P, and the resistance of the air, gradually -deprive it of its motion, and bring it to a state of rest in the -direction P F. - -But for the effects of friction and atmospheric resistance, the body -would continue for ever to oscillate equally from side to side of the -line P F. - -(159.) The phenomenon just developed is only an example of an extensive -class. Whenever the circumstances which restrain the body are of such -a nature that the centre of gravity is prevented from descending below -a certain level, but not, on the other hand, restrained from rising -above it, the body will remain at rest if the centre of gravity be -placed at the lowest limit of its level; any disturbance will cause -it to oscillate around this state, and it cannot return to a state of -rest until friction or some other cause have deprived it of the motion -communicated by the disturbing force. - -(160.) Under the circumstances which we have just described, the body -could not maintain itself in a state of rest in any position except -that in which the centre of gravity is, at the lowest point of the -space in which it is free to move. This, however, is not always the -case. Suppose it were suspended by an inflexible rod instead of a -flexible string; the centre of gravity would then not only be prevented -from receding from the point of suspension, but also from approaching -it; in fact, it would be always kept at the same distance from it. -Thus, instead of being capable of moving anywhere within the sphere, -it is now capable of moving on its surface only. The reasoning used in -the last case may also be applied here, to prove that when the centre -of gravity is on either side of the perpendicular P F, it will -fall towards P F and oscillate, and that if it be placed in the -line P F, it will remain in equilibrium. But in this case there is -another position, in which the centre of gravity may be placed so as to -produce equilibrium. If it be placed at the highest point of the sphere -in which it moves, the whole force acting on it will then be directed -on the point of suspension, perpendicularly downwards, and will be -entirely expended in producing pressure on that point; consequently, -the body will in this case be in equilibrium. But this state of -equilibrium is of a character very different from that in which the -centre of gravity was at the lowest part of the sphere. In the present -case any displacement, however slight, of the centre of gravity, will -carry it to a lower level, and the force of gravity will then prevent -its return to its former state, and will impel it downwards until it -attain the lowest point of the sphere, and round that point it will -oscillate. - -(161.) The two states of equilibrium which have been just noticed, are -called stable and instable equilibrium. The character of the former is, -that any disturbance of the state produces oscillation about it; but -any disturbance of the latter state produces a total overthrow, and -finally causes oscillation around the state of stable equilibrium. - -Let A B, _fig. 45._, be an elliptical board resting on -its edge on an horizontal plane. In the position here represented, -the extremity P of the lesser axis being the point of support, the -board is in stable equilibrium; for any motion on either side must -cause the centre of gravity C to ascend in the directions C O, -and oscillation will ensue. If, however, it rest upon the smaller -end, as in _fig. 46._, the position would still be a state of -equilibrium, because the centre of gravity is directly above the point -of support; but it would be instable equilibrium, because the slightest -displacement of the centre of gravity would cause it to descend. - -Thus an egg or a lemon may be balanced on the end, but the least -disturbance will overthrow it. On the contrary, it will easily rest on -the side, and any disturbance will produce oscillation. - -(162.) When the circumstances under which the body is placed allow the -centre of gravity to move only in an horizontal line, the body is in a -state which may be called _neutral equilibrium_. The slightest force -will move the centre of gravity, but will neither produce oscillation -nor overthrow the body, as in the last two cases. - -An example of this state is furnished by a cylinder placed upon an -horizontal plane. As the cylinder is rolled upon the plane, the centre -of gravity C, _fig. 47._, moves in a line parallel to the plane -A B, and distant from it by the radius of the cylinder. The body -will thus rest indifferently in any position, because the line of -direction always falls upon a point P at which the body rests upon the -plane. - -If the plane were inclined, as in _fig. 48._, a body might be -so shaped, that while it would roll the centre of gravity would move -horizontally. In this case the body would rest indifferently on any -part of the plane, as if it were horizontal, provided the friction be -sufficient to prevent the body from sliding down the plane. - -If the centre of gravity of a cylinder happen not to coincide with -its centre by reason of the want of uniformity in the materials of -which it is composed, it will not be in a state of neutral equilibrium -on an horizontal plane, as in _fig. 47._ In this case let G, -_fig. 49._, be the centre of gravity. In the position here -represented, where the centre of gravity is immediately _below_ the -centre C, the state will be stable equilibrium, because a motion -on either side would cause the centre of gravity to ascend; but in -_fig. 50._, where G is immediately above C, the state is instable -equilibrium, because a motion on either side would cause G to descend, -and the body would turn into the position _fig. 49._ - -(163.) A cylinder of this kind will, under certain circumstances, roll -up an inclined plane. Let A B, _fig. 51._, be the inclined -plane, and let the cylinder be so placed that the line of direction -from G shall be _above_ the point P at which the cylinder rests upon -the plane. The whole weight of the body acting in the direction -G D will obviously cause the cylinder to roll towards A, provided -the friction be sufficient to prevent sliding; but although the -cylinder in this case ascends, the centre of gravity G really descends. - -When G is so placed that the line of direction G D shall fall on -the point P, the cylinder will be in equilibrium, because its weight -acts upon the point on which it rests. There are two cases represented -in _fig. 52._ and _fig. 53._, in which G takes this position. -_Fig. 52._ represents the state of stable, and _fig. 53._ of -instable equilibrium. - -(164.) When a body is placed upon a base, its stability depends upon -the position of the line of direction and the height of the centre of -gravity above the base. If the line of direction fall within the base, -the body will stand firm; if it fall on the edge of the base, it will -be in a state in which the slightest force will overthrow it on that -side at which the line of direction falls; and if the line of direction -fall without the base, the body must turn over that edge which is -nearest to the line of direction. - -In _fig. 54._ and _fig. 55._, the line of direction G P -falls within the base, and it is obvious that the body will stand firm; -for any attempt to turn it over either edge would cause the centre of -gravity to ascend. But in _fig. 56._ the line of direction falls -upon the edge, and if the body be turned over, the centre of gravity -immediately commences to descend. Until it be turned over, however, the -centre of gravity is supported by the edge. - -In _fig. 57._ the line of direction falls outside the base, the -centre of gravity has a tendency to descend from G towards A, and the -body will accordingly fall in that direction. - -(165.) When the line of direction falls within the base, bodies will -always stand firm, but not with the same degree of stability. In -general, the stability depends on the height through which the centre -of gravity must be elevated before the body can be overthrown. The -greater this height is, the greater in the same proportion will be the -stability. - -Let B A C, _fig. 58._, be a pyramid, the centre of -gravity being at G. To turn this over the edge B, the centre of -gravity; must be carried over the arch G E, and must therefore -be raised through the height H E. If, however, the pyramid were -taller relatively to its base, as in _fig. 59._, the height -H E would be proportionally less; and if the base were very small -in reference to the height, as in _fig. 60._, the height H E -would be very small, and a slight force would throw it over the edge B. - -It is obvious that the same observations may be applied to all figures -whatever, the conclusions just deduced depending only on the distance -of the line of direction from the edge of the base, and the height of -the centre of gravity above it. - -(166.) Hence we may perceive the principle on which the stability of -loaded carriages depends. When the load is placed at a considerable -elevation above the wheels, the centre of gravity is elevated, and the -carriage becomes proportionally insecure. In coaches for the conveyance -of passengers, the luggage is therefore sometimes placed below the body -of the coach; light parcels of large bulk may be placed on the top with -impunity. - -When the centre of gravity of a carriage is much elevated, there is -considerable danger of overthrow, if a corner be turned sharply and -with a rapid pace; for the centrifugal force then acting on the centre -of gravity will easily raise it through the small height which is -necessary to turn the carriage over the external wheels (142). - -(167.) The same waggon will have greater stability when loaded with -a heavy substance which occupies a small space, such as metal, than -when it carries the same weight of a lighter substance, such as hay; -because the centre of gravity in the latter case will be much more -elevated. - -[Illustration: _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -If a large table be placed upon a single leg in its centre, it will -be impracticable to make it stand firm; but if the pillar on which it -rests terminate in a tripod, it will have the same stability as if it -had three legs attached to the points directly over the places where -the feet of the tripod rest. - -(168.) When a solid body is supported by more points than one, it is -not necessary for its stability that the line of direction should fall -on one of those points. If there be only two points of support, the -line of direction must fall between them. The body is in this case -supported as effectually as if it rested on an edge coinciding with a -straight line drawn from one point of support to the other. If there -be three points of support, which are not ranged in the same straight -line, the body will be supported in the same manner as it would be by -a base coinciding with the triangle formed by straight lines joining -the three points of support. In the same manner, whatever be the -number of points on which the body may rest, its virtual base will be -found by supposing straight lines drawn, joining the several points -successively. When the line of direction falls within this base, the -body will always stand firm, and otherwise not. The degree of stability -is determined in the same manner as if the base were a continued -surface. - -(169.) Necessity and experience teach an animal to adapt its postures -and motions to the position of the centre of gravity of his body. When -a man stands, the line of direction of his weight must fall within the -base formed by his feet. If A B, C D, _fig. 61._, be the -feet, this base is the space A B D C. It is evident, -that the more his toes are turned outwards, the more contracted the -base will be in the direction E F, and the more liable he will be -to fall backwards or forwards. Also, the closer his feet are together, -the more contracted the base will be in the direction G H, and the -more liable he will be to fall towards either side. - -When a man walks, the legs are alternately lifted from the ground, and -the centre of gravity is either unsupported or thrown from the one -side to the other. The body is also thrown a little forward, in order -that the tendency of the centre of gravity to fall in the direction of -the toes may assist the muscular action in propelling the body. This -forward inclination of the body increases with the speed of the motion. - -But for the flexibility of the knee-joint the labour of walking would -be much greater than it is; for the centre of gravity would be more -elevated by each step. The line of motion of the centre of gravity in -walking is represented by _fig. 62._, and deviates but little -from a regular horizontal line, so that the elevation of the centre -of gravity is subject to very slight variation. But if there were no -knee-joint, as when a man has wooden legs, the centre of gravity would -move as in _fig. 63._, so that at each step the weight of the body -would be lifted through a considerable height, and therefore the labour -of walking would be much increased. - -If a man stand on one leg, the line of direction of his weight must -fall within the space on which his foot treads. The smallness of this -space, compared with the height of the centre of gravity, accounts for -the difficulty of this feat. - -The position of the centre of gravity of the body changes with the -posture and position of the limbs. If the arm be extended from one -side, the centre of gravity is brought nearer to that side than it was -when the arm hung perpendicularly. When dancers, standing on one leg, -extend the other at right angles to it, they must incline the body in -the direction opposite to that in which the leg is extended, in order -to bring the centre of gravity over the foot which supports them. - -When a porter carries a load, his position must be regulated by the -centre of gravity of his body and the load taken together. If he bore -the load on his back, the line of direction would pass beyond his -heels, and he would fall backwards. To bring the centre of gravity -over his feet he accordingly leans forward, _fig. 64._ - -If a nurse carry a child in her arms, she leans back for a like reason. - -When a load is carried on the head, the bearer stands upright, that the -centre of gravity may be over his feet. - -In ascending a hill, we appear to incline forward; and in descending, -to lean backward, but in truth, we are standing upright with respect to -a level plane. This is necessary to keep the line of direction between -the feet, as is evident from _fig. 65._ - -A person sitting on a chair which has no back cannot rise from it -without either stooping forward to bring the centre of gravity over -the feet, or drawing back the feet to bring them under the centre of -gravity. - -A quadruped never raises both feet on the same side simultaneously, -for the centre of gravity would then be unsupported. Let -A B C D, _fig. 66._, be the feet. The base on -which it stands is A B C D, and the centre of gravity -is nearly over the point O, where the diagonals cross each other. The -legs A and C being raised together, the centre of gravity is supported -by the legs B and D, since it falls between them; and when B and D -are raised it is, in like manner, supported by the feet A and C. The -centre of gravity, however, is often unsupported for a moment; for the -leg B is raised from the ground before A comes to it, as is plain from -observing the track of a horse’s feet, the mark of A being upon or -before that of B. In the more rapid paces of all animals the centre of -gravity is at intervals unsupported. - -The feats of rope-dancers are experiments on the management of the -centre of gravity. The evolutions of the performer are found to be -facilitated by holding in his hand a heavy pole. His security in -this case depends, not on the centre of gravity of his body, but on -that of his body and the pole taken together. This point is near the -centre of the pole, so that, in fact, he may be said to hold in his -hands the point on the position of which the facility of his feats -depends. Without the aid of the pole the centre of gravity would be -within the trunk of the body, and its position could not be adapted to -circumstances with the same ease and rapidity. - -(170.) The centre of gravity of a mass of fluid is that point which -would have the properties which have been proved to belong to the -centre of gravity of a solid, if the fluid were solidified without -changing in any respect the quantity or arrangement of its parts. This -point also possesses other properties, in reference to fluids, which -will be investigated in HYDROSTATICS and PNEUMATICS. - -(171.) The centre of gravity of two bodies separated from one another, -is that point which would possess the properties ascribed to the centre -of gravity, if the two bodies were united by an inflexible line, the -weight of which might be neglected. To find this point mathematically -is a very simple problem. Let A and B, _fig. 67._, be the two -bodies, and _a_ and _b_ their centres of gravity. Draw the right line -_a b_, and divide it at C, in such a manner that _a_ C shall have -the same proportion to _b_ C as the mass of the body B has to the mass -of the body A. - -This may easily be verified experimentally. Let A and B be two bodies, -whose weight is considerable, in comparison with that of the rod -_a b_, which joins them. Let a fine silken string, with its -ends attached to them, be hung upon a pin; and on the same pin let a -plumb-line be suspended. In whatever position the bodies may be hung, -it will be observed that the plumb-line will cross the rod _a b_ -at the same point, and that point will divide the line _a b_ into -parts _a_ C and _b_ C, which are in the proportion of the mass of B to -the mass of A. - -(172.) The centre of gravity of three separate bodies is defined in the -same manner as that of two, and may be found by first determining the -centre of gravity of two; and then supposing their masses concentrated -at that point, so as to form one body, and finding the centre of -gravity of that and the third. - -In the same manner the centre of gravity of any number of bodies may be -determined. - -(173.) If a plate of uniform thickness be bounded by straight edges, -its centre of gravity may be found by dividing it into triangles by -diagonal lines, as in _fig. 68._, and having determined by (154) -the centres of gravity of the several triangles, the centre of gravity -of the whole plate will be their common centre of gravity, found as -above. - -(174.) Although the centre of gravity takes its name from the -familiar properties which it has in reference to detached bodies of -inconsiderable magnitude, placed on or near the surface of the earth, -yet it possesses properties of a much more general and not less -important nature. One of the most remarkable of these is, that the -centre of gravity of any number of separate bodies is never affected -by the mutual attraction, impact, or other influence which the bodies -may transmit from one to another. This is a necessary consequence of -the equality of action and reaction explained in Chapter IV. For if A -and B, _fig. 67._, attract each other, and change their places -to A′ and B′, the space a a′ will have to _b b′_ the same -proportion as B has to A, and therefore by what has just been proved -(171) the same proportion as _a_ C has to _b_ C. It follows, that the -remainders _a′_ C and _b′_ C will be in the proportion of B to A, and -that C will continue to be the centre of gravity of the bodies after -they have approached by their mutual attraction. - -Suppose, for example, that A and B were 12lbs. and 8lbs. respectively, -and that _a b_ were 40 feet. The point C must (171) divide -_a b_ into two parts, in the proportion of 8 to 12, or of 2 to -3. Hence it is obvious that _a_ C will be 16 feet, and _b_ C 24 feet. -Now, suppose that A and B attract each other, and that A approaches -B through two feet. Then B must approach A through three feet. Their -distances from C will now be 14 feet and 21 feet, which, being in the -proportion of B to A, the point C will still be their centre of gravity. - -Hence it follows, that if a system of bodies, placed at rest, be -permitted to obey their mutual attractions, although the bodies will -thereby be severally moved, yet their common centre of gravity must -remain quiescent. - -(175.) When one of two bodies is moving in a straight line, the other -being at rest, their common centre of gravity must move in a parallel -straight line. Let A and B, _fig. 69._, be the centres of gravity -of the bodies, and let A move from A to _a_, B remaining at rest. -Draw the lines A B and _a_ B. In every position which the body B -assumes during its motion, the centre of gravity C divides the line -joining them into parts A C, B C, which are in the proportion -of the mass B to the mass A. Now, suppose any number of lines drawn -from B to the line A _a_; a parallel C _c_ to A _a_ through C divides -all these lines in the same proportion; and therefore, while the body A -moves from A to _a_, the common centre of gravity moves from C to _c_. - -If both the bodies A and B moved uniformly in straight lines, the -centre of gravity would have a motion compounded (74) of the two -motions with which it would be affected, if each moved while the other -remained at rest. In the same manner, if there were three bodies, each -moving uniformly in a straight line, their common centre of gravity -would have a motion compounded of that motion which it would have if -one remained at rest while the other two moved, and that which the -motion of the first would give it if the last two remained at rest; and -in the same manner it may be proved, that when any number of bodies -move each in a straight line, their common centre of gravity will have -a motion compounded of the motions which it receives from the bodies -severally. - -It may happen that the several motions which the centre of gravity -receives from the bodies of the system will neutralise each other; and -this does, in fact, take place for such motions as are the consequences -of the mutual action of the bodies upon one another. - -(176.) If a system of bodies be not under the immediate influence of -any forces, and their mutual attraction be conceived to be suspended, -they must severally be either at rest or in uniform rectilinear -motion in virtue of their inertia. Hence, their common centre of -gravity must also be either at rest or in uniform rectilinear motion. -Now, if we suppose their mutual attractions to take effect, the -state of their common centre of gravity will not be changed, but the -bodies will severally receive motions compounded of their previous -uniform rectilinear motions and those which result from their mutual -attractions. The combined effects will cause each body to revolve in -an orbit round the common centre of gravity, or will precipitate it -towards that point. But still that point will maintain its former state -undisturbed. - -This constitutes one of the general laws of mechanical science, and is -of great importance in physical astronomy. It is known by the title -“the conservation of the motion of the centre of gravity.” - -(177.) The solar system is an instance of the class of phenomena to -which we have just referred. All the motions of the bodies which -compose it can be traced to certain uniform rectilinear motions, -received from some former impulse, or from a force whose action has -been suspended, and those motions which necessarily result from the -principle of gravitation. But we shall not here insist further on this -subject, which more properly belongs to another department of the -science. - -(178.) If a solid body suffer an impact in the direction of a line -passing through its centre of gravity, all the particles of the body -will be driven forward with the same velocity in lines parallel to the -direction of the impact, and the whole force of the motion will be -equal to that of the impact. The common velocity of the parts of the -body will in this case be determined by the principles explained in -Chapter IV. The impelling force being equally distributed among all the -parts, the velocity will be found by dividing the numerical value of -that force by the number expressing the mass. - -If any number of impacts be given simultaneously to different points -of a body, a certain complex motion will generally ensue. The mass -will have a relative motion round the centre of gravity as if it were -fixed, while that point will move forward uniformly in a straight line, -carrying the body with it. The relative motion of the mass round the -centre of gravity may be found by considering the centre of gravity -as a fixed point, round which the mass is free to move, and then -determining the motion which the applied forces would produce. This -motion being supposed to continue uninterrupted, let all the forces be -imagined to be applied in their proper directions and quantities to the -centre of gravity. By the principles for the composition of force they -will be mechanically equivalent to a single force through that point. -In the direction of this single force the centre of gravity will move -and have the same velocity as if the whole mass were there concentrated -and received the impelling forces. - -(179.) These general properties, which are entirely independent of -gravity, render the “centre of gravity” an inadequate title for this -important point. Some physical writers have, consequently, called it -the “centre of inertia.” The “centre of gravity,” however, is the name -by which it is still generally designated. - - - - -CHAP. X. - -THE MECHANICAL PROPERTIES OF AN AXIS. - - -(180.) When a body has a motion of rotation, the line round which it -revolves is called an _axis_. Every point of the body must in this -case move in a circle, whose centre lies in the axis, and whose radius -is the distance of the point from the axis. Sometimes while the body -revolves, the axis itself is moveable, and not unfrequently in a state -of actual motion. The motions of the earth and planets, or that of -a common spinning-top, are examples of this. The cases, however, which -will be considered in the present chapter, are chiefly those in which -the axis is immovable, or at least where its motion has no relation to -the phenomena under investigation. Instances of this are so frequent -and obvious, that it seems scarcely necessary to particularise them. -Wheel-work of every description, the moving parts of watches and -clocks, turning lathes, mill-work, doors and lids on hinges, are all -obvious examples. In tools or other instruments which work on joints or -pivots, such as scissors, shears, pincers, although the joint or pivot -be not absolutely fixed, it is to be considered so in reference to the -mechanical effect. - -[Illustration: _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -In some cases, as in most of the wheels of watches and clocks, -fly-wheels and chucks of the turning lathe, and the arms of wind-mills, -the body turns continually in the same direction, and each of its -points traverses a complete circle during every revolution of the -body round its axis. In other instances the motion is alternate or -reciprocating, its direction being at intervals reversed. Such is -the case in pendulums of clocks, balance-wheels of chronometers, the -treddle of the lathe, doors and lids on hinges, scissors, shears, -pincers, &c. When the alternation is constant and regular, it is called -_oscillation_ or _vibration_, as in pendulums and balance-wheels. - -(181.) To explain the properties of an axis of rotation it will be -necessary to consider the different kinds of forces to the action of -which a body moveable on such an axis may be submitted, to show how -this action depends on their several quantities and directions, to -distinguish the cases in which the forces neutralise each other and -mutually equilibrate from those in which motion ensues, to determine -the effect which the axis suffers, and, in the cases where motion is -produced, to estimate the effects of those centrifugal forces (137.) -which are created by the mass of the body whirling round the axis. - -Forces in general have been distinguished by the duration of their -action into instantaneous and continued forces. The effect of an -instantaneous force is produced in an infinitely short time. If -the body which sustains such an action be previously quiescent and -free, it will move with a uniform velocity in the direction of the -impressed force. (93.) If, on the other hand, the body be not free, -but so restrained that the impulse cannot put it in motion, then the -fixed points or lines which resist the motion sustain a corresponding -shock at the moment of the impulse. This effect, which is called -_percussion_, is, like the force which causes it, instantaneous. - -A continued force produces a continued effect. If the body be free and -previously quiescent, this effect is a continual increase of velocity. -If the body be so restrained that the applied force cannot put it in -motion, the effect is a continued pressure on the points or lines which -sustain it. (94.) - -It may happen, however, that although the body be not absolutely free -to move in obedience to the force applied to it, yet still it may not -be altogether so restrained as to resist the effect of that force and -remain at rest. If the point at which a force is applied be free to -move in a certain direction not coinciding with that of the applied -force, that force will be resolved into two elements; one of which is -in the direction in which the point is free to move, and the other at -right angles to that direction. The point will move in obedience to -the former element, and the latter will produce percussion or pressure -on the points or lines which restrain the body. In fact, in such cases -the resistance offered by the circumstances which confine the motion -of the body modifies the motion which it receives, and as every change -of motion must be the consequence of a force applied (44.), the fixed -points or lines which offer the resistance must suffer a corresponding -effect. - -It may happen that the forces impressed on the body, whether they -be continued or instantaneous, are such as, were it free, would -communicate to it a motion which the circumstances which restrain it -do not forbid it to receive. In such a case the fixed points or lines -which restrain the body sustain no force, and the phenomena will be -the same in all respects as if these points or lines were not fixed. - -It will be easy to apply these general reflections to the case in which -a solid body is moveable on a fixed axis. Such a body is susceptible -of no motion except one of rotation on that axis. If it be submitted -to the action of instantaneous forces, one or other of the following -effects must ensue. 1. The axis may resist the forces, and prevent any -motion. 2. The axis may modify the effect of the forces sustaining a -corresponding percussion, and the body receiving a motion of rotation. -3. The forces applied may be such as would cause the body to spin round -the axis even were it not fixed, in which case the body will receive a -motion of rotation, but the axis will suffer no percussion. - -What has been just observed of the effect of instantaneous forces is -likewise applicable to continued ones. 1. The axis may entirely resist -the effect of such forces, in which case it will suffer a pressure -which may be estimated by the rules for the composition of force. 2. -It may modify the effect of the applied forces, in which case it must -also sustain a pressure, and the body must receive a motion of rotation -which is subject to constant variation, owing to the incessant action -of the forces. 3. The forces may be such as would communicate to the -body the same rotatory motion if the axis were not fixed. In this case -the forces will produce no pressure on the axis. - -The impressed forces are not the only causes which affect the axis of -a body during the phenomenon of rotation. This species of motion calls -into action other forces depending on the inertia of the mass, which -produce effects upon the axis, and which play a prominent part in the -theory of rotation. While the body revolves on its axis, the component -particles of its mass move in circles, the centres of which are placed -in the axis. The radius of the circle in which each particle moves is -the line drawn from that particle perpendicular to the axis. It has -been already proved that a particle of matter, moving round a centre, -is attended with a centrifugal force proportionate to the radius of the -circle in which it moves and to the square of its angular velocity. -When a solid body revolves on its axis, all its parts are whirled round -together, each performing a complete revolution in the same time. The -angular velocity is consequently the same for all, and the difference -of the centrifugal forces of different particles must entirely depend -upon their distances from the axis. The tendency of each particle to -fly from the axis, arising from the centrifugal force, is resisted by -the cohesion of the parts of the mass, and in general this tendency is -expended in exciting a pressure or strain upon the axis. It ought to -be recollected, however, that this pressure or strain is altogether -different from that already mentioned, and produced by the forces which -give motion to the body. The latter depends entirely upon the quantity -and directions of the applied forces in relation to the axis: the -former depends on the figure and density of the body, and the velocity -of its motion. - -These very complex effects render a simple and elementary exposition -of the mechanical properties of a fixed axis a matter of considerable -difficulty. Indeed, the complete mathematical development of this -theory long eluded the skill of the most acute geometers, and it was -only at a comparatively late period that it yielded to the searching -analysis of modern science. - -(182.) To commence with the most simple case, we shall consider the -body as submitted to the action of a single force. The effect of this -force will vary according to the relation of its direction to that of -the axis. There are two ways in which a body may be conceived to be -moveable around an axis. 1. By having pivots at two points which rest -in sockets, so that when the body is moved it must revolve round the -right line joining the pivots as an axis. 2. A thin cylindrical rod may -pass through the body, on which it may turn in the same manner as a -wheel upon its axle. - -If the force be applied to the body in the direction of the axis, it -is evident that no motion can ensue, and the effect produced will be a -pressure on that pivot towards which the force is directed. If in this -case the body revolved on a cylindrical rod, the tendency of the force -would be to make it slide along the rod without revolving round it. - -Let us next suppose the force to be applied not in the direction of -the axis itself, but parallel to it. Let A B, _fig. 70._, -be the axis, and let C D be the direction of the force applied. -The pivots being supposed to be at A and B, draw A G and B F -perpendicular to A B. The force C D will be equivalent to -three forces, one acting from B towards A, equal in quantity to the -force C D. This force will evidently produce a corresponding -pressure on the pivot A. The other two forces will act in the -directions A G and B F, and will have respectively to the -force C D the same proportion as A E has to A B. Such -will be the mechanical effect of a force C D parallel to the axis. -And as these effects are all directed on the pivots, no motion can -ensue. - -If the body revolve on a cylindrical rod, the forces A G and -B F would produce a strain upon the axis, while the third force -in the direction B A would have a tendency to make the body slide -along it. - -(183.) If the force applied to the body be directed upon the axis, -and at right angles to it, no motion can be produced. In this case, -if the body be supported by pivots at A and B, the force K L, -perpendicular to the line A B, will be distributed between the -pivots, producing a pressure on each proportional to its distance from -the other. The pressure on A having to the pressure on B the same -proportion as L B has to L A. - -If the force K H be directed obliquely to the axis, it will be -equivalent to two forces (76.), one K L perpendicular to the axis, -and the other K M parallel to it. The effect of each of these may -be investigated as in the preceding cases. - -In all these observations the body has been supposed to be submitted -to the action of one force only. If several forces act upon it, the -direction of each of them crossing the axis either perpendicularly -or obliquely, or taking the direction of the axis or any parallel -direction, their effects may be similarly investigated. In the same -manner we may determine the effects of any number of forces whose -combined results are mechanically equivalent to forces which either -intersect the axis or are parallel to it. - -(184.) If any force be applied whose direction lies in a plane oblique -to the axis, it can always be resolved into two elements (76.), one of -which is parallel to the axis, and the other in a plane perpendicular -to it. The effect of the former has been already determined, and -therefore we shall at present confine our attention to the latter. - -Suppose the axis to be perpendicular to the paper, and to pass through -the point G, _fig. 71._ and let A B C be a section of -the body. It will be convenient to consider the section vertical and -the axis horizontal, omitting, however, any notice of the effect of the -weight of the body. - -Let a weight W be suspended by a cord Q W from any point Q. This -weight will evidently have a tendency to turn the body round in the -direction A B C. Let another cord be attached to any other -point P, and, being carried over a wheel R, let a dish S be attached to -it, and let fine sand be poured into this dish until the tendency of -S to turn the body round the axis in the direction of C B A -balances the opposite tendency of W. Let the weights of W and S be -then exactly ascertained, and also let the distances G I and -G H of the cords from the axis be exactly measured. It will be -found that, if the number of ounces in the weight S be multiplied by -the number of inches in G H, and also the number of ounces in W -by the number of inches in G I, equal products will be obtained. -This experiment may be varied by varying the position of the wheel R, -and thereby changing the direction of the string P R, in which -cases it will be always found necessary to vary the weight of S in -such a manner, that when the number of ounces in it is multiplied by -the number of inches in the distance of the string from the axis, the -product obtained shall be equal to that of the weight W by the distance -G I. We have here used ounces and inches as the measures of weight -and distance; but it is obvious that any other measures would be -equally applicable. - -From what has been just stated it follows, that the energy of the -weight of S to move the body on its axis, does not depend alone upon -the actual amount of that weight, but also upon the distance of the -string from the axis. If, while the position of the string remains -unaltered, the weight of S be increased or diminished, the resisting -weight W must be increased or diminished in the same proportion. But -if, while the weight of S remains unaltered, the distance of the string -P R from the axis G be increased or diminished, it will be found -necessary to increase or diminish the resisting weight W in exactly the -same proportion. It therefore appears that the increase or diminution -of the distance of the direction of a force from the axis has the -same effect upon its power to give rotation as a similar increase -or diminution of the force itself. The power of a force to produce -rotation is, therefore, accurately estimated, not by the force alone, -but by the product found by multiplying the force by the distance of -its direction from the axis. It is frequently necessary in mechanical -science to refer to this power of a force, and, accordingly, the -product just mentioned has received a particular denomination. It is -called the _moment_ of the force round the axis. - -(185.) The distance of the direction of a force from the axis is -sometimes called the _leverage_ of the force. The _moment_ of a force -is therefore found by multiplying the force by its leverage, and the -energy of a given force to turn a body round an axis is proportional to -the leverage of that force. - -From all that has been observed it may easily be inferred that, if -several forces affect a body moveable on an axis, having tendencies -to turn it in different directions, they will mutually neutralise -each other and produce equilibrium, if the sum of the moments of those -forces which tend to turn the body in one direction be equal to the -sum of the moments of those which tend to turn it in the opposite -direction. Thus, if the forces A, B, C, ... tend to turn the body from -right to left, and the distances of their directions from the axis be -_a_, _b_, _c_, ... and the forces A′, B′, C′, ... tend to move it from -left to right, and the distances of their directions from the axis be -_a′_, _b′_, _c′_, ...; then these forces will produce equilibrium, -if the products found by multiplying the ounces in A, B, C, ... -respectively by the inches in _a_, _b_, _c_, ... when added together -be equal to the products found by multiplying the ounces in A′, B′, -C′, ... by the inches in _a′_, _b′_, _c′_, ... respectively when added -together. But if either of these sets of products when added together -exceed the other, the corresponding set of forces will prevail, and the -body will revolve on its axis. - -(186.) When a body receives an impulse in a direction perpendicular to -the axis, but not crossing it, a uniform rotatory motion is produced. -The velocity of this motion depends on the force of the impulse, the -distance of the direction of the impulse from the axis, and the manner -in which the mass of the body is distributed round the axis. It is to -be considered that the whole force of the impulse is shared amongst the -various parts of the mass, and is transmitted to them from the point -where the impulse is applied by reason of the cohesion and tenacity -of the parts, and the impossibility of one part yielding to a force -without carrying all the other parts with it. The force applied acts -upon those particles nearer to the axis than its own direction under -advantageous circumstances; for, according to what has been already -explained, their power to resist the effect of the applied force is -small in the same proportion with their distance. On the other hand, -the applied force acts upon particles of the mass, at a greater -distance than its own direction, under circumstances proportionably -disadvantageous; for their resistance to the applied force is great in -proportion to their distances from the axis. - -Let C D, _fig. 72._, be a section of the body made by a plane -passing through the axis A B. Suppose the impulse to be applied at -P, perpendicular to this plane, and at the distance P O from the -axis. The effect of the impulse being distributed through the mass will -cause the body to revolve on A B, with a uniform velocity. There -is a certain point G, at which, if the whole mass were concentrated, -it would receive from the impulse the same velocity round the axis. -The distance O G is called the _radius of gyration_ of the axis -A B, and the point G is called the _centre of gyration_ relatively -to that axis. The effect of the impulse upon the mass concentrated at -G is great in exactly the same proportion as O G is small. This -easily follows from the property of moments which has been already -explained; from whence it may be inferred, that the greater the radius -of gyration is, the less will be the velocity which the body will -receive from a given impulse. - -(187.) Since the radius of gyration depends on the manner in which the -mass is arranged round the axis, it follows that for different axes -in the same body there will be different radii of gyration. Of all -axes taken in the same body parallel to each other, that which passes -through the centre of gravity has the least radius of gyration. If the -radius of gyration of any axis passing through the centre of gravity be -given, that of any parallel axis can be found; for the square of the -radius of gyration of any axis is equal to the square of the distance -of that axis from the centre of gravity added to the square of the -radius of gyration of the parallel axis through the centre of gravity. - -(188.) The product of the numerical expressions for the mass of the -body and the square of the radius of gyration is a quantity much used -in mechanical science, and has been called the _moment of inertia_. The -moments of inertia, therefore, for different axes in the same body are -proportional to the squares of the corresponding radii of gyration; and -consequently increase as the distances of the axes from the centre of -gravity increase. (187.) - -(189.) From what has been explained in (187.), it follows, that the -moment of inertia of any axis may be computed by common arithmetic, if -the moment of inertia of a parallel axis through the centre of gravity -be previously known. To determine this last, however, would require -analytical processes altogether unsuitable to the nature and objects of -the present treatise. - -The velocity of rotation which a body receives from a given impulse -is great in exactly the same proportion as the moment of inertia is -small. Thus the moment of inertia may be considered in rotatory motion -analogous to the mass of the body in rectilinear motion. - -From what has been explained in (187.) it follows that a given impulse -at a given distance from the axis will communicate the greatest -angular velocity when the axis passes through the centre of gravity, -and that the velocity which it will communicate round other axes -will be diminished in the same proportion as the squares of their -distances from the centre of gravity added to the square of the radius -of gyration for a parallel axis through the centre of gravity are -augmented. - -(190.) If any point whatever be assumed in a body, and right lines -be conceived to diverge in all directions from that point, there are -generally two of these lines, which being taken as axes of rotation, -one has a greater and the other a less moment of inertia than any of -the others. It is a remarkable circumstance, that, whatever be the -nature of the body, whatever be its shape, and whatever be the position -of the point assumed, these two axes of greatest and least moment will -always be at right angles to each other. - -These axes and a third through the same point, and at right angles to -both of them, are called the _principal axes_ of that point from which -they diverge. To form a distinct notion of their relative position, -let the axis of greatest moment be imagined to lie horizontally from -north to south, and the axis of least moment from east to west; then -the third principal axis will be presented perpendicularly upwards and -downwards. The first two being called the principal axes of greatest -and least moment, the third may be called the _intermediate principal -axis_. - -(191.) Although the moments of the three principal axes be in general -unequal, yet bodies may be found having certain axes for which these -moments may be equal. In some cases the moment of the intermediate axis -is equal to that of the principal axis of greatest moment: in others it -is equal to that of the principal axis of least moment, and in others -the moments of all the three principal axes are equal to each other. - -If the moments of any two of three principal axes be equal, the moments -of all axes through the same point and in their plane will also be -equal; and if the moments of the three principal axes through a point -be equal, the moments of all axes whatever, through the same point, -will be equal. - -(192.) If the moments of the principal axes through the centre of -gravity be known, the moments for all other axes through that point may -be easily computed. To effect this it is only necessary to multiply -the moments of the principal axes by the squares of the co-sines of -the angles formed by them respectively with the axis whose moment is -sought. The products being added together will give the required moment. - -(193.) By combining this result with that of (189.), it will be evident -that the moment of all axes whatever may be determined, if those of the -principal axes through the centre of gravity be known. - -(194.) It is obvious that the principal axis of least moment through -the centre of gravity has a less moment of inertia than any other axis -whatever. For it has, by its definition (190.) a less moment of inertia -than any other axis through the centre of gravity, and every other -axis through the centre of gravity has a less moment of inertia than a -parallel axis through any other point (187.) and (189.) - -(195.) If two of the principal axes through the centre of gravity have -equal moments of inertia, all axes in any plane parallel to the plane -of these axes, and passing through the point where a perpendicular from -the centre of gravity meets that plane, must have equal moments of -inertia. For by (191.) all axes in the plane of those two have equal -moments, and by (189.) the axes in the parallel plane have moments -which exceed these by the same quantity, being equally distant from -them. (187.) - -Hence it is obvious that if the three principal axes through the centre -of gravity have equal moments, all axes situated in any given plane, -and passing through the point where the perpendicular from the centre -of gravity meets that plane, will have equal moments, being equally -distant from parallel axes through the centre of gravity. - -(196.) If the three principal axes through the centre of gravity have -unequal moments, there is no point whatever for which all axes will -have equal moments; but if the principal axis of least moment and -the intermediate principal axis through the centre of gravity have -equal moments, then there will be two points on the principal axis -of greatest moment, equally distant at opposite sides of the centre -of gravity, at which all axes will have equal moments. If the three -principal axes through the centre of gravity have equal moments, no -other point of the body can have principal axes of equal moment. - -(197.) When a body revolves on a fixed axis, the parts of its mass are -whirled in circles round the axis; and since they move with a common -angular velocity, they will have centrifugal forces proportional to -their distances from the axis. If the component parts of the mass were -not united together by cohesive forces of energies greater than these -centrifugal forces, they would be separated, and would fly off from -the axis; but their cohesion prevents this, and causes the effects of -the different centrifugal forces, which affect the different parts of -the mass, to be transmitted so as to modify each other, and finally -to produce one or more forces mechanically equivalent to the whole, -and which are exerted upon the axis and resisted by it. We propose -now to explain these effects, as far as it is possible to render them -intelligible without the aid of mathematical language. - -It is obvious that any number of equal parts of the mass, which are -uniformly arranged in a circle round the axis, have equal centrifugal -forces acting from the centre of the circle in every direction. These -mutually neutralise each other, and therefore exert no force on the -axis. The same may be said of all parts of the mass which are regularly -and equally distributed on every side of the axis. - -Also if equal masses be placed at equal distances on opposite sides -of the axis, their centrifugal forces will destroy each other. Hence -it appears that the pressure which the axis of rotation sustains from -the centrifugal forces of the revolving mass, arises from the unequal -distribution of the matter around it. - -From this reasoning it will be easily perceived that in the following -examples the axis of rotation will sustain no pressure. - -A globe revolving on any of its diameters, the density being the same -at equal distances from the centre. - -A spheroid or a cylinder revolving on its axis, the density being equal -at equal distances from the axis. - -A cube revolving on an axis which passes through the centre of two -opposite bases, being of uniform density. - -A circular plate of uniform thickness and density revolving on one of -its diameters as an axis. - -(198.) In all these examples it will be observed that the axis of -rotation passes through the centre of gravity. The general theorem, of -which they are only particular instances, is, “if a body revolve on a -principal axis, passing through the centre of gravity, the axis will -sustain no pressure from the centrifugal force of the revolving mass.” -This is a property in which the principal axes through the centre -of gravity are unique. There is no other axis on which a body could -revolve without pressure. - -If two of the principal axes through the centre of gravity have equal -moments, every axis in their plane has the same moment, and is to be -considered equally as a principal axis. In this case the body would -revolve on any of these axes without pressure. - -A homogeneous spheroid furnishes an example of this. If any of the -diameters of the earth’s equator were a fixed axis, the earth would -revolve on it without producing pressure. - -If the three principal axes through the centre of gravity have equal -moments, all axes through the centre of gravity are to be considered as -principal axes. In this case the body would revolve without pressure on -any axis through the centre of gravity. - -A globe, in which the density of the mass at equal distances from the -centre is the same, is an example of this. Such a body would revolve -without pressure on any axis through its centre. - -(199.) Since no pressure is excited on the axis in these cases, the -state of the body will not be changed, if during its rotation the axis -cease to be fixed. The body will notwithstanding continue to revolve -round the axis, and the axis will maintain its position. - -Thus a spinning-top of homogeneous material and symmetrical form will -revolve steadily in the same position, until the friction of its point -with the surface on which it rests deprives it of motion. This is a -phenomenon which can only be exhibited when the axis of rotation is a -principal axis through the centre of gravity. - -(200.) If the body revolve round any axis through the centre of -gravity, which is not a principal axis, the centrifugal pressure is -represented by two forces, which are equal and parallel, but which act -in opposite directions on different points of the axis. The effect of -these forces is to produce a strain upon the axis, and give the body a -tendency to move round another axis at right angles to the former. - -(201.) If the fixed axis on which a body revolves be a principal axis -through any point different from the centre of gravity, then a pressure -will be produced by the centrifugal force of the revolving mass, and -this pressure will act at right angles to the axis on the point to -which it is a principal axis, and in the plane through that axis and -the centre of gravity. The amount of the pressure will be proportional -to the mass of the body, the distance of the centre of gravity from the -axis, and the square of the velocity of rotation. - -(202.) Since the whole pressure is in this case excited on a single -point, the stability of the axis will not be disturbed, provided that -point alone be fixed. So that even though the axis should be free to -turn on that point, no motion will ensue as long as no external forces -act upon the body. - -(203.) If the axis of rotation be not a principal axis, the centrifugal -forces will produce an effect which cannot be represented by a single -force. The effect may be understood by conceiving two forces to act -on _different points_ of the axis at right angles to it and to each -other. The quantities of these pressures and their directions depend -on the figure and density of the mass and the position of the axis, -in a manner which cannot be explained without the aid of mathematical -language and principles. - -(204.) The effects upon the axis which have been now explained are -those which arise from the motion of rotation, from whatever cause that -motion may have arisen. The forces which produce that motion, however, -are attended with effects on the axis which still remain to be noticed. -When these forces, whether they be of the nature of instantaneous -actions or continued forces, are entirely resisted by the axis, their -directions must severally be in a plane passing through the axis, or -they must, by the principles of the composition of force [(74.) et -seq.], be mechanically equivalent to forces in that plane. In every -other case the impressed forces _must_ produce motion, and, except in -certain cases, must also produce effects upon the axis. - -By the rules for the composition of force it is possible in all cases -to resolve the impressed forces into others which are either in planes -through the axis, or in planes perpendicular to it, or, finally, some -in planes through it, and others in planes perpendicular to it. The -effect of those which are in planes through the axis has been already -explained; and we shall now confine our attention to those impelling -forces which act at right angles to the axis, and which produce motion. - -It will be sufficient to consider the effect of a single force at right -angles to the axis; for whatever be the number of forces which act -either simultaneously or successively, the effect of the whole will -be decided by combining their separate effects. The effect which a -single force produces depends on two circumstances, 1. The position of -the axis with respect to the figure and mass of the body, and 2. The -quantity and direction of the force itself. - -In general the shock which the axis sustains from the impact may be -represented by two impacts applied to it at different points, one -parallel to the impressed force, and the other perpendicular to it, -but both perpendicular to the axis. There are certain circumstances, -however, under which this effect will be modified. - -If the impulse which the body receives be in a direction perpendicular -to a plane through the axis and the centre of gravity, and at a -distance from the axis which bears to the radius of gyration (186.) -the same proportion as that line bears to the distance of the centre -of gravity from the axis, there are certain cases in which the impulse -will produce no percussion. To characterise these cases generally would -require analytical formulæ which cannot conveniently be translated -into ordinary language. That point of the plane, however, where the -direction of the impressed force meets it, when no percussion on the -axis is produced, is called the _centre of percussion_. - -If the axis of rotation be a principal axis, the centre of percussion -must be in the right line drawn through the centre of gravity, -intersecting the axis at right angles, and at the distance from the -axis already explained. - -If the axis of rotation be parallel to a principal axis through the -centre of gravity, the centre of percussion will be determined in the -same manner. - -(205.) There are many positions which the axis may have in which there -will be no centre of percussion; that is, there will be no direction in -which an impulse could be applied without producing a shock upon the -axis. One of these positions is when it is a principal axis through -the centre of gravity. This is the only case of rotation round an axis -in which no effect arises from the centrifugal force; and therefore it -follows that the only case in which the axis sustains no effect from -the motion produced, is one in which it must necessarily suffer an -effect from that which produces the motion. - -If the body be acted upon by continued forces, their effect is at each -instant determined by the general principles for the composition of -force. - - - - -CHAP. XI. - -ON THE PENDULUM. - - -(206.) When a body is placed on a horizontal axis which does not -pass through its centre of gravity, it will remain in permanent -equilibrium only when the centre of gravity is immediately below the -axis. If this point be placed in any other situation, the body will -oscillate from side to side, until the atmospherical resistance and the -friction of the axis destroy its motion. (159, 160.) Such a body is -called a _pendulum_. The swinging motion which it receives is called -_oscillation_ or _vibration_. - -(207.) The use of the pendulum, not only for philosophical purposes, -but in the ordinary economy of life, renders it a subject of -considerable importance. It furnishes the most exact means of measuring -time, and of determining with precision various natural phenomena. By -its means the variation of the force of gravity in different latitudes -is discovered, and the law of that variation experimentally exhibited. -In the present chapter, we propose to explain the general principles -which regulate the oscillation of pendulums. Minute details concerning -their construction will be given in the twenty-first chapter of this -volume. - -(208.) A simple pendulum is composed of a heavy molecule attached -to the end of a flexible thread, and suspended by a fixed point O, -_fig. 73._ When the pendulum is placed in the position O C, -the molecule being vertically below the point of suspension, it will -remain in equilibrium; but if it be drawn into the position O A -and there liberated, it will descend towards C, moving through the arc -A C with accelerated motion. Having arrived at C and acquired -a certain velocity, it will, by reason of its inertia, continue to -move in the same direction. It will therefore commence to ascend the -arc C A′ with the velocity so acquired. During its ascent, the -weight of the molecule retards its motion in exactly the same manner -as it had accelerated it in descending from A to C; and when the -molecule has ascended through the arc C A′ equal to C A, -its entire velocity will be destroyed, and it will cease to move in -that direction. It will thus be placed at A′ in the same manner as in -the first instance it had been placed at A, and consequently it will -descend from A′ to C with accelerated motion, in the same manner as -it first moved from A to C. It will then ascend from C to A, and so -on, continually. In this case the thread, by which the molecule is -suspended, is supposed to be perfectly flexible, inextensible, and -of inconsiderable weight. The point of suspension is supposed to be -without friction, and the atmosphere to offer no resistance to the -motion. - -It is evident from what has been stated, that the times of moving from -A to A′ and from A′ to A are equal, and will continue to be equal so -long as the pendulum continues to vibrate. If the number of vibrations -performed by the pendulum were registered, and the time of each -vibration known, this instrument would become a chronometer. - -The rate at which the motion of the pendulum is accelerated in its -descent towards its lowest position is not uniform, because the force -which impels it is continually decreasing, and altogether disappears -at the point C. The impelling force arises from the effect of gravity -on the suspended molecule, and this effect is always produced in the -vertical direction A V. The greater the angle O A V is, -the less efficient the force of gravity will be in accelerating the -molecule: this angle evidently increases as the molecule approaches -C, which will appear by inspecting _fig. 73._ At C, the force of -gravity acting in the direction C B is totally expended in giving -tension to the thread, and is inefficient in moving the molecule. It -follows, therefore, that the impelling force is greatest at A, and -continually diminishes from A to C, where it altogether vanishes. The -same observations will be applicable to the retarding force from C to -A′, and to the accelerating force from A′ to C, and so on. - -When the length of the thread and the intensity of the force of -gravity are given, the time of vibration depends on the length of the -arc A C, or on the magnitude of the angle A O C. If, -however, this angle do not exceed a certain limit of magnitude, the -time of vibration will be subject to no sensible variation, however -that angle may vary. Thus the time of oscillation will be the same, -whether the angle A O C be 2°, or 1° 30′, or 1°, or any -lesser magnitude. This property of a pendulum is expressed by the word -_isochronism_. The strict demonstration of this property depends on -mathematical principles, the details of which would not be suitable -to the present treatise. It is not difficult, however, to explain -generally how it happens that the same pendulum will swing through -greater and smaller arcs of vibration in the same time. If it swing -from A, the force of gravity at the commencement of its motion impels -it with an effect depending on the obliquity of the lines O A and -A V. If it commence its motion from _a_, the impelling effect from -the force of gravity will be considerably less than at A; consequently, -the pendulum begins to move at a slower rate, when it swings from -_a_ than when it moves from A: the greater magnitude of the swing is -therefore compensated by the increased velocity, so that the greater -and the smaller arcs of vibration are moved through in the same time. - -(209.) To establish this property experimentally, it is only necessary -to suspend a small ball of metal, or other heavy substance, by a -flexible thread, and to put it in a state of vibration, the entire -arc of vibration not exceeding 4° or 5°, the friction on the point -of suspension and other causes will gradually diminish the arc of -vibration, so that after the lapse of some hours it will be so small, -that the motion will scarcely be discerned without microscopic aid. If -the vibration of this pendulum be observed in reference to a correct -timekeeper, at the commencement, at the middle, and towards the end of -its motion, the rate will be found to suffer no sensible change. - -This remarkable law of isochronism was one of the earliest discoveries -of Galileo. It is said, that when very young, he observed a chandelier -suspended from the roof of a church in Pisa swinging with a pendulous -motion, and was struck with the uniformity of the rate even when the -extent of the swing was subject to evident variation. - -(210.) It has been stated in (117.) that the attraction of gravity -affects all bodies equally, and moves them with the same velocity, -whatever be the nature or quantity of the materials of which they are -composed. Since it is the force of gravity which moves the pendulum, we -should therefore expect that the circumstances of that motion should -not be affected either by the quantity or quality of the pendulous -body. And we find this, in fact, to be the case; for if small pieces -of different heavy substances such as lead, brass, ivory, &c., be -suspended by fine threads of equal length, they will vibrate in the -same time, provided their weights bear a considerable proportion to the -atmospherical resistance, or that they be suspended _in vacuo_. - -(211.) Since the time of vibration of a pendulum, which oscillates in -small arcs, depends neither on the magnitude of the arc of vibration -nor on the quality or weight of the pendulous body, it will be -necessary to explain the circumstances on which the variation of this -time depends. - -The first and most striking of these circumstances is the length of -the suspending thread. The rudest experiments will demonstrate the -fact, that every increase in the length of this thread will produce a -corresponding increase in the time of vibration; but according to what -law does this increase proceed? If the length of the thread be doubled -or trebled, will the time of vibration also be increased in a double -or treble proportion? This problem is capable of exact mathematical -solution, and the result shows that the time of vibration increases not -in the proportion of the increased length of the thread, but as the -square root of that length; that is to say, if the length of the thread -be increased in a four-fold proportion, the time of vibration will be -augmented in a two-fold proportion. If the thread be increased to nine -times its length, the time of vibration will be trebled, and so on. -This relation is exactly the same as that which was proved to subsist -between the spaces through which a body falls freely, and the times -of fall. In the table, page 89, if the figures representing the -height be understood to express the length of different pendulums, the -figures immediately above them will express the corresponding times of -vibration. - -This law of the proportion of the lengths of pendulums to the squares -of the time of vibration may be experimentally established in the -following manner:-- - -Let A, B, C, _fig. 74._, be three small pieces of metal each -attached by threads to two points of suspension, and let them be placed -in the same vertical line under the point O; suppose them so adjusted -that the distances O A, O B, and O C shall be in the -proportion of the numbers 1, 4, and 9. Let them be removed from the -vertical in a direction at right angles to the plane of the paper, so -that the threads shall be in the same plane, and therefore the three -pendulums will have the same angle of vibration. Being now liberated, -the pendulum A will immediately gain upon B, and B upon C, so that A -will have completed one vibration before B or C. At the end of the -second vibration of A, the pendulum B will have arrived at the end of -its first vibration, so that the suspending threads of A and B will -then be separated by the whole angle of vibration; at the end of the -fourth vibration of A the suspending threads of A and B will return -to their first position, B having completed two vibrations; thus the -proportion of the times of vibration of B and A will be 2 to 1, the -proportion of their lengths being 4 to 1. At the end of the third -vibration of A, C will have completed one vibration, and the suspending -strings will coincide in the position distant by the whole angle of -vibration from their first position. So that three vibrations of A are -performed in the same time as one of C: the proportion of the time of -vibration of C and A are, therefore, 3 to 1, the proportion of their -lengths being 9 to 1, conformably to the law already explained. - -(212.) In all the preceding observations we have assumed that the -material of the pendulous body is of inconsiderable magnitude, its -whole weight being conceived to be collected in a physical point. -This is generally called a simple pendulum; but since the conditions -of a suspending thread without weight, and a heavy molecule without -magnitude, cannot have practical existence, the simple pendulum must -be considered as imaginary, and merely used to establish hypothetical -theorems, which, though inapplicable in practice, are nevertheless the -means of investigating the laws which govern the real phenomena of -pendulous bodies. - -A pendulous body being of determinate magnitude, its several parts -will be situated at different distances from the axis of suspension. -If each component part of such a body were separately connected with -the axis of suspension by a fine thread, it would, being unconnected -with the other particles, be an independent simple pendulum, and -would oscillate according to the laws already explained. It therefore -follows that those particles of the body which are nearest to the -axis of suspension would, if liberated from their connection with the -others, vibrate more rapidly than those which are more remote. The -connection, however, which the particles of the body have, by reason -of their solidity, compels them all to vibrate in the same time. -Consequently, those particles which are nearer the axis are retarded -by the slower motion of those which are more remote; while the more -remote particles, on the other hand, are urged forward by the greater -tendency of the nearer particles to rapid vibration. This will be more -readily comprehended, if we conceive two particles of matter A and B, -_fig. 75._, to be connected with the same axis O by an inflexible -wire O C, the weight of which may be neglected. If B were removed, -A would vibrate in a certain time depending upon the distance O A. -If A were removed, and B placed upon the wire at a distance B O -equal to four times A O, B would vibrate in twice the former time. -Now if both be placed on the wire at the distances just mentioned, -the tendency of A to vibrate more rapidly will be transmitted to B -by means of the wire, and will urge B forward more quickly than if -A were not present: on the other hand, the tendency of B to vibrate -more slowly will be transmitted by the wire to A, and will cause it to -move more slowly than if B were not present. The inflexible quality -of the connecting wire will in this case compel A and B to vibrate -simultaneously, the time of vibration being greater than that of A, and -less than that of B, if each vibrated unconnected with the other. - -If, instead of supposing two particles of matter placed on the wire, -a greater number were supposed to be placed at various distances from -O, it is evident the same reasoning would be applicable. They would -mutually affect each other’s motion; those placed nearest to point -O accelerating the motion of those more remote, and being themselves -retarded by the latter. Among these particles one would be found -in which all these effects would be mutually neutralised, all the -particles nearer O being retarded in reference to that motion which -they would have if unconnected with the rest, and those more remote -being in the same respect accelerated. The point at which such a -particle is placed is called _the centre of oscillation_. - -What has been here observed of the effects of particles of matter -placed upon rigid wire will be equally applicable to the particles of -a solid body. Those which are nearer to the axis are urged forward by -those which are more remote, and are in their turn retarded by them; -and as with the particles placed upon the wire, there is a certain -particle of the body at which the effects are mutually neutralised, and -which vibrates in the same time as it would if it were unconnected with -the other parts of the body, and simply connected by a fine thread to -the axis. By this centre of oscillation the calculations respecting the -vibration of a solid body are rendered as simple as those of a molecule -of inconsiderable magnitude. All the properties which have been -explained as belonging to a simple pendulum may thus be transferred -to a vibrating body of any magnitude and figure, by considering it as -equivalent to a single particle of matter vibrating at its centre of -oscillation. - -(213.) It follows from this reasoning, that the virtual length of -a pendulum is to be estimated by the distance of its centre of -oscillation from the axis of suspension, and therefore that the times -of vibration of different pendulums are in the same proportion as the -square roots of the distances of their centres of oscillation from -their axes. - -The investigation of the position of the centre of oscillation is, in -most cases, a subject of intricate mathematical calculation. It depends -on the magnitude and figure of the pendulous body, the manner in which -the mass is distributed through its volume, or the density of its -several parts, and the position of the axis on which it swings. - -The place of the centre of oscillation may be determined when the -position of the centre of gravity and the centre of gyration are known; -for the distance of the centre of oscillation from the axis will always -be obtained by dividing the square of the radius of gyration (186.) -by the distance of the centre of gravity from the axis. Thus if 6 be -the radius of gyration, and 9 the distance of gravity from the axis, -36 divided by 9, which is 4, will be the distance of the centre of -oscillation from the axis. Hence it may be inferred generally, that -the greater the proportion which the radius of gyration bears to the -distance of the centre of gravity from the axis, the greater will be -the distance of the centre of oscillation. - -It follows from this reasoning, that the length of a pendulum is not -limited by the dimensions of its volume. If the axis be so placed -that the centre of gravity is near it, and the centre of gyration -comparatively removed from it, the centre of oscillation may be placed -far beyond the limits of the pendulous body. Suppose the centre of -gravity is at a distance of one inch from the axis, and the centre -of gyration 12 inches, the centre of oscillation will then be at the -distance of 144 inches, or 12 feet. Such a pendulum may not in its -greatest dimensions exceed one foot, and yet its time of vibration -would be equal to that of a simple pendulum whose length is 12 feet. - -By these means pendulums of small dimensions may be made to vibrate as -slowly as may be desired. The instruments called _metronomes_, used -for marking the time of musical performances, are constructed on this -principle. - -(214.) The centre of oscillation is distinguished by a very remarkable -property in relation to the axis of suspension. If A, _fig. 76._, -be the point of suspension, and O the corresponding centre of -oscillation, the time of vibration of the pendulum will not be -changed if it be raised from its support, inverted, and suspended from -the point O. It follows, therefore, that if O be taken as the point -of suspension, A will be the corresponding centre of oscillation. -These two points are, therefore, convertible. This property may be -verified experimentally in the following manner. A pendulum being put -into a state of vibration, let a small heavy body be suspended by -a fine thread, the length of which is so adjusted that it vibrates -simultaneously with the pendulum. Let the distance from the point of -suspension to the centre of the vibrating body be measured, and take -this distance on the pendulum from the axis of suspension downwards; -the place of the centre of oscillation will thus be obtained, since -the distance so measured from the axis is the length of the equivalent -simple pendulum. If the pendulum be now raised from its support, -inverted, and suspended from the centre of oscillation thus obtained, -it will be found to vibrate simultaneously with the body suspended by -the thread. - -(215.) This property of the interchangeable nature of the centres -of oscillation and suspension has been, at a late period, adopted -by Captain Kater, as an accurate means of determining the length of -a pendulum. Having ascertained with great accuracy two points of -suspension at which the same body will vibrate in the same time, the -distance between these points being accurately measured, is the length -of the equivalent simple pendulum. See Chapter XXI. - -(216.) The manner in which the time of vibration of a pendulum -depends on its length being explained, we are next to consider how -this time is affected by the attraction of gravity. It is obvious -that, since the pendulum is moved by this attraction, the rapidity -of its motion will be increased, if the impelling force receive any -augmentation; but it still is to be decided, in what exact proportion -the time of oscillation will be diminished by any proposed increase -in the intensity of the earth’s attraction. It can be demonstrated -mathematically, that the time of one vibration of a pendulum has the -same proportion to the time of falling freely in the perpendicular -direction, through a height equal to half the length of the pendulum, -as the circumference of a circle has to its diameter. Since, therefore, -the times of vibration of pendulums are in a fixed proportion to the -times of falling freely through spaces equal to the halves of their -lengths, it follows that these times have the same relation to the -force of attraction as the times of falling freely through their -lengths have to that force. If the intensity of the force of gravity -were increased in a four-fold proportion, the time of falling through -a given height would be diminished in a two-fold proportion; if the -intensity were increased to a nine-fold proportion, the time of falling -through a given space would be diminished in a three-fold proportion, -and so on; the rate of diminution of the time being always as the -square root of the increased force. By what has been just stated this -law will also be applicable to the vibration of pendulums. Any increase -in the intensity of the force of gravity would cause a given pendulum -to vibrate more rapidly, and the increased rapidity of the vibration -would be in the same proportion as the square root of the increased -intensity of the force of gravity. - -(217.) The laws which regulate the times of vibration of pendulums in -relation to one another being well understood, the whole theory of -these instruments will be completed, when the method of ascertaining -the actual time of vibration of any pendulum, in reference to its -length, has been explained. In such an investigation, the two elements -to be determined are, 1. the exact time of a single vibration, and, -2. the exact distance of the centre of oscillation from the point of -suspension. - -The former is ascertained by putting a pendulum in motion in the -presence of a good chronometer, and observing precisely the number of -oscillations which are made in any proposed number of hours. The entire -time during which the pendulum swings, being divided by the number of -oscillations made during that time, the exact time of one oscillation -will be obtained. - -The distance of the centre of oscillation from the point of suspension -may be rendered a matter of easy calculation, by giving a certain -uniform figure and material to the pendulous body. - -(218.) The time of vibration of one pendulum of known length being -thus obtained, we shall be enabled immediately to solve either of the -following problems. - -“To find the length of a pendulum which shall vibrate in a given time.” - -“To find the time of vibration of a pendulum of a given length.” - -The former is solved as follows: the time of vibration of the known -pendulum is to the time of vibration of the required pendulum, as the -square root of the length of the known pendulum is to the square root -of the length of the required pendulum. This length is therefore found -by the ordinary rules of arithmetic. - -The latter may be solved as follows: the length of the known pendulum -is to the length of the proposed pendulum, as the square of the time -of vibration of the known pendulum is to the square of the time of -vibration of the proposed pendulum. The latter time may therefore be -found by arithmetic. - -(219.) Since the rate of a pendulum has a known relation to the -intensity of the earth’s attraction, we are enabled, by this -instrument, not only to detect certain variations in that attraction in -various parts of the earth, but also to discover the actual amount of -the attraction at any given place. - -The actual amount of the earth’s attraction at any given place is -estimated by the height through which a body would fall freely at that -place in any given time, as in one second. To determine this, let the -length of a pendulum which would vibrate in one second at that place -be found. As the circumference of a circle is to its diameter[2] (a -known proportion), so will one second be to the time of falling through -a height equal to half the length of this pendulum. This time is -therefore a matter of arithmetical calculation. It has been proved in -(120.), that the heights, through which a body falls freely, are in the -same proportion as the squares of the times; from whence it follows, -that the square of the time of falling through a height equal to half -the length of the pendulum is to one second as half the length of -that pendulum is to the height through which a body would fall in one -second. This height, therefore, may be immediately computed, and thus -the actual amount of the force of gravity at any given place may be -ascertained. - -[2] This ratio is that of 31,416 to 10,000 very nearly. - -(220.) To compare the force of gravity in different parts of the earth, -it is only necessary to swing the same pendulum in the places under -consideration, and to observe the rapidity of its vibrations. The -proportion of the force of gravity in the several places will be that -of the squares of the velocity of the vibration. Observations to this -effect have been made at several places, by Biot, Kater, Sabine, and -others. - -The earth being a mass of matter of a form nearly spherical, revolving -with considerable velocity on an axis, its component parts are affected -by a centrifugal force; in virtue of which, they have a tendency to fly -off in a direction perpendicular to the axis. This tendency increases -in the same proportion as the distance of any part from the axis -increases, and consequently those parts of the earth which are near the -equator, are more strongly affected by this influence than those near -the pole. It has been already explained (145.) that the figure of the -earth is affected by this cause, and that it has acquired a spheroidal -form. The centrifugal force, acting in opposition to the earth’s -attraction, diminishes its effects; and consequently, where this force -is more efficient, a pendulum will vibrate more slowly. By these means -the rate of vibration of a pendulum becomes an indication of the amount -of the centrifugal force. But this latter varies in proportion to the -distance of the place from the earth’s axis; and thus the rate of a -pendulum indicates the relation of the distances of different parts of -the earth’s surface from its axis. The figure of the earth may be thus -ascertained, and that which theory assigns to it, it may be practically -proved to have. - -This, however, is not the only method by which the figure of the earth -may be determined. The meridians being sections of the earth through -its axis, if their figure were exactly determined, that of the earth -would be known. Measurements of arcs of meridians on a large scale have -been executed, and are still being made in various parts of the earth, -with a view to determine the curvature of a meridian at different -latitudes. This method is independent of every hypothesis concerning -the density and internal structure of the earth, and is considered by -some to be susceptible of more accuracy than that which depends on the -observations of pendulums. - -(221.) It has been stated that, when the arc of vibration of a pendulum -is not very small, a variation in its length will produce a sensible -effect on the time of vibration. To construct a pendulum such that the -time of vibration may be independent of the extent of the swing, was a -favourite speculation of geometers. This problem was solved by Huygens, -who showed that the curve called a _cycloid_, previously discovered and -described by Galileo, possessed the isochronal property; that is, that -a body moving in it by the force of gravity, would vibrate in the same -time, whatever be the length of the arc described. - -Let O A, _fig. 77._, be a horizontal line, and let O B -be a circle placed below this line, and in contact with it. If this -circle be rolled upon the line from O towards A, a point upon its -circumference, which at the beginning of the motion is placed at O, -will during the motion trace the curve O C A. This curve is -called a _cycloid_. If the circle be supposed to roll in the opposite -direction towards A′, the same point will trace another cycloid -O C′ A′. The points C and C′ being the lowest points of the -curves, if the perpendiculars C D and C′ D′ be drawn, they -will respectively be equal to the diameter of the circle. By a known -property of this curve, the arcs O C and O C′ are equal to -twice the diameter of the circle. From the point O suppose a flexible -thread to be suspended, whose length is twice the diameter of the -circle, and which sustains a pendulous body P at its extremity. If -the curves O C and O C′, from the plane of the paper, be -raised so as to form surfaces to which the thread may be applied, the -extremity P will extend to the points C and C′, when the entire thread -has been applied to either of the curves. As the thread is deflected -on either side of its vertical position, it is applied to a greater -or lesser portion of either curve, according to the quantity of its -deflection from the vertical. If it be deflected on each side until -the point P reaches the points C and C′, the extremity would trace a -cycloid C P C′ precisely equal and similar to those already -mentioned. Availing himself of this property of the curve, Huygens -constructed his cycloidal pendulum. The time of vibration was subject -to no variation, however the arc of vibration might change, provided -only that the length of the string O P continued the same. If -small arcs of the cycloid be taken on either side of the point P, they -will not sensibly differ from arcs of a circle described with the -centre O and the radius O P; for, in slight deflections from the -vertical position, the effect of the curves O C and O C′ on -the thread O P is altogether inconsiderable. It is for this reason -that when the arcs of vibration of a circular pendulum are small, they -partake of the property of isochronism peculiar to those of a cycloid. -But when the deflection of P from the vertical is great, the effect of -the curves O C and O C′ on the thread produces a considerable -deviation of the point P from the arc of the circle whose centre is -O and whose radius is O P, and consequently the property of -isochronism will no longer be observed in the circular pendulum. - - - - -CHAP. XII. - -OF SIMPLE MACHINES. - - -(222.) A MACHINE is an instrument by which force or motion may be -transmitted and modified as to its quantity and direction. There are -two ways in which a machine may be applied, and which give rise to -a division of mechanical science into parts denominated STATICS and -DYNAMICS; the one including the theory of equilibrium, and the other -the theory of motion. When a machine is considered statically, it is -viewed as an instrument by which forces of determinate quantities -and direction are made to balance other forces of other quantities -and other directions. If it be viewed dynamically, it is considered -as a means by which certain motions of determinate quantity and -direction may be made to produce other motions in other directions -and quantities. It will not be convenient, however, in the present -treatise, to follow this division of the subject. We shall, on the -other hand, as hitherto, consider the phenomena of equilibrium and -motion together. - -The effects of machinery are too frequently described in such a manner -as to invest them with the appearance of paradox, and to excite -astonishment at what appears to contradict the results of the most -common experience. It will be our object here to take a different -course, and to attempt to show that those effects which have been held -up as matters of astonishment are the necessary, natural, and obvious -results of causes adapted to produce them in a manner analogous to the -objects of most familiar experience. - -(223.) In the application of a machine there are three things to -be considered. 1. The force or resistance which is required to be -sustained, opposed, or overcome. 2. The force which is used to sustain, -support, or overcome that resistance. 3. The machine itself by which -the effect of this latter force is transmitted to the former. Of -whatever nature be the force or the resistance which is to be sustained -or overcome, it is technically called the _weight_, since, whatever it -be, a weight of equivalent effect may always be found. The force which -is employed to sustain or overcome it is technically called the _power_. - -(224.) In expressing the effect of machinery it is usual to say that -the power sustains the weight; but this, in fact, is not the case, and -hence arises that appearance of paradox which has already been alluded -to. If, for example, it is said that a power of one ounce sustains the -weight of one ton, astonishment is not unnaturally excited, because -the fact, as thus stated, if the terms be literally interpreted, is -physically impossible. No power less than a ton can, in the ordinary -acceptation of the word, support the weight of a ton. It will, however, -be asked how it happens that a machine _appears_ to do this? how it -happens that by holding a silken thread, which an ounce weight would -snap, many hundred weight may be sustained? To explain this it will -only be necessary to consider the effect of a machine, when the power -and weight are in equilibrium. - -(225.) In every machine there are some fixed points or props; and the -arrangement of the parts is always such, that the pressure, excited by -the power or weight, or both, is distributed among these props. If the -weight amount to twenty hundred, it is possible so to distribute it, -that any proportion, however great, of it may be thrown on the fixed -points or props of the machine; the remaining part only can properly be -said to be supported by the power, and this part can never be greater -than the power. Considering the effect in this way, it appears that -the power supports just so much of the weight and no more as is equal -to its own force, and that all the remaining part of the weight is -sustained by the machine. The force of these observations will be more -apparent when the nature and properties of the mechanic powers and -other machines have been explained. - -(226.) When a machine is considered dynamically, its effects are -explained on different principles. It is true that, in this case, a -very small power may elevate a very great weight; but nevertheless, -in so doing, whatever be the machine used, the total expenditure of -power, in raising the weight through any height, is never less than -that which would be expended if the power were immediately applied to -the weight without the intervention of any machine. This circumstance -arises from an universal property of machines by which the velocity of -the weight is always less than that of the power, in exactly the same -proportion as the power itself is less than the weight; so that when -a certain power is applied to elevate a weight, the rate at which the -elevation is effected is always slow in the same proportion as the -weight is great. From a due consideration of this remarkable law, it -will easily be understood, that a machine can never diminish the total -expenditure of power necessary to raise any weight or to overcome any -resistance. In such cases, all that a machine ever does or ever can -do, is to enable the power to be expended at a slow rate, and in a -more advantageous direction than if it were immediately applied to the -weight or the resistance. - -Let us suppose that P is a power amounting to an ounce, and that W is -a weight amounting to 50 ounces, and that P elevates W by means of a -machine. In virtue of the property already stated, it follows, that -while P moves through 50 feet, W will be moved through 1 foot; but -in moving P through 50 feet, 50 distinct efforts are made, by each -of which 1 ounce is moved through 1 foot, and by which collectively -50 distinct ounces might be successively raised through 1 foot. But -the weight W is 50 ounces, and has been raised through 1 foot; from -whence it appears, that the expenditure of power is equal to that which -would be necessary to raise the weight without the intervention of any -machine. - -This important principle may be presented under another aspect, which -will perhaps render it more apparent. Suppose the weight W were -actually divided into 50 equal parts, or suppose it were a vessel of -liquid weighing 50 ounces, and containing 50 equal measures; if these -50 measures were successively lifted through a height of 1 foot; the -efforts necessary to accomplish this would be the same as those used -to move the power P through 50 feet, and it is obvious, that the total -expenditure of force would be the same as that which would be necessary -to lift the entire contents of the vessel through 1 foot. - -When the nature and properties of the mechanic powers and other -machines have been explained, the force of these observations will be -more distinctly perceived. The effects of props and fixed points in -sustaining a part of the weight, and sometimes the whole, both of the -weight and power, will then be manifest, and every machine will furnish -a verification of the remarkable proportion between the velocities -of the weight and power, which has enabled us to explain what might -otherwise be paradoxical and difficult of comprehension. - -(227.) The most simple species of machines are those which are commonly -denominated the MECHANIC POWERS. These have been differently enumerated -by different writers. If, however, the object be to arrange in distinct -classes, and in the smallest possible number of them, those machines -which are alike in principle, the mechanic powers may be reduced to -three. - - 1. The lever. - 2. The cord. - 3. The inclined plane. - -To one or other of these classes all simple machines whatever may be -reduced, and all complex machines may be resolved into simple elements -which come under them. - -(228.) The first class includes every machine which is composed of -a solid body revolving on a fixed axis, although the name lever has -been commonly confined to cases where the machine affects certain -particular forms. This is by far the most useful class of machines, and -will require in subsequent chapters very detailed development. The -general principle, upon which equilibrium is established between the -power and weight in machines of this class has been already explained -in (183.) The power and weight are always supposed to be applied in -directions at right angles to the axis. If lines be drawn from the axis -perpendicular to the directions of power and weight, equilibrium will -subsist, provided the power multiplied by the perpendicular distance -of its direction from the axis, be equal to the weight multiplied by -the perpendicular distance of its direction from the axis. This is a -principle to which we shall have occasion to refer in explaining the -various machines of this class. - -(229.) If the moment of the power (184.) be greater than that of the -weight, the effect of the power will prevail over that of the weight, -and elevate it; but if, on the other hand, the moment of the power be -less than that of the weight, the power will be insufficient to support -the weight, and will allow it to fall. - -(230.) The second class of simple machines includes all those cases -in which force is transmitted by means of flexible threads, ropes, -or chains. The principle, by which the effects of these machines are -estimated, is, that the tension throughout the whole length of the same -cord, provided it be perfectly flexible, and free from the effects -of friction, must be the same. Thus, if a force acting at one end be -balanced by a force acting at the other end, however the cord may be -bent, or whatever course it may be compelled to take, by any causes -which may affect it between its ends, these forces must be equal, -provided the cord be free to move over any obstacles which may deflect -it. - -Within this class of machines are included all the various forms of -_pulleys_. - -(231.) The third class of simple machines includes all those cases in -which the weight or resistance is supported or moved on a hard surface -inclined to the vertical direction. - -The effects of such machines are estimated by resolving the whole -weight of the body into two elements by the parallelogram of forces. -One of these elements is perpendicular to the surface, and supported -by its resistance; the other is parallel to the surface, and supported -by the power. The proportion, therefore, of the power to the weight -will always depend on the obliquity of the surface to the direction of -the weight. This will be easily understood by referring to what has -been already explained in Chapter VIII. - -Under this class of machines come the inclined plane, commonly so -called, the wedge, the screw, and various others. - -(232.) In order to simplify the development of the elementary theory -of machines, it is expedient to omit the consideration of many -circumstances, of which, however, a strict account must be taken before -any practically useful application of that theory can be attempted. -A machine, as we must for the present contemplate it, is a thing -which can have no real or practical existence. Its various parts are -considered to be free from friction: all surfaces which move in contact -are supposed to be infinitely smooth and polished. The solid parts are -conceived to be absolutely inflexible. The weight and inertia of the -machine itself are wholly neglected, and we reason upon it as if it -were divested of these qualities. Cords and ropes are supposed to have -no stiffness, to be infinitely flexible. The machine, when it moves, is -supposed to suffer no resistance from the atmosphere, and to be in all -respects circumstanced as if it were _in vacuo_. - -It is scarcely necessary to state, that, all these suppositions -being false, none of the consequences deduced from them can be true. -Nevertheless, as it is the business of art to bring machines as near -to this state of ideal perfection as possible, the conclusions which -are thus obtained, though false in a strict sense, yet deviate from -the truth in but a small degree. Like the first outline of a picture, -they resemble in their general features that truth to which, after many -subsequent corrections, they must finally approximate. - -After a first approximation has been made on the several false -suppositions which have been mentioned, various effects, which have -been previously neglected, are successively taken into account. -Roughness, rigidity, imperfect flexibility, the resistance of air and -other fluids, the effects of the weight and inertia of the machine, -are severally examined, and their laws and properties detected. The -modifications and corrections, thus suggested as necessary to be -introduced into our former conclusions, are applied, and a second -approximation, but still _only_ an approximation, to truth is made. -For, in investigating the laws which regulate the several effects -just mentioned, we are compelled to proceed upon a new group of false -suppositions. To determine the laws which regulate the friction of -surfaces, it is necessary to assume that every part of the surfaces of -contact are uniformly rough; that the solid parts which are imperfectly -rigid, and the cords which are imperfectly flexible, are constituted -throughout their entire dimensions of a uniform material; so that the -imperfection does not prevail more in one part than another. Thus, -all irregularity is left out of account, and a general average of the -effects taken. It is obvious, therefore, that by these means we have -still failed in obtaining a result exactly conformable to the real -state of things; but it is equally obvious, that we have obtained -one much more conformable to that state than had been previously -accomplished, and sufficiently near it for most practical purposes. - -This apparent imperfection in our instruments and powers of -investigation is not peculiar to mechanics: it pervades all departments -of natural science. In astronomy, the motions of the celestial bodies, -and their various changes and appearances as developed by theory, -assisted by observation and experience, are only approximations to the -real motions and appearances which take place in nature. It is true -that these approximations are susceptible of almost unlimited accuracy; -but still they are, and ever will continue to be, only approximations. -Optics and all other branches of natural science are liable to the same -observations. - - - - -CHAP. XIII. - -OF THE LEVER. - - -(233.) An inflexible, straight bar, turning on an axis, is commonly -called a _lever_. The _arms_ of the lever are those parts of the bar -which extend on each side of the axis. - -The axis is called the _fulcrum_ or _prop_. - -(234.) Levers are commonly divided into three kinds, according to the -relative positions of the power, the weight, and the fulcrum. - -In a lever of the first kind, as in _fig. 78._, the fulcrum is -between the power and weight. - -In a lever of the second kind, as in _fig. 79._, the weight is -between the fulcrum and power. - -In a lever of the third kind, as in _fig. 80._, the power is -between the fulcrum and weight. - -(235.) In all these cases, the power will sustain the weight in -equilibrium, provided its moment be equal to that of the weight. (184.) -But the moment of the power is, in this case, equal to the product -obtained by multiplying the power by its distance from the fulcrum; and -the moment of the weight by multiplying the weight by its distance from -the fulcrum. Thus, if the number of ounces in P, being multiplied by -the number of inches in P F, be equal to the number of ounces in -W, multiplied by the number of inches in W F, equilibrium will be -established. It is evident from this, that as the distance of the power -from the fulcrum increases in comparison to the distance of the weight -from the fulcrum, in the same degree exactly will the proportion of the -power to the weight diminish. In other words, the proportion of the -power to the weight will be always the same as that of their distances -from the fulcrum taken in a reverse order. - -In cases where a small power is required to sustain or elevate a great -weight, it will therefore be necessary either to remove the power to a -great distance from the fulcrum, or to bring the weight very near it. - -(236.) Numerous examples of levers of the first kind may be given. A -crow-bar, applied to elevate a stone or other weight, is an instance. -The fulcrum is another stone placed near that which is to be raised, -and the power is the hand placed at the other end of the bar. - -A handspike is a similar example. - -A poker applied to raise fuel is a lever of the first kind, the fulcrum -being the bar of the grate. - -Scissors, shears, nippers, pincers, and other similar instruments are -composed of two levers of the first kind; the fulcrum being the joint -or pivot, and the weight the resistance of the substance to be cut or -seized; the power being the fingers applied at the other end of the -levers. - -The brake of a pump is a lever of the first kind; the pump-rods and -piston being the weight to be raised. - -(237.) Examples of levers of the second kind, though not so frequent as -those just mentioned, are not uncommon. - -An oar is a lever of the second kind. The reaction of the water against -the blade is the fulcrum. The boat is the weight, and the hand of the -boatman the power. - -The rudder of a ship or boat is an example of this kind of lever, and -explained in a similar way. - -The chipping knife is a lever of the second kind. The end attached -to the bench is the fulcrum, and the weight the resistance of the -substance to be cut, placed beneath it. - -A door moved upon its hinges is another example. - -Nut-crackers are two levers of the second kind; the hinge which unites -them being the fulcrum, the resistance of the shell placed between -them being the weight, and the hand applied to the extremity being the -power. - -A wheelbarrow is a lever of the second kind; the fulcrum being the -point at which the wheel presses on the ground, and the weight being -that of the barrow and its load, collected at their centre of gravity. - -The same observation may be applied to all two-wheeled carriages, which -are partly sustained by the animal which draws them. - -(238.) In a lever of the third kind, the weight, being more distant -from the fulcrum than the power, must be proportionably less than -it. In this instrument, therefore, the power acts upon the weight to -a mechanical disadvantage, inasmuch as a greater power is necessary -to support or move the weight than would be required if the power -were immediately applied to the weight, without the intervention of a -machine. We shall, however, hereafter show that the advantage which -is lost in force is gained in despatch, and that in proportion as the -weight is less than the power which moves it, so will the speed of its -motion be greater than that of the power. - -Hence a lever of the third kind is only used in cases where the -exertion of great power is a consideration subordinate to those of -rapidity and despatch. - -The most striking example of levers of the third kind is found in the -animal economy. The limbs of animals are generally levers of this -description. The socket of the bone is the fulcrum; a strong muscle -attached to the bone near the socket is the power; and the weight -of the limb, together with whatever resistance is opposed to its -motion, is the weight. A slight contraction of the muscle in this case -gives a considerable motion to the limb: this effect is particularly -conspicuous in the motion of the arms and legs in the human body; a -very inconsiderable contraction of the muscles at the shoulders and -hips giving the sweep to the limbs from which the body derives so much -activity. - -The treddle of the turning lathe is a lever of the third kind. The -hinge which attaches it to the floor is the fulcrum, the foot applied -to it near the hinge is the power, and the crank upon the axis of the -fly-wheel, with which its extremity is connected, is the weight. - -Tongs are levers of this kind, as also the shears used in shearing -sheep. In these cases the power is the hand placed immediately below -the fulcrum or point where the two levers are connected. - -(239.) When the power is said to support the weight by means of a lever -or any other machine, it is only meant that the power keeps the machine -in equilibrium, and thereby enables it to sustain the weight. It is -necessary to attend to this distinction, to remove the difficulty which -may arise from the paradox of a small power sustaining a great weight. - -In a lever of the first kind, the fulcrum F, _fig. 78._, or axis, -sustains the united forces of the power and weight. - -In a lever of the second kind, if the power be supposed to act over -a wheel R, _fig. 79._, the fulcrum F sustains a pressure equal -to the difference between the power and weight, and the axis of the -wheel R sustains a pressure equal to twice the power; so that the total -pressures on F and R are equivalent to the united forces of the power -and weight. - -In a lever of the third kind similar observations are applicable. The -wheel R, _fig. 80._, sustains a pressure equal to twice the power, -and the fulcrum F sustains a pressure equal to the difference between -the power and weight. - -These facts may be experimentally established by attaching a string -to the lever immediately over the fulcrum, and suspending the lever -by that string from the arm of a balance. The counterpoising weight, -when the fulcrum is removed, will, in the first case, be equal to the -sum of the weight and power, and in the last two cases equal to their -difference. - -(240.) We have hitherto omitted the consideration of the effect of the -weight of the lever itself. If the centre of gravity of the lever be -in the vertical line through the axis, the weight of the instrument -will have no other effect than to increase the pressure on the axis by -its own amount. But if the centre of gravity be on the same side of -the axis with the weight, as at G, it will oppose the effect of the -power, a certain part of which must therefore be allowed to support -it. To ascertain what part of the power is thus expended, it is to -be considered that the moment of the weight of the lever collected -at G, is found by multiplying that weight by the distance G F. -The moment of that part of the power which supports this must be -equal to it; therefore, it is only necessary to find how much of the -power multiplied by P F will be equal to the weight of the lever -multiplied by G F. This is a question in common arithmetic. - -If the centre of gravity of the lever be at a different side of the -axis from the weight, as at G′, the weight of the instrument will -co-operate with the power in sustaining the weight W. To determine what -portion of the weight W is thus sustained by the weight of the lever, -it is only necessary to find how much of W, multiplied by the distance -W F, is equal to the weight of the lever multiplied by G′ F. - -In these cases the pressure on the fulcrum, as already estimated, will -always be increased by the weight of the lever. - -(241.) The sense in which a small power is said to sustain a great -weight, and the manner of accomplishing this, being explained, we -shall now consider how the power is applied in moving the weight. Let -P W, _fig. 81._, be the places of the power and weight, and -F that of the fulcrum, and let the power be depressed to P′ while the -weight is raised to W′. The space P P′ evidently bears the same -proportion to W W′, as the arm P F to W F. Thus if -P F be ten times W F, P P′ will be ten times W W′. -A power of one pound at P being moved from P to P′, will carry a weight -of ten pounds from W to W′. But in this case it ought not to be said, -that a lesser weight moves a greater, for it is not difficult to show, -that the total expenditure of force in the motion of one pound from P -to P′ is exactly the same as in the motion of ten pounds from W to W′. -If the space P P′ be ten inches, the space W W′ will be one -inch. A weight of one pound is therefore moved through ten successive -inches, and in each inch the force expended is that which would be -sufficient to move one pound through one inch. The total expenditure -of force from P to P′ is ten times the force necessary to move one -pound through one inch, or what is the same, it is that which would be -necessary to move ten pounds through one inch. But this is exactly what -is accomplished by the opposite end W of the lever; for the weight W is -ten pounds, and the space W W′ is one inch. - -If the weight W of ten pounds could be conveniently divided into ten -equal parts of one pound each, each part might be separately raised -through one inch, without the intervention of the lever or any other -machine. In this case, the same quantity of power would be expended, -and expended in the same manner as in the case just mentioned. - -It is evident, therefore, that when a machine is applied to raise a -weight or to overcome resistance, as much force must be really used as -if the power were immediately applied to the weight or resistance. All -that is accomplished by the machine is to enable the power to do that -by a succession of distinct efforts which should be otherwise performed -by a single effort. These observations will be found to be applicable -to all machines whatever. - -(242.) Weighing machines of almost every kind, whether used for -commercial or philosophical purposes, are varieties of the lever. The -common balance, which, of all weighing machines, is the most perfect -and best adapted for ordinary use, whether in commerce or experimental -philosophy, is a lever with equal arms. In the steel-yard one weight -serves as a counterpoise and measure of others of different amount, by -receiving a leverage variable according to the varying amount of the -weight against which it acts. A detailed account of such instruments -will be found in Chapter XXI. - -(243.) We have hitherto considered the power and weight as acting on -the lever, in directions perpendicular to its length and parallel to -each other. This does not always happen. Let A B, _fig. 83._, -be a lever whose fulcrum is F, and let A R be the direction of the -power, and B S the direction of the weight. If the lines R A -and S B be continued, and perpendiculars F C and F D -drawn from the fulcrum to those lines, the moment of the power will be -found by multiplying the power by the line F C, and the moment of -the weight by multiplying the weight by F D. If these moments be -equal, the power will sustain the weight in equilibrium. (185). - -It is evident, that the same reasoning will be applicable when the -arms of the lever are not in the same direction. These arms may be of -any figure or shape, and may be placed relatively to each other in any -position. - -(244.) In the rectangular lever the arms are perpendicular to each -other, and the fulcrum F, _fig. 84._, is at the right angle. The -moment of the power, in this case, is P multiplied by A F, and -that of the weight W multiplied by B F. When the instrument is in -equilibrium these moments must be equal. - -When the hammer is used for drawing a nail, it is a lever of this kind: -the claw of the hammer is the shorter arm; the resistance of the nail -is the weight; and the hand applied to the handle the power. - -(245.) When a beam rests on two props A B, _fig. 85._, and -supports, at some intermediate place C, a weight W, this weight is -distributed between the props in a manner which may be determined by -the principles already explained. If the pressure on the prop B be -considered as a power sustaining the weight W, by means of the lever of -the second kind B A, then this power multiplied by B A must -be equal to the weight multiplied by C A. Hence the pressure on -B will be the same fraction of the weight as the part A C is of -A B. In the same manner it may be proved, that the pressure on A -is the same fraction of the weight as B C is of B A. Thus, if -A C be one third, and therefore B C two thirds of B A, -the pressure on B will be one third of the weight, and the pressure on -A two thirds of the weight. - -It follows from this reasoning, that if the weight be in the middle, -equally distant from B and A, each prop will sustain half the weight. -The effect of the weight of the beam itself may be determined by -considering it to be collected at its centre of gravity. If this point, -therefore, be equally distant from the props, the weight of the beam -will be equally distributed between them. - -According to these principles, the manner in which a load borne -on poles between two bearers is distributed between them may be -ascertained. As the efforts of the bearers and the direction of the -weight are always parallel; the position of the poles relatively to the -horizon makes no difference in the distribution of the weights between -the bearers. Whether they ascend or descend, or move on a level plane, -the weight will be similarly shared between them. - -If the beam extend beyond the prop, as in _fig. 86._, and the -weight be suspended at a point not placed between them, the props must -be applied at different sides of the beam. The pressures which they -sustain may be calculated in the same manner as in the former case. -The pressure of the prop B may be considered as a power sustaining the -weight W by means of the lever B C. Hence, the pressure of B, -multiplied by B A, must be equal to the weight W multiplied by -A C. Therefore, the pressure on B bears the same proportion to the -weight as A C does to A B. In the same manner, considering B -as a fulcrum, and the pressure of the prop A as the power, it may be -proved that the pressure of A bears the same proportion to the weight -as the line B C does to A B. It therefore appears, that the -pressure on the prop A is greater than the weight. - -(246.) When great power is required, and it is inconvenient to -construct a long lever, a combination of levers may be used. In -_fig. 87._ such a system of levers is represented, consisting of -three levers of the first kind. The manner in which the effect of the -power is transmitted to the weight may be investigated by considering -the effect of each lever successively. The power at P produces an -upward force at P′, which bears to P the same proportion as P′ F -to P F. Therefore, the effect at P′ is as many times the power -as the line P F is of P′ F. Thus, if P F be ten times -P′ F, the upward force at P′ is ten times the power. The arm -P′ F′ of the second lever is pressed upwards by a force equal -to ten times the power at P. In the same manner this may be shown to -produce an effect at P″ as many times greater than P′ as P′ F′ -is greater than P″ F′. Thus, if P′ F′ be twelve times P″ F′, the -effect at P″ will be twelve times that of P′. But this last was ten -times the power, and therefore the P″ will be one hundred and twenty -times the power. In the same manner it may be shown that the weight is -as many times greater than the effect at P″ as P″ F″ is greater than -W F″. If P″ F″ be five times W F″, the weight will be five -times the effect at P″. But this effect is one hundred and twenty times -the power, and therefore the weight would be six hundred times the -power. - -In the same manner the effect of any compound system of levers may be -ascertained by taking the proportion of the weight to the power in -each lever separately, and multiplying these numbers together. In the -example given, these proportions are 10, 12, and 5, which multiplied -together give 600. In _fig. 87._ the levers composing the system -are of the first kind; but the principles of the calculation will not -be altered if they be of the second or third kind, or some of one kind -and some of another. - -(247.) That number which expresses the proportion of the weight to the -equilibrating power in any machine, we shall call the _power of the -machine_. Thus, if, in a lever, a power of one pound support a weight -of ten pounds, the power of the machine is _ten_. If a power of 2lbs. -support a weight of 11lbs., the power of the machine is 5-1/2, 2 being -contained in 11 5-1/2 times. - -(248.) As the distances of the power and weight from the fulcrum of -a lever may be varied at pleasure, and any assigned proportion given -to them, a lever may always be conceived having a power equal to that -of any given machine. Such a lever may be called, in relation to that -machine, the _equivalent lever_. - -As every complex machine consists of a number of simple machines acting -one upon another, and as each simple machine may be represented by an -equivalent lever, the complex machine will be represented by a compound -system of equivalent levers. From what has been proved in (246.), it -therefore follows that the power of a complex machine may be calculated -by multiplying together the powers of the several simple machines of -which it is composed. - - - - -CHAP. XIV. - -OF WHEEL-WORK. - - -(249.) When a lever is applied to raise a weight, or overcome a -resistance, the space through which it acts at any one time is small, -and the work must be accomplished by a succession of short and -intermitting efforts. In _fig. 81._, after the weight has been -raised from W to W′, the lever must again return to its first position, -to repeat the action. During this return the motion of the weight is -suspended, and it will fall downwards unless some provision be made to -sustain it. The common lever is, therefore, only used in cases where -weights are required to be raised through small spaces, and under these -circumstances its great simplicity strongly recommends it. But where -a continuous motion is to be produced, as in raising ore from the -mine, or in weighing the anchor of a vessel, some contrivance must be -adopted to remove the intermitting action of the lever, and render -it continual. The various forms given to the lever, with a view to -accomplish this, are generally denominated the _wheel and axle_. - -[Illustration: _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -In _fig. 88._, A B is a horizontal axle, which rests in -pivots at its extremities, or is supported in gudgeons, and capable of -revolving. Round this axis a rope is coiled, which sustains the weight -W. On the same axis a wheel C is fixed, round which a rope is coiled -in a contrary direction, to which is appended the power P. The moment -of the power is found by multiplying it by the radius of a wheel, and -the moment of the weight, by multiplying it by the radius of its axle. -If these moments be equal (185.), the machine will be in equilibrium. -Whence it appears that the power of the machine (247.) is expressed by -the proportion which the radius of the wheel bears to the radius of -the axle; or, what is the same, of the diameter of the wheel to the -diameter of the axle. This principle is applicable to the wheel and -axle in every variety of form under which it can be presented. - -(250.) It is evident that as the power descends continually, and the -rope is uncoiled from the wheel, the weight will be raised continually, -the rope by which it is suspended being at the same time coiled upon -the axle. - -When the machine is in equilibrium, the forces of both the weight and -power are sustained by the axle, and distributed between its props, in -the manner explained in (245.) - -When the machine is applied to raise a weight, the velocity with which -the power moves is as many times greater than that with which the -weight rises, as the weight itself is greater than the power. This is -a principle which has already been noticed, and which is common to all -machines whatsoever. It may hence be proved, that in the elevation of -the weight a quantity of power is expended equal to that which would be -necessary to elevate the weight if the power were immediately applied -to it, without the intervention of any machine. This has been explained -in the case of the lever in (241.), and may be explained in the -present instance in nearly the same words. - -In one revolution of the machine the length of rope uncoiled from -the wheel is equal to the circumference of the wheel, and through -this space the power must therefore move. At the same time the length -of rope coiled upon the axle is equal to the circumference of the -axle, and through this space the weight must be raised. The spaces, -therefore, through which the power and weight move in the same time, -are in the proportion of the circumferences of the wheel and axle; but -these circumferences are in the same proportion as their diameters. -Therefore the velocity of the power will bear to the velocity of the -weight the same proportion as the diameter of the wheel bears to the -diameter of the axle, or, what is the same, as the weight bears to the -power (249). - -(251.) We have here omitted the consideration of the thickness of the -rope. When this is considered, the force must be conceived as acting in -the direction of the centre of the rope, and therefore the thickness -of the rope which supports the power ought to be added to the diameter -of the wheel, and the thickness of the rope which supports the weight -to the diameter of the axle. It is the more necessary to attend to -this circumstance, as the strength of the rope necessary to support -the weight causes its thickness to bear a considerable proportion to -the diameter of the axle; while the rope which sustains the power not -requiring the same strength, and being applied to a larger circle, -bears a very inconsiderable proportion to its diameter. - -(252.) In numerous forms of the wheel and axle, the weight or -resistance is applied by a rope coiled upon the axle; but the manner in -which the power is applied is very various, and not often by means of a -rope. The circumference of a wheel sometimes carries projecting pins, -as represented in _fig. 88._, to which the hand is applied to -turn the machine. An instance of this occurs in the wheel used in the -steerage of a vessel. - -In the common _windlass_, the power is applied by means of a _winch_, -which is a rectangular lever, as represented in _fig. 89._ The arm -B C of the winch represents the radius of the wheel, and the power -is applied to C D at right angles to B C. - -In some cases no wheel is attached to the axle; but it is pierced with -holes directed towards its centre, in which long levers are incessantly -inserted, and a continuous action produced by several men working at -the same time; so that while some are transferring the levers from hole -to hole, others are working the windlass. - -The axle is sometimes placed in a vertical position, the wheel or -levers being moved horizontally. The _capstan_ is an example of this: -a vertical axis is fixed in the deck of the ship; the circumference is -pierced with holes presented towards its centre. These holes receive -long levers, as represented in _fig. 90._ The men who work the -capstan walk continually round the axle, pressing forward the levers -near their extremities. - -In some cases the wheel is turned by the weight of animals placed at -its circumference, who move forward as fast as the wheel descends, -so as to maintain their position continually at the extremity of the -horizontal diameter. The _treadmill_, _fig. 91._, and certain -_cranes_, such as _fig. 92._, are examples of this. - -In water-wheels, the power is the weight of water contained in -buckets at the circumference, as in _fig. 93._, which is called -an over-shot wheel: and sometimes by the impulse of water against -float-boards at the circumference, as in the under-shot wheel, -_fig. 94._ Both these principles act in the breast-wheel, -_fig. 95._ - -In the paddle-wheel of a steam-boat, the power is the resistance which -the water offers to the motion of the paddle-boards. - -In windmills, the power is the force of the wind acting on various -parts of the arms, and may be considered as different powers -simultaneously acting on different wheels having the same axle. - -(253.) In most cases in which the wheel and axle is used, the action of -the power is liable to occasional suspension or intermission, in which -case some contrivance is necessary to prevent the recoil of the weight. -A ratchet wheel R, _fig. 88._, is provided for this purpose, which -is a contrivance which permits the wheel to turn in one direction; -but a catch which falls between the teeth of a fixed wheel prevents -its motion in the other direction. The effect of the power or weight -is sometimes transmitted to the wheel or axle by means of a straight -bar, on the edge of which teeth are raised, which engage themselves in -corresponding teeth on the wheel or axle. Such a bar is called a rack; -and an instance of its use may be observed in the manner of working the -pistons of an air-pump. - -(254.) The power of the wheel and axle being expressed by the number -of times the diameter of the axle is contained in that of the wheel, -there are obviously only two ways by which this power may be increased; -viz. either by increasing the diameter of the wheel, or diminishing -that of the axle. In cases where great power is required, each of these -methods is attended with practical inconvenience and difficulty. If the -diameter of the wheel be considerably enlarged, the machine will become -unwieldy, and the power will work through an unmanageable space. If, -on the other hand, the power of the machine be increased by reducing -the thickness of the axle, the strength of the axle will become -insufficient for the support of that weight, the magnitude of which had -rendered the increase of the power of the machine necessary. To combine -the requisite strength with moderate dimensions and great mechanical -power is, therefore, impracticable in the ordinary form of the wheel -and axle. This has, however, been accomplished by giving different -thicknesses to different parts of the axle, and carrying a rope, which -is coiled on the thinner part, through a wheel attached to the weight, -and coiling it in the opposite direction on the thicker part, as in -_fig. 96._ To investigate the proportion of the power to the -weight in this case, let _fig. 97._ represent a section of the -apparatus at right angles to the axis. The weight is equally suspended -by the two parts of the rope, S and S′, and therefore each part is -stretched by a force equal to half the weight. The moment of the force, -which stretches the rope S, is half the weight multiplied by the radius -of the thinner part of the axle. This force being at the same side of -the centre with the power, co-operates with it in supporting the force -which stretches S′, and which acts at the other side of the centre. By -the principle established in (185.), the moments of P and S must be -equal to that of S′; and therefore if P be multiplied by the radius of -the wheel, and added to half the weight multiplied by the radius of the -thinner part of the axle, we must obtain a sum equal to half the weight -multiplied by the radius of the thicker part of the axle. Hence it is -easy to perceive, that the power multiplied by the radius of the wheel -is equal to half the weight multiplied by the difference of the radii -of the thicker and thinner parts of the axle; or, what is the same, the -power multiplied by the diameter of the wheel, is equal to the weight -multiplied by half the difference of the diameters of the thinner and -thicker parts of the axle. - -A wheel and axle constructed in this manner is equivalent to an -ordinary one, in which the wheel has the same diameter, and whose axle -has a diameter equal to half the difference of the diameters of the -thicker and thinner parts. The power of the machine is expressed by the -proportion which the diameter of the wheel bears to half the difference -of these diameters; and therefore this power, when the diameter of the -wheel is given, does not, as in the ordinary wheel and axle, depend -on the smallness of the axle, but on the smallness of the difference -of the thinner and thicker parts of it. The axle may, therefore, -be constructed of such a thickness as to give it all the requisite -strength, and yet the difference of the diameters of its different -parts may be so small as to give it all the requisite power. - -(255.) It often happens that a varying weight is to be raised, or -resistance overcome by a uniform power. If, in such a case, the weight -be raised by a rope coiled upon a uniform axle, the action of the -power would not be uniform, but would vary with the weight. It is, -however, in most cases desirable or necessary that the weight or -resistance, even though it vary, shall be moved uniformly. This will -be accomplished if by any means the leverage of the weight is made -to increase in the same proportion as the weight diminishes, and to -diminish in the same proportion as the weight increases: for in that -case the moment of the weight will never vary, whatever it gains by the -increase of weight being lost by the diminished leverage, and whatever -it loses by the diminished weight being gained by the increased -leverage. An axle, the surface of which is curved in such a manner, -that the thickness on which the rope is coiled continually increased -or diminishes in the same proportion as the weight or resistance -diminishes or increases, will produce this effect. - -It is obvious that all that has been said respecting a variable -weight or resistance, is also applicable to a variable power, which, -therefore, may, by the same means, be made to produce a uniform effect. -An instance of this occurs in a watch, which is moved by a spiral -spring. When the watch has been wound up, this spring acts with its -greatest intensity, and as the watch goes down, the elastic force of -the spring gradually loses its energy. This spring is connected by a -chain with an axle of varying thickness, called a _fusee_. When the -spring is at its greatest intensity, the chain acts upon the thinnest -part of the fusee, and as it is uncoiled it acts upon a part of the -fusee which is continually increasing in thickness, the spring at the -same time losing its elastic power in exactly the same proportion. A -representation of the fusee, and the cylindrical box which contains -the spring, is given in _fig. 98._, and of the spring itself in -_fig. 99._ - -(256.) When great power is required, wheels and axles may be combined -in a manner analogous to a compound system of levers, explained -in (246.) In this case the power acts on the circumference of the -first wheel, and its effect is transmitted to the circumference -of the first axle. That circumference is placed in connection with -the circumference of the second wheel, and the effect is thereby -transmitted to the circumference of the second axle, and so on. It -is obvious from what was proved in (248.), that the power of such a -combination of wheels and axles will be found by multiplying together -the powers of the several wheels of which it is composed. It is -sometimes convenient to compute this power by numbers expressing the -proportions of the circumferences or diameters of the several wheels, -to the circumferences or diameters of the several axles respectively. -This computation is made by first multiplying the numbers together -which express the circumferences or diameters of the wheels, and then -multiplying together the numbers which express the circumferences or -diameters of the several axles. The proportion of the two products -will express the power of the machine. Thus, if the circumferences or -diameters be as the numbers 10, 14, and 15, their product will be 2100; -and if the circumferences or diameters of the axles be expressed by the -numbers 3, 4, and 5, their product will be 60, and the power of the -machine will be expressed by the proportion of 2100 and 60, or 35 to 1. - -[Illustration: _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -(257.) The manner in which the circumferences of the axles act upon -the circumferences of the wheels in compound wheel-work is various. -Sometimes a strap or cord is applied to a groove in the circumference -of the axle, and carried round a similar groove in the circumference -of the succeeding wheel. The friction of this cord or strap with the -groove is sufficient to prevent its sliding and to communicate the -force from the axle to the wheel, or _vice versa_. This method of -connecting wheel-work is represented in _fig. 100._ - -Numerous examples of wheels and axles driven by straps or cords occur -in machinery applied to almost every department of the arts and -manufactures. In the turning lathe, the wheel worked by the treddle is -connected with the mandrel by a catgut cord passing through grooves -in the wheel and axle. In all great factories, revolving shafts are -carried along the apartments, on which, at certain intervals, straps -are attached passing round their circumferences and carried round -the wheels which give motion to the several machines. If the wheels, -connected by straps or cords, are required to revolve in the same -direction, these cords are arranged as in _fig. 100._; but if they -are required to revolve in contrary directions, they are applied as in -_fig. 101._ - -One of the chief advantages of the method of transmitting motion -between wheels and axles by straps or cords, is that the wheel and -axle may be placed at any distance from each other which may be found -convenient, and may be made to turn either in the same or contrary -directions. - -(258.) When the circumference of the wheel acts immediately on the -circumference of the succeeding axle, some means must necessarily be -adopted to prevent the wheel from moving in contact with the axle -without compelling the latter to turn. If the surfaces of both were -perfectly smooth, so that all friction were removed, it is obvious that -either would slide over the surface of the other, without communicating -motion to it. But, on the other hand, if there were any asperities, -however small, upon these surfaces, they would become mutually inserted -among each other, and neither the wheel nor axle could move without -causing the asperities with which its edge is studded to encounter -those asperities which project from the surface of the other; and -thus, until these projections should be broken off, both wheel and -axle must be moved at the same time. It is on this account that if the -surfaces of the wheels and axles are by any means rendered rough, and -pressed together with sufficient force, the motion of either will turn -the other, provided the load or resistance be not greater than the -force necessary to break off these small projections which produce the -friction. - -In cases where great power is not required, motion is communicated in -this way through a train of wheel-work, by rendering the surface of the -wheel and axle rough, either by facing them with buff leather, or with -wood cut across the grain. This method is sometimes used in spinning -machinery, where one large buffed wheel, placed in a horizontal -position, revolves in contact with several small buffed rollers, each -roller communicating motion to a spindle. The position of the wheel W, -and the rollers R R, &c., are represented in _fig. 102._ Each -roller can be thrown out of contact with the wheel, and restored to it -at pleasure. - -The communication of motion between wheels and axles by friction has -the advantage of great smoothness and evenness, and of proceeding with -little noise; but this method can only be used in cases where the -resistance is not very considerable, and therefore is seldom adopted in -works on a large scale. Dr. Gregory mentions an instance of a saw mill -at Southampton, where the wheels act upon each other by the contact of -the end grain of wood. The machinery makes very little noise, and wears -very well, having been used not less than 20 years. - -(259.) The most usual method of transmitting motion through a train of -wheel-work is by the formation of teeth upon their circumferences, so -that these indentures of each wheel fall between the corresponding ones -of that in which it works, and ensure the action so long as the strain -is not so great as to fracture the tooth. - -In the formation of teeth very minute attention must be given to their -figure, in order that the motion may be communicated from wheel to -wheel with smoothness and uniformity. This can only be accomplished -by shaping the teeth according to curves of a peculiar kind, which -mathematicians have invented, and assigned rules for drawing. The ill -consequences of neglecting this will be very apparent, by considering -the nature of the action which would be produced if the teeth were -formed of square projecting pins, as in _fig. 103._ When the -tooth A comes into contact with B, it acts obliquely upon it, and, -as it moves, the corner of B slides upon the plane surface of A in -such a manner as to produce much friction, and to grind away the side -of A and the end of B. As they approach the position C D, they -sustain a jolt the moment their surfaces come into full contact; and -after passing the position of C D, the same scraping and grinding -effect is produced in the opposite direction, until by the revolution -of the wheels the teeth become disengaged. These effects are avoided by -giving to the teeth the curved forms represented in _fig. 104._ -By such means the surfaces of the teeth roll upon each other with very -inconsiderable friction, and the direction in which the pressure is -excited is always that of a line M N, touching the two wheels, and -at right angles to the radii. Thus the pressure being always the same, -and acting with the same leverage, produces a uniform effect. - -(260.) When wheels work together, their teeth must necessarily be of -the same size, and therefore the proportion of their circumferences may -always be estimated by the number of teeth which they carry. Hence it -follows, that in computing the power of compound wheel-work, the number -of teeth may always be used to express the circumferences respectively, -or the diameters which are proportional to these circumferences. When -teeth are raised upon an axle, it is generally called a _pinion_, and -in that case the teeth are called _leaves_. The rule for computing the -train of wheel-work given in (256.) will be expressed as follows: when -the wheel and axle carry teeth, multiply together the number of teeth -in each of the wheels, and next the number of leaves in each of the -pinions; the proportion of the two products will express the power of -the machine. If some of the wheels and axles carry teeth, and others -not, this computation may be made by using for those circumferences -which do not bear teeth the number of teeth which would fill them. -_Fig. 105._ represents a train of three wheels and pinions. The -wheel F which bears the power, and the axle which bears the weight, -have no teeth; but it is easy to find the number of teeth which they -would carry. - -(261.) It is evident that each pinion revolves much more frequently in -a given time than the wheel which it drives. Thus, if the pinion C be -furnished with ten teeth, and the wheel E, which it drives, have sixty -teeth, the pinion C must turn six times, in order to turn the wheel -E once round. The velocities of revolution of every wheel and pinion -which work in one another will therefore have the same proportion as -their number of teeth taken in a reverse order, and by this means the -relative velocity of wheels and pinions may be determined according to -any proposed rate. - -Wheel-work, like all other machinery, is used to transmit and modify -force in every department of the arts and manufactures; but it is also -used in cases where motion alone, and not force, is the object to be -attained. The most remarkable example of this occurs in watch and -clock-work, where the object is merely to produce uniform motions of -rotation, having certain proportions, and without any regard to the -elevation of weights, or the overcoming of resistances. - -(262.) A _crane_ is an example of combination of wheel-work used for -the purpose of raising or lowering great weights. _Fig. 106._ -represents a machine of this kind. A B is a strong vertical beam, -resting on a pivot, and secured in its position by beams in the floor. -It is capable, however, of turning on its axis, being confined between -rollers attached to the beams and fixed in the floor. C D is a -projecting arm called a _gib_, formed of beams which are mortised into -A B. The wheel-work is mounted in two cast-iron crosses, bolted on -each side of the beams, one of which appears at E F G H. -The winch at which the power is applied is at I. This carries a pinion -immediately behind H. This pinion works in a wheel K, which carries -another pinion upon its axle. This last pinion works in a larger wheel -L, which carries upon its axis a barrel M, on which a chain or rope -is coiled. The chain passes over a pulley D at the top of the gib. At -the end of the chain a hook O is attached, to support the weight W. -During the elevation of the weight it is convenient that its recoil -should be hindered in case of any occasional suspension of the power. -This is accomplished by a ratchet wheel attached to the barrel M, as -explained in (253.); but when the weight W is to be lowered, the catch -must be removed from this ratchet wheel. In this case the too rapid -descent of the weight is in some cases checked by pressure excited on -some part of the wheel-work, so as to produce sufficient friction to -retard the descent in any required degree, or even to suspend it, if -necessary. The vertical beam at B resting on a pivot, and being fixed -between rollers, allows the gib to be turned round in any direction; so -that a weight raised from one side of the crane may be carried round, -and deposited on another side, at any distance within the range of the -gib. Thus, if a crane be placed upon a wharf near a vessel, weights may -be raised, and when elevated, the gib may be turned round so as to let -them descend into the hold. - -The power of this machine may be computed upon the principles already -explained. The magnitude of the circle, in which the power at I moves, -may be determined by the radius of the winch, and therefore the number -of teeth which a wheel of that size would carry may be found. In -like manner we may determine the number of leaves in a pinion whose -magnitude would be equal to the barrel M. Let the first number be -multiplied by the number of teeth in the wheel K, and that product -by the number of teeth in the wheel L. Next let the number of leaves -in the pinion H be multiplied by the number of leaves in the pinion -attached to the axle of the wheel K, and let that product be multiplied -by the number of leaves in a pinion, whose diameter is equal to that of -the barrel M. These two products will express the power of the machine. - -(263.) Toothed wheels are of three kinds, distinguished by the position -which the teeth bear with respect to the axis of the wheel. When they -are raised upon the edge of the wheel as in _fig. 105._, they are -called _spur wheels_, or _spur gear_. When they are raised parallel to -the axis, as in _fig. 107._, it is called a _crown wheel_. When -the teeth are raised on a surface inclined to the plane of the wheel, -as in _fig. 108._, they are called _bevelled wheels_. - -[Illustration: _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -If a motion round one axis is to be communicated to another axis -parallel to it, spur gear is generally used. Thus, in _fig. 105._, -the three axes are parallel to each other. If a motion round one -axis is to be communicated to another at right angles to it, a crown -wheel, working in a spur pinion, as in _fig. 107._, will serve. -Or the same object may be obtained by two bevelled wheels, as in -_fig. 108._ - -If a motion round one axis is required to be communicated to another -inclined to it at any proposed angle, two bevelled wheels can always be -used. In _fig. 109._ let A B and A C be the two axles; -two bevelled wheels, such as D E and E F, on these axles will -transmit the motion or rotation from one to the other, and the relative -velocity may, as usual, be regulated by the proportional magnitude of -the wheels. - -(264.) In order to equalise the wear of the teeth of a wheel and -pinion, which work in one another, it is necessary that every leaf -of the pinion should work in succession through every tooth of the -wheel, and not continually act upon the same set of teeth. If the -teeth could be accurately shaped according to mathematical principles, -and the materials of which they are formed be perfectly uniform, this -precaution would be less necessary; but as slight inequalities, both -of material and form, must necessarily exist, the effects of these -should be as far as possible equalised, by distributing them through -every part of the wheel. For this purpose it is usual, especially -in mill-work, where considerable force is used, so to regulate the -proportion of the number of teeth in the wheel and pinion, that the -same leaf of the pinion shall not be engaged twice with any one tooth -of the wheel, until after the action of a number of teeth, expressed -by the product of the number of teeth in the wheel and pinion. Let us -suppose that the pinion contains ten leaves, which we shall denominate -by the numbers 1, 2, 3, &c., and that the wheel contains 60 teeth -similarly denominated. At the commencement of the motion suppose the -leaf 1 of the pinion engages the tooth 1 of the wheel; then after -one revolution the leaf 1 of the pinion will engage the tooth 11 of -the wheel, and after two revolutions the leaf 1 of the pinion will -engage the tooth 21 of the wheel; and in like manner, after 3, 4, and -5 revolutions of the pinion, the leaf 1 will engage successively the -teeth 31, 41, and 51 of the wheel. After the sixth revolution, the -leaf 1 of the pinion will again engage the tooth 1 of the wheel. Thus -it is evident, that in the case here supposed the leaf 1 of the pinion -will continually be engaged with the teeth 1, 11, 21, 31, 41, and 51 -of the wheel, and no others. The like may be said of every leaf of the -pinion. Thus the leaf 2 of the pinion will be successively engaged with -the teeth 2, 12, 22, 32, 42, and 52 of the wheel, and no others. Any -accidental inequalities of these teeth will therefore continually act -upon each other, until the circumference of the wheel be divided into -parts of ten teeth each, unequally worn. This effect would be avoided -by giving either the wheel or pinion one tooth more or one tooth less. -Thus, suppose the wheel, instead of having sixty teeth, had sixty-one, -then after six revolutions of the pinion the leaf 1 of the pinion would -be engaged with the tooth 61 of the wheel; and after one revolution of -the wheel, the leaf 2 of the pinion would be engaged with the tooth 1 -of the wheel. Thus, during the first revolution of the wheel the leaf -1 of the pinion would be successively engaged with the teeth 1, 11, -21, 31, 41, 51, and 61 of the wheel: at the commencement of the second -revolution of the wheel the leaf 2 of the pinion would be engaged with -the tooth 1 of the wheel; and during the second revolution of the wheel -the leaf 1 of the pinion would be successively engaged with the teeth -10, 20, 30, 40, 50, and 60 of the wheel. In the same manner it may be -shown, that in the third revolution of the wheel the leaf 1 of the -pinion would be successively engaged with the teeth 9, 19, 29, 39, 49, -and 59 of the wheel: during the fourth revolution of the wheel the -leaf 1 of the pinion would be successively engaged with the teeth 8, -18, 28, 38, 48, and 58 of the wheel. By continuing this reasoning it -will appear, that during the tenth revolution of the wheel the leaf -1 of the pinion will be engaged successively with the teeth 2, 12, -22, 32, 42, and 52 of the wheel. At the commencement of the eleventh -revolution of the wheel the leaf 1 of the pinion will be engaged with -the tooth 1 of the wheel, as at the beginning of the motion. It is -evident, therefore, that during the first ten revolutions of the wheel -each leaf of the pinion has been successively engaged with every tooth -of the wheel, and that during these ten revolutions the pinion has -revolved sixty-one times. Thus the leaves of the pinion have acted six -hundred and ten times upon the teeth of the wheel, before two teeth can -have acted twice upon each other. - -The odd tooth which produces this effect is called by millwrights the -_hunting cog_. - -(265.) The most familiar case in which wheel-work is used to produce -and regulate motion merely, without any reference to weights to be -raised or resistances to be overcome, is that of chronometers. In watch -and clock work the object is to cause a wheel to revolve with a uniform -velocity, and at a certain rate. The motion of this wheel is indicated -by an index or hand placed upon its axis, and carried round with it. -In proportion to the length of the hand the circle over which its -extremity plays is enlarged, and its motion becomes more perceptible. -This circle is divided, so that very small fractions of a revolution -of the hand may be accurately observed. In most chronometers it is -required to give motion to two hands, and sometimes to three. These -motions proceed at different rates, according to the subdivisions of -time generally adopted. One wheel revolves in a minute, bearing a -hand which plays round a circle divided into sixty equal parts; the -motion of the hand over each part indicating one second, and a complete -revolution of the hand being performed in one minute. Another wheel -revolves once, while the former revolves sixty times; consequently the -hand carried by this wheel revolves once in sixty minutes, or one hour. -The circle on which it plays is, like the former, divided into sixty -equal parts, and the motion of the hand over each division is performed -in one minute. This is generally called the _minute hand_, and the -former the _second hand_. - -A third wheel revolves once, while that which carries the minute hand -revolves twelve times; consequently this last wheel, which carries -the _hour hand_, revolves at a rate twelve times less than that of -the minute hand, and therefore seven hundred and twenty times less -than the second hand. We shall now endeavour to explain the manner in -which these motions are produced and regulated. Let A, B, C, D, E, -_fig. 110._, represent a train of wheels, and _a_, _b_, _c_, _d_ -represent their pinions, _e_ being a cylinder on the axis of the wheel -E, round which a rope is coiled, sustaining a weight W. Let the effect -of this weight transmitted through the train of wheels be opposed by -a power P acting upon the wheel A, and let this power be supposed -to be of such a nature as to cause the weight W to descend with a -uniform velocity, and at any proposed rate. The wheel E carries on its -circumference eighty-four teeth. The wheel D carries eighty teeth; -the wheel C is also furnished with eighty teeth, and the wheel B with -seventy-five. The pinions _d_ and _c_ are each furnished with twelve -leaves, and the pinions _b_ and _a_ with ten. - -If the power at P be so regulated as to allow the wheel A to revolve -once in a minute, with a uniform velocity, a hand attached to the axis -of this wheel will serve as the _second hand_. The pinion _a_ carrying -ten teeth must revolve seven times and a half to produce one revolution -of B, consequently fifteen revolutions of the wheel A will produce two -revolutions of the wheel B; the wheel B, therefore, revolves twice in -fifteen minutes. The pinion _b_ must revolve eight times to produce -one revolution of the wheel C, and therefore the wheel C must revolve -once in four quarters of an hour, or in one hour. If a hand be attached -to the axis of this wheel, it will have the motion necessary for the -minute hand. The pinion _c_ must revolve six times and two thirds to -produce one revolution of the wheel D, and therefore this wheel must -revolve once in six hours and two thirds. The pinion _d_ revolves seven -times for one revolution of the wheel E, and therefore the wheel E will -revolve once in forty-six hours and two thirds. - -On the axis of the wheel C a second pinion may be placed, furnished -with seven leaves, which may lead a wheel of eighty-four teeth, so -that this wheel shall turn once during twelve turns of the wheel C. If -a hand be fixed upon the axis, this hand will revolve once for twelve -revolutions of the minute hand fixed upon the axis of the wheel C; -that is, it will revolve once in twelve hours. If it play upon a dial -divided into twelve equal parts, it will move over each part in an -hour, and will serve the purpose of the hour hand of the chronometer. - -We have here supposed that the second hand, the minute hand, and the -hour hand move on separate dials. This, however, is not necessary. The -axis of the hour hand is commonly a tube, inclosing within it that of -the minute hand, so that the same dial serves for both. The second -hand, however, is generally furnished with a separate dial. - -(266.) We shall now explain the manner in which a power is applied -to the wheel A, so as to regulate and equalise the effect of the -weight W. Suppose the wheel A furnished with thirty teeth, as in -_fig. 111._; if nothing check the motion, the weight W would -descend with an accelerated velocity, and would communicate an -accelerated motion to the wheel A. This effect, however, is interrupted -by the following contrivance:--L M is a pendulum vibrating on the -centre L, and so regulated that the time of its oscillation is one -second. The pallets I and K are connected with the pendulum, so as to -oscillate with it. In the position of the pendulum represented in the -figure, the pallet I stops the motion of the wheel A, and entirely -suspends the action of the weight W, _fig. 110._, so that for -a moment the entire machine is motionless. The weight M, however, -falls by its gravity towards the lowest position, and disengages the -pallet I from the tooth of the wheel. The weight W begins then to take -effect, and the wheel A turns from A towards B. Meanwhile the pendulum -M oscillates to the other side, and the pallet K falls under a tooth -of the wheel A, and checks for a moment its further motion. On the -returning vibration the pallet K becomes again disengaged, and allows -the tooth of the wheel to escape, and by the influence of the weight W -another tooth passes before the motion of the wheel A is again checked -by the interposition of the pallet I. - -From this explanation it will appear that, in two vibrations of the -pendulum, one tooth of the wheel A passes the pallet I, and therefore, -if the wheel A be furnished with 30 teeth, it will be allowed to make -one revolution during 60 vibrations of the pendulum. If, therefore, the -pendulum be regulated so as to vibrate seconds, this wheel will revolve -once in a minute. From the action of the pallets in checking the motion -of the wheel A, and allowing its teeth alternately to _escape_, this -has been called the _escapement_ wheel; and the wheel and pallets -together are generally called the _escapement_, or _’scapement_. - -We have already explained, that by reason of the friction on the -points of support, and other causes, the swing of the pendulum would -gradually diminish, and its vibration at length cease. This, however, -is prevented by the action of the teeth of the scapement wheel upon the -pallets, which is just sufficient to communicate that quantity of force -to the pendulum which is necessary to counteract the retarding effects, -and to maintain its motion. It thus appears, that although the effect -of the gravity of the weight W in giving motion to the machine is at -intervals suspended, yet this part of the force is not lost, being, -during these intervals, employed in giving to the pendulum all that -motion which it would lose by the resistances to which it is inevitably -exposed. - -In stationary clocks, and in other cases in which the bulk of the -machine is not an objection, a descending weight is used as the -moving power. But in watches and portable chronometers, this would be -attended with evident inconvenience. In such cases, a spiral spring, -called the _mainspring_, is the moving power. The manner in which this -spring communicates rotation to an axis, and the ingenious method of -equalising the effect of its variable elasticity by giving to it a -leverage, which increases as the elastic force diminishes, have been -already explained. (255.) - -A similar objection lies against the use of a pendulum in portable -chronometers. A spiral spring of a similar kind, but infinitely -more delicate, called a _hair spring_, is substituted in its place. -This spring is connected with a nicely-balanced wheel, called _the -balance wheel_, which plays in pivots. When this wheel is turned to -a certain extent in one direction, the hair spring is coiled up, and -its elasticity causes the wheel to recoil, and return to a position -in which the energy of the spring acts in the opposite direction. -The balance wheel then returns, and continually vibrates in the same -manner. The axis of this wheel is furnished with pallets similar to -those of the pendulum, which are alternately engaged with the teeth of -a crown wheel, which takes the place of the scapement wheel already -described. - -A general view of the work of a common watch is represented in -_fig. 111._ _bis._ A is the balance wheel bearing pallets _p_ -_p_ upon its axis; C is the crown wheel, whose teeth are suffered to -escape alternately by those pallets in the manner already described -in the scapement of a clock. On the axis of the crown wheel is placed -a pinion _d_, which drives another crown wheel K. On the axis of this -is placed the pinion _c_, which plays in the teeth of the third wheel -L. The pinion _b_ on the axis of L is engaged with the wheel M, called -the centre wheel. The axle of this wheel is carried up through the -centre of the dial. A pinion _a_ is placed upon it, which works in -the great wheel N. On this wheel the mainspring immediately acts. -O P is the mainspring stripped of its barrel. The axis of the -wheel M passing through the centre of the dial is squared at the end -to receive the minute hand. A second pinion Q is placed upon this -axle which drives a wheel T. On the axle of this wheel a pinion _g_ -is placed, which drives the hour wheel V. This wheel is placed upon a -tubular axis, which incloses within it the axis of the wheel M. This -tubular axis passing through the centre of the dial, carries the hour -hand. The wheels A, B, C, D, E, _fig. 110._, correspond to the wheels -C, K, L, M, N, _fig. 112._; and the pinions _a_, _b_, _c_, _d_, -_e_, _fig. 109._, correspond to the pinions _d_, _c_, _b_, _a_, -_fig. 111_. From what has already been explained of these wheels, -it will be obvious that the wheel M, _fig. 111._, revolves once -in an hour, causing the minute hand to move round the dial once in -that time. This wheel at the same time turns the pinion Q which leads -the wheel T. This wheel again turns the pinion _g_ which leads the -hour wheel V. The leaves and teeth of these pinions and wheels are -proportioned, as already explained, so that the wheel V revolves once -during twelve revolutions of the wheel M. The hour hand, therefore, -which is carried by the tubular axle of the wheel V, moves once round -the dial in twelve hours. - -Our object here has not been to give a detailed account of watch and -clock work, a subject for which we must refer the reader to the proper -department of this work. Such a general account has only been attempted -as may explain how tooth and pinion work may be applied to regulate -motion. - -[Illustration: _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - - - - -CHAP. XV. - -OF THE PULLEY. - - -(267.) The next class of simple machines, which present themselves -to our attention, is that which we have called the _cord_. If a rope -were perfectly flexible, and were capable of being bent over a sharp -edge, and of moving upon it without friction, we should be enabled by -its means to make a force in any one direction overcome resistance, or -communicate motion in any other direction. Thus if P, _fig. 112._, -be such an edge, a perfectly flexible rope passing over it would be -capable of transmitting a force S F to a resistance Q R, -so as to support or overcome R, or by a motion in the direction of -S F to produce another motion in the direction R Q. But as -no materials of which ropes can be constructed can give them perfect -flexibility, and as in proportion to the strength by which they are -enabled to transmit force their rigidity increases, it is necessary, -in practice, to adopt means to remove or mitigate those effects which -attend imperfect flexibility, and which would otherwise render cords -practically inapplicable as machines. - -When a cord is used to transmit a force from one direction to another, -its stiffness renders some force necessary in bending it over the -angle P, which the two directions form; and if the angle be sharp, -the exertion of such a force may be attended with the rupture of the -cord. If, instead of bending the rope at one point over a single angle, -the change of direction were produced by successively deflecting it -over several angles, each of which would be less sharp than a single -one could be, the force requisite for the deflection, as well as the -liability of rupturing the cord, would be considerably diminished. But -this end will be still more perfectly attained if the deflection of the -cord be produced by bending it over the surface of a curve. - -If a rope were applied only to sustain, and not to move a weight, -this would be sufficient to remove the inconveniences arising from -its rigidity. But when motion is to be produced, the rope, in passing -over the curved surface, would be subject to excessive friction, and -consequently to rapid wear. This inconvenience is removed by causing -the surface on which the rope runs to move with it, so that no more -friction is produced than would arise from the curved surface rolling -upon the rope. - -(268.) All these ends are attained by the common pulley, which consists -of a wheel called a _sheave_, fixed in a block and turning on a pivot. -A groove is formed in the edge of the wheel in which the rope runs, -the wheel revolving with it. Such an apparatus is represented in -_fig. 113._ - -We shall, for the present, omit the consideration of that part of the -effects of the stiffness and friction of the machine which is not -removed by the contrivance just explained, and shall consider the rope -as perfectly flexible and moving without friction. - -From the definition of a flexible cord, it follows, that its tension, -or the force by which it is stretched throughout its entire length, -must be uniform. From this principle, and this alone, all the -mechanical properties of pulleys may be derived. - -Although, as already explained, the whole mechanical efficacy of this -machine depends on the qualities of the cord, and not on those of the -block and sheave, which are only introduced to remove the accidental -effects of stiffness and friction; yet it has been usual to give the -name pulley to the block and sheave, and a combination of blocks, -sheaves, and ropes is called a _tackle_. - -(269.) When the rope passes over a single wheel, which is fixed in -its position, as in _fig. 113._, the machine is called a _fixed -pulley_. Since the tension of the cord is uniform throughout its -length, it follows, that in this machine the power and weight are -equal. For the weight stretches that part of the cord which is between -the weight and pulley, and the power stretches that part between the -power and the pulley. And since the tension throughout the whole length -is the same, the weight must be equal to the power. - -Hence it appears that no mechanical advantage is gained by this -machine. Nevertheless, there is scarcely any engine, simple or complex, -attended with more convenience. In the application of power, whether -of men or animals, or arising from natural forces, there are always -some directions in which it may be exerted to much greater convenience -and advantage than others, and in many cases the exertion of these -powers is limited to a single direction. A machine, therefore, which -enables us to give the most advantageous direction to the moving power, -whatever be the direction of the resistance opposed to it, contributes -as much practical convenience as one which enables a small power to -balance or overcome a great weight. In directing the power against the -resistance, it is often necessary to use two fixed pulleys. Thus, in -elevating a weight A, _fig. 114._, to the summit of a building, -by the strength of a horse moving below, two fixed pulleys B and C may -be used. The rope is carried from A over the pulley B; and, passing -downwards, is brought under C, and finally drawn by the animal on -the horizontal plane. In the same manner sails are spread, and flags -hoisted on the yards and masts of a ship, by sailors pulling a rope on -the deck. - -By means of the fixed pulley a man may raise himself to a considerable -height, or descend to any proposed depth. If he be placed in a chair -or bucket attached to one end of a rope which is carried over a fixed -pulley, by laying hold of this rope on the other side, as represented -in _fig. 115._, he may, at will, descend to a depth equal to half -of the entire length of the rope, by continually yielding rope on the -one side, and depressing the bucket or chair by his weight on the -other. Fire-escapes have been constructed on this principle, the fixed -pulley being attached to some part of the building. - -(270.) A _single moveable pulley_ is represented in _fig. 116._ -A cord is carried from a fixed point F, and passing through a block -B, attached to a weight W, passes over a fixed pulley C, the power -being applied at P. We shall first suppose the parts of the cord on -each side the wheel B to be parallel; in this case, the whole weight W -being sustained by the parts of the cords B C and B F, and -these parts being equally stretched (268.), each must sustain half the -weight, which is therefore the tension of the cord. This tension is -resisted by the power at P, which must, therefore, be equal to half the -weight. In this machine, therefore, the weight is twice the power. - -(271.) If the parts of the cord B C and B F be not parallel, -as in _fig. 117._, a greater power than half the weight is -therefore necessary to sustain it. To determine the power necessary -to support a given weight, in this case take the line B A in the -vertical direction, consisting of as many inches as the weight consists -of ounces; from A draw A D parallel to B C, and A E -parallel to B F; the force of the weight represented by A B -will be equivalent to two forces represented by B D and B E. -(74.) The number of inches in these lines respectively will represent -the number of ounces which are equivalent to the tensions of the parts -B F and B C of the cord. But as these tensions are equal, -B D and B E must be equal, and each will express the amount -of the power P, which stretches the cord at P C. - -It is evident that the four lines, A E, E B, B D, and -D A, are equal. And as each of them represents the power, the -weight which is represented by A B must be less than twice the -power which is represented by A E and E B taken together. It -follows, therefore, that as parts of the ropes which support the weight -depart from parallelism the machine becomes less and less efficacious; -and there are certain obliquities at which the equilibrating power -would be much greater than the weight. - -(272.) The mechanical power of pulleys admits of being almost -indefinitely increased by combination. Systems of pulleys may be -divided into two classes; those in which a single rope is used, and -those which consist of several distinct ropes. _Fig. 118._ and -_119._ represent two systems of pulleys, each having a single rope. -The weight is in each case attached to a moveable block, B, in which -are fixed two or more wheels; A is a fixed block, and the rope is -successively passed over the wheels above and below, and, after passing -over the last wheel above, is attached to the power. The tension of -that part of the cord to which the power is attached is produced by -the power, and therefore equivalent to it, and the same tension must -extend throughout its whole length. The weight is sustained by all -those parts of the cord which pass from the lower block, and as the -force which stretches them all is the same, viz. that of the power, -the effect of the weight must be equally distributed among them, their -directions being supposed to be parallel. It will be evident, from -this reasoning, that the weight will be as many times greater than the -power as the number of cords which support the lower block. Thus, if -there be six cords, each cord will support a sixth part of the weight, -that is, the weight will be six times the tension of the cord, or six -times the power. In _fig. 118._ the cord is represented as being -finally attached to a hook on the upper block. But it may be carried -over an additional wheel fixed in that block, and finally attached -to a hook in the lower block, as in _fig. 119._, by which one -will be added to the power of the machine, the number of cords at -the lower block being increased by one. In the system represented in -_fig. 118._ the wheels are placed in the blocks one above the -other; in _fig. 119._ they are placed side by side. In all systems -of pulleys of this class, the weight of the lower block is to be -considered as a part of the weight to be raised, and in estimating the -power of the machine, this should always be attended to. - -(273.) When the power of the machine, and therefore the number of -wheels, is considerable, some difficulty arises in the arrangement of -the wheels and cords. The celebrated Smeaton contrived a tackle, which -takes its name from him, in which there are ten wheels in each block: -five large wheels placed side by side, and five smaller ones similarly -placed above them in the lower block, and below them in the upper. -_Fig. 120._ represents Smeaton’s blocks without the rope. The -wheels are marked with the numbers 1, 2, 3, &c., in the order in which -the rope is to be passed over them. As in this pulley 20 distinct parts -of the rope support the lower block, the weight, including the lower -block, will be 20 times the equilibrating power. - -(274.) In all these systems of pulleys, every wheel has a separate -axle, and there is a distinct wheel for every turn of the rope at each -block. Each wheel is attended with friction on its axle, and also with -friction between the sheave and block. The machine is by this means -robbed of a great part of its efficacy, since, to overcome the friction -alone, a considerable power is in most cases necessary. - -An ingenious contrivance has been suggested, by which all the advantage -of a large number of wheels may be obtained without the multiplied -friction of distinct sheaves and axles. To comprehend the excellence -of this contrivance, it will be necessary to consider the rate at -which the rope passes over the several wheels of such a system, as -_fig. 118._ If one foot of the rope G F pass over the -pulley F, two feet must pass over the pulley E, because the distance -between F and E being shortened one foot, the total length of the rope -G F E must be shortened two feet. These two feet of rope -must pass in the direction E D, and the wheel D, rising one foot, -three feet of rope must consequently pass over it. These three feet of -rope passing in the direction D C, and the rope D C being -also shortened one foot by the ascent of the lower block, four feet of -rope must pass over the wheel C. In the same way it may be shown that -five feet must pass over B, and six feet over A. Thus, whatever be -the number of wheels in the upper and lower blocks, the parts of the -rope which pass in the same time over the wheels in the lower block -are in the proportion of the odd numbers 1, 3, 5, &c.; and those which -pass over the wheels in the upper block in the same time, are as the -even numbers 2, 4, 6, &c. If the wheels were all of equal size, as in -_fig. 119._, they would revolve with velocities proportional to -the rate at which the rope passes over them. So that, while the first -wheel below revolves once, the first wheel above will revolve twice; -the second wheel below three times; the second wheel above, four times, -and so on. If, however, the wheels differed in size in proportion to -the quantity of rope which must pass over them, they would evidently -revolve in the same time. Thus, if the first wheel above were twice the -size of the first wheel below, one revolution would throw off twice the -quantity of rope. Again, if the second wheel below were thrice the size -of the first wheel below, it would throw off in one revolution thrice -the quantity of rope, and so on. Wheels thus proportioned, revolving -in exactly the same time, might be all placed on one axle, and would -partake of one common motion, or, what is to the same effect, several -grooves might be cut upon the face of one solid wheel, with diameters -in the proportion of the odd numbers 1, 3, and 5, &c., for the lower -pulley, and corresponding grooves on the face of another solid wheel -represented by the even numbers 2, 4, 6, &c., for the upper pulley. The -rope being passed successively over the grooves of such wheels, would -be thrown off exactly in the same manner as if every groove were upon a -separate wheel, and every wheel revolved independently of the others. -Such is White’s pulley, represented in _fig. 121._ - -The advantage of this machine, when accurately constructed, is very -considerable. The friction, even when great resistances are to be -opposed, is very trifling; but, on the other hand, it has corresponding -disadvantages which greatly circumscribe its practical utility. In the -workmanship of the grooves great difficulty is found in giving them -the exact proportions. In doing which, the thickness of the rope must -be accurately allowed for; and consequently it follows, that the same -pulley can never act except with a rope of a particular diameter. A -very slight deviation from the true proportion of the grooves will -cause the rope to be unequally stretched, and will throw on some parts -of it an undue proportion of the weight, while other parts become -nearly, and sometimes altogether slack. Besides these defects, the rope -is so liable to derangement by being thrown out of the grooves, that -the pulley can scarcely be considered portable. - -For these and other reasons, this machine, ingenious as it -unquestionably is, has never been extensively used. - -(275.) In the several systems of pulleys just explained, the hook to -which the fixed block is attached supports the entire of both the power -and weight. When the machine is in equilibrium, the power only supports -so much of the weight as is equal to the tension of the cord, all the -remainder of the weight being thrown on the fixed point, according to -what was observed in (225.) - -If the power be moved so as to raise the weight, it will move with a -velocity as many times greater than that of the weight as the weight -itself is greater than the power. Thus in _fig. 118._ if the -weight attached to the lower block ascend one foot, six feet of line -will pass over the pulley A, according to what has been already proved. -Thus, the power will descend through six feet, while the weight rises -one foot. But, in this case, the weight is six times the power. All the -observations in (226.) will therefore be applicable to the cases of -great weights raised by small powers by means of the system of pulleys -just described. - -(276.) When two or more ropes are used, pulleys may be combined in -various ways so as to produce any degree of mechanical effect. If -to any of the systems already described a single moveable pulley -be added, the power of the machine would be doubled. In this case, -the second rope is attached to the hook of the lower block, as in -_fig. 122._, and being carried through a moveable pulley -attached to the weight, it is finally brought up to a fixed point. The -tension of the second cord is equal to half the weight (270.); and -therefore the power P, by means of the first cord, will have only half -the tension which it would have if the weight were attached to the -lower block. A moveable pulley thus applied is called a _runner_. - -[Illustration: _C. Varley, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -(277.) Two systems of pulleys, called _Spanish bartons_, having -each two ropes, are represented in _fig. 123._ The tension of -the rope P A B C in the first system is equal to -the power; and therefore the parts B A and B C support a -portion of the weight equal to twice the power. The rope E A -supports the tensions of A P and A B; and therefore the -tension of A E D is twice the power. Thus, the united -tensions of the ropes which support the pulley B is four times the -power, which is therefore the amount of the weight. In the second -system, the rope P A D is stretched by the power. The rope -A E B C acts against the united tensions A P and -A D; and therefore the tension of A E or E B is twice -the power. Thus, the weight acts against three tensions; two of which -are equal to twice the power, and the remaining one is equal to the -power. The weight is therefore equal to five times the power. - -A single rope may be so arranged with one moveable pulley as to support -a weight equal to three times the power. In _fig. 124._ this -arrangement is represented, where the numbers sufficiently indicate the -tension of the rope, and the proportion of the weight and power. In -_fig. 125._ another method of producing the same effect with two -ropes is represented. - -(278.) If several single moveable pulleys be made successively to act -upon each other, the effect is doubled by every additional pulley: -such a system as this is represented in _fig. 126._ The tension -of the first rope is equal to the power; the second rope acts against -twice the tension of the first, and therefore it is stretched with -a force equal to twice the power: the third rope acts against twice -this tension, and therefore it is stretched with a force equal -to four times the power, and so on. In the system represented in -_fig. 126._ there are three ropes, and the weight is eight times -the power. Another rope would render it sixteen times the power, and so -on. - -In this system, it is obvious that the ropes will require to have -different degrees of strength, since the tension to which they are -subject increases in a double proportion from the power to the weight. - -(279.) If each of the ropes, instead of being attached to fixed points -at the top, are carried over fixed pulleys, and attached to the several -moveable pulleys respectively, as in _fig. 127._, the power of -the machine will be greatly increased; for in that case the forces -which stretch the successive ropes increase in a treble instead of a -double proportion, as will be evident by attending to the numbers which -express the tensions in the figure. One rope would render the weight -three times the power, two ropes nine times, three ropes twenty-seven -times, and so on. An arrangement of pulleys is represented in _fig. -128._, by which each rope, instead of being finally attached to a fixed -point, as in _fig. 126._, is attached to the weight. The weight -is in this case supported by three ropes; one stretched with a force -equal to the power; another with a force equal to twice the power; -and a third with a force equal to four times the power. The weight is -therefore, in this case, seven times the power. - -(280.) If the ropes, instead of being attached to the weight, pass -through wheels, as in _fig. 129._, and are finally attached to the -pulleys above, the power of the machine will be considerably increased. -In the system here represented the weight is twenty-six times the power. - -(281.) In considering these several combinations of pulleys, we have -omitted to estimate the effects produced by the weights of the sheaves -and blocks. Without entering into the details of this computation, -it may be observed generally, that in the systems represented in -_figs. 126._, _127._ the weight of the wheel and blocks acts -against the power; but that in _figs. 128._ and _129._ they -assist the powers in supporting the weight. In the systems represented -in _fig. 123._ the weight of the pulleys, to a certain extent, -neutralise each other. - -(282.) It will in all cases be found, that that quantity by which the -weight exceeds the power is supported by fixed points; and therefore, -although it be commonly stated that a small power supports a great -weight, yet in the pulley, as in all other machines, the power supports -no more of the weight than is exactly equal to its own amount. It -will not be necessary to establish this in each of the examples which -have been given: having explained it in one instance, the student -will find no difficulty in applying the same reasoning to others. In -_fig. 126._, the fixed pulley sustains a force equal to twice the -power, and by it the power giving tension to the first rope sustains a -part of the weight equal to itself. The first hook sustains a portion -of the weight equal to the tension of the first string, or to the -power. The second hook sustains a force equal to twice the power; and -the third hook sustains a force equal to four times the power. The -three hooks therefore sustain a portion of the weight equal to seven -times the power; and the weight itself being eight times the power, it -is evident that the part of the weight which remains to be supported by -the power is equal to the power itself. - -(283.) When a weight is raised by any of the systems of pulleys which -have been last described, the proportion between the velocity of -the weight and the velocity of the power, so frequently noticed in -other machines, will always be observed. In the system of pulleys -represented in _fig. 126._ the weight being eight times the power, -the velocity of the power will be eight times that of the weight. If -the power be moved through eight feet, that part of the rope between -the fixed pulley and the first moveable pulley will be shortened by -eight feet. And since the two parts which lie above the first moveable -pulley must be equally shortened, each will be diminished by four feet; -therefore the first pulley will rise through four feet while the power -moves through eight feet. In the same way it may be shown, that while -the first pulley moves through four feet, the second moves through two; -and while the second moves through two, the third, to which the weight -is attached, is raised through one foot. While the power, therefore, is -carried through eight feet, the weight is moved through one foot. - -By reasoning similar to this, it may be shown that the space through -which the power is moved in every case is as many times greater than -the height through which the weight is raised, as the weight is greater -than the power. - -(284.) From its portable form, cheapness of construction, and the -facility with which it may be applied in almost every situation, -the pulley is one of the most useful of the simple machines. The -mechanical advantage, however, which it appears in theory to possess -is considerably diminished in practice, owing to the stiffness of the -cordage, and the friction of the wheels and blocks. By this means it -is computed that in most cases so great a proportion as two thirds of -the power is lost. The pulley is much used in building, where weights -are to be elevated to great heights. But its most extensive application -is found in the rigging of ships, where almost every motion is -accomplished by its means. - -(285.) In all the examples of pulleys, we have supposed the parts of -the rope sustaining the weight and each of the moveable pulleys to be -parallel to each other. If they be subject to considerable obliquity, -the relative tensions of the different ropes must be estimated -according to the principle applied in (271.) - - - - -CHAP. XVI. - -ON THE INCLINED PLANE, WEDGE, AND SCREW. - - -(286.) The inclined plane is the most simple of all machines. It is -a hard plane surface forming some angle with a horizontal plane, -that angle not being a right angle. When a weight is placed on such -a plane, a two-fold effect is produced. A part of the effect of the -weight is resisted by the plane, and produces a pressure upon it; and -the remainder urges the weight down the plane, and would produce a -pressure against any surface resisting its motion placed in a direction -perpendicular to the plane (131.) - -Let A B, _fig. 130._, be such a plane, B C its -horizontal base, A C its height, and A B C its angle -of elevation. Let W be a weight placed upon it. This weight acts in -the vertical direction W D, and is equivalent to two forces, -W F perpendicular to the plane, and W E directed down the -plane (74.) If a plane be placed at right angles to the inclined -plane below W, it will resist the descent of the weight, and sustain -a pressure expressed by W E. Thus, the weight W resting in the -corner, instead of producing one pressure in the direction W D, -will produce two pressures, one expressed by W F upon the inclined -plane, and the other expressed by W E upon the resisting plane. -These pressures respectively have the same proportion to the entire -weight as W F and W E have to W D, or as D E and -W E have to W D, because D E is equal to W F. Now -the triangle W E D is in all respects similar to the triangle -A B C, the one differing from the other only in the scale on -which it is constructed. Therefore, the three lines A C, C B, -and B A, are in the same proportion to each other as the lines -W E, E D, and W D. Hence, A B has to A C the -same proportion as the whole weight has to the pressure directed toward -B, and A B has to B C the same proportion as the whole -weight has to the pressure on the inclined plane. - -We have here supposed the weight to be sustained upon the inclined -plane by a hard plane fixed at right angles to it. But the power -necessary to sustain the weight will be the same in whatever way it is -applied, provided it act in the direction of the plane. Thus, a cord -may be attached to the weight, and stretched towards A, or the hands of -men may be applied to the weight below it, so as to resist its descent -towards B. But in whatever way it be applied, the amount of the power -will be determined in the same manner. Suppose the weight to consist -of as many pounds as there are inches in A B, then the power -requisite to sustain it upon the plane will consist of as many pounds -as there are inches in A C, and the pressure on the plane will -amount to as many pounds as there are inches in B C. - -From what has been stated it may easily be inferred that the less the -elevation of the plane is, the less will be the power requisite to -sustain a given weight upon it, and the greater will be the pressure -upon it. Suppose the inclined plane A B to turn upon a hinge -at B, and to be depressed so that its angle of elevation shall be -diminished, it is evident that as this angle decreases the height of -the plane decreases, and its base increases. Thus, when it takes the -position B A′, the height A′ C′ is less than the former -height A C, while the base B C′ is greater than the former -base B C. The power requisite to support the weight upon the plane -in the position B A′ is represented by A′ C′, and is as much -less than the power requisite to sustain it upon the plane A B, -as the height A′ C′ is less than the height A C. On the -other hand, the pressure upon the plane in the position B A′ is -as much greater than the pressure upon the plane B A, as the base -B C′ is greater than the base B C. - -(287.) The power of an inclined plane, considered as a machine, is -therefore estimated by the proportion which its length bears to its -height. This power is always increased by diminishing the elevation of -the plane. - -Roads which are not level may be regarded as inclined planes, and -loads drawn upon them in carriages, considered in reference to the -powers which impel them, are subject to all the conditions which have -been established for inclined planes. The inclination of the road is -estimated by the height corresponding to some proposed length. Thus it -is said to rise one foot in fifteen, one foot in twenty, &c., meaning -that if fifteen or twenty feet of the road be taken as the length of -an inclined plane, such as A B, the corresponding height will be -one foot. Or the same may be expressed thus: that if fifteen or twenty -feet be measured upon the road, the difference of the levels of the two -extremities of the distance measured is one foot. According to this -method of estimating the inclination of roads, the power requisite to -sustain a load upon them (setting aside the effect of friction), is -always proportional to that elevation. Thus, if a road rise one foot in -twenty, a power of one ton will be sufficient to sustain twenty tons, -and so on. - -On a horizontal plane the only resistance which the power has -to overcome is the friction of the load with the plane, and the -consideration of this being for the present omitted, a weight once put -in motion would continue moving for ever, without any further action of -the power. But if the plane be inclined, the power will be expended in -raising the weight through the perpendicular height of the plane. Thus, -in a road which rises one foot in ten, the power is expended in raising -the weight through one perpendicular foot for every ten feet of the -road over which it is moved. As the expenditure of power depends upon -the rate at which the weight is raised perpendicularly, it is evident -that the greater the inclination of the road is, the slower the motion -must be with the same force. If the energy of the power be such as to -raise the weight at the rate of one foot per minute, the weight may be -moved in each minute through that length of the road which corresponds -to a rise of one foot. Thus, if two roads rise one at the rate of a -foot in fifteen feet, and the other at the rate of one foot in twenty -feet, the same expenditure of power will move the weight through -fifteen feet of the one, and twenty feet of the other at the same rate. - -From such considerations as these, it will readily appear that it may -often be more expedient to carry a road through a circuitous route -than to continue it in the most direct course; for though the measured -length of road may be considerably greater than in the former case, yet -more may be gained in speed with the same expenditure of power than is -lost by the increase of distance. By attending to these circumstances, -modern road-makers have greatly facilitated and expedited the -intercourse between distant places. - -(288.) If the power act obliquely to the plane, it will have a twofold -effect; a part being expended in supporting or drawing the weight, -and a part in diminishing or increasing the pressure upon the plane. -Let W P, _fig. 130._, be the power. This will be equivalent -to two forces, W F′, perpendicular to the plane, and W E′ -in the direction of the plane. (74.) In order that the power should -sustain the weight, it is necessary that that part W E′ of the -power which acts in the direction of the plane should be equal to that -part W E, _fig. 130._, of the weight which acts down the -plane. The other part W F′ of the power acting perpendicular to -the plane is immediately opposed to that part W F of the weight -which produces pressure. The pressure upon the plane will therefore -be diminished by the amount of W F′. The amount of the power -which will equilibrate with the weight may, in this case, be found -as follows. Take W E′ equal to W E, and draw E′ P -perpendicular to the plane, and meeting the direction of the power. -The proportion of the power to the weight will be that of W P to -W D. And the proportion of the pressure to the weight will be that -of the difference between W F and W F′ to W D. If the -amount of the power have a less proportion to the weight than W P -has to W D, it will not support the body on the plane, but will -allow it to descend. And if it have a greater proportion, it will draw -the weight up the plane towards A. - -(289.) It sometimes happens that a weight upon one inclined plane is -raised or supported by another weight upon another inclined plane. -Thus, if A B and A B′, _fig. 131._, be two inclined -planes forming an angle at A, and W W′ be two weights placed -upon these planes, and connected by a cord passing over a pulley at -A, the one weight will either sustain the other, or one will descend, -drawing the other up. To determine the circumstances under which these -effects will ensue, draw the lines W D and W′ D′ in the -vertical direction, and take upon them as many inches as there are -ounces in the weights respectively. W D and W′ D′ being the -lengths thus taken, and therefore representing the weights, the lines -W E and W′ E′ will represent the effects of these weights -respectively down the planes. If W E and W′ E′ be equal, the -weights will sustain each other without motion. But if W E be -greater than W′ E′, the weight W will descend, drawing the weight -W′ up. And if W′ E′ be greater than W E, the weight W′ will -descend, drawing the weight W up. In every case the lines W F and -W′ F′ will represent the pressures upon the planes respectively. - -It is not necessary, for the effect just described, that the inclined -planes should, as represented in the figure, form an angle with each -other. They may be parallel, or in any other position, the rope being -carried over a sufficient number of wheels placed so as to give it the -necessary deflection. This method of moving loads is frequently applied -in great public works where rail-roads are used. Loaded waggons descend -one inclined plane, while other waggons, either empty or so loaded as -to permit the descent of those with which they are connected, are drawn -up the other. - -(290.) In the application of the inclined plane which we have hitherto -noticed, the machine itself is supposed to be fixed in its position, -while the weight or load is moved upon it. But it frequently happens -that resistances are to be overcome which do not admit of being thus -moved. In such cases, instead of moving the load upon the planes, -the plane is to be moved under or against the load. Let D E, -_fig. 132._, be a heavy beam secured in a vertical position -between guides F G and H I, so that it is free to move -upwards and downwards, but not laterally. Let A B C be an -inclined plane, the extremity of which is placed beneath the end of -the beam. A force applied to the back of this plane A C, in the -direction C B, will urge the plane under the beam so as to raise -the beam to the position represented in _fig. 133._ Thus, while -the inclined plane is moved through the distance C B, the beam is -raised through the height C A. - -(291.) When the inclined plane is applied in this manner, it is called -a _wedge_. And if the power applied to the back were a continued -pressure, its proportion to the weight would be that of A C to -C B. It follows, therefore, that the more acute the angle B is, -the more powerful will be the wedge. - -In some cases, the wedge is formed of two inclined planes, placed base -to base, as represented in _fig. 134._ The theoretical estimation -of the power of this machine is not applicable in practice with any -degree of accuracy. This is in part owing to the enormous proportion -which the friction in most cases bears to the theoretical value of -the power, but still more to the nature of the power generally used. -The force of a blow is of a nature so wholly different from continued -forces, such as the pressure of weights, or the resistance offered by -the cohesion of bodies, that it admits of no numerical comparison with -them. Hence we cannot properly state the proportion which the force -of a blow bears to the amount of a weight or resistance. The wedge is -almost invariably urged by percussion; while the resistances which it -has to overcome are as constantly forces of the other kind. Although, -however, no exact numerical comparison can be made, yet it may be -stated in a general way that the wedge is more and more powerful as its -angle is more acute. - -[Illustration: _C. Varley, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -In the arts and manufactures, wedges are used where enormous force -is to be exerted through a very small space. Thus it is resorted to -for splitting masses of timber or stone. Ships are raised in docks -by wedges driven under their keels. The wedge is the principal agent -in the oil-mill. The seeds from which the oil is to be extracted are -introduced into hair bags, and placed between planes of hard wood. -Wedges inserted between the bags are driven by allowing heavy beams to -fall on them. The pressure thus excited is so intense, that the seeds -in the bags are formed into a mass nearly as solid as wood. Instances -have occurred in which the wedge has been used to restore a tottering -edifice to its perpendicular position. - -All cutting and piercing instruments, such as knives, razors, scissors, -chisels, &c., nails, pins, needles, awls, &c. are wedges. The angle -of the wedge, in these cases, is more or less acute, according to -the purpose to which it is to be applied. In determining this, two -things are to be considered--the mechanical power, which is increased -by diminishing the angle of the wedge; and the strength of the tool, -which is always diminished by the same cause. There is, therefore, -a practical limit to the increase of the power, and that degree of -sharpness only is to be given to the tool which is consistent with the -strength requisite for the purpose to which it is to be applied. In -tools intended for cutting wood, the angle is generally about 30°. For -iron it is from 50° to 60°; and for brass, from 80° to 90°. Tools which -act by pressure may be made more acute than those which are driven by a -blow; and in general the softer and more yielding the substance to be -divided is, and the less the power required to act upon it, the more -acute the wedge may be constructed. - -In many cases the utility of the wedge depends on that which is -entirely omitted in its theory, viz. the friction which arises between -its surface and the substance which it divides. This is the case when -pins, bolts, or nails are used for binding the parts of structures -together; in which case, were it not for the friction, they would -recoil from their places, and fail to produce the desired effect. Even -when the wedge is used as a mechanical engine, the presence of friction -is absolutely indispensable to its practical utility. The power, as -has already been stated, generally acts by successive blows, and is -therefore subject to constant intermission, and but for the friction -the wedge would recoil between the intervals of the blows with as much -force as it had been driven forward. Thus the object of the labour -would be continually frustrated. The friction in this case is of the -same use as a ratchet wheel, but is much more necessary, as the power -applied to the wedge is more liable to intermission than in the cases -where ratchet wheels are generally used. - -(292.) When a road directly ascends the side of a hill, it is to be -considered as an inclined plane; but it will not lose its mechanical -character, if, instead of directly ascending towards the top of the -hill, it winds successively round it, and gradually ascends so as after -several revolutions to reach the top. In the same manner a path may be -conceived to surround a pillar by which the ascent may be facilitated -upon the principle of the inclined plane. Winding stairs constructed in -the interior of great columns partake of this character; for although -the ascent be produced by successive steps, yet if a floor could -be made sufficiently rough to prevent the feet from slipping, the -ascent would be accomplished with equal facility. In such a case the -winding path would be equivalent to an inclined plane, bent into such -a form as to accommodate it to the peculiar circumstances in which it -would be required to be used. It will not be difficult to trace the -resemblance between such an adaptation of the inclined plane and the -appearances presented by the thread of a _screw_: and it may hence be -easily understood that a screw is nothing more than an inclined plane -constructed upon the surface of a cylinder. - -This will, perhaps, be more apparent by the following contrivance: -Let A B, _fig. 135._, be a common round ruler, and let -C D E be a piece of white paper cut in the form of an -inclined plane, whose height C D is equal to the length of the -ruler A B, and let the edge C E of the paper be marked with -a broad black line: let the edge C D be applied to the ruler -A B, and being attached thereto, let the paper be rolled round -the ruler; the ruler will then present the appearance of a screw, -_fig. 136._ the thread of the screw being marked by the black -line C E, winding continually round the ruler. Let D F, -_fig. 135._, be equal to the circumference of the ruler, and draw -F G parallel to D C, and G H parallel to D E, the -part C G F D of the paper will exactly surround the -ruler once: the part C G will form one convolution of the thread, -and may be considered as the length of one inclined plane surrounding -the cylinder, C H being the corresponding height, and G H -the base. The power of the screw does not, as in the ordinary cases of -the inclined plane, act parallel to the plane or thread, but at right -angles to the length of the cylinder A B, or, what is to the same -effect, parallel to the base H G; therefore the proportion of the -power to the weight will be, according to principles already explained, -the same as that of C H to the space through which the power -moves parallel to H G in one revolution of the screw. H C is -evidently the distance between the successive positions of the thread -as it winds round the cylinder; and it appears from what has been just -stated, that the less this distance is, or, in other words, the finer -the thread is, the more powerful the machine will be. - -(293.) In the application of the screw the weight or resistance is -not, as in the inclined plane and wedge, placed upon the surface of -the plane or thread. The power is usually transmitted by causing the -screw to move in a concave cylinder, on the interior surface of which -a spiral cavity is cut, corresponding exactly to the thread of the -screw, and in which the thread will move by turning round the screw -continually in the same direction. This hollow cylinder is usually -called the _nut_ or _concave screw_. The screw surrounded by its -spiral thread is represented in _fig. 137._; and a section of the -same playing in the nut is represented in _fig. 138._ - -There are several ways in which the effect of the power may be conveyed -to the resistance by this apparatus. - -First, let us suppose that the nut A B is fixed. If the screw be -continually turned on its axis, by a lever E F inserted in one -end of it, it will be moved in the direction C D, advancing every -revolution through a space equal to the distance between two contiguous -threads. By turning the lever in an opposite direction, the screw will -be moved in the direction D C. - -If the screw be fixed, so as to be incapable either of moving -longitudinally or revolving on its axis, the nut A B may be turned -upon the screw by a lever, and will move on the screw towards C or -towards D, according to the direction in which the lever is turned. - -In the former case we have supposed the nut to be absolutely -immoveable, and in the latter case the screw to be absolutely -immoveable. It may happen, however, that the nut, though capable of -revolving, is incapable of moving longitudinally; and that the screw, -though incapable of revolving, is capable of moving longitudinally. In -that case, by turning the nut A B upon the screw by the lever, the -screw will be urged in the direction C D or D C, according to -the way in which the nut is turned. - -The apparatus may, on the contrary, be so arranged, that the nut, -though incapable of revolving, is capable of moving longitudinally; -and the screw, though capable of revolving, is incapable of moving -longitudinally. In this case, by turning the screw in the one direction -or in the other, the nut A B will be urged in the direction -C D or D C. - -All these various arrangements may be observed in different -applications to the machine. - -(294.) A screw may be cut upon a cylinder by placing the cylinder in -a turning lathe, and giving it a rotatory motion upon its axis. The -cutting point is then presented to the cylinder, and moved in the -direction of its length, at such a rate as to be carried through the -distance between the intended thread, while the cylinder revolves -once. The relative motions of the cutting point and the cylinder being -preserved with perfect uniformity, the thread will be cut from one end -to the other. The shape of the threads may be either square, as in -_fig. 137._, or triangular, as in _fig. 139._ - -(295.) The screw is generally used in cases where severe pressure -is to be excited through small spaces; it is therefore the agent in -most presses. In _fig. 140._, the nut is fixed, and by turning -the lever, which passes through the head of the screw, a pressure is -excited upon any substance placed upon the plate immediately under -the end of the screw. In _fig. 141._, the screw is incapable of -revolving, but is capable of advancing in the direction of its length. -On the other hand, the nut is capable of revolving, but does not -advance in the direction of the screw. When the nut is turned by means -of the screw inserted in it, the screw advances in the direction of its -length, and urges the board which is attached to it upwards, so as to -press any substance placed between it and the fixed board above. - -In cases where liquids or juices are to be expressed from solid bodies, -the screw is the agent generally employed. It is also used in coining, -where the impression of a die is to be made upon a piece of metal, and -in the same way in producing the impression of a seal upon wax or other -substance adapted to receive it. When soft and light materials, such -as cotton, are to be reduced to a convenient bulk for transportation, -the screw is used to compress them, and they are thus reduced into hard -dense masses. In printing, the paper is urged by a severe and sudden -pressure upon the types, by means of a screw. - -(296.) As the mechanical power of the screw depends upon the relative -magnitude of the circumference through which the power revolves, and -the distance between the threads, it is evident, that, to increase -the efficacy of the machine, we must either increase the length -of the lever by which the power acts, or diminish the magnitude of -the thread. Although there is no limit in theory to the increase of -the mechanical efficacy by these means, yet practical inconvenience -arises which effectually prevents that increase being carried beyond -a certain extent. If the lever by which the power acts be increased, -the same difficulty arises as was already explained in the wheel and -axle (254.); the space through which the power should act would be so -unwieldy, that its application would become impracticable. If, on the -other hand, the power of the machine be increased by diminishing the -size of the thread, the strength of the thread will be so diminished, -that a slight resistance will tear it from the cylinder. The cases -in which it is necessary to increase the power of the machine, being -those in which the greatest resistances are to be overcome, the object -will evidently be defeated, if the means chosen to increase that power -deprive the machine of the strength which is necessary to sustain the -force to which it is to be submitted. - -(297.) These inconveniences are removed by a contrivance of Mr. Hunter, -which, while it gives to the machine all the requisite strength and -compactness, allows it to have an almost unlimited degree of mechanical -efficacy. - -This contrivance consists in the use of two screws, the threads of -which may have any strength and magnitude, but which have a very small -difference of breadth. While the working point is urged forward by -that which has the greater thread, it is drawn back by that which has -the less; so that during each revolution of the screw, instead of -being advanced through a space equal to the magnitude of either of -the threads, it moves through a space equal to their difference. The -mechanical power of such a machine will be the same as that of a single -screw having a thread, whose magnitude is equal to the difference of -the magnitudes of the two threads just mentioned. - -Thus, without inconveniently increasing the sweep of the power, on the -one hand, or, on the other, diminishing the thread until the necessary -strength is lost, the machine will acquire an efficacy limited by -nothing but the smallness of the difference between the two threads. - -This principle was first applied in the manner represented in -_fig. 142._ A is the greater thread, playing in the fixed nut; B -is the lesser thread, cut upon a smaller cylinder, and playing in a -concave screw, cut within the greater cylinder. During every revolution -of the screw, the cylinder A descends through a space equal to the -distance between its threads. At the same time the smaller cylinder -B ascends through a space equal to the distance between the threads -cut upon it: the effect is, that the board D descends through a space -equal to the difference between the threads upon A and the threads upon -B, and the machine has a power proportionate to the smallness of this -difference. - -Thus, suppose the screw A has twenty threads in an inch, while the -screw B has twenty-one; during one revolution, the screw A will -descend through a space equal to the 20th part of an inch. If, during -this motion, the screw B did not turn within A, the board D would be -advanced through the 20th of an inch; but because the hollow screw -within A turns upon B, the screw B will, relatively to A, be raised in -one revolution through a space equal to the 21st part of an inch. Thus, -while the board D is depressed through the 20th of an inch by the screw -A, it is raised through the 21st of an inch by the screw B. It is, -therefore, on the whole, depressed through a space equal to the excess -of the 20th of an inch above the 21st of an inch, that is, through the -420th of an inch. - -The power of this machine will, therefore, be expressed by the number -of times the 420th of an inch is contained in the circumference through -which the power moves. - -(298.) In the practical application of this principle at present the -arrangement is somewhat different. The two threads are usually cut -on different parts of the same cylinder. If nuts be supposed to be -placed upon these, which are capable of moving in the direction of -the length, but not of revolving, it is evident that by turning the -screw once round, each nut will be advanced through a space equal to -the breadth of the respective threads. By this means the two nuts -will either approach each other, or mutually recede, according to the -direction in which the screw is turned, through a space equal to the -difference of the breadth of the threads, and they will exert a force -either in compressing or extending any substance placed between them, -proportionate to the smallness of that difference. - -(299.) A toothed wheel is sometimes used instead of a nut, so that -the same quality by which the revolution of the screw urges the nut -forward is applied to make the wheel revolve. The screw is in this -case called an endless screw, because its action upon the wheel may be -continued without limit. This application of the screw is represented -in _fig. 143._ P is the winch to which the power is applied; and -its effect at the circumference of the wheel is estimated in the same -manner as the effect of the screw upon the nut. This effect is to be -considered as a power acting upon the circumference of the wheel; and -its proportion to the weight or resistance is to be calculated in the -same manner as the proportion of the power to the weight in the wheel -and axle. - -(300.) We have hitherto considered the screw as an engine used to -overcome great resistances. It is also eminently useful in several -departments of experimental science, for the measurement of very -minute motions and spaces, the magnitude of which could scarcely be -ascertained by any other means. The very slow motion which may be -imparted to the end of a screw, by a very considerable motion in the -power, renders it peculiarly well adapted for this purpose. To explain -the manner in which it is applied--suppose a screw to be so cut as -to have fifty threads in an inch, each revolution of the screw will -advance its point through the fiftieth part of an inch. Now, suppose -the head of the screw to be a circle, whose diameter is an inch, the -circumference of the head will be something more than three inches: -this may be easily divided into a hundred equal parts distinctly -visible. If a fixed index be presented to this graduated circumference, -the hundredth part of a revolution of the screw may be observed, by -noting the passage of one division of the head under the index. Since -one entire revolution of the head moves the point through the fiftieth -of an inch, one division will correspond to the five thousandth of an -inch. In order to observe the motion of the point of the screw in this -case, a fine wire is attached to it, which is carried across the field -of view of a powerful microscope, by which the motion is so magnified -as to be distinctly perceptible. - -A screw used for such purposes is called a _micrometer screw_. Such an -apparatus is usually attached to the limbs of graduated instruments, -for the purposes of astronomical and other observation. Without the -aid of this apparatus, no observation could be taken with greater -accuracy than the amount of the smallest division upon the limb. Thus, -if an instrument for measuring angles were divided into small arcs of -one minute, and an angle were observed which brought the index of the -instrument to some point between two divisions, we could only conclude -that the observed angle must consist of a certain number of degrees and -minutes, together with an additional number of seconds, which would -be unknown, inasmuch as there would be no means of ascertaining the -fraction of a minute between the index and the adjacent division of -the instrument. But if a screw be provided, the point of which moves -through a space equal to one division of the instrument, with sixty -revolutions of the head, and that the head itself be divided into -one hundred equal parts, each complete revolution of the screw will -correspond to the sixtieth part of a minute, or to one second, and each -division on the head of the screw will correspond to the hundredth part -of a second. The index being attached to this screw, let the head -be turned until the index be moved from its observed position to the -adjacent division of the limb. The number of complete revolutions of -the screw necessary to accomplish this will be the number of seconds; -and the number of parts of a revolution over the complete number of -revolutions will be the hundredth parts of a second necessary to be -added to the degrees and minutes primarily observed. - -It is not, however, only to such instruments that the micrometer screw -is applicable; any spaces whatever may be measured by it. An instance -of its mechanical application may be mentioned in a steel-yard, -an instrument for ascertaining the amount of weights by a given -weight, sliding on a long graduated arm of a lever. The distance from -the fulcrum, at which this weight counterpoises the weight to be -ascertained, serves as a measure to the amount of that weight. When the -sliding weight happens to be placed between two divisions of the arm, a -micrometer screw is used to ascertain the fraction of the division. - -Hunter’s screw, already described, seems to be well adapted to -micrometrical purposes; since the motion of the point may be rendered -indefinitely slow, without requiring an exquisitely fine thread, such -as in the single screw would be necessary. - - - - -CHAP. XVII. - -ON THE REGULATION AND ACCUMULATION OF FORCE. - - -(301.) It is frequently indispensable, and always desirable, that the -operation of a machine should be regular and uniform. Sudden changes -in its velocity, and desultory variations in the effective energy -of its power, are often injurious or destructive to the apparatus -itself, and when applied to manufactures never fail to produce -unevenness in the work. To invent methods for insuring the regular -motion of machinery, by removing those causes of inequality which may -be avoided, and by compensating others, has therefore been a problem to -which much attention and ingenuity have been directed. This is chiefly -accomplished by controlling, and, as it were, measuring out the power -according to the exigencies of the machine, and causing its effective -energy to be always commensurate with the resistance which it has to -overcome. - -[Illustration: _C. Varley, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -Irregularity in the motion of machinery may proceed from one or more -of the following causes:--1. irregularity in the prime mover; 2. -occasional variation in the amount of the load or resistance; and, 3. -because, in the various positions which the parts of the machine assume -during its motion, the power may not be transmitted with equal effect -to the working point. - -The energy of the prime mover is seldom if ever regular. The force of -water varies with the copiousness of the stream. The power which impels -the windmill is proverbially capricious. The pressure of steam varies -with the intensity of the furnace. Animal power, the result of will, -temper, and health is difficult of control. Human labour is most of -all unmanageable; hence no machine works so irregularly as one which -is manipulated. In some cases the moving force is subject, by the very -conditions of its existence, to constant variation, as in the example -of a spring, which gradually loses its energy as it recoils. (255.) In -many instances the prime mover is liable to regular intermission, and -is actually suspended for certain intervals of time. This is the case -in the single acting steam-engine, where the pressure of the steam -urges the descent of the piston, but is suspended during its ascent. - -The load or resistance to which the machine is applied is not less -fluctuating. In mills there are a multiplicity of parts which are -severally liable to be occasionally disengaged, and to have their -operation suspended. In large factories for spinning, weaving, -printing, &c. a great number of separate spinning machines, looms, -presses, or other engines, are usually worked by one common mover, such -as a water-wheel or steam-engine. In these cases the number of machines -employed from time to time necessarily varies with the fluctuating -demand for the articles produced, and from other causes. Under such -circumstances the velocity with which every part of the machinery is -moved would suffer corresponding changes, increasing its rapidity with -every augmentation of the moving power or diminution of the resistance, -or being retarded in its speed by the contrary circumstances. - -But even when the prime mover and the resistance are both regular, or -rendered so by proper contrivances, still it will rarely happen that -the machine by which the energy of the one is transmitted to the other -conveys this with unimpaired effect in all the phases of its operation. -To give a general notion of this cause of inequality to those who have -not been familiar with machinery would not be easy, without having -recourse to an example. For the present we shall merely state, that the -several moving parts of every machine assume in succession a variety of -positions; that at regular periods they return to their first position, -and again undergo the same succession of changes. In the different -positions through which they are carried in every period of motion, -the efficacy of the machine to transmit the power to the resistance -is different, and thus the effective energy of the machine in acting -upon the resistance would be subject to continual fluctuation. This -will be more clearly understood when we come to explain the methods of -counteracting the defect or equalising the action of the power upon the -resistance. - -Such are the chief causes of the inequalities incidental to the -motion of machinery, and we now propose to describe a few of the many -ingenious contrivances which the skill of engineers has produced to -remove the consequent inconveniences. - -(302.) Setting aside, for the present, the last cause of inequality, -and considering the machinery, whatever it be, to transmit the power to -the resistance without irregular interruption, it is evident that every -contrivance, having for its object to render the velocity uniform, -can only accomplish this by causing the variations of the power and -resistance to be proportionate to each other. This may be done either -by increasing or diminishing the power as the resistance increases or -diminishes; or by increasing or diminishing the resistance as the power -increases or diminishes. - -According to the facilities or convenience presented by the peculiar -circumstances of the case either of these methods is adopted. - -The contrivances for effecting this are called _regulators_. Most -regulators act upon that part of the machine which commands the supply -of the power by means of levers, or some other mechanical contrivance, -so as to check the quantity of the moving principle conveyed to the -machine when the velocity has a tendency to increase; and, on the -other hand, to increase that supply upon any undue abatement of its -speed. In a water-mill this is done by acting upon the shuttle; in a -wind-mill, by an adjustment of the sail-cloth; and in a steam-engine, -by opening or closing, in a greater or less degree, the valve by which -the cylinder is supplied with steam. - -(303.) Of all the contrivances for regulating machinery, that which is -best known and most commonly used is the _governor_. This regulator, -which had been long in use in mill-work and other machinery, has of -late years attracted more general notice by its beautiful adaptation -in the steam-engines of Watt. It consists of heavy balls B B, -_fig. 144._, attached to the extremities of rods B F. -These rods play upon a joint at E, passing through a mortise in the -vertical stem D D′. At F they are united by joints to the short -rods F H, which are again connected by joints at H to a ring -which slides upon the vertical shaft D D′. From this description -it will be apparent that when the balls B are drawn from the axis, -their upper arms E F are caused to increase their divergence in -the same manner as the blades of a scissors are opened by separating -the handles. These, acting upon the ring by means of the short links -F H, draw it down the vertical axis from D towards E. A contrary -effect is produced when the balls B are brought closer to the axis, and -the divergence of the rods B E diminished. A horizontal wheel W is -attached to the vertical axis D D′, having a groove to receive a -rope or strap upon its rim. This strap passes round the wheel or axis -by which motion is transmitted to the machinery to be regulated, so -that the spindle or shaft D D′ will always be made to revolve with -a speed proportionate to that of the machinery. - -As the shaft D D′ revolves, the balls B are carried round it -with a circular motion, and consequently acquire a centrifugal force -which causes them to recede from the axle, and therefore to depress -the ring H. On the edge or rim of this ring is formed a groove, which -is embraced by the prongs of a fork I, at the extremity of one arm of -a lever whose fulcrum is at G. The extremity K of the other arm is -connected by some means with the part of the machine which supplies -the power. In the present instance we shall suppose it a steam-engine, -in which case the rod K I communicates with a flat circular valve -V, placed in the principal steam-pipe, and so arranged that, when K is -elevated as far as by their divergence the balls B have power over it, -the passage of the pipe will be closed by the valve V, and the passage -of steam entirely stopped; and, on the other hand, when the balls -subside to their lowest position, the valve will be presented with its -edge in the direction of the tube, so as to intercept no part of the -steam. - -The property which renders this instrument so admirably adapted to -the purpose to which it is applied is, that when the divergence of -the balls is not very considerable, they must always revolve with the -same velocity, whether they move at a greater or lesser distance from -the vertical axis. If any circumstance increases that velocity, the -balls instantly recede from the axis, and closing the valve V, check -the supply of steam, and thereby diminishing the speed of the motion, -restore the machine to its former rate. If, on the contrary, that -fixed velocity be diminished, the centrifugal force being no longer -sufficient to support the balls, they descend towards the axle, open -the valve V, and, increasing the supply of steam, restore the proper -velocity of the machine. - -When the governor is applied to a water-wheel it is made to act upon -the shuttle through which the water flows, and controls its quantity as -effectually, and upon the same principle, as has just been explained in -reference to the steam-engine. When applied to a windmill it regulates -the sail-cloth so as to diminish the efficacy of the power upon the -arms as the force of the wind increases, or _vice versâ_. - -In cases where the resistance admits of easy and convenient change, the -governor may act so as to accommodate it to the varying energy of the -power. This is often done in corn-mills, where it acts upon the shuttle -which metes out the corn to the millstones. When the power which drives -the mill increases, a proportionally increased feed of corn is given -to the stones, so that the resistance being varied in the ratio of the -power, the same velocity will be maintained. - -(304.) In some cases the centrifugal force of the revolving balls is -not sufficiently great to control the power or the resistance, and -regulators of a different kind must be resorted to. The following -contrivance is called the _water-regulator_:-- - -A common pump is worked by the machine, whose motion is to be -regulated, and water is thus raised and discharged into a cistern. -It is allowed to flow from this cistern through a pipe of a given -magnitude. When the water is pumped up with the same velocity as it is -discharged by this pipe, it is evident that the level of the water in -the cistern will be stationary, since it receives from the pump the -exact quantity which it discharges from the pipe. But if the pump -throw in more water in a given time than is discharged by the pipe, -the cistern will begin to be filled, and the level of the water will -rise. If, on the other hand, the supply from the pump be less than -the discharge from the pipe, the level of the water in the cistern -will subside. Since the rate at which water is supplied from the -pump will always be proportional to the velocity of the machine, it -follows that every fluctuation in this velocity will be indicated by -the rising or subsiding of the level of the water in the cistern, and -that level never can remain stationary, except at that exact velocity -which supplies the quantity of water discharged by the pipe. This pipe -may be constructed so as by an adjustment to discharge the water at -any required rate; and thus the cistern may be adapted to indicate a -constant velocity of any proposed amount. - -If the cistern were constantly watched by an attendant, the velocity of -the machine might be abated by regulating the power when the level of -the water is observed to rise, or increased when it falls; but this is -much more effectually and regularly performed by causing the surface -of the water itself to perform the duty. A float or large hollow metal -ball is placed upon the surface of the water in the cistern. This ball -is connected with a lever acting upon some part of the machinery, which -controls the power or regulates the amount of resistance, as already -explained in the case of the governor. When the level of the water -rises, the buoyancy of the ball causes it to rise also with a force -equal to the difference between its own weight and the weight of as -much water as it displaces. By enlarging the floating ball, a force may -be obtained sufficiently great to move those parts of the machinery -which act upon the power or resistance, and thus either to diminish -the supply of the moving principle or to increase the amount of the -resistance, and thereby retard the motion and reduce the velocity to -its proper limit. When the level of the water in the cistern falls, -the floating ball being no longer supported on the liquid surface, -descends with the force of its own weight, and producing an effect upon -the power or resistance contrary to the former, increases the effective -energy of the one, or diminishes that of the other, until the velocity -proper to the machine be restored. - -The sensibility of these regulators is increased by making the surface -of water in the cistern as small as possible; for then a small change -in the rate at which the water is supplied by the pump will produce a -considerable change in the level of the water in the cistern. - -Instead of using a float, the cistern itself may be suspended from -the lever which controls the supply of the power, and in this case a -sliding weight may be placed on the other arm, so that it will balance -the cistern when it contains that quantity of water which corresponds -to the fixed level already explained. If the quantity of water in -the cistern be increased by an undue velocity of the machine, the -weight of the cistern will preponderate, draw down the arm of the -lever, and check the supply of the power. If, on the other hand, the -supply of water be too small, the cistern will no longer balance the -counterpoise, the arm by which it is suspended will be raised, and the -energy of the power will be increased. - -(305.) In the steam-engine the self-regulating principle is carried -to an astonishing pitch of perfection. The machine itself raises in -due quantity the cold water necessary to condense the steam. It pumps -off the hot water produced by the steam, which has been cooled, and -lodges it in a reservoir for the supply of the boiler. It carries from -this reservoir exactly that quantity of water which is necessary to -supply the wants of the boiler, and lodges it therein according as it -is required. It breathes the boiler of redundant steam, and preserves -that which remains fit, both in quantity and quality, for the use of -the engine. It blows its own fire, maintaining its intensity, and -increasing or diminishing it according to the quantity of steam which -it is necessary to raise; so that when much work is expected from the -engine, the fire is proportionally brisk and vivid. It breaks and -prepares its own fuel, and scatters it upon the bars at proper times -and in due quantity. It opens and closes its several valves at the -proper moments, works its own pumps, turns its own wheels, and is only -not alive. Among so many beautiful examples of the self-regulating -principle, it is difficult to select. We shall, however, mention one or -two, and for others refer the reader to our treatise on this subject.[3] - -[3] Lardner on the Steam-Engine, Steam-Navigation, Roads, and Railways. -8th edition. 1851. - -It is necessary in this machine that the water in the boiler be -maintained constantly at the same level, and, therefore, that as -much be supplied, from time to time, as is consumed by evaporation. -A pump which is wrought by the engine itself supplies a cistern C, -_fig. 145._, with hot water. At the bottom of this cistern is a -valve V opening into a tube which descends into the boiler. This valve -is connected by a wire with the arm of a lever on the fulcrum D, the -other arm E of which is also connected by a wire with a stone float F, -which is partially immersed in the water of the boiler, and is balanced -by a sliding weight A. The weight A only counterpoises the stone float -F by the aid of its buoyance in the water; for if the water be removed, -the stone F will preponderate, and raise the weight A. When the water -in the boiler is at its proper level, the length of the wire connecting -the valve V with the lever is so adjusted that this valve shall be -closed, the wire at the same time being fully extended. When, by -evaporation, the water in the boiler begins to be diminished, the level -falls, and the stone weight F, being no longer supported, overcomes -the counterpoise A, raises the arm of the lever, and, pulling the -wire, opens the valve V. The water in the cistern C then flows through -the tube into the boiler, and continues to flow until the level be so -raised that the stone weight F is again elevated, the valve V closed, -and the further supply of water from the cistern C suspended. - -In order to render the operation of this apparatus easily -intelligible, we have here supposed an imperfection which does not -exist. According to what has just been stated, the level of the water -in the boiler descends from its proper height, and subsequently returns -to it. But, in fact, this does not happen. The float F and valve V -adjust themselves, so that a constant supply of water passes through -the valve, which proceeds exactly at the same rate as that at which the -water in the boiler is consumed. - -(306.) In the same machine there occurs a singularly happy example of -self-adjustment, in the method by which the strength of the fire is -regulated. The governor regulates the supply of steam to the engine, -and proportions it to the work to be done. With this work, therefore, -the demands upon the boiler increase or diminish, and with these -demands the production of steam in the boiler ought to vary. In fact, -the rate at which steam is generated in the boiler, ought to be equal -to that at which it is consumed in the engine, otherwise one of two -effects must ensue: either the boiler will fail to supply the engine -with steam, or steam will accumulate in the boiler, being produced in -undue quantity, and, escaping at the safety valve, will thus be wasted. -It is, therefore, necessary to control the agent which generates the -steam, namely, the fire, and to vary its intensity from time to time, -proportioning it to the demands of the engine. To accomplish this, -the following contrivance has been adopted:--Let T, _fig. 146._, -be a tube inserted in the top of the boiler, and descending nearly -to the bottom. The pressure of the steam confined in the boiler, -acting upon the surface of the water, forces it to a certain height -in the tube T. A weight F, half immersed in the water in the tube, is -suspended by a chain, which passes over the wheels P P′, and is -balanced by a metal plate D, in the same manner as the stone float, -_fig. 145._, is balanced by the weight A. The plate D passes -through the mouth of the flue E as it issues finally from the boiler; -so that when the plate D falls it stops the flue, suspending thereby -the draught of air through the furnace, mitigating the intensity of -the fire, and checking the production of steam. If, on the contrary, -the plate D be drawn up, the draught is increased, the fire is rendered -more active, and the production of steam in the boiler is stimulated. -Now, suppose that the boiler produces steam faster than the engine -consumes it, either because the load on the engine has been diminished, -and, therefore, its consumption of steam reduced, or because the -fire has become too intense; the consequence is, that the steam, -beginning to accumulate in the boiler, will press upon the surface of -the water with increased force, and the water will be raised in the -tube T. The weight F will, therefore, be lifted, and the plate D will -descend, diminish, or stop the draught, mitigate the fire, and retard -the production of steam, and will continue to do so until the rate -at which steam is produced shall be commensurate to the wants of the -engine. If, on the other hand, the production of steam be inadequate -to the exigency of the machine, either because of an increased load, -or of the insufficient force of the fire, the steam in the boiler will -lose its elasticity, and the surface of the water not sustaining its -wonted pressure, the water in the tube T will fall; consequently the -weight F will descend, and the plate D will be raised. The flue being -thus opened, the draught will be increased, and the fire rendered -more intense. Thus the production of steam becomes more rapid, and is -rendered sufficiently abundant for the purposes of the engine. This -apparatus is called the _self-acting damper_. - -(307.) When a perfectly uniform rate of motion has not been attained, -it is often necessary to indicate small variations of velocity. The -following contrivance, called a _tachometer_[4], has been invented -to accomplish this. A cup, _fig. 147._, is filled to the level -C D with quicksilver, and is attached to a spindle, which is -whirled by the machine in the same manner as the governor already -described. It is well known that the centrifugal force produced by this -whirling motion will cause the mercury to recede from the centre and -rise upon the sides of the cup, so that its surface will assume the -concave appearance represented in _fig. 148._ In this case the -centre of the surface will obviously have fallen below its original -level, _fig. 147._, and the edges will have risen above that -level. As this effect is produced by the velocity of the machine, so -it is proportionate to that velocity, and subject to corresponding -variations. Any method of rendering visible small changes in the -central level of the surface of the quicksilver will indicate minute -variations in the velocity of the machine. - -[4] From the Greek words _tachos_ speed, and _metron_ measure. - -A glass tube A, open at both ends, and expanding at one extremity into -a bell B, is immersed with its wider end in the mercury, the surface -of which will stand at the same level in the bell B, and in the cup -C D. The tube is so suspended as to be unconnected with the cup. -This tube is then filled to a certain height A, with spirits tinged -with some colouring matter, to render it easily observable. When the -cup is whirled by the machine to which it is attached, the level of -the quicksilver in the bell falls, leaving more space for the spirits, -which, therefore, descends in the tube. As the motion is continued, -every change of velocity causes a corresponding change in the level -of the mercury, and, therefore, also in the level A of the spirits. -It will be observed, that, in consequence of the capacity of the bell -B being much greater than that of the tube A, a very small change in -the level of the quicksilver in the bell will produce a considerable -change in the height of the spirits in the tube. Thus this ingenious -instrument becomes a very delicate indicator of variations in the -motion of machinery. - -(308.) The governor, and other methods of regulating the motion of -machinery which have been just described, are adapted principally to -cases in which the proportion of the resistance to the load is subject -to certain fluctuations or gradual changes, or at least to cases in -which the resistance is not at any time entirely withdrawn, nor the -energy of the power actually suspended. Circumstances, however, -frequently occur in which, while the power remains in full activity, -the resistance is at intervals suddenly removed and as suddenly again -returns. On the other hand, cases also present themselves, in which, -while the resistance is continued, the impelling power is subject to -intermission at regular periods. In the former case, the machine would -be driven with a ruinous rapidity during those periods at which it is -relieved from its load, and on the return of the load every part would -suffer a violent strain, from its endeavour to retain the velocity -which it had acquired, and the speedy destruction of the engine could -not fail to ensue. In the latter case, the motion would be greatly -retarded or entirely suspended during those periods at which the moving -power is deprived of its activity, and, consequently, the motion which -it would communicate would be so irregular as to be useless for the -purposes of manufactures. - -It is also frequently desirable, by means of a weak but continued -power, to produce a severe but instantaneous effect. Thus a blow may -be required to be given by the muscular action of a man’s arm with a -force to which, unaided by mechanical contrivance, its strength would -be entirely inadequate. - -In all these cases, it is evident that the object to be attained is, -an effectual method of accumulating the energy of the power so as -to make it available after the action by which it has been produced -has ceased. Thus, in the case in which the load is at periodical -intervals withdrawn from the machine, if the force of the power could -be imparted to something by which it would be preserved, so as to be -brought against the load when it again returned, the inconvenience -would be removed. In like manner, in the case where the power itself -is subject to intermission, if a part of the force which it exerts in -its intervals of action could be accumulated and preserved, it might -be brought to bear upon the machine during its periods of suspension. -By the same means of accumulating force, the strength of an infant, -by repeated efforts, might produce effects which would be vainly -attempted by the single and momentary action of the strongest man. - -(309.) The property of inertia, explained and illustrated in the third -and fourth chapters of this volume furnishes an easy and effectual -method of accomplishing this. A mass of matter retains, by virtue of -its inertia, the whole of any force which may have been given to it, -except that part of which friction and the atmospheric resistance -deprives it. By contrivances which are well known and present no -difficulty, the part of the moving force thus lost may be rendered -comparatively small, and the moving mass may be regarded as retaining -nearly the whole of the force impressed upon it. To render this method -of accumulating force fully intelligible, let us first imagine a -polished level plane on which a heavy globe of metal, also polished, is -placed. It is evident that the globe will remain at rest on any part of -the plane without a tendency to move in any direction. As the friction -is nearly removed by the polish of the surfaces, the globe will be -easily moved by the least force applied to it. Suppose a slight impulse -given to it, which will cause it to move at the rate of one foot in -a second. Setting aside the effects of friction, it will continue to -move at this rate for any length of time. The same impulse repeated -will increase its speed to two feet per second. A third impulse to -three feet, and so on. Thus 10,000 repetitions of the impulse will -cause it to move at the rate of 10,000 feet per second. If the body to -which these impulses were communicated were a cannon ball, it might, -by a constant repetition of the impelling force, be at length made to -move with as much force as if it were projected from the most powerful -piece of ordnance. The force with which the ball in such a case would -strike a building might be sufficient to reduce it to ruins, and yet -such force would be nothing more than the accumulation of a number -of weak efforts not beyond the power of a child to exert, which are -stored up, and preserved, as it were, by the moving mass, and thereby -brought to bear, at the same moment, upon the point to which the force -is directed. It is the sum of a number of actions exerted successively, -and, during a long interval, brought into operation at one and the same -moment. - -But the case which is here supposed cannot actually occur; because -we have not usually any practical means of moving a body for any -considerable time in the same direction without much friction, and -without encountering numerous obstacles which would impede its -progress. It is not, however, essential to the effect which is to be -produced, that the motion should be in a straight line. If a leaden -weight be attached to the end of a light rod or cord, and be whirled by -the force of the arm in a circle, it will gradually acquire increased -speed and force, and at length may receive an impetus which would -cause it to penetrate a piece of board as effectually as if it were -discharged from a musket. - -The force of a hammer or sledge depends partly on its weight, but much -more on the principle just explained. Were it allowed merely to fall -by the force of its weight upon the head of a nail, or upon a bar of -heated iron which is to be flattened, an inconsiderable effect would be -produced. But when it is wielded by the arm of a man, it receives at -every moment of its motion increased force, which is finally expended -in a single instant on the head of the nail, or on the bar of iron. - -The effects of flails in threshing, of clubs, whips, canes, and -instruments for striking, axes, hatchets, cleavers, and all instruments -which cut by a blow, depend on the same principle, and are similarly -explained. - -The bow-string which impels the arrow does not produce its effect at -once. It continues to act upon the shaft until it resumes its straight -position, and then the arrow takes flight with the force accumulated -during the continuance of the action of the string, from the moment it -was disengaged from the finger of the bow-man. - -Fire-arms themselves act upon a similar principle, as also the air-gun -and steam-gun. In these instruments the ball is placed in a tube, and -suddenly exposed to the pressure of a highly elastic fluid, either -produced by explosion as in fire-arms, by previous condensation as in -the air-gun, or by the evaporation of highly heated liquids as in the -steam-gun. But in every case this pressure continues to act upon it -until it leaves the mouth of the tube, and then it departs with the -whole force communicated to it during its passage along the tube. - -(310.) From all these considerations it will easily be perceived that -a mass of inert matter may be regarded as a magazine in which force -may be deposited and accumulated, to be used in any way which may be -necessary. For many reasons, which will be sufficiently obvious, the -form commonly given to the mass of matter used for this purpose in -machinery is that of a wheel, in the rim of which it is principally -collected. Conceive a massive ring of metal, _fig. 149._, -connected with a central box or nave by light spokes, and turning on -an axis with little friction. Such an apparatus is called a fly-wheel. -If any force be applied to it, with that force (making some slight -deduction for friction) it will move, and will continue to move until -some obstacle be opposed to its motion, which will receive from it a -part of the force it has acquired. The uses of this apparatus will be -easily understood by examples of its application. - -Suppose that a heavy stamper or hammer is to be raised to a certain -height, and thence to be allowed to fall, and that the power used -for this purpose is a water-wheel. While the stamper ascends, the -power of the wheel is nearly balanced by its weight, and the motion -of the machine is slow. But the moment the stamper is disengaged and -allowed to fall, the power of the wheel, having no resistance, nor any -object on which to expend itself, suddenly accelerates the machine, -which moves with a speed proportioned to the amount of the power, -until it again engages the stamper, when its velocity is as suddenly -checked. Every part suffers a strain, and the machine moves again -slowly until it discharges its load, when it is again accelerated, -and so on. In this case, besides the certainty of injury and wear, and -the probability of fracture from the sudden and frequent changes of -velocity, nearly the whole force exerted by the power in the intervals -between the commencement of each descent of the stamper and the next -ascent is lost. These defects are removed by a fly-wheel. When the -stamper is discharged, the energy of the power is expended in moving -the wheel, which, by reason of its great mass, will not receive an -undue velocity. In the interval between the descent and ascent of the -stamper, the force of the power is lodged in the heavy rim of the -fly-wheel. When the stamper is again taken up by the machine, this -force is brought to bear upon it, combined with the immediate power -of the water-wheel, and the stamper is elevated with nearly the same -velocity as that with which the machine moved in the interval of its -descent. - -(311.) In many cases, when the moving power is not subject to -variation, the efficacy of the machine to transmit it to the working -point is subject to continual change. The several parts of every -machine have certain periods of motion, in which they pass through a -variety of positions, to which they continually return after stated -intervals. In these different positions the effect of the power -transmitted to the working point is different; and cases even occur -in which this effect is altogether annihilated, and the machine is -brought into a predicament in which the power loses all influence -over the weight. In such cases the aid of a fly-wheel is effectual -and indispensable. In those phases of the machine, which are most -favourable to the transmission of force, the fly-wheel shares the -effect of the power with the load, and retaining the force thus -received directs it upon the load at the moments when the transmission -of power by the machine is either feeble or altogether suspended. These -general observations will, perhaps, be more clearly apprehended by an -example of an application of the fly-wheel, in a case such as those now -alluded to. - -Let A B C D E F, _fig. 150._, be a -_crank_, which is a double winch ((252.) and _fig. 89._), by -which an axle, A B E F, is to be turned. Attached -to the middle of C D by a joint is a rod, which is connected -with a beam, worked with an alternate motion on a centre, like -the brake of a pump, and driven by any constant power, such as a -steam-engine. The bar C D is to be carried with a circular motion -round the axis A E. Let the machine, viewed in the direction -A B E F of the axis, be conceived to be represented -in _fig. 151._, where A represents the centre round which the -motion is to be produced, and G the point where the connecting rod -G H is attached to the arm of the crank. The circle through which -G is to be urged by the rod is represented by the dotted line. In -the position represented in _fig. 151._, the rod acting in the -direction H G has its full power to turn the crank G A round -the centre A. As the crank comes into the position represented in -_fig. 152._, this power is diminished, and when the point G comes -immediately below A, as in _fig. 153._, the force in the direction -H G has no effect in turning the crank round A, but, on the -contrary, is entirely expended in pulling the crank in the direction -A G, and, therefore, only acts upon the pivots or gudgeons which -support the axle. At this crisis of the motion, therefore, the whole -effective energy of the power is annihilated. - -After the crank has passed to the position represented in -_fig. 154._, the direction of the force which acts upon the -connecting rod is changed, and now the crank is drawn upward in the -direction G H. In this position the moving force has some efficacy -to produce rotation round A, which efficacy continually increases -until the crank attains the position shown in _fig. 155._, when -its power is greatest. Passing from this position its efficacy is -continually diminished, until the point G comes immediately above the -axis A, _fig. 156._ Here again the power loses all its efficacy -to turn the axle. The force in the direction G H or H G can -obviously produce no other effect than a strain upon the pivots or -gudgeons. - -In the critical situations represented in _fig. 153._, and -_fig. 156._, the machine would be incapable of moving, were -the immediate force of the power the only impelling principle. But -having been previously in motion by virtue of the inertia of its -various parts, it has a tendency to continue in motion; and if the -resistance of the load and the effects of friction be not too great, -this disposition to preserve its state of motion will extricate the -machine from the dilemma in which it is involved in the cases just -mentioned, by the peculiar arrangement of its parts. In many cases, -however, the force thus acquired during the phases of the machine, in -which the power is active, is insufficient to carry it through the -dead points (_fig. 153._ and _fig. 156._); and in all cases -the motion would be very unequal, being continually retarded as it -approached these points, and continually accelerated after it passed -them. A fly-wheel attached to the axis A, or to some other part of the -machinery, will effectually remove this defect. When the crank assumes -the positions in _fig. 151._ and _fig. 155._, the power is -in full play upon it, and a share of the effect is imparted to the -massive rim of the fly-wheel. When the crank gets into the predicament -exhibited in _fig. 153._ and _fig. 156._, the momentum which -the fly-wheel received when the crank acted with most advantage, -immediately extricates the machine, and, carrying the crank beyond the -dead point, brings the power again to bear upon it. - -The astonishing effects of a fly-wheel, as an accumulator of force, -have led some into the error of supposing that such an apparatus -increases the actual power of a machine. It is hoped, however, that -after what has been explained respecting the inertia of matter and the -true effects of machines, the reader will not be liable to a similar -mistake. On the contrary, as a fly cannot act without friction, and as -the amount of the friction, like that of inertia, is in proportion to -the weight, a portion of the actual moving force must unavoidably be -lost by the use of a fly. In cases, however, where a fly is properly -applied this loss of power is inconsiderable, compared with the -advantageous distribution of what remains. - -[Illustration: _C. Varley, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -As an accumulator of force, a fly can never have more force than has -been applied to put it in motion. In this respect it is analogous to -an elastic spring, or the force of condensed air, or any other power -which derives its existence from causes purely mechanical. In bending -a spring a gradual expenditure of power is necessary. On the recoil -this power is exerted in a much shorter time than that consumed in its -production, but its total amount is not altered. Air is condensed by a -succession of manual efforts, one of which alone would be incapable of -projecting a leaden ball with any considerable force, and all of which -could not be immediately applied to the ball at the same instant. But -the reservoir of condensed air is a magazine in which a great number of -such efforts are stored up, so as to be brought at once into action. -If a ball be exposed to their effect, it may be projected with a -destructive force. - -In mills for rolling metal the fly-wheel is used in this way. The -water-wheel or other moving power is allowed for some time to act upon -the fly-wheel alone, no load being placed upon the machine. A force -is thus gained which is sufficient to roll a large piece of metal, to -which without such means the mill would be quite inadequate. In the -same manner a force may be gained by the arm of a man acting on a fly -for a few seconds, sufficient to impress an image on a piece of metal -by an instantaneous stroke. The fly is, therefore, the principal agent -in coining presses. - -(312.) The power of a fly is often transmitted to the working point -by means of a screw. At the extremities of the cross arm A B, -_fig. 157._, which works the screw, two heavy balls of metal are -placed. When the arm A B is whirled round, those masses of metal -acquire a momentum, by which the screw, being driven downward, urges -the die with an immense force against the substance destined to receive -the impression. - -Some engines used in coining have flies with arms four feet long, -bearing one hundred weight at each of their extremities. By turning -such an arm at the rate of one entire circumference in a second, the -die will be driven against the metal with the same force as that with -which 7500 pounds weight would fall from the height of 16 feet; an -enormous power, if the simplicity and compactness of the machine be -considered. - -The place to be assigned to a fly-wheel relatively to the other parts -of the machinery is determined by the purpose for which it is used. If -it be intended to equalise the action, it should be near the working -point. Thus, in a steam-engine, it is placed on the crank which turns -the axle by which the power of the engine is transmitted to the object -it is finally designed to affect. On the contrary, in handmills, such -as those commonly used for grinding coffee, &c., it is placed upon the -axis of the winch by which the machine is worked. - -The open work of fenders, fire-grates, and similar ornamental articles -constructed in metal, is produced by the action of a fly, in the -manner already described. The cutting tool, shaped according to the -pattern to be executed, is attached to the end of the screw; and the -metal being held in a proper position beneath it, the fly is made to -urge the tool downwards with such force as to stamp out pieces of the -required figure. When the pattern is complicated, and it is necessary -to preserve with exactness the relative situation of its different -parts, a number of punches are impelled together, so as to strike the -entire piece of metal at the same instant, and in this manner the most -elaborate open work is executed by a single stroke. - - - - -CHAP. XVIII. - -MECHANICAL CONTRIVANCES FOR MODIFYING MOTION. - - -(313.) The classes of simple machines denominated mechanic powers, -have relation chiefly to the peculiar principle which determines the -action of the power on the weight or resistance. In explaining this -arrangement various other reflections have been incidentally mixed up -with our investigations; yet still much remains to be unfolded before -the student can form a just notion of those means by which the complex -machinery used in the arts and manufactures so effectually attains the -ends, to the accomplishment of which it is directed. - -By a power of a given energy to oppose a resistance of a different -energy, or by a moving principle having a given velocity to generate -another velocity of a different amount, is only one of the many objects -to be effected by a machine. In the arts and manufactures the _kind_ -of motion produced is generally of greater importance than its _rate_. -The latter may affect the quantity of work done in a given time, but -the former is essential to the performance of the work in any quantity -whatever. In the practical application of machines, the object to be -attained is generally to communicate to the working point some peculiar -sort of motion suitable to the uses for which the machine is intended; -but it rarely happens that the moving power has this sort of motion. -Hence, the machine must be so contrived that, while that part on which -this power acts is capable of moving in obedience to it, its connection -with the other parts shall be such that the working point may receive -that motion which is necessary for the purposes to which the machine is -applied. - -To give a perfect solution of this problem it would be necessary to -explain, first, all the varieties of moving powers which are at -our disposal; secondly, all the variety of motions which it may be -necessary to produce; and, thirdly, to show all the methods by which -each variety of prime mover may be made to produce the several species -of motion in the working point. It is obvious that such an enumeration -would be impracticable, and even an approximation to it would be -unsuitable to the present treatise. Nevertheless, so much ingenuity has -been displayed in many of the contrivances for modifying motion, and an -acquaintance with some of them is so essential to a clear comprehension -of the nature and operation of complex machines, that it would be -improper to omit some account of those at least which most frequently -occur in machinery, or which are most conspicuous for elegance and -simplicity. - -The varieties of motion which most commonly present themselves in the -practical application of mechanics may be divided into _rectilinear_ -and _rotatory_. In rectilinear motion the several parts of the -moving body proceed in parallel straight lines with the same speed. -In rotatory motion the several points revolve round an axis, each -performing a complete circle, or similar parts of a circle, in the same -time. - -Each of these may again be resolved into continued and reciprocating. -In a continued motion, whether rectilinear or rotatory, the parts move -constantly in the same direction, whether that be in parallel straight -lines, or in rotation on an axis. In reciprocating motion the several -parts move alternately in opposite directions, tracing the same spaces -from end to end continually. Thus, there are four principal species of -motion which more frequently than any others act upon, or are required -to be transmitted by, machines:-- - -1. _Continued rectilinear motion._ - -2. _Reciprocating rectilinear motion._ - -3. _Continued circular motion._ - -4. _Reciprocating circular motion._ - -These will be more clearly understood by examples of each kind. - -Continued rectilinear motion is observed in the flowing of a river, in -a fall of water, in the blowing of the wind, in the motion of an animal -upon a straight road, in the perpendicular fall of a heavy body, in the -motion of a body down an inclined plane. - -Reciprocating rectilinear motion is seen in the piston of a common -syringe, in the rod of a common pump, in the hammer of a pavier, the -piston of a steam-engine, the stampers of a fulling mill. - -Continued circular motion is exhibited in all kinds of wheel-work, and -is so common, that to particularise it is needless. - -Reciprocating circular motion is seen in the pendulum of a clock, and -in the balance-wheel of a watch. - -We shall now explain some of the contrivances by which a power having -one of these motions may be made to communicate either the same species -of motion changed in its velocity or direction, or any of the other -three kinds of motion. - -(314.) By a continued rectilinear motion another continued rectilinear -motion in a different direction may be produced, by one or more fixed -pulleys. A cord passed over these, one end of it being moved by the -power, will transmit the same motion unchanged to the other end. If the -directions of the two motions cross each other, one fixed pulley will -be sufficient; see _fig. 113._, where the hand takes the direction -of the one motion, and the weight that of the other. In this case the -pulley must be placed in the angle at which the directions of the two -motions cross each other. If this angle be distant from the places at -which the objects in motion are situate, an inconvenient length of rope -may be necessary. In this case the same may be effected by the use of -two pulleys, as in _fig. 158._ - -If the directions of the two motions be parallel, two pulleys must -be used as in _fig. 158._, where P′ A′ is one motion, and -B W the other. In these cases the axles of the two wheels are -parallel. - -It may so happen that the directions of the two motions neither cross -each other nor are parallel. This would happen, for example, if the -direction of one were upon the paper in the line P A, while the -other were perpendicular to the paper from the point O. In this case -two pulleys should be used, the axle of one O′ being perpendicular to -the paper, while the axle of the other O should be on the paper. This -will be evident by a little reflection. - -In general, the axle of each pulley must be perpendicular to the -two directions in which the rope passes from its groove; and by due -attention to this condition it will be perceived, that a continued -rectilinear motion may be transferred from any one direction to any -other direction, by means of a cord and two pulleys, without changing -its velocity. - -If it be necessary to change the velocity, any of the systems of -pulleys described in chap. XV. may be used in addition to the fixed -pulleys. - -By the wheel and axle any one continued rectilinear motion may be -made to produce another in any other direction, and with any other -velocity. It has been already explained (250.) that the proportion of -the velocity of the power to that of the weight is as the diameter of -the wheel to the diameter of the axle. The thickness of the axle being -therefore regulated in relation to the size of the wheel, so that -their diameters shall have that proportion which subsists between the -proposed velocities, one condition of the problem will be fulfilled. -The rope coiled upon the axle may be carried, by means of one or more -fixed pulleys, into the direction of one of the proposed motions, while -that which surrounds the wheel is carried into the direction of the -other by similar means. - -(315.) By the wheel and axle a continued rectilinear motion may be made -to produce a continued rotatory motion, or _vice versâ_. If the power -be applied by a rope coiled upon the wheel, the continued motion of -the power in a straight line will cause the machine to have a rotatory -motion. Again, if the weight be applied by a rope coiled upon the -axle, a power having a rotatory motion applied to the wheel will cause -the continued ascent of the weight in a straight line. - -Continued rectilinear and rotatory motions may be made to produce each -other, by causing a toothed wheel to work in a straight bar, called a -_rack_, carrying teeth upon its edge. Such an apparatus is represented -in _fig. 159._ - -In some cases the teeth of the wheel work in the links of a chain. The -wheel is then called a _rag-wheel_, _fig. 160._ - -Straps, bands, or ropes, may communicate rotation to a wheel, by their -friction in a groove upon its edge. - -A continued rectilinear motion is produced by a continued circular -motion in the case of a screw. The lever which turns the screw has -a continued circular motion, while the screw itself advances with a -continued rectilinear motion. - -The continued rectilinear motion of a stream of water acting upon a -wheel produces continued circular motion in the wheel, _fig. 93_, -_94_, _95_. In like manner the continued rectilinear motion of the wind -produces a continued circular motion in the arms of a windmill. - -Cranes for raising and lowering heavy weights convert a circular motion -of the power into a continued rectilinear motion of the weight. - -(316.) Continued circular motion may produce reciprocating rectilinear -motion, by a great variety of ingenious contrivances. - -Reciprocating rectilinear motion is used when heavy stampers are to -be raised to a certain height, and allowed to fall upon some object -placed beneath them. This may be accomplished by a wheel bearing on -its edge curved teeth, called _wipers_. The stamper is furnished with -a projecting arm or peg, beneath which the wipers are successively -brought by the revolution of the wheel. As the wheel revolves the -wiper raises the stamper, until its extremity passes the extremity of -the projecting arm of the stamper, when the latter immediately falls -by its own weight. It is then taken up by the next wiper, and so the -process is continued. - -A similar effect is produced if the wheel be partially furnished with -teeth, and the stamper carry a rack in which these teeth work. Such an -apparatus is represented in _fig. 161._ - -It is sometimes necessary that the reciprocating rectilinear motion -shall be performed at a certain varying rate in both directions. This -may be accomplished by the machine represented in _fig. 162._ -A wheel upon the axle C turns uniformly in the direction -A B D E. A rod _mn_ moves in guides, which only permit -it to ascend and descend perpendicularly. Its extremity _m_ rests -upon a path or groove raised from the face of the wheel, and shaped -into such a curve that as the wheel revolves the rod _mn_ shall be -moved alternately in opposite directions through the guides, with the -required velocity. The manner in which the velocity varies will depend -on the form given to the groove or channel raised upon the face of the -wheel, and this may be shaped so as to give any variation to the motion -of the rod _mn_ which may be required for the purpose to which it is to -be applied. - -The _rose-engine_ in the turning-lathe is constructed on this -principle. It is also used in spinning machinery. - -It is often necessary that the rod to which reciprocating motion is -communicated shall be urged by the same force in both directions. A -wheel partially furnished with teeth, acting on two racks placed on -different sides of it, and both connected with the bar or rod to which -the reciprocating motion is to be communicated, will accomplish this. -Such an apparatus is represented in _fig. 163._, and needs no -further explanation. - -Another contrivance for the same purpose is shown in _fig. 164._, -where A is a wheel turned by a winch H, and connected with a rod or -beam moving in guides by the joint _ab_. As the wheel A is turned -by the winch H the beam is moved between the guides alternately -in opposite directions, the extent of its range being governed by -the length of the diameter of the wheel. Such an apparatus is used -for grinding and polishing plane surfaces, and also occurs in silk -machinery. - -An apparatus applied by M. Zureda in a machine for pricking holes in -leather is represented in _fig. 165._ The wheel A B has its -circumference formed into teeth, the shape of which may be varied -according to the circumstances under which it is to be applied. One -extremity of the rod _ab_ rests upon the teeth of the wheel upon which -it is pressed by a spring at the other extremity. When the wheel -revolves, it communicates to this rod a reciprocating rectilinear -motion. - -Leupold has applied this mechanism to move the pistons of pumps.[5] -Upon the vertical axis of a horizontal hydraulic wheel is fixed another -horizontal wheel, which is furnished with seven teeth in the manner -of a crown wheel (263.). These teeth are shaped like inclined planes, -the intervals between them being equal to the length of the planes. -Projecting arms attached to the piston rods rest upon the crown of this -wheel; and, as it revolves, the inclined surfaces of the teeth, being -forced under the arm, raise the rod upon the principle of the wedge. -To diminish the obstruction arising from friction, the projecting arms -of the piston rods are provided with rollers, which run upon the teeth -of the wheel. In one revolution of the wheel each piston makes as many -ascents and descents as there are teeth. - -[5] Theatrum Machinarum, tom. ii. pl. 36. fig. 3. - -(317.) Wheel-work furnishes numerous examples of continued circular -motion round one axis, producing continued circular motion round -another. If the axles be in parallel directions, and not too distant, -rotation may be transmitted from one to the other by two spur-wheels -(263.); and the relative velocities may be determined by giving a -corresponding proportion to the diameter of the wheels. - -If a rotary motion is to be communicated from one axis to another -parallel to it, and at any considerable distance, it cannot in -practice be accomplished by wheels alone, for their diameters would -be too large. In this case a strap or chain is carried round the -circumferences of both wheels. If they are intended to turn in the same -direction, the strap is arranged as in _fig. 100._; but if in -contrary directions it is crossed, as in _fig. 101._ In this case, -as with toothed wheels, the relative velocities are determined by the -proportion of the diameters of the wheels. - -If the axles be distant and not parallel, the cord, by which the motion -is transmitted, must be passed over grooved wheels, or fixed pulleys, -properly placed between the two axles. - -It may happen that the strain upon the wheel, to which the motion is to -be transmitted, is too great to allow of a strap or cord being used. In -this case, a shaft extending from the one axis to another, and carrying -two bevelled wheels (263.), will accomplish the object. One of these -bevelled wheels is placed upon the shaft near to, and in connection -with, the wheel from which the motion is to be taken, and the other at -a part of it near to, and in connection with, that wheel to which the -motion is to be conveyed, _fig. 166._ - -The methods of transmitting rotation from one axis to another -perpendicular to it, by crown and by bevelled wheels, have been -explained in (263.). - -The endless screw (299.) is a machine by which a rotatory motion round -one axis may communicate a rotatory motion round another perpendicular -to it. The power revolves round an axis coinciding with the length of -the screw, and the axis of the wheel driven by the screw is at right -angles to this. - -The axis to which rotation is to be given, or from which it is to -be taken, is sometimes variable in its position. In such cases, an -ingenious contrivance, called a _universal joint_, invented by the -celebrated Dr. Hooke, may be used. The two shafts or axles A B, -_fig. 167._, between which the motion is to be communicated, -terminate in semicircles, the diameters of which, C D and -E F, are fixed in the form of a cross, their extremities moving -freely in bushes placed in the extremities of the semicircles. -Thus, while the central cross remains unmoved, the shaft A and its -semicircular end may revolve round C D as an axis; and the shaft -B and its semicircular end may revolve round E F as an axis. If -the shaft A be made to revolve without changing its direction, the -points C D will move in a circle whose centre is at the middle of -the cross. The motion thus given to the cross will cause the points -E F to move in another circle round the same centre, and hence the -shaft B will be made to revolve. - -[Illustration: _C. Varley, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -This instrument will not transmit the motion if the angle under the -directions of the shafts be less than 140°. In this case a double -joint, as represented in _fig. 168._, will answer the purpose. -This consists of four semicircles united by two crosses, and its -principle and operation is the same as in the last case. - -Universal joints are of great use in adjusting the position of large -telescopes, where, while the observer continues to look through the -tube, it is necessary to turn endless screws or wheels whose axes are -not in an accessible position. - -The cross is not indispensably necessary in the universal joint. A -hoop, with four pins projecting from it at four points equally distant -from each other, or dividing the circle of the hoop into four equal -arcs, will answer the purpose. These pins play in the bushes of the -semicircles in the same manner as those of the cross. - -The universal joint is much used in cotton-mills, where shafts are -carried to a considerable distance from the prime mover, and great -advantage is gained by dividing them into convenient lengths, connected -by a joint of this kind. - -(318.) In the practical application of machinery, it is often necessary -to connect a part having a continued circular motion with another which -has a reciprocating or alternate motion, so that either may move the -other. There are many contrivances by which this may be effected. - -One of the most remarkable examples of it is presented in the -scapements of watches and clocks. In this case, however, it can -scarcely be said with strict propriety that it is the rotation of -the scapement-wheel (266.) which _communicates_ the vibration to the -balance-wheel or pendulum. That vibration is produced in the one case -by the peculiar nature of the spiral spring fixed upon the axis of the -balance-wheel, and in the other case by the gravity of the pendulum. -The force of the scapement-wheel only _maintains_ the vibration, -and prevents its decay by friction and atmospheric resistance. -Nevertheless, between the two parts thus moving there exists a -mechanical connection, which is generally brought within the class of -contrivances now under consideration. - -A beam vibrating on an axis, and driven by the piston of a -steam-engine, or any other power, may communicate rotary motion to -an axis by a connector and a crank. This apparatus has been already -described in (311.). Every steam-engine which works by a beam affords -an example of this. The working beam is generally placed over the -engine, the piston rod being attached to one end of it, while the -connecting rod unites the other end with the crank. In boat-engines, -however, this position would be inconvenient, requiring more room than -could easily be spared. The piston rod, in these cases, is, therefore, -connected with the end of the beam by long rods, and the beam is placed -beside and below the engine. The use of a fly-wheel here would also -be objectionable. The effect of the dead points explained in (311.) -is avoided without the aid of a fly, by placing two cranks upon the -revolving axle, and working them by two pistons. The cranks are so -placed that when either is at its dead point, the other is in its most -favourable position. - -A wheel A, _fig. 169._, armed with wipers, acting upon a -sledge-hammer B, fixed upon a centre or axle C, will, by a continued -rotatory motion, give the hammer the reciprocating motion necessary -for the purposes to which it is applied. The manner in which this acts -must be evident on inspecting the figure. - -The treddle of the lathe furnishes an obvious example of a vibrating -circular motion producing a continued circular one. The treddle acts -upon a crank, which gives motion to the principal wheel, in the same -manner as already described in reference to the working beam and crank -in the steam-engine. - -By the following ingenious mechanism an alternate or vibrating force -may be made to communicate a circular motion continually in the same -direction. Let A B, _fig. 170._, be an axis receiving an -alternate motion from some force applied to it, such as a swinging -weight. Two ratchet wheels (253.) _m_ and _n_ are fixed on this axle, -their teeth being inclined in opposite directions. Two toothed wheels C -and D are likewise placed upon it, but so arranged that they turn upon -the axle with a little friction. These wheels carry two catches _p_, -_q_, which fall into the teeth of the ratchet wheels _m_, _n_, but fall -on opposite sides conformably to the inclination of the teeth already -mentioned. The effect of these catches is, that if the axis be made -to revolve in one direction, one of the two toothed wheels is always -compelled (by the catch _against_ which the motion is directed) to -revolve with it, while the other is permitted to remain stationary in -obedience to any force sufficiently great to overcome its friction with -the axle on which it is placed. The wheels C and D are both engaged by -bevelled teeth (263.) with the wheel E. - -According to this arrangement, in whichever direction the axis -A B is made to revolve, the wheel E will continually turn in -the same direction, and, therefore, if the axle A B be made to -turn alternately in the one direction and the other, the wheel E will -not change the direction of its motion. Let us suppose that the axle -A B is turned against the catch _p_. The wheel C will then be -made to turn with the axle. This will drive the wheel E in the same -direction. The teeth on the opposite side of the wheel E being engaged -with those of the wheel D, the latter will be turned upon the axle, -the friction, which alone resists its motion in that direction, being -overcome. Let the motion of the axle A B be now reversed. Since -the teeth of the ratchet wheel _n_ are moved against the catch _q_, the -wheel D will be compelled to revolve with the axle. The wheel E will be -driven in the same direction as before, and the wheel C will be moved -on the axle A B, and in a contrary direction to the motion of the -axle, the friction being overcome by the force of the wheel E. Thus, -while the axle A B is turned alternately in the one direction and -the other, the wheel E is constantly moved in the same direction. - -It is evident that the direction in which the wheel E moves may be -reversed by changing the position of the ratchet wheels and catches. - -(319.) It is often necessary to communicate an alternate circular -motion, like that of a pendulum, by means of an alternate motion in -a straight line. A remarkable instance of this occurs in the steam -engine. The moving force in this machine is the pressure of steam, -which impels a piston from end to end alternately in a cylinder. The -force of this piston is communicated to the working beam by a strong -rod, which passes through a collar in one end of the piston. Since it -is necessary that the steam included in the cylinder should not escape -between the piston rod and the collar through which it moves, and yet, -that it should move as freely and be subject to as little resistance as -possible, the rod is turned so as to be truly cylindrical, and is well -polished. It is evident that, under these circumstances, it must not be -subject to any lateral or cross strain, which would bend it towards one -side or the other of the cylinder. But the end of the beam to which it -communicates motion, if connected immediately with the rod by a joint, -would draw it alternately to the one side and the other, since it moves -in the arc of a circle, the centre of which is at the centre of the -beam. It is necessary, therefore, to contrive some method of connecting -the rod and the end of the beam, so that while the one shall ascend -and descend in a straight line, the other may move in the circular arc. - -The method which first suggests itself to accomplish this is, to -construct an arch-head upon the end of the beam, as in _fig. 171._ -Let C be the centre on which the beam works, and let B D be an -arch attached to the end of the beam, being a part of a circle having C -for its centre. To the highest point B of the arch a chain is attached, -which is carried upon the face of the arch B A, and the other end -of which is attached to the piston rod. Under these circumstances it is -evident, that when the force of the steam impels the piston downwards, -the chain P A B will draw the end of the beam down, and will, -therefore, elevate the other end. - -When the steam-engine is used for certain purposes, such as pumping, -this arrangement is sufficient. The piston in that case is not forced -upwards by the pressure of steam. During its ascent it is not subject -to the action of any force of steam, and the other end of the beam -falls by the weight of the pump-rods drawing the piston, at the -opposite end A, to the top of the cylinder. Thus the machine is in fact -passive during the ascent of the piston, and exerts its power only -during the descent. - -If the machine, however, be applied to purposes in which a constant -action of the moving force is necessary, as is always the case in -manufactures, the force of the piston must drive the beam in its ascent -as well as in its descent. The arrangement just described cannot effect -this; for although a chain is capable of transmitting any force, by -which its extremities are drawn in opposite directions, yet it is, from -its flexibility, incapable of communicating a force which drives one -extremity of it towards the other. In the one case the piston first -_pulls_ down the beam, and then the beam _pulls_ up the piston. The -chain, because it is inextensible, is perfectly capable of both these -actions; and being flexible, it applies itself to the arch-head of the -beam, so as to maintain the direction of its force upon the piston -continually in the same straight line. But when the piston acts upon -the beam in both ways, in pulling it down and pushing it up, the chain -becomes inefficient, being from its flexibility incapable of the latter -action. - -The problem might be solved by extending the length of the piston -rod, so that its extremity shall be above the beam, and using two -chains; one connecting the highest point of the rod with the lowest -point of the arch-head, and the other connecting the highest point -of the arch-head with a point on the rod below the point which -meets the arch-head when the piston is at the top of the cylinder, -_fig. 172._ - -The connection required may also be made by arming the arch-head with -teeth, _fig. 173._, and causing the piston rod to terminate in a -rack. In cases where, as in the steam-engine, smoothness of motion is -essential, this method is objectionable; and under any circumstances -such an apparatus is liable to rapid wear. - -The method contrived by Watt, for connecting the motion of the piston -with that of the beam, is one of the most ingenious and elegant -solutions ever proposed for a mechanical problem. He conceived the -motion of two straight rods A B, C D, _fig. 174._, -moving on centres or pivots A and C, so that the extremities B and D -would move in the arcs of circles having their centres at A and C. The -extremities B and D of these rods he conceived to be connected with a -third rod B D united with them by pivots on which it could turn -freely. To the system of rods thus connected let an alternate motion -on the centres A and C be communicated: the points B and D will move -upwards and downwards in the arcs expressed by the dotted lines, but -the middle point P of the connecting rod B D will move upwards and -downwards without any sensible deviation from a straight line. - -To prove this demonstratively would require some abstruse mathematical -investigation. It may, however, be rendered in some degree apparent -by reasoning of a looser and more popular nature. As the point B is -raised to E, it is also drawn aside towards the right. At the same -time the other extremity D of the rod B D is raised to E′, and -is drawn aside towards the left. The ends of the rod B D being -thus at the same time drawn equally towards opposite sides, its middle -point P will suffer no lateral derangement, and will move directly -upwards. On the other hand, if B be moved downwards to F, it will be -drawn laterally to the right, while D being moved to F′ will be drawn -to the left. Hence, as before, the middle point P sustains no lateral -derangement, but merely descends. Thus, as the extremities B and D move -upwards and downwards in circles, the middle point P moves upwards and -downwards in a straight line.[6] - -[6] In a strictly mathematical sense, the path of the point P is a -curve, and not a straight line; but in the play given to it in its -application to the steam-engine, it moves through a part only of its -entire locus, and this part extending equally on each side of a point -of inflection, the radius of curvature is infinite, so that in practice -the deviation from a straight line, when proper proportions are -observed in the rods, is imperceptible. - -The application of this geometrical principle in the steam-engine -evinces much ingenuity. The same arm of the beam usually works two -pistons, that of the cylinder and that of the _air-pump_. The apparatus -is represented on the arm of the beam in _fig. 175._ The beam -moves alternately upwards and downwards on its axis A. Every point of -it, therefore, describes a part of a circle of which A is the centre. -Let B be the point which divides the arm A G into two equal parts -A B and B G; and let C D be a straight rod, equal in -length to G B, and fixed on a centre or pivot C on which it is at -liberty to play. The end D of this rod is connected by a straight bar -with the point B, by pivots on which the rod B D turns freely. If -the beam be now supposed to rise and fall alternately, the points B and -D will move upwards and downwards in circular arcs, and, as already -explained with respect to the points B D, _fig. 174._, the -middle point P of the connecting rod B D will move upwards and -downwards without lateral deflection. To this point one of the piston -rods which are to be worked is attached. - -To comprehend the method of working the other piston, conceive a rod -G P′, equal in length to B D, to be attached to the end G of -the beam by a pivot on which it moves freely; and let its extremity -P′ be connected with D by another rod P′ D, equal in length to -G B, and playing on points at P′ and D. The piston rod of the -cylinder is attached to the point P′, and this point has a motion -precisely similar to that of P, without any lateral derangement, but -with a range in the perpendicular direction twice as great. This will -be apparent by conceiving a straight line drawn from the centre A -of the beam to P′, which will also pass through P. Since G P′ -is always parallel to B P, it is evident that the triangle -P′ G A is always similar to P B A, and has its -sides and angles similarly placed, but those sides are each twice the -magnitude of the corresponding sides of the other triangle. Hence -the point P′ must be subject to the same changes of position as the -point P, with this difference only, that in the same time it moves -over a space of twice the magnitude. In fact, the line traced by P′ -is the same as that traced by P, but on a scale twice as large. This -contrivance is usually called the _parallel motion_, but the same name -is generally applied to all contrivances by which a circular motion is -made to produce a rectilinear one. - - - - -CHAP. XIX. - -OF FRICTION AND THE RIGIDITY OF CORDAGE. - - -(320.) With a view to the simplification of the elementary theory of -machines, the consideration of several mechanical effects of great -practical importance has been postponed, and the attention of the -student has been directed exclusively to the way in which the moving -power is modified in being transmitted to the resistance independently -of such effects. A machine has been regarded as an instrument by -which a moving principle, inapplicable in its existing state to the -purpose for which it is required, may be changed either in its velocity -or direction, or in some other character, so as to be adapted to that -purpose. But in accomplishing this, the several parts of the machine -have been considered as possessing in a perfect degree qualities which -they enjoy only in an imperfect degree; and accordingly the conclusions -to which by such reasoning we are conducted are infected with errors, -the amount of which will depend on the degree in which the machinery -falls short of perfection in those qualities which theoretically are -imputed to it. - -[Illustration: _C. Varley, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -Of the several parts of a machine, some are designed to move, while -others are fixed; and of those which move, some have motions differing -in quantity and direction from those of others. The several parts, -whether fixed or movable, are subject to various strains and pressures, -which they are intended to resist. These forces not only vary according -to the load which the machine has to overcome, but also according to -the peculiar form and structure of the machine itself. During the -operation the surfaces of the movable parts move in immediate contact -with the surfaces either of fixed parts or of parts having other -motions. If these surfaces were endued with perfect smoothness or -polish, and the several parts subject to strains possessed perfect -inflexibility and infinite strength, then the effects of machinery -might be practically investigated by the principles already explained. -But the materials of which every machine is formed are endued with -limited strength, and therefore the load which is placed upon it must -be restricted accordingly, else it will be liable to be distorted by -the flexure, or even to be destroyed by the fracture of those parts -which are submitted to an undue strain. The surfaces of the movable -parts, and those surfaces with which they move in contact, cannot in -practice be rendered so smooth but that such roughness and inequality -will remain as sensibly to impede the motion. To overcome such an -impediment requires no inconsiderable part of the moving power. This -part is, therefore, intercepted before its arrival at the working -point, and the resistance to be finally overcome is deprived of it. The -property thus depending on the imperfect smoothness of surfaces, and -impeding the motion of bodies whose surfaces are in immediate contact, -is called _friction_. Before we can form a just estimate of the effects -of machinery, it is necessary to determine the force lost by this -impediment, and the laws which under different circumstances regulate -that loss. - -When cordage is engaged in the formation of any part of a machine, it -has hitherto been considered as possessing perfect flexibility. This is -not the case in practice; and the want of perfect flexibility, which -is called _rigidity_, renders a certain quantity of force necessary -to bend a cord or rope over the surface of an axle or the groove of -a wheel. During the motion of the rope a different part of it must -thus be continually bent, and the force which is expended in producing -the necessary flexure must be derived from the moving power, and is -thus intercepted on its way to the working point. In calculating the -effects of cordage, due regard must be had to this waste of power; -and therefore it is necessary to enquire into the laws which govern -the flexure of imperfectly flexible ropes, and the way in which these -affect the machines in which ropes are commonly used. - -To complete, therefore, the elementary theory of machinery, we propose -in the present and following chapter to explain the principal laws -which determine the effects of friction, the rigidity of cordage, and -the strength of materials. - -(321.) If a horizontal plane surface were perfectly smooth, and free -from the smallest inequalities, and a body having a flat surface also -perfectly smooth were placed upon it, any force applied to the latter -would put it in motion, and that motion would continue undiminished -as long as the body would remain upon the smooth horizontal surface. -But if this surface, instead of being every where perfectly even, had -in particular places small projecting eminences, a certain quantity -of force would be necessary to carry the moving body over these, and -a proportional diminution in its rate of motion would ensue. Thus, if -such eminences were of frequent occurrence, each would deprive the -body of a part of its speed, so that between that and the next it -would move with a less velocity than it had between the same and the -preceding one. This decrease being continued by a sufficient number of -such eminences encountering the body in succession, the velocity would -at last be so much diminished that the body would not have sufficient -force to carry it over the next eminence, and its motion would thus -altogether cease. - -Now, instead of the eminences being at a considerable distance asunder, -suppose them to be contiguous, and to be spread in every direction -over the horizontal plane, and also suppose corresponding eminences to -be upon the surface of the moving body; these projections incessantly -encountering one another will continually obstruct the motion of the -body, and will gradually diminish its velocity, until it be reduced to -a state of rest. - -Such is the cause of friction. The amount of this resisting force -increases with the magnitude of these asperities, or with the roughness -of the surfaces; but it does not solely depend on this. The surfaces -remaining the same, a little reflection on the method of illustration -just adopted, will show that the amount of friction ought also to -depend upon the force with which the surfaces moving one upon the -other are pressed together. It is evident, that as the weight of -the body supposed to move upon the horizontal plane is increased, a -proportionally greater force will be necessary to carry it over the -obstacles which it encounters, and therefore it will the more speedily -be deprived of its velocity and reduced to a state of rest. - -(322.) Thus we might predict with probability, that which accurate -experimental enquiry proves to be true, that the resistance from -friction depends conjointly on the roughness of the surfaces and the -force of the pressure. When the surfaces are the same, a double -pressure will produce a double amount of friction, a treble pressure a -treble amount of friction, and so on. - -Experiment also, however, gives a result which, at least at first view, -might not have been anticipated from the mode of illustration we have -adopted. It is found that the resistance arising from friction does not -at all depend on the magnitude of the surface of contact; but provided -the nature of the surfaces and the amount of pressure remain the same, -this resistance will be equal, whether the surfaces which move one upon -the other be great or small. Thus, if the moving body be a flat block -of wood, the face of which is equal to a square foot in magnitude, and -the edge of which does not exceed a square inch, it will be subject to -the same amount of friction, whether it move upon its broad face or -upon its narrow edge. If we consider the effect of the pressure in each -case, we shall be able to perceive why this must be the case. Let us -suppose the weight of the block to be 144 ounces. When it rests upon -its face, a pressure to this amount acts upon a surface of 144 square -inches, so that a pressure of one ounce acts upon each square inch. The -total resistance arising from friction will, therefore, be 144 times -that resistance which would be produced by a surface of one square inch -under a pressure of one ounce. Now, suppose the block placed upon its -edge, there is then a pressure of 144 ounces upon a surface equal to -one square inch. But it has been already shown, that when the surface -is the same, the friction must increase in proportion to the pressure. -Hence we infer that the friction produced in the present case is 144 -times the friction which would be produced by a pressure of one ounce -acting on one square inch of surface, which is the same resistance as -that which the body was proved to be subject to when resting on its -face. - -These two laws, that friction is independent of the magnitude of the -surface, and is proportional to the pressure when the quality of the -surfaces is the same, are useful in practice, and _generally_ true. In -very extreme cases they are, however, in error. When the pressure is -very intense, in proportion to the surface, the friction is somewhat -_less_ than it would be by these laws; and when it is very small in -proportion to the surface, it is somewhat _greater_. - -(323.) There are two methods of establishing by experiment the laws of -friction, which have been just explained. - -First. The surfaces between which the friction is to be determined -being rendered perfectly flat, let one be fixed in the horizontal -position on a table T T′, _fig. 176._; and let the other be -attached to the bottom of a box B C, adapted to receive weights, -so as to vary the pressure. Let a silken cord S P, attached to the -box, be carried parallel to the table over a wheel at P, and let a dish -D be suspended from it. If no friction existed between the surfaces, -the smallest weight appended to the cord would draw the box towards -P with a continually increasing speed. But the friction which always -exists interrupts this effect, and a small weight may act upon the -string without moving the box at all. Let weights be put in the dish D, -until a sufficient force is obtained to overcome the friction without -giving the box an accelerated motion. Such a weight is equivalent to -the amount of the friction. - -The amount of the weight of the box being previously ascertained, let -this weight be now doubled by placing additional weights in the box. -The pressure will thus be doubled, and it will be found that the weight -of the dish D and its load, which before was able to overcome the -friction, is now altogether inadequate to it. Let additional weights -be placed in the dish until the friction be counteracted as before, -and it will be observed, that the whole weight necessary to produce -this effect is exactly twice the weight which produced it in the former -case. Thus it appears that a double amount of pressure produces a -double amount of friction; and in a similar way it may be proved, that -any proposed increase or decrease of the pressure will be attended with -a proportionate variation in the amount of the friction. - -Second. Let one of the surfaces be attached to a flat plane A B, -_fig. 177._, which can be placed at any inclination with an -horizontal plane B C, the other surface being, as before, attached -to the box adapted to receive weights. The box being placed upon the -plane, let the latter be slightly elevated. The tendency of the box -to descend upon A B, will bear the same proportion to its entire -weight as the perpendicular A E bears to the length of the plane -A B (286.). Thus if the length A B be 36 inches, and the -height A E be three inches, that is a twelfth part of the length, -then the tendency of the weight to move down the plane is equal to a -twelfth part of its whole amount. If the weight were twelve ounces, and -the surfaces perfectly smooth, a force of one ounce acting up the plane -would be necessary to prevent the descent of the weight. - -In this case also the pressure on the plane will be represented by -the length of the base B E (286.), that is, it will bear the -same proportion to the whole weight as B E bears to B A. -The relative amounts of the weight, the tendency to descend, and the -pressure, will always be exhibited by the relative lengths of A B, -A E, and B E. - -This being premised, let the elevation of the plane A B be -gradually increased until the tendency of the weight to descend just -overcomes the friction, but not so much as to allow the box to descend -with accelerated speed. The proportion of the whole weight, which then -acts down the plane, will be found by measuring the height A E, -and the pressure will be determined by measuring the base B E. Now -let the weight in the box be increased, and it will be found that the -same elevation is necessary to overcome the friction; nor will this -elevation suffer any change, however the pressure or the magnitude of -the surfaces which move in contact may be varied. - -Since, therefore, in all these cases, the height A E and the base -B E remain the same, it follows that the proportion between the -friction and pressure is undisturbed. - -(324.) The law that friction is proportional to the pressure, has been -questioned by the late professor Vince of Cambridge, who deduced from -a series of experiments, that although the friction increases with the -pressure, yet that it increases in a somewhat less ratio; and from -this it would follow, that the variation of the surface of contact -must produce some effect upon the amount of friction. The law, as we -have explained it, however, is sufficiently near the truth for most -practical purposes. - -(325.) There are several circumstances regarding the quality of the -surfaces which produce important effects on the quantity of friction, -and which ought to be noticed here. - -This resistance is different in the surfaces of different substances. -When the surfaces are those of wood newly planed, it amounts to about -half the pressure, but is different in different kinds of wood. The -friction of metallic surfaces is about one fourth of the pressure. - -In general the friction between the surfaces of bodies of different -kinds is less than between those of the same kind. Thus, between wood -and metal the friction is about one fifth of the pressure. - -It is evident that the smoother the surfaces are the less will be the -friction. On this account, the friction of surfaces, when first brought -into contact, is often greater than after their attrition has been -continued for a certain time, because that process has a tendency to -remove and rub off those minute asperities and projections on which the -friction depends. But this has a limit, and after a certain quantity -of attrition the friction ceases to decrease. Newly planed surfaces -of wood have at first a degree of friction which is equal to half the -entire pressure, but after they are worn by attrition it is reduced to -a third. - -If the surfaces in contact be placed with their grains in the same -direction, the friction will be greater than if the grains cross each -other. - -Smearing the surfaces with unctuous matter diminishes the friction, -probably by filling the cavities between the minute projections which -produce the friction. - -When the surfaces are first placed in contact, the friction is less -than when they are suffered to rest so for some time; this is proved -by observing the force which in each case is necessary to move the one -upon the other, that force being less if applied at the first moment -of contact than when the contact has continued. This, however, has a -limit. There is a certain time, different in different substances, -within which this resistance attains its greatest amount. In surfaces -of wood this takes place in about two minutes; in metals the time -is imperceptibly short; and when a surface of wood is placed upon a -surface of metal, it continues to increase for several days. The limit -is larger when the surfaces are great, and belong to substances of -different kinds. - -The velocity with which the surfaces move upon one another produces but -little effect upon the friction. - -(326.) There are several ways in which bodies may move one upon the -other, in which friction will produce different effects. The principal -of these are, first, the case where one body _slides_ over another; the -second, where a body having a round form _rolls_ upon another; and, -_thirdly_, where an axis revolves within a hollow cylinder, or the -hollow cylinder revolves upon the axis. - -With the same amount of pressure and a like quality of surface, the -quantity of friction is greatest in the first case and least in the -second. The friction in the second case also depends on the diameter -of the body which rolls, and is small in proportion as that diameter -is great. Thus a carriage with large wheels is less impeded by the -friction of the road than one with small wheels. - -In the third case, the leverage of the wheel aids the power in -overcoming the friction. Let _fig. 178._ represent a section of -the wheel and axle; let C be the centre of the axle, and let B E -be the hollow cylinder in the nave of the wheel in which the axle is -inserted. If B be the part on which the axle presses, and the wheel -turn in the direction N D M, the friction will act at B in -the direction B F, and with the leverage B C. The power acts -against this at D in the direction D A, and with the leverage -D C. It is therefore evident, that as D C is greater than -B C, in the same proportion does the power act with mechanical -advantage on the friction. - -(327.) Contrivances for diminishing the effects of friction depend on -the properties just explained, the motion of rolling being as much -as possible substituted for that of sliding; and where the motion of -rolling cannot be applied, that of a wheel upon its axle is used. In -some cases both these motions are combined. - -If a heavy load be drawn upon a plane in the manner of a sledge, the -motion will be that of sliding, the species which is attended with -the greatest quantity of friction; but if the load be placed upon -cylindrical rollers, the nature of the motion is changed, and becomes -that in which there is the least quantity of friction. Thus large -blocks of stone, or heavy beams of timber, which would require an -enormous power to move them on a level road, are easily advanced when -rollers are put under them. - -When very heavy weights are to be moved through small spaces, this -method is used with advantage; but when loads ore to be transported to -considerable distances, the process is inconvenient and slow, owing to -the necessity of continually replacing the rollers in front of the load -as they are left behind by its progressive advancement. - -The wheels of carriages may be regarded as rollers which are -continually carried forward with the load. In addition to the friction -of the rolling motion on the road, they have, it is true, the friction -of the axle in the nave; but, on the other hand, they are free from -the friction of the rollers with the under surface of the load, or -the carriage in which the load is transported. The advantages of -wheel carriages in diminishing the effects of friction is sometimes -attributed to the slowness with which that axle moves within the box, -compared with the rate at which the wheel moves over the road; but -this is erroneous. The quantity of friction does not in any case vary -considerably with the velocity of the motion, but least of all does it -in that particular kind of motion here considered. - -In certain cases, where it is of great importance to remove the -effects of friction, a contrivance called _friction-wheels_, or -friction-rollers, is used. The axle of a friction-wheel, instead of -revolving within a hollow cylinder, which is fixed, rests upon the -edges of wheels which revolve with it; the species of motion thus -becomes that in which the friction is of least amount. - -Let A B and D C, _fig. 179._, be two wheels revolving -on pivots P Q with as little friction as possible, and so placed -that the axle O of a third wheel E F may rest between their edges. -As the wheel E F revolves, the axle O, instead of grinding its -surface on the surface on which it presses, carries that surface with -it, causing the wheels A B, C D, to revolve. - -In wheel carriages, the roughness of the road is more easily overcome -by large wheels than by small ones. The cause of this arises partly -from the large wheels not being so liable to sink into holes as small -ones, but more because, in surmounting obstacles, the load is elevated -less abruptly. This will be easily understood by observing the curves -in _fig. 180._, which represent the elevation of the axle in each -case. - -(328.) If a carriage were capable of moving on a road without friction, -the most advantageous direction in which a force could be applied to -draw it would be parallel to the road. When the motion is impeded by -friction, it is better, however, that the line of draught should be -inclined to the road, so that the drawing force may be expended partly -in lessening the pressure on the road, and partly in advancing the load. - -Let W, _fig. 181._, be a load which is to be moved upon the plane -surface A B. If the drawing force be applied in the direction -C D, parallel to the plane A B, it will have to overcome the -friction produced by the pressure of the whole weight of the load upon -the plane; but if it be inclined upwards in the direction C E, -it will be equivalent to two forces expressed (74.) by C G and -C F. The part C G has the effect of lightening the pressure -of the carriage upon the road, and therefore of diminishing the -friction in the same proportion. The part C F draws the load along -the plane. Since C F is less than C E or C D the whole -moving force, it is evident that a part of the force of draught is -lost by this obliquity; but, on the other hand, a part of the opposing -resistance is also removed. If the latter exceed the former, an -advantage will be gained by the obliquity; but if the former exceed the -latter, force will be lost. - -By mathematical reasoning, founded on these considerations, it is -proved that the best angle of draught is exactly that obliquity which -should be given to the road in order to enable the carriage to move of -itself. This obliquity is sometimes called the _angle of repose_, and -is that angle which determines the proportion of the friction to the -pressure in the second method, explained in (323.). The more rough the -road is, the greater will this angle be; and therefore it follows, that -on bad roads the obliquity of the traces to the road should be greater -than on good ones. On a smooth Macadamised way, a very slight declivity -would cause a carriage to roll by its own weight: hence, in this case, -the traces should be nearly parallel to the road. - -In rail roads, for like reasons, the line of draught should be parallel -to the road, or nearly so. - -(329.) When ropes or cords form a part of machinery, the effects of -their imperfect flexibility are in a certain degree counteracted by -bending them over the grooves of wheels. But although this so far -diminishes these effects as to render ropes practically useful, yet -still, in calculating the powers of machinery, it is necessary to take -into account some consequences of the rigidity of cordage which even -by these means are not removed. - -To explain the way in which the stiffness of a rope modifies the -operation of a machine, we shall suppose it bent over a wheel and -stretched by weights A B, _fig. 182._, at its extremities. -The weights A and B being equal, and acting at C and D in opposite -ways, balance the wheel. If the weight A receive an addition, it will -overcome the resistance of B, and turn the wheel in the direction -D E C. Now, for the present, let us suppose that the rope -is perfectly inflexible; the wheel and weights will be turned into -the position represented in _fig. 183._ The leverage by which -A acts will be diminished, and will become O F, having been -before O C; and the leverage by which B acts will be increased to -O G, having been before O D. - -But the rope not being inflexible will yield partially to the effects -of the weights A and B, and the parts A C and B D will -be bent into the forms represented in _fig. 184._ The form of -the curvature which the rope on each side of the wheel receives is -still such that the descending weight A works with a diminished -leverage F O, while the ascending weight resists it with an -increased leverage G O. Thus so much of the moving power is lost, -by the stiffness of the rope, as is necessary to compensate this -disadvantageous change in the power of the machine. - - - - -CHAP. XX. - -ON THE STRENGTH OF MATERIALS. - - -(330.) Experimental enquiries into the laws which regulate the strength -of solid bodies, or their power to resist forces variously applied -to tear or break them, are obstructed by practical difficulties, the -nature and extent of which are so discouraging that few have ventured -to encounter them at all, and still fewer have had the steadiness to -persevere until any result showing a general law has been obtained. -These difficulties arise, partly from the great forces which must be -applied, but more from the peculiar nature of the objects of those -experiments. The end to which such an enquiry must be directed is -the development of a _general law_; that is, such a rule as would be -rigidly observed if the materials, the strength of which is the object -of enquiry, were perfectly uniform in their texture, and subject to no -casual inequalities. In proportion as these inequalities are frequent, -experiments must be multiplied, that a long average may embrace cases -varying in both extremes, so as to eliminate each other’s effects in -the final result. - -The materials of which structures and works of art are composed are -liable to so many and so considerable inequalities of texture, that -any rule which can be deduced, even by the most extensive series of -experiments, must be regarded as a mean result, from which individual -examples will be found to vary in so great a degree, that more than -usual caution must be observed in its practical application. The -details of this subject belong to engineering, more properly than -to the elements of mechanics. Nevertheless, a general view of the -most important principles which have been established respecting the -strength of materials will not be misplaced in this treatise. - -A piece of solid matter may be submitted to the action of a force -tending to separate its parts in several ways; the principal of which -are,-- - -1. To a _direct pull_,--as when a rope or wire is stretched by a -weight. When a tie-beam resists the separation of the sides of a -structure, &c. - -2. To a direct pressure or thrust,--as when a weight rests upon a -pillar. - -3. To a transverse strain,--as when weights on the ends of a lever -press it on the fulcrum. - -(331.) If a solid be submitted to a force which draws it in the -direction of its length, having a tendency to pull its ends in -opposite directions, its strength or power to resist such a force is -proportional to the magnitude of its transverse section. Thus, suppose -a square rod of metal A B, _fig. 185._, of the breadth and -thickness of one inch, be pulled by a force in the direction A B, -and that a certain force is found sufficient to tear it; a rod of the -same metal of twice the breadth and the same thickness will require -double the force to break it; one of treble the breadth and the same -thickness will require treble the force to break it, and so on. - -The reason of this is evident. A rod of double or treble the thickness, -in this case, is equivalent to two or three equal and similar rods -which equally and separately resist the drawing force, and therefore -possess a degree of strength proportionate to their number. - -It will easily be perceived, that whatever be the section, the same -reasoning will be applicable, and the power of resistance will, in -general, be proportional to its magnitude or area. - -If the material were perfectly uniform throughout its dimensions, the -resistance to a direct pull would not be affected by the length of the -rod. In practice, however, the increase of length is found to lessen -the strength. This is to be attributed to the increased chance of -inequality. - -(332.) No satisfactory results have been obtained either by theory or -experiment respecting the laws by which solids resist compression. -The power of a perpendicular pillar to support a weight placed upon -it evidently depends on its thickness, or the magnitude of its base, -and on its height. It is certain that when the height is the same, -the strength increases with every increase of the base, but it seems -doubtful whether the strength be exactly proportional to the base. That -is, if two columns of the same material have equal heights, and the -base of one be double the base of the other, the strength of one will -be greater, but it is not certain whether it will exactly double that -of the other. According to the theory of Euler, which is in a certain -degree verified by the experiments of Musschenbrock, the strength will -be increased in a greater proportion than the base, so that, if the -base be doubled, the strength will be more than doubled. - -When the base is the same, the strength is diminished by increasing the -height, and this decrease of strength is proportionally greater than -the increase of height. According to Euler’s theory, the decrease of -strength is proportional to the square of the height; that is, when -the height is increased in a two-fold proportion, the strength is -diminished in a four-fold proportion. - -(333.) The strain to which solids forming the parts of structures of -every kind are most commonly exposed is the lateral or transverse -strain, or that which acts at right angles to their lengths. If any -strain act obliquely to the direction of their length it may be -resolved into two forces (76.), one in the direction of the length, and -the other at right angles to the length. That part which acts in the -direction of the length will produce either compression or a direct -pull, and its effect must be investigated accordingly. - -Although the results of theory, as well as those of experimental -investigations, present great discordances respecting the transverse -strength of solids, yet there are some particulars, in which they, for -the most part, agree; to this it is our object here to confine our -observations, declining all details relating to disputed points. - -Let A B C D, _fig. 186._, be a beam, supported -at its ends A and B. Its strength to support a weight at E pressing -downwards at right angles to its length is evidently proportional to -its breadth, the other things being the same. For a beam of double or -treble breadth, and of the same thickness, is equivalent to two or -three equal and similar beams placed side by side. Since each of these -would possess the same strength, the whole taken together would possess -double or treble the strength of any one of them. - -When the breadth and length are the same the strength obviously -increases with the depth, but not in the same proportion. The -increase of strength is found to be much greater in proportion than -the increase of depth. By the theory of Galileo, a double or treble -thickness ought to increase the strength in a four-fold or nine-fold -proportion, and experiments in most cases do not materially vary from -this rule. - -If while the breadth and depth remain the same, the length of the -beam, or rather, the distance between the points of support, vary, the -strength will vary accordingly, decreasing in the same proportion as -the length increases. - -From these observations it appears, that the transverse strength of -a beam depends more on its thickness than its breadth. Hence we find -that a broad thin board is much stronger when its edge is presented -upwards. On this principle the joists or rafters of floors and roofs -are constructed. - -If two beams be in all respects similar, their strengths will be in the -proportion of the squares of their lengths. Let the length, breadth, -and depth of the one be respectively double the length, breadth, -and depth of the other. By the double breadth the beam doubles its -strength, but by doubling the length half this strength is lost. Thus -the increase of length and breadth counteract each other’s effects, and -as far as they are concerned the strength of the beam is not changed. -But by doubling the thickness the strength is increased in a four-fold -proportion, that is, as the square of the length. In the same manner it -may be shown, that when all the dimensions are trebled, the strength is -increased in a nine-fold proportion, and so on. - -(334.) In all structures the materials have to support their own -weight, and therefore their available strength is to be estimated -by the excess of their absolute strength above that degree of -strength which is just sufficient to support their own weight. This -consideration leads to some conclusions, of which numerous and striking -illustrations are presented in the works of nature and art. - -We have seen that the absolute strength with which a lateral strain is -resisted is in the proportion of the square of the linear dimensions of -similar parts of a structure, and therefore the amount of this strength -increases rapidly with every increase of the dimensions of a body. But -at the same time the weight of the body increases in a still more rapid -proportion. Thus, if the several dimensions be doubled, the strength -will be increased in a four-fold but the weight in an eight-fold -proportion. If the dimensions be trebled, the strength will be -multiplied nine times, but the weight twenty-seven times. Again, if the -dimensions be multiplied four times, the strength will be multiplied -sixteen times, and the weight sixty-four times, and so on. - -Hence it is obvious, that although the strength of a body of small -dimensions may greatly exceed its weight, and, therefore, it may be -able to support a load many times its own weight; yet by a great -increase in the dimensions the weight increasing in a much greater -degree the available strength may be much diminished, and such a -magnitude may be assigned, that the weight of the body must exceed its -strength, and it not only would be unable to support any load, but -would actually fall to pieces by its own weight. - -The strength of a structure of any kind is not, therefore, to be -determined by that of its model, which will always be much stronger in -proportion to its size. All works natural and artificial have limits -of magnitude which, while their materials remain the same, they cannot -surpass. - -In conformity with what has just been explained, it has been observed, -that small animals are stronger in proportion than large ones; that the -young plant has more available strength in proportion than the large -forest tree; that children are less liable to injury from accident -than men, &c. But although to a certain extent these observations are -just, yet it ought not to be forgotten, that the mechanical conclusions -which they are brought to illustrate are founded on the supposition, -that the smaller and greater bodies which are compared are composed -of precisely similar materials. This is not the case in any of the -examples here adduced. - - - - -CHAP. XXI. - -ON BALANCES AND PENDULUMS. - - -(335.) The preceding chapters have been confined almost wholly to -the consideration of the laws of mechanics, without entering into a -particular description of the machinery and instruments dependant upon -those laws. Such descriptions would have interfered too much with the -regular progress of the subject, and it therefore appeared preferable -to devote a chapter exclusively to this portion of the work. - -Perhaps there are no ideas which man receives through the medium of -sense which may not be referred ultimately to matter and motion. In -proportion, therefore, as he becomes acquainted with the properties -of the one and the laws of the other, his knowledge is extended, his -comforts are multiplied; he is enabled to bend the powers of nature to -his will, and to construct machinery which effects with ease that which -the united labour of thousands would in vain be exerted to accomplish. - -Of the properties of matter, one of the most important is its weight, -and the element which mingles inseparably with the laws of motion is -time. - -In the present chapter it is our intention to describe such instruments -as are usually employed for determining the weight of bodies. To -attempt a description of the various machines which are used for the -measurement of time, would lead us into too wide a field for the -present occasion, and we shall, therefore, confine ourselves to an -account of the methods which have been practised to perfect, to perfect -that instrument which affords the most correct means of measuring time, -the pendulum. - -The instrument by which we are enabled to determine, with greater -accuracy than by any other means, the relative weight of a body, -compared with the weight of another body assumed as a standard, is the -balance. - -[Illustration: _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - - -_Of the Balance._ - -The balance may be described as consisting of an inflexible rod or -lever, called the beam, furnished with three axes; one, the fulcrum or -centre of motion situated in the middle, upon which the beam turns, -and the other two near the extremities, and at equal distances from -the middle. These last are called the points of support, and serve to -sustain the pans or scales. - -The points of support and the fulcrum are in the same right line, and -the centre of gravity of the whole should be a little below the fulcrum -when the position of the beam is horizontal. - -The arms of the lever being equal, it follows that if equal weights be -put into the scales no effect will be produced on the position of the -balance, and the beam will remain horizontal. - -If a small addition be made to the weight in one of the scales, the -horizontality of the beam will be disturbed; and after oscillating -for some time, it will, on attaining a state of rest, form an angle -with the horizon, the extent of which is a measure of the delicacy or -sensibility of the balance. - -As the sensibility of a balance is of the utmost importance in -nice scientific enquiries, we shall enter somewhat at large into a -consideration of the circumstances by which this property is influenced. - -In _fig. 187._ let A B represent the beam drawn from the -horizontal position by a very small weight placed in the scale -suspended from the point of support B; then the force tending to draw -the beam from the horizontal position may be expressed by P B, -multiplied by such very small weight acting upon the point B. - -Let the centre of gravity of the whole be at G; then the force acting -against the former will be G P multiplied into the weight of the -beam and scales, and when these forces are equal, the beam will rest -in an inclined position. Hence we may perceive that as the centre of -gravity is nearer to or further from the fulcrum S, (every thing else -remaining the same) the sensibility of the balance will be increased or -diminished. - -For, suppose the centre of gravity were removed to _g_, then to produce -an opposing force equal to that acting upon the extremity of the beam, -the distance _g p_ from the perpendicular line must be increased -until it becomes nearly equal to G P; but for this purpose -the end of the beam B must descend, which will increase the angle -H S B. - -As all weights placed in the scales are referred to the line joining -the points of support, and as this line is above the centre of gravity -of the beam when not loaded, such weights will raise the centre of -gravity; but it will be seen that the sensibility of the balance, as -far as it depends upon this cause, will remain unaltered. - -For, calling the distance S G unity, the distance of the centre of -gravity from the point S (to which the weight which has been added is -referred) will be expressed by the reciprocal of the weight of the beam -so increased; that is, if the weight of the beam be doubled by weights -placed in the scales, S _g_ will be one half of S G; and if the -weight of the beam be in like manner trebled, S _g_ will be one third -of S G, and so on. And as G P varies as S G, _g p_ -will be inversely proportionate to the increased weight of the beam, -and consequently, the product obtained by multiplying _g p_ by the -weight of the beam and its load will be a constant quantity, and the -sensibility of the balance, as before stated, will suffer no alteration. - -We will now suppose that the fulcrum S, _fig. 188._, is situated -below the line joining the points of support, and that the centre of -gravity of the beam when not loaded is at G. Also that when a very -small weight is placed in the scale suspended from the point B, the -beam is drawn from its horizontal position, the deviation being a -measure of the sensibility of the balance. Then, as before stated, -G P multiplied by the weight of the beam will be equal to -P′ B multiplied by the very small additional weight acting on the -point B. - -Now if we place equal weights in both scales, such additional weights -will be referred to the point W, and the resulting distance of the -centre of gravity from the point W, calling W G unity, will be -expressed as before by the reciprocal of the increased weight of the -loaded beam. But G P will decrease in a greater proportion than -W G: thus, supposing the weight of the beam to be doubled, W _g_ -would be one half of W G; but _g p_, as will be evident on -an inspection of the figure, will be less than half of G P; and -the same small weight which was before applied to the point B, if -now added, would depress the point B, until the distance _g p_ -became such as that, when multiplied by the weight of the whole, the -product would be as before equal to P′ B, multiplied by the before -mentioned very small added weight. The sensibility of the balance, -therefore, in this case would be increased. - -If the beam be sufficiently loaded, the centre of gravity will at -length be raised to the fulcrum S, and the beam will rest indifferently -in any position. If more weight be then added, the centre of gravity -will be raised above the fulcrum, and the beam will turn over. - -Lastly, if the fulcrum S, _fig. 189._, is above the line joining -the two points of support, as any additional weights placed in the -scales will be referred to the point W, in the line joining A and B, -if the weight of the beam be doubled by such added weights, and the -centre of gravity be consequently raised to _g_, W _g_ will become -equal to half of W G. But _g p_, being greater than one half -of G P, the end of the beam B will rise until _g p_ becomes -such as to be equal, when multiplied by the whole increased weight of -the beam, to P B, multiplied by the small weight, which we suppose -to have been placed as in the preceding examples, in the scale. - -From what has been said it will be seen that there are three positions -of the fulcrum which influence the sensibility of the balance: first, -when the fulcrum and the points of support are in a right line, when -the sensibility of the balance will remain the same, though the weight -with which the beam is loaded should be varied: secondly, when the -fulcrum is below the line joining the two points of support, in which -case the sensibility of the balance will be increased by additional -weights, until at length the centre of gravity is raised above the -fulcrum, when the beam will turn over; and, thirdly, when the fulcrum -is above the line joining the two points of support, in which case the -sensibility of the balance will be diminished as the weight with which -the beam is loaded is increased. - -The sensibility of a balance, as here defined, is the angular deviation -of the beam occasioned by placing an additional constant small weight -in one of the scales; but it is frequently expressed by the proportion -which such small additional weight bears to the weight of the beam -and its load, and sometimes to the weight the value of which is to be -determined. - -This proportion, however, will evidently vary with different weights, -except in the case where the centre of gravity of the beam is in the -line joining the points supporting the scales, the fulcrum being -above this line, and it is therefore necessary, in every other case, -when speaking of the sensibility of the balance, to designate the -weight with which it is loaded: thus, if a balance has a troy pound in -each scale, and the horizontality of the beam varies a certain small -quantity, just perceptible on the addition of one hundredth of a grain, -we say that the balance is sensible to 1/1152000 part of its load with -a pound in each scale, or that it will determine the weight of a troy -pound within 1/576000 part of the whole. - -The nearer the centre of gravity of a balance is to its fulcrum -the slower will be the oscillations of the beam. The number of -oscillations, therefore, made by the beam in a given time (a minute -for example), affords the most accurate method of judging of the -sensibility of the balance, which will be the greater as the -oscillations are fewer. - -Balances of the most perfect kind, and of such only it is our present -object to treat, are usually furnished with adjustments, by means of -which the length of the arms, or the distances of the fulcrum from the -points of support, may be equalised, and the fulcrum and the two points -of support be placed in a right line; but these adjustments, as will -hereafter be seen, are not absolutely necessary. - -The beam is variously constructed, according to the purposes to -which the balance is to be applied. Sometimes it is made of a rod of -solid steel; sometimes of two hollow cones joined at their bases; -and, in some balances, the beam is a frame in the form of a rhombus: -the principal object in all, however, is to combine strength and -inflexibility with lightness. - -A balance of the best kind, made by Troughton, is so contrived as to -be contained, when not in use, in a drawer below the case; and when -in use, it is protected from any disturbance from currents of air, by -being enclosed in the case above the drawer, the back and front of -which are of plate glass. There are doors in the sides, through which -the scale-pans are loaded, and there is a door at the top through which -the beam may be taken out. - -A strong brass pillar, in the centre of the box, supports a square -piece, on the front and back of which rise two arches, nearly -semicircular, on which are fixed two horizontal planes of agate, -intended to support the fulcrum. Within the pillar is a cylindrical -tube, which slides up and down by means of a handle on the outside -of the case. To the top of this interior tube is fixed an arch, the -terminations of which pass beneath and outside of the two arches before -described. These terminations are formed into Y _s_, destined to -receive the ends of the fulcrum, which are made cylindrical for this -purpose, when the interior tube is elevated in order to relieve the -axis when the balance is not in use. On depressing the interior tube, -the Y _s_ quit the axis, and leave it in its proper position on the -agate planes. The beam is about eighteen inches long, and is formed of -two hollow cones of brass, joined at their bases. The thickness of the -brass does not exceed 0·02 of an inch, but by means of circular rings -driven into the cones at intervals they are rendered almost inflexible. -Across the middle of the beam passes a cylinder of steel, the lower -side of which is formed into an edge, having an angle of about thirty -degrees, which, being hardened and well polished, constitutes the -fulcrum, and rests upon the agate planes for the length of about 0·05 -of an inch. - -Each point of suspension is formed of an axis having two sharp concave -edges, upon which rest at right angles two other sharp concave -edges formed in the spur-shaped piece to which the strings carrying -the scale-pan are attached. The two points are adjustable, the one -horizontally, for the purpose of equalising the arms of the beam, and -the other vertically, for bringing the points of suspension and the -fulcrum into a right line. - -Such is the form of Troughton’s balance: we shall now give the -description of a balance as constructed by Mr. Robinson of Devonshire -Street, Portland Place:-- - -The beam of this balance is only ten inches long. It is a frame of -bell-metal in the form of a rhombus. The fulcrum is an equilateral -triangular prism of steel one inch in length; but the edge on which -the beam vibrates is formed to an angle of 120°, in order to prevent -any injury from the weight with which it may be loaded. The chief -peculiarity in this balance consists in the knife-edge which forms -the fulcrum bearing upon an agate plane throughout its whole length, -whereas we have seen in the balance before described that the whole -weight is supported by portions only of the knife-edge, amounting -together to one tenth of an inch. The supports for the scales are -knife-edges each six tenths of an inch long. These are each furnished -with two pressing screws, by means of which they may be made parallel -to the central knife-edge. - -Each end of the beam is sprung obliquely upwards and towards the -middle, so as to form a spring through which a pushing screw passes, -which serves to vary the distance of the point of support from the -fulcrum, and, at the same time, by its oblique action to raise or -depress it, so as to furnish a means of bringing the points of support -and the fulcrum into a right line. - -A piece of wire, four inches long, on which a screw is cut, proceeds -from the middle of the beam downwards. This is pointed to serve as -an index, and a small brass ball moves on the screw, by changing the -situation of which the place of the centre of gravity may be varied at -pleasure. - -The fulcrum, as before remarked, rests upon an agate plane throughout -its whole length, and the scale-pans are attached to planes of agate -which rest upon the knife-edges forming the points of support. This -method of supporting the scale-pans, we have reason to believe, is -due to Mr. Cavendish. Upon the lower half of the pillar to which the -agate plane is fixed, a tube slides up and down by means of a lever -which passes to the outside of the case. From the top of this tube -arms proceed obliquely towards the ends of the balance, serving to -support a horizontal piece, carrying at each extremity two sets of Y -_s_, one a little above the other. The upper Y _s_ are destined to -receive the agate planes to which the scale-pans are attached, and thus -to relieve the knife-edges from their pressure; the lower to receive -the knife-edges which, form the points of support, consequently these -latter Y _s_, when in action, sustain the whole beam. - -When the lever is freed from a notch in which it is lodged, a spring -is allowed to act upon the tube we have mentioned, and to elevate it. -The upper Y _s_ first meet the agate planes carrying the scale-pans -and free them from the knife-edges. The lower Y _s_ then come into -action and raise the whole beam, elevating the central knife-edge above -the agate plane. This is the usual state of the balance when not in -use: when it is to be brought into action, the reverse of what we -have described takes place. On pressing down the lever, the central -knife-edge first meets the agate plane, and afterwards the two agate -planes carrying the scale-pans are deposited upon their supporting -knife-edges. - -A balance of this construction was employed by the writer of this -article in adjusting the national standard pound. With a pound troy in -each scale, the addition of one hundredth of a grain caused the index -to vary one division, equal to one tenth of an inch, and Mr. Robinson -adjusts these balances so that with one thousand grains in each scale, -the index varies perceptibly on the addition of one thousandth of a -grain, or of one-millionth part of the weight to be determined. - -It may not be uninteresting to subjoin, from the Philosophical -Transactions for 1826, the description of a balance perhaps the most -sensible that has yet been made, constructed for verifying the national -standard bushel. The author says,-- - -“The weight of the bushel measure, together with the 80 lbs. of -water it should contain, was about 250 lbs.; and as I could find -no balance capable of determining so large a weight with sufficient -accuracy, I was under the necessity of constructing one for this -express purpose. - -“I first tried cast iron; but though the beam was made as light as was -consistent with the requisite degree of strength, the inertia of such -a mass appeared to be so considerable, that much time must have been -lost before the balance would have answered to the small differences I -wished to ascertain. Lightness was a property essentially necessary, -and bulk was very desirable, in order to preclude such errors as might -arise from the beam being partially affected by sudden alterations of -temperature. I therefore determined to employ wood, a material in which -the requisites I sought were combined. The beam was made of a plank -of mahogany, about 7O inches long, 22 inches wide, and 2-1/4 thick, -tapering from the middle to the extremities. An opening was cut in the -centre, and strong blocks screwed to each side of the plank, to form a -bearing for the back of a knife-edge which passed through the centre. -Blocks were also screwed to each side at the extremities of the beam on -which rested the backs of the knife-edges for supporting the pans. The -opening in the centre was made sufficiently large to admit the support -hereafter to be described, upon which the knife-edge rested. - -“In all beams which I have seen, with the exception of those made by -Mr. Robinson, the whole weight is sustained by short portions at the -extremities of the knife-edge; and the weight being thus thrown upon a -few points, the knife-edge becomes more liable to change its figure and -to suffer injury. - -“To remedy this defect, the central knife-edge of the beam I am -describing was made 6 inches, and the two others 5 inches long. They -were triangular prisms with equal sides of three fourths of an inch, -very carefully finished, and the edges ultimately formed to an angle of -120°. - -“Each knife-edge was screwed to a thick plate of brass, the surfaces in -contact having been previously ground together; and these plates were -screwed to the beam, the knife-edges being placed in the same plane, -and as nearly equidistant and parallel to each other as could be done -by construction. - -“The support upon which the central knife-edge rested throughout its -whole length was formed of a plate of polished hard steel, screwed to -a block of cast iron. This block was passed through the opening before -mentioned in the centre of the beam, and properly attached to a frame -of cast iron. - -“The stirrups to which the scales were hooked rested upon plates of -polished steel to which they were attached, and the under surfaces of -which were formed by careful grinding into cylindrical segments. These -were in contact with the knife-edges their whole length, and were -known to be in their proper position by the correspondence of their -extremities with those of the knife-edges. A well imagined contrivance -was applied by Mr. Bate for raising the beam when loaded, in order -to prevent unnecessary wear of the knife-edge, and for the purpose of -adjusting the place of the centre of gravity, when the beam was loaded -with the weight required to be determined, a screw carrying a movable -ball projected vertically from the middle of die beam. - -“The performance of this balance fully equalled my expectations. With -two hundred and fifty pounds in each scale, the addition of a single -grain occasioned an immediate variation in the index of one twentieth -of an inch, the radius being fifty inches.” - -From the preceding account it appears that this balance is sensible to -1/1750000 part of the weight which was to be determined. - -We shall now describe the method to be pursued in adjusting a balance. - -1. To bring the points of suspension and the fulcrum into a right line. - -Make the vibrations of the balance very slow by moving the weight which -influences the centre of gravity, and bring the beam into a horizontal -position, by means of small bits of paper thrown into the scales. Then -load the scales with nearly the greatest weight the beam is fitted to -carry. If the vibrations are performed in the same time as before, no -further adjustment is necessary; but if the beam vibrates quicker, -or if it oversets, cause it to vibrate in the same time as at first, -by moving the adjusting weight, and note the distance through which -the weight has passed. Move the weight then in the contrary direction -through double this distance, and then produce the former slow motion -by means of the screw acting vertically on the point of support. Repeat -this operation until the adjustment is perfect. - -2. To make the arms of the beam of an equal length. - -Put weights in the scales as before; bring the beam as nearly as -possible to a horizontal position, and note the division at which the -index stands; unhook the scales, and transfer them with their weights -to the other ends of the beam, when, if the index points to the same -division, the arms are of an equal length; but if not, bring the index -to the division which had been noted, by placing small weights in one -or the other scale. Take away half these weights, and bring the index -again to the observed division by the adjusting screw, which acts -horizontally on the point of support. If the scale-pans are known to be -of the same weight, it will not be necessary to change the scales, but -merely to transfer the weights from one scale-pan to the other. - - -_Of the Use of the Balance._ - -Though we have described the method of adjusting the balance, these -adjustments, as we have before remarked, may be dispensed with. -Indeed, in all delicate scientific operations, it is advisable never -to rely upon adjustments, which, after every care has been employed in -effecting them, can only be considered as approximations to the truth. -We shall, therefore, now describe the best method of ascertaining the -weight of a body, and which does not depend on the accuracy of these -adjustments. - -Having levelled the case which contains the balance, and thrown the -beam out of action, place a weight in each scale-pan nearly equal to -the weight which is to be determined. Lower the beam very gently till -it is in action, and by means of the adjustment for raising or lowering -the centre of gravity, cause the beam to vibrate very slowly. Remove -these weights, and place the substance, the weight of which is to be -determined in one of the scale-pans; carefully counterpoise it by means -of any convenient substances put into the other scale-pan, and observe -the division at which the index stands; remove the body, the weight of -which is to be ascertained, and substitute standard weights for it so -as to bring the index to the same division as before. These weights -will be equal to the weight of the body. - -If it be required to compare two weights together which are intended -to be equal, and to ascertain their difference, if any, the method -of proceeding will be nearly the same. The standard weight is to be -carefully counterpoised, and the division at which the index stands, -noted. And now it will be convenient to add in either of the scales -some small weight, such as one or two hundredths of a grain, and mark -the number of divisions passed over in consequence by the index, by -which the value of one division of the scale will be known. This should -be repeated a few times, and the mean taken for greater certainty. - -Having noted the division at which the index rests, the standard -weight is to be removed, and the weight which is to be compared with -it substituted for it. The index is then again to be noted, and the -difference between this and the former indication will give the -difference between the weights in parts of a grain. - -If the balance is adjusted so as to be very sensible, it will be long -before it comes to a state of rest. It may, therefore, sometimes be -advisable to take the mean of the extent of the vibrations of the index -as the point where it would rest, and this may be repeated several -times for greater accuracy. It must, however, be remembered, that it is -not safe to do this when the extent of the vibrations is beyond one or -two divisions of the scale; but with this limitation it is, perhaps, as -good a method as can be pursued. - -Many precautions are necessary to ensure a satisfactory result. The -weights should never be touched by the hand; for not only would -this oxydate the weight, but by raising its temperature it would -appear lighter, when placed in the scale-pan, than it should do, in -consequence of the ascent of the heated air. For the larger weights a -wooden fork or tongs, according to the form of the weight, should be -employed; and for the smaller, a pair of forceps made of copper will be -found the most convenient. This metal possessing sufficient elasticity -to open the forceps on their being released from pressure, and yet not -opposing a resistance sufficient to interfere with that delicacy of -touch which is desirable in such operations. - - -_Of Weights._ - -It must be obvious, that the excellence of the balance would be of -little use, unless the weights employed were equally to be depended -upon. The weights may either be accurately adjusted, or the difference -between each weight and the standard may be determined, and, -consequently, its true value ascertained. It has been already shown how -the latter may be effected, in the instructions which have been given -for comparing two weights together; and we shall now show the readiest -mode of adjusting weights to an exact equality with a given standard. - -The material of the weight may be either brass or platina, and its form -may be cylindrical: the diameter being nearly twice the height. A small -spherical knob is screwed into the centre, a space being left under -the screw to receive the portions of fine wire used in the adjustment. -It will be convenient to form a cavity in the bottom of each weight to -receive the knob of the weight upon which it may be placed. - -Each weight is now to be compared with the standard, and should it -be too heavy, it is to be reduced till it becomes in a very small -degree too light, when the amount of the deficiency is to be carefully -determined. - -Some very fine silver wire is now to be taken, and the weight of three -or four feet of it ascertained. From this it will be known what length -of the wire is equal to the error of the weight to be adjusted; and -this length being cut off is to be enclosed under the screw. To guard -against any possible error, it will be advisable before the screw -is firmly fixed in its place, again to compare the weight with the -standard. - -The most approved method of making weights expressing the decimal parts -of a grain, is to determine, as before, with great care, the weight of -a certain length of fine wire, and then to cut off such portions as are -equal to the weights required. - -Before we conclude this article we shall give a description, from the -Annals of Philosophy for 1825, of “a very sensible balance,” used by -the late Dr. Black:-- - -“A thin piece of fir wood, not thicker than a shilling, and a foot -long, three tenths of an inch broad in the middle, and one tenth and -a half at each end, is divided by transverse lines into twenty parts; -that is, ten parts on each side of the middle. These are the principal -divisions, and each of them is subdivided into halves and quarters. -Across the middle is fixed one of the smallest needles I could procure, -to serve as an axis, and it is fixed in its place by means of a little -sealing wax. The numeration of the divisions is from the middle to each -end of the beam. The fulcrum is a bit of plate brass, the middle of -which lies flat on my table when I use the balance, and the two ends -are bent up to a right angle so as to stand upright. These two ends -are ground at the same time on a flat hone, that the extreme surfaces -of them may be in the same plane; and their distance is such that the -needle, when laid across them, rests on them at a small distance from -the sides of the beam. They rise above the surface of the table only -one tenth and a half or two tenths of an inch, so that the beam is very -limited in its play. See _fig. 190._ - -“The weights I use are one globule of gold, which weighs one grain, and -two or three others which weigh one tenth of a grain each; and also -a number of small rings of fine brass wire, made in the manner first -mentioned by Mr. Lewis, by appending a weight to the wire, and coiling -it with the tension of that weight round a thicker brass wire in a -close spiral, after which, the extremity of the spiral being tied hard -with waxed thread, I put the covered wire into a vice, and applying -a sharp knife, which is struck with a hammer, I cut through a great -number of the coils at one stroke, and find them as exactly equal to -one another as can be desired. Those I use happen to be the 1/30 part -of a grain each, or 300 of them weigh ten grains; but I have others -much lighter. - -“You will perceive that by means of these weights placed on different -parts of the beam, I can learn the weight of any little mass from one -grain, or a little more, to the 1/1200 of a grain. For if the thing to -be weighed weighs one grain, it will, when placed on one extremity of -the beam, counterpoise the large gold weight at the other extremity. -If it weighs half a grain it will counterpoise the heavy gold weight -placed at 5. If it weigh 6/10 of a grain, you must place the heavy -gold weight at 5, and one of the lighter ones at the extremity to -counterpoise it, and if it weighs only one or two, or three or four -hundredths of a grain, it will be counterpoised by one of the small -gold weights placed at the first or second, or third or fourth -division. If, on the contrary, it weighs one grain and a fraction, it -will be counterpoised by the heavy gold weight at the extremity, and -one or more of the lighter ones placed in some other part of the beam. - -“This beam has served me hitherto for every purpose; but had I occasion -for a more delicate one, I could make it easily by taking a much -thinner and lighter slip of wood, and grinding the needle to give it an -edge. It would also be easy to make it carry small scales of paper for -particular purposes.” - -The writer of this article has used a balance of this kind, and -finds that it is sensible to 1/1000 of a grain when loaded with ten -grains. It is necessary, however, where accuracy is required, to -employ a scale-pan. This may be made of thin card paper, shaped as in -_fig. 191._ - -A thread is to be passed through the two ends, by tightening which they -may be brought near each other. - -The most convenient weights for this beam appear to be two of one grain -each, and one of one tenth of a grain. They should be made of straight -wire; and if the beam be notched at the divisions, they may be lodged -in these notches very conveniently. Ten divisions on each side of the -middle will be sufficient. The weight of the scale-pan must first be -carefully ascertained, in order that it may be deducted from the -weight, afterwards determined, of the scale-pan and the substance it -may contain. - -If the scale-pan be placed at the tenth division of the beam, it is -evident that by means of the two grain weights, a greater weight cannot -be determined than one grain and nine tenths; but if the scale-pan be -placed at any other division of the beam, the resulting apparent weight -must be increased by multiplying it by ten, and dividing by the number -of the division at which the scale-pan is placed; and in this manner it -is evident that if the scale-pan be placed at the division numbered 1, -a weight amounting to nineteen grains may be determined. - -We have been tempted to describe this little apparatus, because it is -extremely simple in its construction, may be easily made, and may be -very usefully employed on many occasions where extreme accuracy is not -necessary. - - -_Description of the Steelyard._ - -The steelyard is a lever, having unequal arms; and in its most simple -form it is so arranged, that one weight alone serves to determine a -great variety of others, by sliding it along the longer arm of the -lever, and thus varying its distance from the fulcrum. - -It has been demonstrated, chapter xiii., that in the lever the -proportion of the power to the weight will be always the same as -that of their distances from the fulcrum, taken in a reverse order; -consequently, when a constant weight is used, and an equilibrium -established by sliding this weight on the longer arm of the lever, the -relative weight of the substance weighed, to the constant weight, will -be in the same proportion as the distance of the constant weight from -the fulcrum is to the length of the shorter arm. - -Thus, suppose the length of the shorter arm, or the distance of -the fulcrum from the point from which the weight to be determined -is suspended, to be one inch; let the longer arm of the lever be -divided into parts of one inch each, beginning at the fulcrum. Now -let the constant weight be equal to one pound, and let the steelyard -be so constructed that the shorter arm shall be sufficiently heavy -to counterpoise the longer when the bar is unloaded. Then suppose a -substance, the weight of which is five pounds, to be suspended from the -shorter arm. It will be found that when the constant weight is placed -at the distance of five inches from the fulcrum, the weights will be in -equilibrium, and the bar consequently horizontal. In this steelyard, -therefore, the distance of each inch from the fulcrum indicates a -weight of one pound. An instrument of this form was used by the Romans, -and it is usually described as the Roman statera or steelyard. A -representation of it is given at _fig. 192._ - -The steelyard is in very general use for the coarser purposes of -commerce, but constructed differently from that which we have -described. The beam with the scales or hooks is seldom in equilibrium -upon the point F, when the weight P is removed; but the longer arm -usually preponderates, and the commencement of the graduations, -therefore, is not at F, but at some point between B and F. The common -steelyard, which we have represented at _fig. 193._, is usually -furnished with two points, from either of which the substance, the -weight of which is to be determined, may be suspended. The value of -the divisions is in this case increased in proportion as the length -of the shorter arm is decreased. Thus, in the steelyard which we have -described, if there be a second point of suspension at the distance of -half an inch from the fulcrum, each division of the longer arm will -indicate two pounds instead of one, and these divisions are usually -marked upon the opposite edge of the steelyard, which is made to turn -over. - -This instrument is very convenient, because it requires but one weight; -and the pressure on the fulcrum is less than in the balance, when the -substance to be weighed is heavier than the constant weight. But, -on the contrary, when the constant weight exceeds the substance to -be weighed, the pressure on the fulcrum is greater in the steelyard -than in the balance, and the balance is, therefore, preferable in -determining small weights. There is also an advantage in the balance, -because the subdivision of weights can be effected with a greater -degree of precision than the subdivision of the arm of the steelyard. - - -_C. Paul’s Steelyard._ - -A steelyard has been constructed by Mr. C. Paul, inspector of weights -and measures at Geneva, which is much to be preferred to that in -common use. Mr. C. Paul states, that steelyards have two advantages -over balances: 1. That their axis of suspension is not loaded with -any other weight than that of the merchandise, the constant weight of -the apparatus itself excepted; while the axis of the balance, besides -the weight of the instrument, sustains a weight double to that of -the merchandise. 2. The use of the balance requires a considerable -assortment of weights, which causes a proportional increase in the -price of the apparatus, independently of the chances of error which it -multiplies, and of the time employed in producing an equilibrium. - -1. In C. Paul’s steelyard the centres of the movement of suspension, or -the two constant centres, are placed on the exact line of the divisions -of the beam; an elevation almost imperceptible in the axis of the beam, -destined to compensate for the very slight flexion of the bar, alone -excepted. - -2. The apparatus, by the construction of the beam, is balanced below -its centre of motion, so that when no weight is suspended the beam -naturally remains horizontal, and resumes that position when removed -from it, as also when the steelyard is loaded, and the weight is at -the division which ought to show how much the merchandise weighs. The -horizontal situation in this steelyard, as well as in the others, is -known by means of the tongue which rises vertically above the axis of -suspension. - -3. It may be discovered, that the steelyard is deranged if, when not -loaded, the beam does not remain horizontal. - -4. The advantage of a great and a small side (which in the other -augments the extent of their power of weighing) is supplied by a very -simple process, which accomplishes the same end with some additional -advantages. This process is to employ on the same division different -weights. The numbers of the divisions on the bar, point out the degree -of heaviness expressed by the corresponding weights. For example, when -the large weight of the large steelyard weighs 16 lbs., each -division it passes over on the bar is equivalent to a pound; the small -weight, weighing sixteen times less than the large one, will represent -on each of these divisions the sixteenth part of a pound, or one ounce; -and the opposite face of the bar is marked by pounds at each sixteenth -division. In this construction, therefore, we have the advantage -of being able, by employing both weights at once, to ascertain, -for example, almost within an ounce, the weight of 500 pounds of -merchandise. It will be sufficient to add what is indicated by the -small weight in ounces, to that of the large one in pounds, after an -equilibrium has been obtained by the position of the two weights, viz. -the large one placed at the next pound below its real weight, and the -small one at the division which determines the number of ounces to be -added. - -5. As the beam is graduated only on one edge, it may have the form of -a thin bar, which renders it much less susceptible of being bent by -the action of the weight, and affords room for making the figures more -visible on both the faces. - -6. In these steelyards the disposition of the axes is not only such -that the beam represents a mathematical lever without weight, but in -the principle of its division, the interval between every two divisions -is a determined and aliquot part of the distance between the two fixed -points of suspension; and each of the two weights employed has for its -absolute weight the unity of the weight it represents, multiplied by -the number of the divisions contained in the interval between the two -centres of motion. - -Thus, supposing the arms of the steelyard divided in such a manner -that ten divisions are exactly contained in the distance between the -two constant centres of motion, a weight to express the pounds on each -division of the beam must really weigh ten pounds; that to point out -the ounces on the same divisions must weigh ten ounces, &c. So that the -same steelyard may be adapted to any system of measures whatever, and -in particular to the decimal system, by varying the absolute heaviness -of the weights, and their relation with each other. - -But to trace out, in a few words, the advantages of the steelyards -constructed by C. Paul for commercial purposes, we shall only observe,-- - -1. That the buyer and seller are certain of the correctness of the -instrument, if the beam remains horizontal when it is unloaded and in -its usual position. 2. That these steelyards have one suspension less -than the old ones, and are so much more simple. 3. That by these means -we obtain, with the greatest facility, by employing two weights, the -exact weight of merchandise, with all the approximation that can be -desired, and even with a greater precision than that given by common -balances. There are few of these which, when loaded with 500 pounds -at each end, give decided indication of an ounce variation; and the -steelyards of C. Paul possess that advantage, and cost one half less -than balances of equal dominion. 4. In the last place, we may verify at -pleasure the justness of the weights, by the transposition which their -ratio to each other will permit; for example, by observing whether, -when the weight of one pound is brought back one division, and the -weight of one ounce carried forward sixteen divisions, the equilibrium -still remains. - -It is on this simple and advantageous principle that C. Paul has -constructed his universal steelyard. It serves for weighing in the -usual manner, and according to any system of weights, all ponderable -bodies to the precision of half a grain in the weight of a hundred -ounces; that is to say, of a ten-thousandth part. It is employed, -besides, for ascertaining the specific gravity of solids, of liquids, -and of air, by processes extremely simple, and which do not require -many subdivisions in the weights. - -We think the description above given will be sufficiently intelligible -without a representation of this instrument. An account of its -application to the determination of specific gravities will be found in -vol. iii. of the Philosophical Magazine. - - -_The Chinese Steelyard._ - -This instrument is used in China and the East Indies for weighing -gems, precious metals, &c. The beam is a small rod of ivory, about -a foot in length. Upon this are three lines of divisions, marked by -fine silver studs, all beginning from the end of the beam, whence the -first is extended 8 inches, the second 6-1/2, and the third 8-1/2. The -first is European weight, and the other two Chinese. At the other end -of the beam hangs a round scale, and at three several distances from -this end are holes, through which pass so many fine strings, serving as -different points of suspension. The first distance makes 1-3/5 inches, -the second 3-1/5, or double the former, and the third 4-4/5, or triple -the same. The instrument, when used, is held by one of the strings, -and a sealed weight of about 1-1/4 oz. troy, is slid upon the -beam until an equilibrium is produced; the weight of the body is then -indicated by the graduated scale above mentioned. - - -_The Danish Balance._ - -The Danish balance is a straight bar or lever, having a heavy weight -fixed to one end, and a hook or scale-pan to receive the substance, -the weight of which is to be determined, suspended from the other -end. The fulcrum is moveable, and is made to slide upon the bar, till -the beam rests in a horizontal position, when the place of the fulcrum -indicates the weight required. In order to construct a balance of this -kind, let the distance of the centre of gravity from that point to -which the substance to be weighed is suspended be found by experiment, -when the beam is unloaded. Multiply this distance by the weight of the -whole apparatus, and divide the product by the weight of the apparatus -increased by the weight of the body. This will give the distance from -the point of suspension, at which the fulcrum being placed, the whole -will be in equilibrio: for example, supposing the distance of the -centre of gravity from the point of suspension to be 10 inches, and -the weight of the whole apparatus to be ten pounds; suppose, also, it -were required to mark the divisions which should indicate weights of -one, two, or three pounds, &c. First, for the place of the division -indicating one pound we have (10 × 10)/(10 + 1) = 100/(10 + 1) = 9-1/11 -inches, the place of the division marking one pound. For two pounds we -have 100/(10 + 2) = 8-1/3 inches, the place of the division indicating -two pounds; and for three pounds 100/(10 + 3) = 7-9/13 inches for the -place of the division indicating three pounds, and so on. - -This balance is subject to the inconvenience of the divisions becoming -much shorter as the weight increases. The distance between the -divisions indicating one and two pounds being, in the example we have -given, about seven tenths of an inch, whilst that between 20 and 21 -pounds is only one tenth of an inch; consequently a very small error -in the place of the divisions indicating the larger weights would -occasion very inaccurate results. The Danish balance is represented at -_fig. 194._ - - -_The Bent Lever Balance._ - -This instrument is represented at _fig. 195._ The weight at C, is -fixed at the end of the bent lever A B C, which is supported -by its axis B on the pillar I H. A scale-pan E, is suspended from -the other end of the lever at A. Through the centre of motion B draw -the horizontal line K B G, upon which, from A and C let fall -the perpendiculars A K and C D. Then if B K and B D -are reciprocally proportional to the weights at A and C, they will be -in equilibrio, but if not, the weight C will move upwards or downwards -along the arc F G till that ratio is obtained. If the lever be so -bent that when A coincides with the line G K, C coincides with -the vertical B H, then as C moves from F to G, its momentum will -increase while that of the weight in the scale-pan E will decrease. -Hence the weight in E, corresponding to different positions of the -balance, may be expressed on the graduated arc F G. - - -_Brady’s Balance, or Weighing Apparatus._ - -This partakes of the properties both of the bent lever balance and of -the steelyard. It is represented, at _fig. 196._ A B C -is a frame of cast iron having a great part of its weight towards A. F -is a fulcrum, and E a moveable suspender, having a scale and hook -at its lower extremity. E K G are three distinct places, to -which the suspender E may be applied, and to which belong respectively -the three graduated scales of division expressing weights, _f_ C, -_c d_, and _a b_. When the scale and suspender are applied at -G, the apparatus is in equilibrio, with the edge A B horizontal, -and the suspender cuts the zero on the scale _a b_. Now, any -substance, the weight of which is to be ascertained, being put into -the scale, the whole apparatus turns about F, and the part towards B -descends till the equilibrium is again established, when the weight -of the body is read off from the scale _a b_, which registers to -ounces and extends to two pounds. If the weight of the body exceed two -pounds, and be less than eleven pounds, the suspender is placed at K; -and when the scale is empty, the number 2 is found to the right of the -index of the suspender. If now weights exceeding two pounds be placed -in the scale, the whole again turns about F, and the weight of the -body is shown on the graduated arc _c d_, which extends to eleven -pounds, and registers to every two ounces. - -If the weight of the body exceed eleven pounds, the suspender is hung -on at E, and the weights are ascertained in the same manner on the -scale _f_ C to thirty pounds, the subdivisions being on this scale -quarters of pounds. The same principles would obviously apply to -weights greater or less than the above. To prevent mistake, the three -points of support G, K, E, are numbered 1, 2, 3; and the corresponding -arcs are respectively numbered in the same manner. When the hook is -used instead of the scale, the latter is turned upwards, there being a -joint at _m_ for that purpose. - - -_The Weighing Machine for Turnpike Roads._ - -This machine is for the purpose of ascertaining the weight of heavy -bodies, such as wheel carriages. It consists of a wooden platform -placed over a pit made in the line of the road, and which contains the -machinery. The pit is walled withinside, and the platform is fitted to -the walls of the pit, but without touching them, and it is therefore -at liberty to move freely up and down. The platform is supported by -levers placed beneath it, and is exactly level with the surface of the -road, so that a carriage is easily drawn on it, the wheels being upon -the platform whilst the horses are upon the solid ground beyond it. The -construction of this machine will be readily understood by reference to -_fig. 197._, in which the platform is supposed to be transparent -so as to allow of the levers being seen below it. - -A, B, C, D, represent four levers tending towards the centre of the -platform, and each moveable on its fulcrum at A, B, C, D; the fulcrum -of each rests upon a piece securely fixed in the corner of the pit. -The platform is supported upon the cross pins _a_, _b_, _c_, _d_, -by means of pieces of iron which project from it near its corners, and -which are represented in the plate by the short dark lines crossing the -pins _a_, _b_, _c_, _d_. The four levers are connected under the centre -of the platform, but not so as to prevent their free motion, and are -supported by a long lever at the point F, the fulcrum of which rests -upon a piece of masonry at E: the end of this last lever passes below -the surface of the road into the turnpike house, and is there attached -to one arm of a balance, or, as in Salmon’s patent weighing machine, to -a strap passing round a cylinder which winds up a small weight round a -spiral, and indicates, by means of an index, the weight placed upon the -platform. - -[Illustration: _Captn. Kater, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -Suppose the distance from A to F to be ten times as great as that from -A to _a_, then a force of one pound applied beneath F would balance -ten pounds applied at _a_, or upon the platform. Again: let the -distance from E to G be also ten times greater than the distance from -the fulcrum E to F; then a force of one pound applied to raise up the -end of the lever G would counterpoise a weight of ten pounds placed -upon F. Now, as we gain ten times the power by the first levers, and -ten times more by the lever E G, it follows, that a force of one -pound tending to elevate G, would balance 100 lbs. placed on the -platform; so that if the end of the lever G be attached to one arm of -a balance, a weight of 10 lbs. placed in a scale suspended from -the other arm, will express the value of 1000 lbs. placed upon -the platform. The levers are counterpoised, when the platform is not -loaded, by a weight H applied to the end of the last lever, continued -beyond the fulcrum for that purpose. - - -_Of Instruments for weighing by means of a Spring._ - -The spring is well adapted to the construction of a weighing machine, -from the property it possesses of yielding in proportion to the force -impressed, and consequently giving a scale of equal parts for equal -additions of weight. It is liable, however, to suffer injury, unless -the steel of which it is composed be very well tempered, from a want -of perfect elasticity, and, consequently, from not returning to its -original place after it has been forcibly compressed. This, however, -must be considered to arise, in a great measure, from imperfection -of workmanship, or of the material employed, or from its having been -subjected to too great a force. - - -_The Spring Steelyard._ - -The little instrument known by this name is in very general use, and -is particularly convenient where great accuracy is not necessary, as -a spring which will ascertain weights from one pound to fifty, is -contained in a cylinder only 4 inches long and 3/4 inch diameter. - -This instrument is represented at _fig. 198._ It consists of a -tube of iron, of the dimensions just stated, closed at the bottom, -to which is attached an iron hook for supporting the substance to be -weighed; a rod of iron _a b_, four tenths of an inch wide and one -tenth thick, is firmly fixed in the circular plate _c d_, which -slides smoothly in the iron tube. - -A strong steel spring is also fastened to this plate, and passed round -the rod _a b_ without touching it, and without coming in contact -with the interior of the cylindrical tube. The tube is closed at the -top by a circular piece of iron through which the piece _a b_ -passes. - -Upon the face of _a b_ the weight is expressed by divisions, -each of which indicates one pound, and five of such divisions in the -instrument now before us occupy two tenths of an inch. The divisions, -notwithstanding, are of sufficient size to enable them to be subdivided -by the eye. - -To use this instrument, the substance to be weighed is suspended by the -hook, the instrument being held by a ring passing through the rod at -the other end. The spring then suffers a compression proportionate to -the weight, and the number of pounds is indicated by the division on -the rod which is cut by the top of the cylindrical tube. - - -_Salter’s improved Spring Balance._ - -A very neat form of the instrument last described has been recently -brought before the public by Mr. Salter, under the name of the Improved -Spring Balance. It is represented at _fig. 199._ The spring is -contained in the upper half of a cylinder behind the brass plate -forming the face of the instrument; and the rod is fixed to the lower -extremity of the spring, which is consequently extended, instead of -being compressed, by the application of the weight. The divisions, each -indicating half a pound, are engraved upon the face of the brass plate, -and are pointed out by an index attached to the rod. - - -_Marriott’s Patent Dial Weighing Machine._ - -The exterior of this instrument is represented at _fig. 200._, and -the interior at _fig. 201._ A B C is a shallow brass -box, having a solid piece as represented at A, to which the spring -D E F is firmly fixed by a nut at D. The other end of the -spring at F is pinned to the brass piece G H, to the part of which -at G is also fixed the iron racked plate I. A screw L serves as a stop -to keep this rack in its place. The teeth of the rack fit into those -of the pinion M, the axis of which passes through the centre of the -dial-plate, and carries an index which points out the weight. The brass -piece G H is merely a plate where it passes over the spring, and -the tail piece H, to which the weight is suspended, passes through an -opening in the side of the box. - - -_Of the Dynamometer._ - -This is an important instrument in mechanics, calculated to measure -the muscular strength exerted by men and animals. It consists -essentially of a spring steelyard, such as that we first described. -This is sometimes employed alone, and sometimes in combination with -various levers, which allow of the spring being made more delicate, -and consequently increase the extent of the divisions indicating the -weight. - -The first instrument of this kind appears to have been invented by Mr. -Graham, but it was too bulky and inconvenient for use. M. le Roy made -one of a more simple construction. It consisted of a metal tube, about -a foot long, placed vertically upon a stand, and containing in the -inside a spiral spring, having above it a graduated rod terminating in -a globe. This rod entered the tube more or less in proportion to the -force applied to the globe, and the divisions indicated the quantity of -this force. Therefore, when a man pressed upon the globe with all his -strength, the divisions upon the rod showed the number of pounds weight -to which it was equal. - -An instrument of this kind for determining the force of a blow struck -by a man with his fist was lately exhibited at the National Repository. -It was fixed to a wall, from which it projected horizontally. In -place of the globe there was a cushion to receive the blow, and as -the suddenness with which the spring returned rendered it impossible -to read the division upon the rod, another rod similarly divided was -forced in by the plate forming the basis of the cushion, and remained -stationary when the spring returned. The common spring steelyard, -however, which we first described, is in principle the same as M. le -Roy’s dynamometer, and is much more conveniently constructed for the -purpose we are considering. The ring at one end may be fixed to an -immovable object, and the hook at the other attached to a man, or to -an animal, and the extent to which the graduated rod is drawn out of -the cylinder shows at once the force which is applied. Though this is -perhaps the best, and certainly the most simple dynamometer, others -have been contrived, which are, however, but modifications of the -spring steelyard. One of these is represented at _fig. 202._ The -spiral spring acts in the manner before described, but its divisions -are increased in size, and therefore rendered more perceptible by means -of a rack fixed to the plate, acting against the spiral spring, the -teeth of which move a pinion upon which the arm I is fixed, pointing to -the graduated arc K. - -Another dynamometer has been invented by Mr. Salmon; it is represented -at _fig. 203._ and is a combination of levers with the spring. -By means of these levers a much more delicate spring, and which is -therefore more sensible, may be employed than in the dynamometer last -described. - -The manner in which these levers and spring act will be readily -understood by an inspection of the figure. Like the weighing machine -for carriages, the fulcrum of each lever is at one end, and the force -is diminished in passing to the spring, in the ratio of the length of -its arms. The spring moves a pinion by means of a rack, upon which -pinion a hand is placed, indicating by divisions upon a circular -dial-plate, the amount of the force employed. - -The spring used in this machine is calculated to weigh only about -50 lbs. instead of about 5 cwt., as in the last described; -but by means of the levers which intervene between it and the force -applied, it will serve to estimate a force equal to 6 cwt., and -might obviously be made to go to a much greater extent, by varying the -ratio of the length of the arms of the levers. - - -ON COMPENSATION PENDULUMS. - -(336.) It is said of Galileo that, when very young, he observed a -lamp suspended from the roof of a church at Pisa, swinging backwards -and forwards with a pendulous motion. This, if it had been remarked -at all by an uneducated mind, would, most probably, have been passed -by as a common occurrence, unworthy of the slightest notice; but to -the mind imbued with science no incident is insignificant; and a -circumstance apparently the most trivial, when subjected to the giant -force of expanded intellect, may become of immense importance to the -improvement and to the well-being of man. The fall of an apple, it is -said, suggested to Newton the theory of gravitation, and his powerful -mind speedily extended to all creation that great law which brings an -apple to the ground. The swinging of a lamp in a church at Pisa, viewed -by the piercing intellect of Galileo, gave rise to an instrument which -affords the most perfect measure of time, which serves to determine the -figure of the earth, and which is inseparably connected with all the -refinements of modern astronomy. - -The properties of the pendulum, and the manner in which it serves -to measure time, have been fully explained in chapter xi.; and if -a substance could be found not susceptible of any change in its -dimensions from a change of temperature, nothing more would be -necessary, as the centre of oscillation would always remain at the -same distance from the point of suspension. As every known substance, -however, expands with heat, and contracts with cold, the length of the -pendulum will vary with every alteration of temperature, and thus the -time of its vibration will suffer a corresponding change. The effect -of a difference of temperature of 25°, or that which usually occurs -between winter and summer, would occasion a clock furnished with a -pendulum having an iron rod to gain or lose six seconds in twenty-four -hours. - -It became, then, highly important to discover some means of -counteracting this variation to which the length of the pendulum was -liable, or, in other words, to devise a method by which the centre of -oscillation should, under every change of temperature, remain at the -same distance from the point of suspension: happily, the difference in -the rate of expansion of different metals presented a ready means of -effecting this. - -Graham, in the year 1715, made several experiments to ascertain the -relative expansions of various metals, with a view of availing himself -of the difference of the expansions of two or more of them when opposed -to each other, to construct a compensating pendulum. But the difference -he found was so small, that he gave up all hope of being able to -accomplish his object in that way. Knowing, however, that mercury was -much more affected by a given change of temperature than any other -substance, he saw that if the mercury could be made to ascend while -the rod of the pendulum became longer, and _vice versâ_, the centre of -oscillation might always be kept at the same distance from the point -of suspension. This idea happily gave birth to the mercurial pendulum, -which is now in very general use. - -[Illustration: _Captn. Kater, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -In the mean time, Graham’s suggestion excited the ingenuity of -Harrison, originally a carpenter at Barton in Lincolnshire, who, in -1726, produced a pendulum formed of parallel brass and steel rods, -known by the name of the gridiron pendulum. - -In the mercurial pendulum, the bob or weight is the material affording -the compensation; but in the gridiron pendulum the object is attained -by the greater expansion of the brass rods, which raise the bob upwards -towards the point of suspension as much as the steel rods elongate -downwards. - -In the present article, we shall describe such compensation pendulums -as appear to us likely to answer best in practice; and we trust we -shall be able to simplify the subject so as to render a knowledge -of mathematics in the construction of this important instrument -unnecessary. - -The following table contains the linear expansion of various substances -in parts of their length, occasioned by a change of temperature -amounting to one degree. We have taken the liberty of extracting -it from a very valuable paper by F. Bailey, Esq., on the mercurial -compensation pendulum, published in the Memoirs of the Astronomical -Society of London for 1824. - - -TABLE I. - -_Linear Expansion of various Substances for One Degree of Fahrenheit’s -Thermometer._ - - +----------------------+-------------+-----------------------+ - | Substances. | Expansions. | Authors. | - +----------------------+-------------+-----------------------+ - |White Deal, { | ·0000022685 | Captain Kater. | - | { | ·0000028444 | Dr. Struve. | - |English Flint Glass, | ·0000047887 | Dulong and Petit. | - |Iron (cast), { | ·0000061700 | General Roy. | - | { | ·0000065668 | Dulong and Petit. | - |Iron (wire), | ·0000068613 | Lavoisier and L. | - |Iron (bar), | ·0000069844 | Hasslar. | - |Steel (rod), | ·0000063596 | General Roy. | - | | | {Commissioners of | - |Brass, | ·0000104400 | {Weights and Measures | - | | | {--mean of several | - | | | {experiments. | - |Lead, | ·0000159259 | Smeaton. | - |Zinc, | ·0000163426 | Ditto. | - |Zinc (hammered), | ·0000172685 | Ditto. | - |Mercury _in bulk_, | ·00010010 | Dulong and Petit. | - +----------------------+-------------+-----------------------+ - -From this table it is easy to determine the length of a rod of any -substance the expansion of which shall be equal to that of a rod of -given length of any other substance. - -The lengths of such rods will be inversely proportionate to their -expansions. If, therefore, we divide the lesser expansion by the -greater (supposing the rod the length of which is given to be made of -the lesser expansible material), and multiply the given length by this -quotient, we shall have the required length of a rod, the expansion -of which will be equal to that of the rod given. For example:--The -expansion of a rod of steel being, from the above table, ·0000063596, -and that of brass, ·0000104400; if it were required to determine the -length of a rod of brass which should expand as much as a rod of steel -of 39 inches in length, we have ·0000063596/·0000104400 = ·6091, which, -multiplied by 39, gives 23·75 inches for the length of brass required. - -We shall here, in order to facilitate calculation, give the ratio of -the lengths of such substances as may be employed in the construction -of compensation pendulums. - - -TABLE II. - - +---------------------------------------------------+ - | Steel rod and brass compensation, as 1: ·6091 | - | Iron wire rod and lead compensation, ·4308 | - | Steel rod and lead compensation, ·3993 | - | Iron wire rod and zinc compensation, ·3973 | - | Steel rod and zinc compensation, ·3682 | - | Glass rod and lead compensation, ·3007 | - | Glass rod and zinc compensation, ·2773 | - | Deal rod and lead compensation, ·1427 | - | Deal rod and zinc compensation, ·1313 | - | Steel rod and mercury in a steel cylinder, ·0728 | - | Steel rod and mercury in a glass cylinder, ·0703 | - | Glass rod and mercury in a glass cylinder, ·0529 | - +---------------------------------------------------+ - -It is evident that in this table the decimals express the length of a -rod of the compensating material, the expansion of which is equal to -that of a pendulum rod whose length is unity. - -As we are not aware of the existence of any work which contains -instructions that might enable an artist or an amateur to make a -compensation pendulum, we shall endeavour to give such detailed -information as may free the subject from every difficulty. - -The pendulum of a clock is generally suspended by a spring, fixed -to its upper extremity, and passing through a slit made in a piece -which is called the cock of the pendulum. The point of suspension is, -therefore, that part of the spring which meets the lower surface of the -cock. Now the distance of the centre of oscillation of the pendulum -from this point may be varied in two ways; the one by drawing up the -spring through this slit, and the other by raising the bob of the -pendulum. Either of these methods may be practised in the compensation -pendulum, but the former is subject to objections from which the latter -is exempt. - -Suppose it were required to compensate a pendulum of 39 inches in -length, of steel, by means of the expansion of a brass rod. Here, -referring to _fig. 204._, we have S C 39 inches (which is -to remain constant) of steel; the pendulum spring, passing through -the cock at S, is attached to another rod of steel, which is fixed to -the cross piece R A at A. The other end of the cross piece at R -is fastened to a brass rod, the lower extremity of which is fixed to -the cock of the pendulum at B. Now the brass rod B R must expand -upwards, as much as the steel rod A C expands downwards; and the -length of the brass must be such as to effect this, leaving 39 inches -of the steel rod below the cock of the pendulum. - -Let us first try 80 inches of steel. Multiplying this by ·6091, we have -48·73 inches for the length of brass, which compensates 80 inches of -steel. But as 48·73 inches of the steel, equal in length to the brass, -would in this case be above the cock of the pendulum, it would leave -only 31·27 inches below it, instead of 39 inches. - -Let us now try 100 inches of steel. This, multiplied as before by -·6091, gives 60·91 inches, according to the expansions which we have -used, for the length of the brass rod, and leaves 39·09 inches below -the cock of the pendulum, which is sufficiently near for our present -purpose. - -From what has been said we may perceive that the total length of the -material of which the pendulum rod is composed must be always equal to -the length of the pendulum added to the length of the compensation. - -In this instance we have effected our object, by drawing the -pendulum-spring through the slit; but we will now show how the -same thing may be done by moving the bob of the pendulum. At -_fig. 205._, let S C, as before, be equal to 39 inches. Let -the steel rod S D turn off at right angles at D, and let a rod -of brass B R, of 61 inches in length, ascend perpendicularly from -this cross piece to R. To the upper part of the brass rod fix another -cross piece R A, and from the extremity A let a steel rod descend -to E, bending it as in the figure till it reaches C. Now the total -length of the pieces of steel expanding downwards is equal to S D, -D F, and F C (amounting together to 39 inches), to which must -be added a length of steel equal to that of the brass rod B R, (61 -inches), making together 100 inches of steel as before, the expansion -of which downwards is compensated by that of the brass rod, of 61 -inches in length, expanding upwards. - -This form, however, is evidently inconvenient, from the great length -of brass and steel which is carried above the cock of the pendulum; -but it is the same thing whether the brass and steel be each in one -piece, or divided into several, provided the pieces of steel be all -so arranged as to expand downwards, and those of brass upwards. Thus, -at _fig. 206._, the portions of steel expanding downwards are -together equal, as before, to 100 inches, and the two brass pieces -expanding upwards are together equal to 61 inches. So that, in fact, -the two last forms of compensation which we have described differ in -no respect from each other in principle, but only in the arrangement -of the materials. The last is the half of the gridiron pendulum, the -remaining bars being merely duplicates of those we have described, and -serving no other purpose but to form a secure frame-work. - - -_Harrison’s Gridiron Pendulum._ - -After what has been said, little more is necessary than to give a -representation of this pendulum. This is done at _fig. 207._, in -which the darker lines represent the steel rods, and the lighter those -of brass. The central rod is fixed at its lower extremity to the middle -of the third cross piece from the bottom, and passes freely through -holes in the cross pieces which are above, whilst the other rods are -secured near their extremities to the cross pieces by pins passing -through them. In order to render the whole more secure, the bars pass -freely through holes made in two other cross pieces, the extremities of -which are fixed to the exterior steel wires. As different kinds of the -same metal vary in their rate of expansion, the pendulum when finished -may be found upon trial to be not duly compensated. In this case one or -more of the cross pieces is shifted higher or lower upon the bars, and -secured by pins passed through fresh holes. - - -_Troughton’s Tubular Pendulum._ - -This is an admirable modification of Harrison’s gridiron pendulum. -It is represented at _fig. 208._, where it may be seen that it -has the appearance of a simple pendulum, as the whole compensation is -concealed within a tube six tenths of an inch in diameter. - -A steel wire, about one tenth of an inch in diameter, is fixed in the -usual manner to the spring by which the pendulum is suspended. This -wire passes to the bottom of an interior brass tube, in the centre of -which it is firmly screwed. The top of this tube is closed, the steel -rod passing freely through a hole in the centre. Into the top of this -interior tube two steel wires, of one tenth of an inch in diameter, -are screwed into holes made in that diameter, which is at right angles -to the motion of the pendulum. These wires pass down the tube without -touching either it or the central rod, through holes made in the piece -which closes the bottom of the interior tube. The lower extremities of -these wires, which project a little beyond the inner tube, are securely -fixed in a piece which closes the bottom of an exterior brass tube, -which is of such a diameter as just to allow the interior tube to pass -freely through it, and of a sufficient length to extend a little above -it. The top of the exterior tube is closed like that of the interior, -having also a hole in its centre, to allow the first steel rod to pass -freely through it. Into the top of the exterior tube, in that diameter -which coincides with the motion of the pendulum, a second pair of -steel wires of the same diameter as the former are screwed, their -distance from the central rod being equal to the distance of each from -the first pair. They consequently pass down within the interior tube, -and through holes made in the pieces closing the lower ends of both the -interior and exterior tubes. The lower ends of these wires are fastened -to a short cylindrical piece of brass of the same diameter as the -exterior tube, to which the bob is suspended by its centre. - -[Illustration: _Captn. Kater, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -_Fig. 209._ is a full sized section of the rod; the three -concentric circles represent the two tubes, and the rectangular -position of the two pair of wires round the middle one is shown by the -five small circles. - -_Fig. 210._ is the part which closes the upper end of the interior -tube. The two small circles are the two wires which proceed from it, -and the three large circles show the holes through which the middle -wire and the other pair of wires pass. - -_Fig. 211._ is the bottom of the interior tube. The small circle -in the centre is where the central rod is fastened to it, the others -the holes for the other four wires to pass through. - -_Fig. 212._ is the part which closes the top of the external tube. -In the large circle in the centre a small brass tube is fixed, which -serves as a covering for the upper part of the middle wire, and the two -small circles are to receive the wires of the last expansion. - -_Fig. 213._ represents the bottom of the exterior tube, in which -the small circles show the places where the wires of the second -expansion are fastened, and the larger ones the holes for the other -pair of wires to pass through. - -_Fig. 214._ is a cylindrical piece of brass, showing the manner in -which the lower ends of the wires of the last expansion are fastened -to it, and the hole in the middle is that by which it is pinned to -the centre of the bob. The upper ends of the two pair of wires are, -as we have observed, fastened by screwing them into the pieces which -stop up the ends of the tubes, but at the lower ends they are all -fixed as represented in _fig. 214._ The pieces represented by -_figs. 213._ and _214._ have each a jointed motion, by means -of which the fellow wires of each pair would be equally stretched, -although they were not exactly of the same length. - -The action of this pendulum is evidently the same as that of the -gridiron pendulum, as we have three lengths of steel expanding -downwards, and two of brass expanding upwards. The weight of the -pendulum has a tendency to straighten the steel rods, and the tubular -form of the brass compensation effectually precludes the fear of its -bending; an advantage not possessed by the gridiron pendulum, in which -brass rods are employed. - -Mr. Troughton, to the account he has given of this pendulum in -Nicholson’s Journal, for December, 1804, has added the lengths of -the different parts of which it was composed, and the expansions of -brass and steel from which these lengths were computed. The length of -the interior tube was 31·9 inches, and that of the exterior one 32·8 -inches, to which must be added 0·4, the quantity by which in this -pendulum the centre of oscillation is higher than the centre of the -bob. These are all of brass. The parts which are of steel are,--the -middle wire, which, including 0·6, the length of the suspension spring, -is 39·3 inches. The first pair of wires 32·5 inches; and the second -pair, 33·2 inches. The expansions used were, for brass ·00001666, -and for steel ·00000661, in parts of their length for one degree of -temperature. - - -_Benzenberg’s Pendulum._ - -This pendulum is mentioned in Nicholson’s Journal for April, 1804, and -is taken from Voigt’s Magazin für den Neuesten Zustande der Naturkunde, -vol. iv. p. 787. The compensation appears to have been -effected by a single rod of lead in the centre, of about half an inch -thick; the descending rods were made of the best thick iron wire. - -As this pendulum deserves attention from the ease with which it may -be made, and as others which have since been produced resemble it in -principle, we have given a representation of it at _fig. 215._, -where A B C D are two rods of iron wire riveted into the -cross pieces A C B D. E F is a rod of lead pinned -to the middle of the piece B D, and also at its upper extremity -to the cross piece G H, into which the second pair of iron wires -are fixed, which pass downwards freely through holes made in the cross -piece B D. The lower extremities of these last iron wires are -fastened into the piece K L, which carries the bob of the pendulum. - -To determine the length of lead necessary for the compensation, we must -recollect, as before, that the distance from the point of suspension -to the centre of the bob (speaking always of a pendulum intended to -vibrate seconds) must be 39 inches. Let us suppose the total length -of the iron wire to be 60 inches; then, from the table which we have -given, we have ·4308 for the length of a rod of lead, the expansion -of which is equivalent to that of an iron rod whose length is unity. -Multiplying 60 inches by ·4308, we have 25·84 inches of lead, which -would compensate 60 inches of iron; but this, taken from 60 inches, -leaves only 34·16 instead of 39 inches. Trying again, in like manner, -68·5 inches of iron, we find 29·5 inches of lead for the length, -affording an equivalent compensation, and which, taken from 68·5 -inches, leaves 39 inches. - -The length of the rod of lead then required as a compensation in this -pendulum is about 29-1/2 inches. - -The writer of this article would suggest another form for this -pendulum, which has the advantage of greater simplicity of construction. - -S A, _fig. 216._, is a rod of iron wire, to which the -pendulum spring is attached. Upon this passes a cylindrical tube of -lead, 29-1/2 inches long, which is either pinned at its lower extremity -to the end of the iron rod S A, or rests upon a nut firmly screwed -upon the extremity of this rod. - -A tube of sheet iron passes over the tube of lead, and is furnished at -top with a flanche, by which it is supported upon the leaden tube; or -it may be fastened to the top of this tube in any manner that may be -thought convenient. - -The bob of the pendulum may be either passed upon the iron tube -(continued to a sufficient length) and secured by a pin passing through -the centre of the bob, or the iron tube may be terminated by an iron -wire serving the same purpose. - -Here we have evidently the same expansions upwards and downwards as in -the gridiron form, given to this pendulum by Mr. Benzenberg, joined to -the compactness of Troughton’s tubular pendulum. - - -_Ward’s Compensation Pendulum._ - -In the year 1806, Mr. Henry Ward, of Blandford in Dorsetshire, received -the silver medal of the Society of Arts for the compensation pendulum -which we are about to describe. - -_Fig. 217._ is a side view of the pendulum rod when together. -H H and I I are two flat rods of iron about an eighth of -an inch thick. K K is a bar of zinc placed between them, and is -nearly a quarter of an inch thick. The corners of the iron bars are -bevelled off, which gives them a much lighter appearance. These bars -are kept together by means of three screws, O O O, which -pass through oblong holes in the bars H H and K K, and screw -into the rod I I. The bar H H is fastened to the bar of zinc -K K, by the screw _m_, which is called the adjusting screw. This -screw is tapped into H H, and passes just through K K; but -that part of the screw which passes K K has its threads turned -off. The iron bar I I has a shoulder at its upper end, and rests -on the top of the zinc bar K K and is wholly supported by it. -There are several holes for the screw _m_, in order to adjust the -compensation. - -The action of this pendulum is similar to that last described, the -zinc expanding upwards as much as the iron rods expand downwards, and -consequently the instance from the point of suspension to the centre -of oscillation remains the same. - -[Illustration: _Captn. Kater, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -Mr. Ward states that the expansion of the zinc he used (hammered zinc) -was greater than that given in the tables. He found that the true -length of the zinc bar should be about 23 inches; our computation would -make it nearly 26. - - -_The Compensation Tube of Julien le Roy._ - -We mention this merely to state that it is similar in principal to the -apparatus represented at _fig. 204._, with merely this difference, -that, instead of the steel rod being fixed to a cross piece proceeding -from the brass bar B R, it is attached to a cap fixed upon a brass -tube (through which it passes) of the same length as that of the brass -rod B R. Cassini spoke well of this pendulum, and it was used in -the observatory of Cluny about the year 1748. - - -_Deparcieux’s Compensation._ - -This was contrived in the same year as that invented by Julien le Roy. -It is represented at _fig. 218._, where A B D F -is a steel bar, the ends of which are to be fixed to the lower -sides of pieces forming a part of the cock of the pendulum. -G E I H is of brass, and stands with its extremities -resting on the horizontal part B D of the steel frame. The upper -part E I of the brass frame passes above the cock of the pendulum, -and admits the tapped wire K, to which the pendulum spring is fixed -through a squared hole in the middle. A nut upon this tapped wire gives -the adjustment for time. The spring passes through the slit in the cock -in the usual manner. - -It may be easily perceived that this pendulum is in principle the -same as that of Le Roy; the expansion of the total length of steel -A B S C downwards being compensated by the equivalent -expansion of the brass bar G E upwards. It is, however, preferable -to Le Roy’s, because the compensation is contained in the clock case. - -Deparcieux had previously published, in the year 1739, an improvement -of an imperfectly compensating pendulum, proposed in the year 1733 by -Regnauld, a clockmaker of Chalons. In this pendulum Deparcieux employed -a lever with unequal arms to increase the effect of the expansion of -the brass rod, which was too short. - -We may here remark, that all fixed compensations are liable to the same -objection, namely, that of not moving with the pendulum, and therefore -not taking precisely the same temperature. - - -_Captain Kater’s Compensation Pendulum._ - -In Nicholson’s Journal, for July, 1808, is the description of a -compensation pendulum by the writer of this article. In this pendulum -the rod is of white deal, three quarters of an inch wide, and a quarter -of an inch thick. It was placed in an oven, and suffered to remain -there for a long time until it became a little charred. The ends were -then soaked in melted sealing-wax; and the rod, being cleaned, was -coated several times with copal varnish. To the lower extremity of the -rod a cap of brass was firmly fixed, from which a strong steel screw -proceeded for the purpose of regulating the pendulum for time in the -usual manner. - -A square tube of zinc was cast, seven inches long and three quarters -of an inch square; the internal dimensions being four tenths of an -inch. The lower part of the pendulum rod was cut away on the two sides, -so as to slide with perfect freedom within the tube of zinc. To the -bottom of this zinc tube a piece of brass a quarter of an inch thick -was soldered, in which a circular hole was made nearly four tenths of -an inch in diameter, having a screw on the inside. A cylinder of zinc, -furnished with a corresponding screw on its surface, fitted into this -aperture, and a thin plate of brass screwed upon the cylinder, served -as a clamp to prevent any shake after the length of zinc necessary for -compensation should have been determined. A hole was made through the -axis of the cylinder, through which passed the steel screw terminating -the pendulum rod. - -An opening was made through the bob of the pendulum, extending to its -centre, to admit the square tube of zinc which was fixed at its upper -extremity to the centre of the bob. The pendulum rod passed through the -bob in the usual manner, and the whole was supported by a nut on the -steel screw at the extremity. - -In this form the compensation acts immediately upon the centre of the -bob, elevating it along the rod as much as the rod elongates downwards: -the method of calculating the length of the required compensation is -precisely the same as that we have before given. - -Assuming the length of the deal rod to be 43 inches, and multiplying -this by ·1313 from Table II., we have 5·64 inches for the length of -the zinc necessary to counteract the expansion of the deal. The length -of the steel screw between the termination of the pendulum rod and the -nut was two inches, and that of the suspension spring one inch. Now, -3 inches of steel multiplied by ·3682 would give 1·10 inches for the -length of zinc which would compensate the steel, and, adding this to -5·64 inches, we have 6·74 inches for the whole length of zinc required. - -In this pendulum, the length of the compensating part may be varied by -means of the zinc cylinder furnished with a screw for that purpose. -The bob of this pendulum and its compensation are represented at -_fig. 219._ - -It has been objected to the use of wooden pendulum rods, that it -is difficult, if not impossible, to secure them from the action of -moisture, which would at once be fatal to their correct performance. -The pendulum now before us has, however, been going with but little -intermission since it was first constructed: it is attached to a -sidereal clock, not of a superior description, and exposed to very -considerable variations of moisture and dryness; yet the change in its -rate has been so very trifling as to authorize the belief that moisture -has little or no effect upon a wooden rod prepared in the manner we -have described. Its rate, under different temperatures, shows that it -is over-compensated; the length of the zinc remaining, as stated in -Nicholson’s Journal 7·42 inches, instead of which it appears, by our -present compensation, that it should be 6·78 inches. - - -_Reid’s Compensation Pendulum._ - -Mr. Adam Reid of Woolwich presented to the Society of Arts, in 1809, a -compensation pendulum, for which he was rewarded with fifteen guineas. -This pendulum is the same in principle with that last described; -the rod, however, is of steel instead of wood, and the compensation -possesses no means of adjustment. This pendulum is represented at -_fig. 220._, where S B is the steel rod, a little thicker -where it enters the bob C, and of a lozenge shape to prevent the bob -turning, but above and below it is cylindrical. - -A tube of zinc D passes to the centre of the bob from below, and the -bob is supported upon it by a piece which crosses its centre, and which -meets the upper end of the tube. - -The rod being passed through the bob and zinc tube, a nut is applied -upon a screw at the lower extremity of the rod in the usual manner. If -the compensation should be too much, the zinc tube is to be shortened -until it is correct. - -The length of the zinc tube will be the same in this pendulum as in -that of Mr. Ward--about 23 inches, if his experiments are to be relied -upon. - -The objection to this pendulum appears to be its great length, which -amounts to 62 inches. We conceive it would be preferable to place the -zinc above the bob, as in the modification which we have suggested of -Benzenberg’s pendulum. - - -_Ellicott’s Pendulum._ - -It appears that the idea of combining the expansions of different -metals with a lever, so as to form a compensation pendulum, -originated with Mr. Graham; for Mr. Short, in the Philosophical -Transactions for 1752, states that he was informed by Mr. Shelton, that -Mr. Graham, in the year 1737, made a pendulum, consisting of three -bars, one of steel between two of brass; and that the steel bar acted -upon a lever so as to raise the pendulum when lengthened by heat, and -to let it down when shortened by cold. - -[Illustration: _Captn. Kater, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -This pendulum, however, was found upon trial to move by jerks, and was -therefore laid aside by the inventor to make way for the mercurial -pendulum. - -Mr. Short also says that Mr. Fotheringham, a quaker of Lincolnshire, -caused a pendulum to be made, in the year 1738 or 1739, consisting of -two bars, one of brass and the other of steel, fastened together by -screws with levers to raise or let down the bob, and that these levers -were placed above the bob. - -Mr. John Ellicott of London had made very, accurate experiments on the -relative expansions of seven different metals, which, however, will be -found to differ more or less from the results of the experiments of -others. It is not, however, from this to be concluded that Ellicott’s -determinations were erroneous; for the expansion of a metal will suffer -considerable change even by the processes to which it is necessarily -subjected in the construction of a pendulum. It is therefore desirable, -whenever a compensation pendulum is to be made, that the expansions -of the materials employed should be determined after the processes of -drilling, filing, and hammering have been gone through. - -It had been objected to Harrison’s gridiron pendulum, that the -adjustments of the rods was inconvenient, and that the expansion of the -bob supported at its lower edge would, unless taken into the account, -vitiate the compensation. These considerations, it is supposed, gave -rise to Ellicott’s pendulum, which is nearly similar to those we have -just mentioned. - -Ellicott’s pendulum is thus constructed:--A bar of brass and a bar of -iron are firmly fixed together at their upper ends, the bar of brass -lying upon the bar of iron, which is the rod of the pendulum. These -bars are held near each other by screws passing through oblong holes in -the brass, and tapped into the iron, and thus the brass is allowed to -expand or contract freely upon the iron with any change of temperature. -The brass bar passes to the centre of the bob of the pendulum, a little -above and below which the iron is left broader for the purpose of -attaching the levers to it, and the iron is made of a sufficient length -to pass quite through the bob of the pendulum. - -The pivots of two strong steel levers turn in two holes drilled in -the broad part of the iron bar. The short arms of these levers are in -contact with the lower extremity of the brass bar, and their longer -arms support the bob of the pendulum by meeting the heads of two -screws which pass horizontally from each side of the bob towards its -centre. By advancing these screws towards the centre of the bob, the -longer arms of the lever are shortened, and thus the compensation may -be readily adjusted. At the lower end of the iron rod, under the bob, -a strong double spring is fixed, to support the greater part of the -weight of the bob by its pressure upwards against two points at equal -distances from the pendulum rod. Mr. Ellicott gave a description of -this pendulum to the Royal Society in 1752, but he says the thought was -executed in 1738. As this pendulum is very seldom met with, we think it -unnecessary to give a representation of it. - - -_Compensation by means of a Compound Bar of Steel and Brass._ - -Several compensations for pendulums have been proposed, by means of -a compound bar formed of steel and brass soldered together. In a -bar of this description, the brass expanding more than the steel, -the bar becomes curved by a change of temperature, the brass side -becoming convex and the steel concave with heat. Now, if a bar of -this description have its ends resting on supports on each side the -cock of the pendulum, the bar passing above the cock with the brass -uppermost, if the pendulum spring be attached to the middle of the bar, -and it pass in the usual manner through the slit of the cock, it is -evident that, by an increase of temperature, the bar will become curved -upwards, and the pendulum spring be drawn upwards through the slit, and -thus the elongation of the pendulum downwards will be compensated. The -compensation may be adjusted by varying the distance of the points of -support from the middle of the bar. - -Such was one of the modes of compensation proposed by Nicholson. Others -of the same description (that is, with compound bars) have been brought -before the public by Mr. Thomas Doughty and Mr. David Ritchie; but as -they are supposed to be liable to many practical objections, we do not -think it requisite to describe them more particularly. - -There is, however, a mode of compensation by means of a compound bar, -described by M. Biot in the first volume of his Traité de Physique, -which appears to possess considerable merit, of which he mentions -having first witnessed the successful employment by the inventor, a -clockmaker named Martin. At _fig. 221._, S C, is the rod of -the pendulum, made, in the usual manner, of iron or steel; this rod -passes through the middle of a compound bar of brass and steel (the -brass being undermost), which should be furnished with a short tube and -screws, by means of which, or by passing a pin through the tube and -rod, it may be securely fixed at any part of the pendulum rod. - -Two small equal weights W W slide along the compound bar, and, -when their proper position has been determined, may be securely clamped. - -The manner in which this compensation acts is thus:--Suppose the -temperature to increase, the brass expanding more than the steel, the -bar becomes curved, and its extremities carrying the weights W and W -are elevated, and thus the place of the centre of oscillation is made -to approach the point of suspension as much, when the compensation is -properly adjusted, as it had receded from it by the elongation of the -pendulum rod. - -There are three methods of adjusting this compensation: the first, by -increasing or diminishing the weights W and W; the second, by varying -the distance of the weights W and W from the middle of the bar; and -the third, by varying the distance of the bar from the bob of the -pendulum, taking care not to pass the middle of the rod. The effect of -the compensation is greater as the weights W and W are greater or more -distant from the centre of the bar, and also as the bar is nearer to -the bob of the pendulum. - -M. Biot says that he and M. Matthieu employed a pendulum of this kind -for a long time in making astronomical observations in which they were -desirous of attaining an extreme degree of precision, and that they -found its rate to be always perfectly regular. - -In all the pendulums which we have described, the bob is supposed -to be fixed to the rod by a pin passing through its centre, and the -adjustment for time is to be made by means of a small weight sliding -upon the rod. - - -_Of the Mercurial Pendulum._ - -We have been guided, in our arrangement of the pendulums which we have -described, by the similarity in the mode of compensation employed; and -we have now to treat of that method of compensation which is effected -by the expansion of the material of which the bob itself of the -pendulum is composed. - -On this subject, as we have before observed, an admirable paper, from -the pen of Mr. Francis Baily, may be found in the Memoirs of the -Astronomical Society of London, which leaves nothing to be desired -by the mathematical reader. But as our object is to simplify, and -to render our subjects as popular as may be, we must endeavour to -substitute for the perfect accuracy which Mr. Baily’s paper presents, -such rules as may be found not only readily intelligible, but -practically applicable, within the limits of those inevitable errors -which arise from a want of knowledge of the exact expansion of the -materials employed. - -At _fig. 222._, let S B represent the rod of a pendulum, and -F C B a metallic tube or cylinder, supported by a nut -at the extremity of the pendulum rod, in the usual manner, and having -a greater expansibility than that of the rod. Now C, the centre of -gravity, supposing the rod to be without weight, will be in the middle -of the cylinder; and if C B, or half the cylinder, be of such -a length as to expand upwards as much as the pendulum rod S B -expands downwards, it is evident that the centre of gravity C will -remain, under any change of temperature, at the same distance from the -point of suspension S. M. Biot imagined that, in effecting this, a -compensation sufficiently accurate would be obtained; but Mr. Baily has -shown that this is by no means the fact. - -Let us suppose the place of the centre of oscillation to be at O, -about three or four tenths of an inch, in a pendulum of the usual -construction, below the centre of gravity. Now, the object of the -compensation is to preserve the distance from S to O invariable, and -not the distance from S to C. - -The distance of the centre of oscillation varies with the length of the -cylinder F B, and hence suffers an alteration in its distance from -the point of suspension by the elongation of the cylinder, although -the distance of the centre of gravity C from the point of suspension -remains unaltered. - -We shall endeavour to render this perfectly familiar. Suppose a -metallic cylinder, 6 inches long, to be suspended by a thread 36 inches -long, thus forming a pendulum in which the distance of the centre -of gravity from the point of suspension is 39 inches: the centre of -oscillation in such a pendulum will be nearly one tenth of an inch -below the centre of gravity. Now let us imagine cylindrical portions of -equal lengths to be added to each end of the cylinder, until it reaches -the point of suspension; we shall then have a cylinder of 78 inches in -length, the centre of gravity of which will still be at the distance of -39 inches from the point of suspension. But it is well known that the -centre of oscillation of such a cylinder is at the distance of about -two thirds of its length from the point of suspension. The centre of -oscillation, therefore, has been removed, by the elongation of the -cylinder, about 13 inches below the centre of gravity, whilst the -centre of gravity has remained stationary. - -Now the same thing as that which we have just described takes place, -though in a very minor degree, with our former cylinder, employed as a -compensating bob to a pendulum. The rod expands downwards, the centre -of gravity remains at the same distance from the point of suspension, -and the cylinder elongates both above and below this point; the -consequence of which is, that though the centre of gravity has remained -stationary, the distance of the centre of oscillation from the point -of suspension has increased. It is, therefore, evident that the length -of the compensation must be such as to carry the centre of gravity -a little nearer to the point of suspension than it was before the -expansion took place; by which means the centre of oscillation will be -restored to its former distance from the point of suspension. - -Let us suppose the expansions to have taken place, and that the -centre of gravity, remaining at the same distance from the point of -suspension, the centre of oscillation is removed to a greater distance, -as we have before explained. It is well known that the product obtained -by multiplying the distance from the point of suspension to the centre -of gravity, by the distance from the centre of gravity to the centre -of oscillation, is a constant quantity; if, therefore, the distance -from the centre of gravity to the point of suspension be lessened, the -distance from the centre of gravity to the centre of oscillation will -be proportionally, though not equally, increased, and the centre of -oscillation will, therefore, be elevated. We see, then, if we elevate -the centre of gravity precisely the requisite quantity, by employing -a sufficient length of the compensating material, that although the -distance from the centre of gravity to the point of suspension is -lessened, yet the distance from the point of suspension to the centre -of oscillation will suffer no change. - -The following rule for finding the length of the compensating material -in a pendulum of the kind we have been considering will be found -sufficiently accurate for all practical purposes:-- - -_Find in the manner before directed the length of the compensating -material, the expansion of which will be equal to that of the rod of -the pendulum. Double this length, and increase the product by its -one-tenth part, which will give the total length required._ We shall -give examples of this as we proceed. - - -_Graham’s Mercurial Pendulum._ - -It was in the year 1721 that Graham first put up a pendulum of this -description, and subjected it to the test of experiment; but it appears -to have been afterwards set aside to make way for Harrison’s gridiron -pendulum, or for others of a similar description. For some years past, -however, its merits have been more generally known, and it is not -surprising that it should be considered as preferable to others, both -from the simplicity of its construction, and the perfect ease with -which the compensation may be adjusted. - -We have already alluded to Mr. Baily’s very able paper on this -pendulum, and we shall take the liberty of extracting from it the -following description:-- - -At _fig. 223._ is a drawing of the mercurial pendulum, as -constructed in the manner proposed by Mr. Baily. - -“The rod S F is made of steel, and perfectly straight; its form -may be either cylindrical, of about a quarter of an inch in diameter, -or a flat bar, three eighths of an inch wide, and one eighth of an inch -thick: its length from S to F, that is, from the bottom of the spring -to the bottom of the rod at F, should be 34 inches. The lower part of -this rod, which passes through the top of the stirrup, and about half -an inch above and below the same, must be formed into a _coarse_ and -_deep_ screw, about two tenths of an inch in diameter, and having about -thirty turns in an inch. A steel nut with a milled head must be placed -at the end of the rod, in order to support the stirrup; and a similar -nut must also be placed on the rod _above_ the head of the stirrup, -in order to screw firmly down on the same, and thus secure it in its -position, after it has been adjusted _nearly_ to the required rate. -These nuts are represented at B and C. A small slit is cut in the rod, -where it passes through the head of the stirrup, through which a steel -pin E is screwed, in order to keep the stirrup from turning round on -the rod. The stirrup itself is also made of steel, and the side pieces -should be of the same form as the rod, in order that they may readily -acquire the same temperature. The top of the stirrup consists of a flat -piece of steel, shaped as in the drawing, somewhat more than three -eighths of an inch thick. Through the middle of the top (which at this -part is about one inch deep) a hole must be drilled sufficiently large -to enable the screw of the rod to pass _freely_, but without _shaking_. -The inside height of the stirrup from A to D may be 8-1/2 inches, and -the inside width between the bars about three inches. The bottom piece -should be about three eighths of an inch thick, and hollowed out nearly -a quarter of an inch deep, so as to admit the glass cylinder freely. -This glass cylinder should have a brass or iron cover G, which should -fit the mouth of it freely, with a shoulder projecting on each side, by -means of which it should be screwed to the side bars of the stirrup, -and thus be secured always in the same position. This cap should not -_press_ on the glass cylinder, so as to prevent its expansion. The -measures above given may require a slight modification, according to -the weight of the mercury employed, and the magnitude of the cylinder: -the final adjustment, however, may be safely left to the artist. Some -persons have recommended that a circular piece of thick plate glass -should float on the mercury, in order to preserve its surface uniformly -level.[7] The part at the bottom marked H is a piece of brass fastened -with screws to the front of the bottom of the stirrup, through a small -hole, in which a steel wire or common needle is passed, in order to -indicate (on a scale affixed to the case of the clock) the arc of -vibration. This wire should merely rest in the hole, whereby it may -be easily removed when it is required to detach the pendulum from the -clock, in order that the stirrup might then stand securely on its base. -One of the screw holes should be rather larger than the body of the -screw, in order to admit of a small adjustment, in case the steel wire -should not stand exactly perpendicular to the axis of motion. The scale -should be divided into _degrees_, and not _inches_, observing that with -a radius of 44 inches (the estimated distance from the bend of the -spring to the end of the steel wire) the length of each degree on the -scale must be 0·768 inch.” - -[7] The variation produced in the height of the column of mercury -(supposed to be 6-1/2 inches high) by an alteration of ± 16° in the -temperature will be only ± 1/100 of an inch, or in other words, 1/100 -of an inch will be the total variation from its _mean_ state, by an -alteration of 32° in the temperature. It is therefore probable that, in -most cases of moderate alteration in the temperature, the _centre_ only -of the column of mercury is subject to elevation and depression, whilst -the exterior parts remain attached to the sides of the glass vessel. It -was with a view to obviate this inconvenience that Henry Browne, Esq. -of Portland Place (I believe) first suggested the piece of floating -glass. - -In order to determine the length of the mercurial column necessary -to form the compensation for this pendulum, we must proceed in the -following manner:-- - -Let us suppose the length of the steel rod and stirrup together to be -42 inches. The absolute expansion of the mercury is ·00010010; but it -is not the absolute expansion, but the vertical expansion in a glass -cylinder, which is required, and this will evidently be influenced by -the expansion of the base of this cylinder. It is easily demonstrable -that, if we multiply the linear expansion of any substance (always -supposed to be a very small part of its length) by 3, we may in all -cases take the result for the cubical or absolute expansion of such -substance. In like manner, if we multiply the linear expansion by 2, we -shall have the superficial expansion. - -If we want the apparent expansion of mercury, the absolute or cubical -expansion of the glass vessel must be deducted from the absolute -expansion of the mercury, which will leave its excess or apparent -expansion. In like manner, deducting the superficial expansion of glass -from the absolute expansion of mercury, we shall have its relative -vertical expansion. Now, taking the rate of expansion of glass to be -·00000479, and multiplying it by 2, the relative vertical expansion -of the mercury in the glass cylinder will be ·00010010 - ·00000958 = -·00009052. - -The expansion of a steel rod, according to our table, is ·0000063596; -which, divided by ·00009052, gives ·0703 for the length of a column of -mercury, the expansion of which is equal to that of a steel rod whose -length is unity. - -We have now to multiply 42 inches by ·0703, which gives 2·95 inches; -and this, deducted from 42, leaves 39·1 inches; so that the length -of rod we have chosen is sufficiently near the truth. Now, double -2·95 inches, and add one tenth of its product, and we shall have 6·49 -inches for the length of the mercurial column forming the requisite -compensation. Mr. Baily’s more accurate calculation gives 6·31 inches. - -A mercurial compensation pendulum may be formed, having a cylinder -of steel or iron, with its top constructed in the same manner as the -top of the stirrup, so as to receive the screw of the rod. To find -the length of the mercurial column necessary in a pendulum of this -description (that is, with a cylinder made of steel), we must double -the linear expansion of steel, and take it from the absolute expansion -of mercury to obtain the relative vertical expansion of the mercury. -This will be ·00010010 - ·00001272 = ·00008738; and, proceeding as -before, we have ·0000063596/·00008738 = ·07279. - -Let the length of the steel rod be, as before, 42 inches. Multiplying -this by ·07279, we have 3·057, which being doubled, and one tenth -of the product added, we obtain 6·72 inches for the length of the -compensating mercurial column; which Mr. Baily states to be 6·59. - -A mercurial compensation pendulum having a rod of glass has been -employed by the writer of this article, who has had reason to think -well of its performance. Its cheapness and simplicity much recommend -it. It is merely a cylinder of glass of about 7 inches in depth, and -2-1/2 inches diameter, terminated by a long neck, which forms the -rod of the pendulum, the whole blown in one piece. A cap of brass is -clamped by means of screws to the top of the rod, and to this the -pendulum spring is pinned. - -We have unquestionable authority for saying, that the mercurial -pendulum of the usual construction, that is, with a steel rod and glass -cylinder, is not affected by a change of temperature simultaneously in -all its parts. Now, the pendulum of which we are treating being formed -throughout of the same material in a single piece, and in every part -of the same thickness, it is presumed it cannot expand in a linear -direction, until the temperature has penetrated to the whole interior -surface of the glass, when it is rapidly diffused through the mass of -mercury. M. Biot mentions that a pendulum of this kind was formerly -used in France, and expresses his surprise that it was no longer -employed, as he had heard it very highly spoken of. The writer of this -article has also used a pendulum with a glass rod, which differs from -that we have just mentioned, in having the lower end of the rod firmly -fixed in a socket attached to the centre of a circular iron plate, on -the circumference of which a screw is cut, which fits into a collar of -iron, supporting the cylinder (to which it is cemented) by means of a -circular lip. - -This arrangement, though perhaps less perfect than that we have just -described, the pendulum not being in one piece, has the advantage of -allowing a circular plate of glass to be placed upon the surface of -the mercury, as practised by Mr. Browne. To determine the length of -a column of mercury for a glass pendulum, let us suppose the glass, -including the cylinder, to be 41 inches in length. Multiplying this -by ·0529, the number taken from Table II. for a glass rod and mercury -in a glass cylinder, we have 2·17 inches for the uncorrected length -of mercury, which compensates 41 inches of glass. Suppose the steel -spring to be one inch and a half long: multiplying this by ·0703, the -appropriate decimal taken from Table II., we have 0·1, the length of -mercury due to the steel, making with the former 2·27 inches, which, -being doubled, and the product increased by its one-tenth part, we -obtain five inches for the length of the required column of mercury. - - -_Compensation Pendulum of Wood and Lead, on the Principle of the -Mercurial Pendulum._ - -If by any contrivance wood could be rendered impervious to moisture, -it would afford one of the most convenient substances known for a -compensation pendulum. It does not appear that sufficient experiments -have been made upon this subject to decide the question. Mr. Browne -of Portland Place, who has devoted much of his time and attention to -the most delicate enquiries of this kind, has, we believe, found that -if a teak rod is well gilded, it will not afterwards be affected by -moisture. At all events, it makes a far superior pendulum, when thus -prepared, to what it does when such preparation is omitted. - -Mr. Baily, in the paper we have before alluded to, proposes an -economical pendulum to be constructed by means of a leaden cylinder and -a deal rod. He prefers lead to zinc, on account of its inferior price, -and the ease with which it may be formed into the required shape; and -as there is no considerable difference in their rates of expansion, it -is equally applicable to the purpose. - -Let the length of the deal rod be taken at 46 inches. Then, to find the -length of the cylinder of lead to compensate this, we have, in Table -II., ·1427 for such a pendulum; which, being multiplied by 46, the -product doubled, and one tenth of the result added to it, gives 14·44 -inches for the length of the leaden cylinder. Mr. Baily’s compensation -gives 14·3 inches. - -[Illustration: _Captn. Kater, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -The rod is recommended to be made of about three eighths of an inch -in diameter: the leaden cylinder is to be cast with a hole through -its centre, which will admit with perfect freedom the cylindrical end -of the rod. The cylinder is supported upon a nut, which screws on the -end of the rod in the usual manner. This pendulum is represented at -_fig. 224._ - -Mr. Baily proposes that the pendulum should be adjusted nearly to the -given rate by means of the screw at the bottom, and that the final -adjustment be made by means of a slider moving along the rod. Indeed, -this is a means of adjustment which we would recommend to be employed -in every pendulum. - - -_Smeaton’s Pendulum._ - -We shall conclude our account of compensation pendulums with a -description of that invented by Mr. Smeaton. The compensation for -temperature in this pendulum is effected by combining the two modes, -which have been so fully described in the preceding part of this -article. - -The pendulum rod is of solid glass, and is furnished with a steel screw -and nut at the bottom in the usual manner. Upon the glass rod a hollow -cylinder of zinc, about the eighth of an inch thick, and about 12 -inches long, passes freely, and rests upon the nut at the bottom of the -pendulum rod. - -Over the zinc cylinder passes a tube made of sheet-iron. The edge of -this tube at the top is turned inwards, and is notched so as to allow -of this being effected. A flanche is thus formed, by which the iron -tube is supported, upon the zinc cylinder. The lower edge of the iron -tube is turned outwards, so as to form a base destined to support a -leaden cylinder, which we are about to describe. - -A cylinder of lead, rather more than 12 inches long, is cast with a -hole through its axis, of such a diameter as to allow of its sliding -freely, but without shake, upon the iron tube over which it passes, and -by the lower extremity of which it is supported. - -Now the zinc, resting upon the nut and expanding upwards, will raise -the whole of the remaining part of the compensation. This expansion -upwards will be slightly counteracted by the lesser expansion downwards -of the iron tube, which carries with it the leaden cylinder. The -cylinder of lead now acts upon the principle of the mercurial pendulum, -and, expanding upwards, contributes that which was wanting to restore -the centre of oscillation to its proper distance from the point of -suspension. - -This pendulum, we have been informed, does well in practice, and we are -not aware that any description of it has been before published. - -The method of calculating the length of the tubes required to form the -compensation is very simple; nothing more is necessary than to find the -length of zinc, the expansion of which is equal to that of the pendulum -rod. - -Let the pendulum rod be composed of 43 inches of glass, the spring -being an inch and a half long, and the screw between the end of the -glass rod and the nut half an inch, making in the whole two inches of -steel and 43 inches of glass. - -Now to find the length of zinc that will compensate the glass, we have, -from Table II., for glass and zinc ·2773, which, multiplied by 43, -gives 11·92 inches. In like manner we obtain as a compensation for two -inches of steel 0·74 of zinc, which, added to 11·92, gives 12·66 inches -for the total length of the zinc cylinder. - -Now if the iron tube and the lead cylinder be each made of the same -length as the zinc, and arranged as we have described, the compensation -will be perfect. - -To prove this, find, by means of the expansions given in Table I., the -actual expansion of each of the substances employed in the pendulum, -and we shall have the following results:-- - - The expansion of 12·66 inches of zinc expanding - upwards is ·0002186 - - Deduct that of 12·66 inches of iron expanding - downwards ·0000869 - -------- - Remaining effect of expansion upwards, referred - to the lower extremity of the iron tube ·0001317 - - Now, for the lead.--On the principle of the - mercurial compensation, subtract one tenth part - of the length of the cylinder, and take half - the remainder, and we shall have six inches of - lead, the expansion of which upwards is ·0000955 - -------- - Total expansion of the compensation upwards ·0002272 - -------- - To find the expansion of the rod, we have - the expansion of 43 inches of glass ·0002059 - - Of two inches of steel ·0000127 - -------- - Total expansion of the pendulum rod ·0002186 - -Agreeing near enough with that of the compensation before found. - -As we conceive we have been sufficiently explicit in our description -of this pendulum, in the construction of which no difficulty presents -itself, we think an engraved representation of it would be superfluous. - -We have hitherto treated only of compensations for temperature; but -there is another kind of error, which has been sometimes insisted upon, -arising from a variation in the density of the atmosphere. If the -density of the atmosphere be increased, the pendulum will experience -a greater resistance, the arc of vibration will in consequence be -diminished, and the pendulum will vibrate faster. This, however, is in -some measure counteracted by the increased buoyancy of the atmosphere, -which, acting in opposition to gravity, occasions the pendulum to -vibrate slower. If the one effect exactly equalled the other, it is -evident no error would arise; and in a paper by Mr. Davies Gilbert, -President of the Royal Society of London, published in the Quarterly -Journal for 1826, he has proved that, by a happy chance, the arc in -which pendulums of clocks are usually made to vibrate is the arc at -which this compensation of error takes place. This arc, for a pendulum -having a brass bob, is 1° 56′ 30″ on each side of the perpendicular; -and for a mercurial pendulum, 1° 31′ 44″, or about one degree and a -half. - -It is well known that, if a pendulum vibrates in a circular arc, the -times of vibration will vary nearly as the squares of the arcs; but -if the pendulum could be made to vibrate in a cycloid, the time of -its vibration in arcs of different extent would then remain the same. -Huygens and others, therefore, endeavoured to effect this by placing -the spring of the pendulum between cheeks of a cycloidal form. - -When escapements are employed which do not insure an unvarying impulse -to the pendulum, the force may be unequally transmitted through the -train of the clock in consequence of unavoidable imperfections of -workmanship, and the arc of vibration may suffer some increase or -diminution from this cause. To discover a remedy for this is certainly -desirable. - -The writer of this article some years ago imagined a mode, which he -believes has also been suggested by others, by which he conceived a -pendulum might be made to describe an arc approaching in form to that -of a cycloid. The pendulum spring was of a triangular form, and the -point or vertex was pinned into the top of the pendulum rod, the base -of the triangle forming the axis of suspension. Now it is evident that -when the pendulum is in motion, the spring will resist bending at the -axis of suspension, with a force in some sort proportionate to the base -of the triangle. - -Suppose the pendulum to have arrived at the extent of its vibrations; -the spring will present a curved appearance; and if the distance from -the point of suspension to the centre of oscillation be then measured, -it will evidently, in consequence of the curvature of the spring, be -shorter than the distance from the point of suspension to the centre of -oscillation, measured when the pendulum is in a perpendicular position, -and consequently when the spring is perfectly straight. - -The base of the triangle may be diminished, or the spring be made -thinner; either of which will lessen its effect. We cannot say how this -plan might answer upon further trial, as sufficient experiments were -not made at the time to authorize a decisive conclusion. - -We have thus completed our account of compensation pendulums; but -before we conclude, it may not be unacceptable if we offer a few -remarks on some points which may be found of practical utility. - -The cock of the pendulum should be firmly fixed either to the wall or -to the case of the clock, and not to the clock itself, as is sometimes -done, and which has occasioned much irregularity in its rate, from the -motion communicated to the point of suspension. We prefer a bracket or -shelf of cast iron or brass, upon which the clock may be fixed, and the -cock carrying the pendulum attached to its perpendicular back. This -bracket may either be screwed to the back of the clock-case, or, which -is the better mode, securely fixed to the wall; and if the latter be -adopted, the whole may be defended from the atmosphere, or from dust, -by the clock-case, which thus has no connection either with the clock -or with the pendulum. - -The point of suspension should be distinctly defined and immovable. -This may be readily effected, after the pendulum shall have taken the -direction of gravity, by means of a strong screw entering the cock -(which should be very stout) on one side, and pressing a flat piece of -brass into firm contact with the spring. - -The impulse should be given in that plane of the rod which coincides -with the plane of vibration passing through the axis of the rod. If the -impulse be given at any point either before or behind this plane, the -probable result will be a tremulous unsteady motion of the pendulum. - -A few rough trials, and moving the weight, will bring the pendulum near -its intended time of vibration, which should be left a little too slow; -when the bob should be firmly fixed to the rod, if the form of the -pendulum will admit of it, by a pin or screw passing through its centre. - -The more delicate adjustment may be completed by shifting the place of -the slider with which the pendulum is supposed to be furnished on the -rod. - -Mr. Browne (of whom we have before spoken) practises the following very -delicate mode of adjustment for rate, which will be found extremely -convenient, as it is not necessary to stop the pendulum in order to -make the required alteration. Having ascertained, by experiment, the -effect produced on the rate of the clock, by placing a weight upon the -bob equal to a given number of grains, he prepares certain smaller -weights of sheet-lead, which are turned up at the corners, that they -may be conveniently laid hold of by a pair of forceps, and the effect -of these small weights on the rate of the clock will be, of course, -known by proportion. The rate being supposed to be in defect, the -weights necessary to correct this may be deposited, without difficulty, -upon the bob of the pendulum, or upon some convenient plane surface, -placed in order to receive them: and should it be necessary to remove -any one of the weights, this may readily be done by employing a -delicate pair of forceps, without producing the slightest disturbance -in the motion of the pendulum. - - - - -INDEX. - - - A. - - Action and reaction, 34. - - Aeriform fluids, 26. - - Animalcules, 12. - - Atmosphere, impenetrability of, 22. - Compressibility and elasticity of, 23. - - Atoms, 6. - Coherence of, 7. - - Attraction, magnetic, of gravitation, 8, 50, 64. - Molecular or atomic, 69. - Cohesion, 70. - - Attwood, machine of, 92. - - Axes, principal, 138. - - Axis, mechanical properties of, 128. - - - B. - - Balance, 279. - Of Bates, 288. - Use of, 289. - Danish, 299. - Bent-lever of Brady, 301. - - Bodies, 2. - Lines, surfaces, edges, area, length of, 4. - Figure, volume, shape of, 5. - Porosity of, 17. - Compressibility of, 18. - Elasticity, dilatibility of, 19. - Inertia of, 27. - Rule for determining velocity of; motion of two bodies after - impact, 38. - - - C. - - Capillary attraction, 73. - - Capstan, 179. - - Cause and effect, 7. - - Circle of curvature, 99. - - Cog, hunting, 191. - - Components, 51. - - Cord, 163. - - Cordage, friction and rigidity of, 260. - - Crank, 241. - - Crystallisation, 14. - - Cycloid, 158. - - - D. - - Damper, self-acting, 234. - - Deparcieux’s compensation pendulum, 319. - - Diagonal, 51. - - Dynamics, 160. - - Dynamometer, 305. - - - E. - - Electricity, 76. - - Electro-magnetism, 76. - - Equilibrium, neutral, instable, and stable, 118. - - - F. - - Figure, 5. - - Fly-wheel, 239. - - Force, 6. - Composition and resolution of, 49. - Centrifugal, 98. - Moment of; leverage of, 135. - Regulation and accumulation of, 224. - - Friction, effects of, 96. - Laws of, 264. - - - G. - - Governor, 227. - - Gravitation, attraction of, 77. - Terrestrial, 84. - - Gravity, centre of, 107. - - Gyration, radius of, centre of, 137. - - - H. - - Hooke’s universal joint, 252. - - Hydrophane, porosity of, 18. - - - I. - - Impact, 39. - - Impulse, 65. - - Inclined plane, 163–209. - - Inclined roads, 211. - - Inertia, 27. - Laws of, 32. - Moment of, 137. - - - J. - - Julien le Roy, compensation tube of, 319. - - - L. - - Lever, 163. - Fulcrum of; three kinds of, 167. - Equivalent, 176. - - Line of direction, 110. - - Liquids, compressibility of, 24. - - Loadstone, 68. - - - M. - - Machines, simple, 160. - Power of, 175. - Regulation of, 225. - - Magnet, 68. - - Magnetic attraction, 8. - - Magnetism, 76. - - Magnitude, 4. - - Marriott’s patent weighing machine, 305. - - Materials, strength of, 272. - - Matter, properties of, 2. - Impenetrability of, 4. - Atoms of; molecules of, 6. - Divisibility of, 9. - Examples of the subtilty of, 12. - Limit to the divisibility of, 13. - Porosity of; density of, 17. - Compressibility of, 18. - Elasticity and dilatability of, 19. - Impenetrability of, 22. - Inertia of, 27. - - Mechanical science, foundation of, 16. - - Metronomes, principles of, 153. - - Molecules, 6. - - Motion, laws of, 46. - Uniformly accelerated, 87. - Table illustrative of, 90. - Retarded; of bodies on inclined planes and curves, 94. - Rotary and progressive, 127. - Mechanical contrivances for the modification of, 245. - Continued rectilinear; reciprocatory rectilinear; continued - circular; reciprocating circular, 246. - - - N. - - Newton, method of, for determining the thickness of transparent - substances, 10. - Laws of motion of, 46. - - - O. - - Oscillation, 129. - Of the pendulum, 145. - Centre of, 152. - - - P. - - Parallelogram, 51. - - Particle, 6. - - Pendulum, oscillation or vibration of, 145. - Isochronism of, 147. - Centre of oscillation of, 152. - Of Troughton, 284. - Compensation, 307. - Of Harrison, 313. - Tubular, of Troughton, 314. - Of Benzenberg, 316. - Ward’s compensation, 318. - Captain Kater’s compensation, 320. - Reid’s; Ellicott’s compensation, 322. - Steel and brass compensation, 324. - Mercurial, 326. - Graham’s mercurial, 329. - Wood and lead, 334. - Smeaton’s, 335. - - Percussion, 130. - Centre of, 144. - - Planes of cleavage, 15. - - Porosity, 17. - - Power, 161. - - Properties, 2. - - Projectiles, curvilinear path of, 82. - - Pulley, 164. - Tackle; fixed, 198. - Single moveable, 200. - Called a runner; Spanish bartons, 205. - - - R. - - Rail-roads, 213. - - Regulating damper, 233. - - Regulators, 227. - - Repulsion, 8. - Molecular, 74. - - Resultant, 51. - - Rose-engine, 250. - - - S. - - Salters, spring balance of, 305. - - Screw, 209. - Concave, 217. - Micrometer, 223. - - Shape, 5. - - Siphon, capillary, 73. - - Spring, 304. - - Statics, 160. - - Steelyard, 294. - C. Paul’s, 296. - Chinese, 299. - - - T. - - Table, whirling, 99. - - Tachometer, 234. - - Tread-mill, 179. - - - V. - - Velocity, angular, 99. - - Vibration, 129. - Of the pendulum, 145. - Centre of, 152. - - Volume, 5–17. - - - W. - - Watch, mainspring of; balance wheel of, 195. - - Water regulator, 229. - - Wedge, 209. - Use of, 215. - - Weight, 161–291. - - Weighing machines, 278. - For turnpike roads, 302. - By means of a spring, 303. - - Wheels, spur, crown, bevelled, 189. - Escapement, 194. - - Wheel and axle, 177. - - Wheel-work, 176. - - Winch, 179. - - Windlass, 178. - - Wollaston’s wire, 10. - - - Z. - - Zureda, apparatus of; Leupold’s application of, 251. - - - END OF MECHANICS. - - LONDON: - SPOTTISWOODES and SHAW - New-street-Square. - -*** END OF THE PROJECT GUTENBERG EBOOK A TREATISE ON MECHANICS *** - -Updated editions will replace the previous one--the old editions will -be renamed. - -Creating the works from print editions not protected by U.S. copyright -law means that no one owns a United States copyright in these works, -so the Foundation (and you!) can copy and distribute it in the -United States without permission and without paying copyright -royalties. 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- text-decoration: underline; -} - -a:link:hover { - background: aqua; -} - -/*fractions*/ -.fraction { - display: inline-block; - vertical-align: middle; - position: relative; - top: 0.1em; - text-align: center; - font-size: 70%; - line-height: 0.4em; -} - -.fraction2 { - display: inline-block; - vertical-align: middle; - position: relative; - top: 0.1em; - text-align: center; - font-size: 100%; - line-height: 1em; -} - -.fraction>span, .fraction2>span { - display: block; - padding: 0.1em; -} - -.fnum { - position: relative; - top: -0.25em -} - -.fraction span.bar, .fraction2 span.bar { - display: none; -} - -.fraction span.fden { - border-top: thin solid black; - line-height: 0.6em; - padding-top: 0.2em; -} - -.fraction2 span.fden2 { - border-top: thin solid black; - line-height: 1em; -/* padding-top: 0.1em;*/ -} - -/* Illustrations */ -img { - max-width: 100%; /* no image to be wider than screen or containing div */ - height: auto; /* keep height in proportion to width */ -} - -div.figcenter { - clear: both; - margin: 2em auto; - text-align: center; - max-width: 100%; /* div no wider than screen, even when screen is narrow */ -} - -.caption { - font-size: 70%; - text-align: center; -} - -.figcenter div { /* make caption a block item with div/div to sit below image (not inline) */ - text-align: justify; /* and to justify text */ -} - -.x-ebookmaker .figcenter div { - float: none; - margin-left: 15%; /* reduce caption width in epubs */ - margin-right: 15%; - display: block; -} - -/* Footnotes */ -.footnotes { - margin-top: 4em; - border: dashed 1px; - padding-bottom: 2em; -} - -.footnote { - margin-left: 10%; - margin-right: 10%; -} - -.footnote p { - font-size: 0.9em; - text-indent: 0em; -} - -.footnote .label { - position: absolute; - right: 84%; - text-align: right; - font-size: 0.9em; -} - -.label:hover { - background: aqua; -} - -.fnanchor { - vertical-align: baseline; - position: relative; - top: -0.4em; - margin-left: 0.05em; - font-size: 0.7em; - font-weight: normal; - font-style: normal; - white-space: nowrap; -} - -/* Transcriber's notes */ -.transnote { - background-color: #F2F2F2; - color: black; - font-size: 85%; - padding: 0.5em; - margin-bottom: 5em; - font-family: sans-serif, serif; -} - - </style> - </head> -<body> - -<div style='text-align:center; font-size:1.2em; font-weight:bold'>The Project Gutenberg eBook of A Treatise on Mechanics, by Henry Kater</div> - -<div style='display:block; margin:1em 0'> -This eBook is for the use of anyone anywhere in the United States and -most other parts of the world at no cost and with almost no restrictions -whatsoever. You may copy it, give it away or re-use it under the terms -of the Project Gutenberg License included with this eBook or online -at <a href="https://www.gutenberg.org">www.gutenberg.org</a>. If you -are not located in the United States, you will have to check the laws of the -country where you are located before using this eBook. -</div> - -<p style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Title: A Treatise on Mechanics</p> - -<div style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Author: Henry Kater and Dionysius Lardner</div> - -<div style='display:block; margin:1em 0'>Release Date: August 17, 2021 [eBook #66078]</div> - -<div style='display:block; margin:1em 0'>Language: English</div> - -<div style='display:block; margin:1em 0'>Character set encoding: UTF-8</div> - -<div style='display:block; margin-left:2em; text-indent:-2em'>Produced by: Thiers Halliwell, deaurider and the Online Distributed Proofreading Team at https://www.pgdp.net (This file was produced from images generously made available by The Internet Archive)</div> - -<div style='margin-top:2em; margin-bottom:4em'>*** START OF THE PROJECT GUTENBERG EBOOK A TREATISE ON MECHANICS ***</div> - - -<div class="transnote"> <p><b>Transcriber’s notes</b>:</p> - -<p>The text of this e-book has mostly been preserved in its original -form including some inconsistency of hyphenation and use of diacritics -(aeriform/aëriform). Three spelling typos have been corrected -(arrangment → arrangement, pully → pulley, dye → die) as have typos -in equations on pages 40 and 43. And some missing punctuation has been -corrected silently (periods, commas, incorrect quotes). To assist -the reader, hyperlinks have been added to the table of contents, -index and footnotes, as well as to the numerous cross-references -within the text. <span class="htmlonly">Page numbers are shown in -the right margin and footnotes are located at the end.</span> <span -class="epubonly">Footnotes are located at the end.</span></p> - -<p class="epubonly">The cover image of the book was created by the -transcriber and is placed in the public domain.</p> -</div> - - - - -<div class="titlepage"> - -<h1><span class="fs70">A</span><br /> - -TREATISE <span class="lowercase smcap">ON</span> MECHANICS,</h1> - -<div class="tp1">BY</div> - -<div class="tp2">CAPTAIN HENRY KATER, V. PRES: R.S.</div> - -<div class="tp3">─── and ───</div> - -<div class="tp2">DIONYSIUS LARDNER, D.C.L. F.R.S. &c. &c.</div> - -<div class="tp4">A NEW EDITION REVISED & CORRECTED.<br /> -1852.</div> - -<div class="figcenter" id="i_f001" style="max-width: 23.375em;"> - <img src="images/i_f001.jpg" alt="" /> - <div class="caption"><p><span class="l-align"><i>H. Corbould del.</i></span><span class="r-align"><i>E. Finder fc.</i></span></p></div> -</div> - - -<div class="tp5"><b>London:</b><br /> -PRINTED FOR LONGMAN, BROWN, GREEN & LONGMANS. PATERNOSTER ROW: -</div> -</div> - -<hr class="chap x-ebookmaker-drop" /> - -<p class="tac">ADVERTISEMENT.</p> - - -<p>This Treatise on Mechanics, which was originally -published in 1830, is the work of Dr. Lardner, with -the exception of the twenty-first chapter, which was -written by the late Captain Kater. The present edition -has been revised and corrected by Dr. Lardner.</p> - -<p class="ml1em fs90"><i>London, January, 1852.</i></p> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_v">v</span></p> - -<h2 class="nobreak" id="CONTENTS">CONTENTS.</h2> -</div> - - -<div class="center"> -<table class="fs85" width="600" summary="Table of Contents"> -<tr><td class="tac pt1"><div>CHAP. I.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>PROPERTIES OF MATTER.</div></td></tr> - -<tr><td class="taj pl1hi1">Organs of Sense.—Sensations.—Properties -or Qualities.—Observation. —Comparison and -Generalisation.—Particular and general Qualities.—Magnitude. -—Size.—Volume.—Lines.—Surfaces.—Edges.—Area.—Length. -—Impenetrability.—Apparent Penetration.—Figure.—Different -from Volume. —Atoms.—Molecules.—Matter -separable.—Particles.—Force.—Cohesion of Atoms.—Hypothetical Phrases -unnecessary.—Attraction.</td><td class="pl1 vab tar"><div><a href="#Page_1">1</a></div></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. II.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>PROPERTIES OF MATTER, CONTINUED.</div></td></tr> - -<tr><td class="taj pl1hi1">Divisibility.—Unlimited Divisibility.—Wollaston’s -micrometric Wire. —Method of making it.—Thickness of a Soap -Bubble.—Wings of Insects.—Gilding of Wire for Embroidery.—Globules -of the Blood.—Animalcules.—Their minute Organisation.—Ultimate -Atoms.—Crystals.—Porosity.—Volume.—Density. —Quicksilver -passing through Pores of Wood.—Filtration.—Porosity of Hydrophane. -—Compressibility.—Elasticity.—Dilatability.—Heat.—Contraction -of Metal used to restore the Perpendicular to Walls of a Building.—Impenetrability -of Air. —Compressibility of it.—Elasticity of it.—Liquids not absolutely -incompressible. —Experiments.—Elasticity of Fluids.—Aeriform -Fluids.—Domestic Fire Box.— Evolution of Heat by compressed Air.</td><td -class="pl1 vab tar"><div><a href="#Page_9">9</a></div></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. III.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>INERTIA.</div></td></tr> - -<tr><td class="taj pl1hi1">Inertia.—Matter Incapable of spontaneous -Change.—Impediments to Motion.—Motion of the Solar System.—Law of -Nature.—Language used to express Inertia sometimes faulty.—Familiar Examples of -Inertia.</td><td class="pl1 vab tar"><div><a href="#Page_27">27</a></div></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. IV.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>ACTION AND REACTION.</div></td></tr> - -<tr><td class="taj pl1hi1">Inertia in a single Body.—Consequences of Inertia in two or -more Bodies.— Examples.—Effects of Impact.—Motion not estimated by Speed or -Velocity alone.—Examples.—Rule for estimating the Quantity of Motion.—Action -and Reaction.—Examples of.—Velocity of two Bodies after Impact.—Rule for -finding the common Velocity after Impact.—Magnet and Iron.—Feather and Cannon Ball -impinging.—Newton’s Laws of Motion.—Inutility of.—Familiar Effects -resulting from Consequences of Inertia.</td><td class="pl1 vab tar"><div><a href="#Page_34">34</a></div><span class="pagenum" -id="Page_vi">vi</span></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. V.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>COMPOSITION AND RESOLUTION OF FORCE.</div></td></tr> - -<tr><td class="taj pl1hi1">Motion and -Pressure.—Force.—Attraction.—Parallelogram of -Forces.—Resultant.—Components.—Composition of Force.—Resolution of -Force.—Illustrative Experiments.—Composition of Pressures.—Theorems -regulating Pressures also regulate Motion.—Examples.—Resolution -of Motion.—Forces in Equilibrium.—Composition of Motion and -Pressure.—Illustrations.—Boat in a Current.—Motions of Fishes.—Flight -of Birds.—Sails of a Vessel.—Tacking.—Equestrian Feats.—Absolute and -relative Motion.</td><td class="pl1 vab tar"><div><a href="#Page_48">48</a></div></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. VI.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>ATTRACTION.</div></td></tr> - -<tr><td class="taj pl1hi1">Impulse.—Mechanical State of Bodies.—Absolute -Rest.—Uniform and rectilinear Motion.—Attractions.—Molecular -or atomic.—Interstitial Spaces in Bodies.—Repulsion and -Attraction.—Cohesion.—In Solids and Fluids.—Manufacture of -Shot.—Capillary Attractions.—Shortening of Rope by Moisture.—Suspension -of Liquids in capillary Tubes.—Capillary Siphon.—Affinity between Quicksilver -and Gold.—Examples of Affinity.—Sulphuric Acid and Water.—Oxygen and -Hydrogen. —Oxygen and Quicksilver.—Magnetism.—Electricity and -Electro-Magnetism.—Gravitation.—Its Law.—Examples of.—Depends -on the Mass.—Attraction between the Earth and detached Bodies on its -Surface.—Weight.—Gravitation of the Earth.—Illustrated by Projectiles. -—Plumb-Line.—Cavendish’s Experiments.</td><td class="pl1 vab tar"><div><a href="#Page_63">63</a></div></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. VII.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>TERRESTRIAL GRAVITY.</div></td></tr> - -<tr><td class="taj pl1hi1">Phenomena of falling Bodies.—Gravity greater at the Poles -than Equator.—Heavy and light Bodies fall with equal Speed to the Earth.— -Experiment.—Increased Velocity of falling Bodies.—Principles of uniformly -accelerated Motion.—Relations between the Height, Time, and Velocity.—Attwood’s -Machine.—Retarded Motion.</td><td class="pl1 vab tar"><div><a href="#Page_84">84</a></div></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. VIII.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>OF THE MOTION OF BODIES ON INCLINED PLANES AND CURVES.</div></td></tr> - -<tr><td class="taj pl1hi1">Force perpendicular to a Plane.—Oblique -Force.—Inclined Plane.—Weight produces Pressure and Motion.—Motion -uniformly accelerated.—Space moved through in a given Time.—Increased -Elevation produces increased Force.—Perpendicular and horizontal -Plane.—Final Velocity.—Motion down a Curve.—Depends upon Velocity -and Curvature.—Centrifugal Force.—Circle of Curvature.—Radius of -Curvature.—Whirling Table.—Experiments.—Solar System.—Examples of -centrifugal Force.</td><td class="pl1 vab tar"><div><a href="#Page_85">85</a></div></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. IX.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>THE CENTRE OF GRAVITY.</div></td></tr> - -<tr><td class="taj pl1hi1">Terrestrial Attraction the combined Action of parallel -Forces.—Single equivalent Force.—Examples.—Method of finding -the Centre of<span class="pagenum" id="Page_vii">vii</span> Gravity.—Line of -Direction.—Globe.—Oblate Spheroid.—Prolate Spheroid.—Cube. -—Straight Wand.—Flat Plate.—Triangular Plate.—Centre of Gravity -not always within the Body.—A Ring.—Experiments.—Stable, instable, and -neutral Equilibrium. —Motion and Position of the Arms and Feet.—Effect of -the Knee-Joint.—Positions of a Dancer.—Porter under a Load.—Motion of a -Quadruped.—Rope Dancing.—Centre of Gravity of two Bodies separated from each -other.—Mathematical and experimental Examples. —The Conservation of the Motion -of the Centre of Gravity.—Solar System.—Centre of Gravity sometimes called Centre of -Inertia.</td><td class="pl1 vab tar"><div><a href="#Page_107">107</a></div></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. X.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>THE MECHANICAL PROPERTIES OF AN AXIS.</div></td></tr> - -<tr><td class="taj pl1hi1">An Axis.—Planets and common spinning Top.—Oscillation -or Vibration.—Instantaneous and continued Forces.—Percussion.—Continued -Force.—Rotation.—Impressed Forces.—Properties of a fixed -Axis.—Movement of the Force round the Axis.—Leverage of the Force.—Impulse -perpendicular to, but not crossing, the Axis.—Radius of Gyration.—Centre of -Gyration.—Moment of Inertia.—Principal Axes.—Centre of Percussion.</td><td -class="pl1 vab tar"><div><a href="#Page_128">128</a></div></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. XI.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>OF THE PENDULUM.</div></td></tr> - -<tr><td class="taj pl1hi1">Isochronism.—Experiments.—Simple Pendulum.—Examples -illustrative of.—Length of.—Experiments of Kater, Biot, Sabine, and -others.—Huygens’ Cycloidal Pendulum.</td><td class="pl1 vab tar"><div><a href="#Page_145">145</a></div></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. XII.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>OF SIMPLE MACHINES.</div></td></tr> - -<tr><td class="taj -pl1hi1">Statics.—Dynamics.—Force.—Power.—Weight.—Lever.—Cord.—Inclined -Plane.</td><td class="pl1 vab tar"><div><a href="#Page_160">160</a></div></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. XIII.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>OF THE LEVER.</div></td></tr> - -<tr><td class="taj pl1hi1">Arms.—Fulcrum.—Three Kinds of Levers.—Crow -Bar.—Handspike. —Oar.—Nutcrackers.—Turning -Lathe.—Steelyard.—Rectangular Lever.—Hammer.—Load between two -Bearers.—Combination of Levers.—Equivalent Lever.</td><td class="pl1 vab tar"> -<div><a href="#Page_167">167</a></div></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. XIV.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>OF WHEEL-WORK.</div></td></tr> - -<tr><td class="taj pl1hi1">Wheel and Axle.—Thickness of the Rope.—Ways of applying -the Power.—Projecting Pins.—Windlass.—Winch.—Axle.—Horizontal -Wheel.—Tread-Mill.—Cranes.—Water-Wheels. -—Paddle-Wheel.—Rachet-Wheel.—Rack.—Spring -of a Watch.—Fusee.—Straps or Cords.—Examples -of.—Turning Lathe.—Revolving Shafts.—Spinning -Machinery.—Saw-Mill.—Pinion.—Leaves. -—Crane.—Spur-Wheels.—Crown-Wheels.—Bevelled -Wheels.—Hunting-Cog.—Chronometers. —Hair-Spring.—Balance-Wheel.</td><td -class="pl1 vab tar"><div><a href="#Page_178">178</a></div><span class="pagenum" id="Page_viii">viii</span></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. XV.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>OF THE PULLEY.</div></td></tr> - -<tr><td class="taj pl1hi1">Cord.—Sheave.—Fixed Pulley.—Fire Escapes.—Single -moveable Pulley.—Systems of Pulleys.—Smeaton’s Tackle.—White’s -Pulley.—Advantage of.—Runner.—Spanish Bartons.</td><td class="pl1 vab tar"> -<div><a href="#Page_199">199</a></div></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. XVI.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>ON THE INCLINED PLANE, WEDGE, AND SCREW.</div></td></tr> - -<tr><td class="taj pl1hi1">Inclined Plane.—Effect of a Weight on.—Power -of.—Roads.—Power Oblique to the Plane.—Plane sometimes moves under the -Weight.—Wedge.—Sometimes formed of two inclined Planes.—More powerful as -its Angle is acute.—Where used.—Limits to the Angle.—Screw.—Hunter’s -Screw.—Examples.—Micrometer Screw.</td><td class="pl1 vab tar"><div><a href="#Page_209">209</a></div></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. XVII.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>ON THE REGULATION AND ACCUMULATION OF FORCE.</div></td></tr> - -<tr><td class="taj pl1hi1">Uniformity of Operation.—Irregularity of prime -Mover.—Water-Mill.—Wind-Mill.—Steam Pressure.—Animal -Power.—Spring.—Regulators.—Steam-Engine.—Governor.—Self-acting -Damper.—Tachometer.—Accumulation of -Power.—Examples.—Hammer.—Flail.—Bow-string.—Fire -Arms.—Air-Gun.—Steam-Gun.—Inert Matter a Magazine for -Force.—Fly-Wheel.—Condensed Air.—Rolling Metal.—Coining-Press.</td><td -class="pl1 vab tar"><div><a href="#Page_224">224</a></div></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. XVIII.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>MECHANICAL CONTRIVANCES FOR MODIFYING MOTION.</div></td></tr> - -<tr><td class="taj pl1hi1">Division of Motion into rectilinear and -rotatory.—Continued and reciprocating.—Examples.—Flowing -Water.—Wind.—Animal Motion.—Falling of a -Body.—Syringe-Pump.—Hammer.—Steam-Engine.—Fulling -Mill.—Rose-Engine.—Apparatus of Zureda.—Leupold’s Application -of it.—Hooke’s universal Joint.—Circular and alternate -Motion.—Examples.—Watt’s Methods of connecting the Motion of the Piston with that of -the Beam.—Parallel Motion.</td><td class="pl1 vab tar"><div><a href="#Page_245">245</a></div></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. XIX.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>OF FRICTION AND THE RIGIDITY OF CORDAGE.</div></td></tr> - -<tr><td class="taj pl1hi1">Friction and Rigidity.—Laws of Friction.—Rigidity of -Cordage.—Strength of Materials.—Resistance from Friction.—Independent -of the Magnitude of Surfaces.—Examples.—Vince’s Experiments.—Effect -of Velocity.—Means for diminishing Friction.—Friction Wheels.—Angle -of Repose.—Best Angle of Draught.—Rail-Roads.—Stiffness of Ropes.</td><td -class="pl1 vab tar"><div><a href="#Page_260">260</a></div></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. XX.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>ON THE STRENGTH OF MATERIALS.</div></td></tr> - -<tr><td class="taj pl1hi1">Difficulty of determining the Laws which govern the Strength -of Materials.—Forces tending to separate the Parts of a Solid.—Laws by -which<span class="pagenum" id="Page_ix">ix</span> Solids resist Compression.—Euler’s -theory.—Transverse Strength of Solids.—Strength diminished by the Increase of -Height.—Lateral or Transverse Strain.—Limits of Magnitude.—Relative -Strength of small Animals greater than large ones.</td><td class="pl1 vab tar"><div><a href="#Page_272">272</a></div></td></tr> - - -<tr><td class="tac pt1"><div>CHAP. XXI.</div></td></tr> - -<tr><td class="tac fs80 ptb08"><div>ON BALANCES AND PENDULUMS.</div></td></tr> - -<tr><td class="taj pl1hi1">Weight.—Time.—The Balance.—Fulcrum.—Centre -of Gravity of.—Sensibility of.—Positions of the Fulcrum.—Beam -variously constructed.—Troughton’s Balance.—Robinson’s -Balance.—Kater’s Balance.—Method of adjusting a Balance.—Use of -it.—Precautions necessary.—Of Weights.—Adjustment of.—Dr. -Black’s Balance.—Steelyard.—Roman Statera or Steelyard.—Convenience -of.—C. Paul’s Steelyard.—Chinese Steel-yard.—Danish Balance.—Bent Lever -Balance.—Brady’s Balance.—Weighing Machine for Turnpike Roads.—Instruments -for Weighing by means of a Spring.—Spring Steelyard.—Salter’s Spring -Balance.—Marriott’s Dial Weighing Machine.—Dynamometer.—Compensation -Pendulums.—Barton’s Gridiron Pendulum.—Table of linear Expansion.—Second -Table.—Harrison’s Pendulum.—Troughton’s Pendulum.—Benzenberg’s -Pendulum.—Ward’s Compensation Pendulum.—Compensation Tube of Julien -le Roy.—Deparcieux’s Compensation.—Kater’s Pendulum.—Reed’s -Pendulum.—Ellicott’s Pendulum.—Mercurial Pendulum.—Graham’s -Pendulum.—Compensation Pendulum of Wood and Lead.—Smeaton’s -Pendulum.—Brown’s Mode of Adjustment.</td><td class="pl1 vab tar"><div><a href="#Page_278">278</a></div></td></tr> -</table> -</div> - -<p><span class="pagenum hide" id="Page_1">1</span></p> -<hr class="chap x-ebookmaker-drop" /> - -<p class="tac">THE</p> - -<p class="tac fs160 ws03">ELEMENTS OF MECHANICS.</p> - -<hr class="r20 x-ebookmaker-drop" /> - -<div class="chapter"> -<h2 class="nobreak" id="CHAP_I">CHAP. I.<br /> - -<span class="title">PROPERTIES OF MATTER—MAGNITUDE—IMPENETRABILITY—FIGURE—FORCE.</span></h2> -</div> - - -<p id="p1">(1.) <span class="smcap">Placed</span> in the material world, Man is continually -exposed to the action of an infinite variety of objects by -which he is surrounded. The body, to which the thinking -and living principles have been united, is an apparatus -exquisitely contrived to receive and to transmit -impressions. Its various parts are organised with obvious -reference to the several external agents by which -it is to be effected. Each organ is designed to convey -to the mind immediate notice of some peculiar action, -and is accordingly endued with a corresponding susceptibility. -This adaptation of such organs to the particular -influences of material agents, is rendered still more conspicuous -when we consider that, however delicate its -structure, each organ is wholly insensible to every influence -except that to which it appears to be specially -appropriated. The eye, so intensely susceptible of -impressions from light, is not at all affected by those -of sound; while the fine mechanism of the ear, so sensitively -alive to every effect of the latter class, is altogether -insensible to the former. The splendour of excessive -light may occasion blindness, and deafness may -result from the roar of a cannonade; but neither the -sight nor the hearing can be injured by the most ex<span class="pagenum" id="Page_2">2</span>treme -action of that principle which is designed to affect -the other.</p> - -<p>Thus the organs of sense are instruments by which -the mind is enabled to determine the existence and the -qualities of external things. The effects which these -objects produce upon the mind through the organs, are -called <i>sensations</i>, and these sensations are the immediate -elements of all human knowledge. <span class="smcap">Matter</span> is the -general name which has been given to that substance, -which, under forms infinitely various, affects the senses. -Metaphysicians have differed in defining this principle. -Some have even doubted of its existence. But these -discussions are beyond the sphere of mechanical philosophy, -the conclusions of which are in nowise affected -by them. Our investigations here relate, not to matter -as an abstract existence, but to those qualities which we -discover in it by the senses, and of the existence of -which we are sure, however the question as to matter -itself may be decided. When we speak of “bodies,” -we mean those things, whatever they be, which excite -in our minds certain sensations; and the powers to -excite those sensations are called “properties,” or -“qualities.”</p> - -<p id="p2">(2.) To ascertain by observation the properties of -bodies, is the first step towards obtaining a knowledge -of nature. Hence man becomes a natural philosopher -the moment he begins to feel and to perceive. The -first stage of life is a state of constant and curious excitement. -Observation and attention, ever awake, are -engaged upon a succession of objects new and wonderful. -The large repository of the memory is opened, and -every hour pours into it unbounded stores of natural -facts and appearances, the rich materials of future knowledge. -The keen appetite for discovery implanted in -the mind for the highest ends, continually stimulated -by the presence of what is novel, renders torpid every -other faculty, and the powers of reflection and comparison -are lost in the incessant activity and unexhausted -vigour of observation. After a season, however, the<span class="pagenum" id="Page_3">3</span> -more ordinary classes of phenomena cease to excite by -their novelty. Attention is drawn from the discovery -of what is new, to the examination of what is familiar. -From the external world the mind turns in upon itself, -and the feverish astonishment of childhood gives place -to the more calm contemplation of incipient maturity. -The vast and heterogeneous mass of phenomena collected -by past experience is brought under review. The great -work of comparison begins. Memory produces her -stores, and reason arranges them. Then succeed those -first attempts at generalisation which mark the dawn -of science in the mind.</p> - -<p>To compare, to classify, to generalise, seem to be -instinctive propensities peculiar to man. They separate -him from inferior animals by a wide chasm. It is -to these powers that all the higher mental attributes -may be traced, and it is from their right application -that all progress in science must arise. Without these -powers, the phenomena of nature would continue a -confused heap of crude facts, with which the memory -might be loaded, but from which the intellect would -derive no advantage. Comparison and generalisation -are the great digestive organs of the mind, by which -only nutrition can be extracted from this mass of intellectual -food, and without which, observation the most -extensive, and attention the most unremitting, can be -productive of no real or useful advancement in knowledge.</p> - -<p id="p3">(3.) Upon reviewing those properties of bodies which -the senses most frequently present to us, we observe -that very few of them are essential to, and inseparable -from, matter. The greater number may be called <i>particular</i> -or <i>peculiar qualities</i>, being found in some bodies -but not in others. Thus the property of attracting -iron is peculiar to the loadstone, and not observable in -other substances. One body excites the sensation of -green, another of red, and a third is deprived of all -colour. A few characteristic and essential qualities are, -however, inseparable from matter in whatever state, or<span class="pagenum" id="Page_4">4</span> -under whatever form it exist. Such properties alone -can be considered as tests of materiality. Where their -presence is neither manifest to sense, nor demonstrable -by reason, <i>there</i> matter is not. The principal of these -qualities are <i>magnitude</i> and <i>impenetrability</i>.</p> - -<p id="p4">(4.) <i>Magnitude.</i>—Every body occupies space, that is, -it has magnitude. This is a property observable by the -senses in all bodies which are not so minute as to elude -them, and which the understanding can trace to the -smallest particle of matter. It is impossible, by any -stretch of imagination, even to conceive a portion of -matter so minute as to have no magnitude.</p> - -<p>The <i>quantity</i> of space which a body occupies is sometimes -called its <i>magnitude</i>. In colloquial phraseology, -the word <i>size</i> is used to express this notion; but the -most correct term, and that which we shall generally -adopt is <i>volume</i>. Thus we say, the volume of the earth -is so many cubic miles, the volume of this room is so -many cubic feet.</p> - -<p>The external limits of the magnitude of a body are -<i>lines</i> and <i>surfaces</i>, lines being the limits which separate -the several surfaces of the same body. The linear -limits of a body are also called <i>edges</i>. Thus the line -which separates the top of a chest from one of its sides -is called an edge.</p> - -<p>The <i>quantity</i> of a surface is called its <i>area</i>, and the -<i>quantity</i> of a line is called its <i>length</i>. Thus we say, the -<i>area</i> of a field is so many acres, the <i>length</i> of a rope is so -many yards. The word “magnitude” is, however, often -used indifferently for volume, area, and length. If the -objects of investigation were of a more complex and subtle -character, as in metaphysics, this unsteady application -of terms might be productive of confusion, and even -of error; but in this science the meaning of the term -is evident, from the way in which it is applied, and no -inconvenience is found to arise.</p> - -<p id="p5">(5.) <i>Impenetrability.</i>—This property will be most -clearly explained by defining the positive quality from -which it takes its name, and of which it merely signifies<span class="pagenum" id="Page_5">5</span> -the absence. A substance would be <i>penetrable</i> if it were -such as to allow another to pass through the space which -it occupies, without disturbing its component parts. Thus, -if a comet striking the earth could enter it at one side, -and, passing through it, emerge from the other without -separating or deranging any bodies on or within the -earth, then the earth would be penetrable by the comet. -When bodies are said to be impenetrable, it is therefore -meant that one cannot pass through another without -displacing some or all of the component parts of that -other. There are many instances of apparent penetration; -but in all these, the parts of the body which -seem to be penetrated are displaced. Thus, if the -point of a needle be plunged in a vessel of water, all the -water which previously filled the space into which the -needle enters will be displaced, and the level of the -water will rise in the vessel to the same height as it -would by pouring in so much more water as would fill -the space occupied by the needle.</p> - -<p id="p6">(6.) <i>Figure.</i>—If the hand be placed upon a solid body, -we become sensible of its impenetrability, by the obstruction -which it opposes to the entrance of the hand within -its dimensions. We are also sensible that this obstruction -commences at certain places; that it has certain determinate -limits; that these limitations are placed in certain -directions relatively to each other. The mutual relation -which is found to subsist between these boundaries of a -body, gives us the notion of its <i>figure</i>. The <i>figure</i> and -<i>volume</i> of a body should be carefully distinguished. -Each is entirely independent of the other. Bodies having -very different <i>volumes</i> may have the same <i>figure</i>; -and in like manner bodies differing in <i>figure</i> may have -the same <i>volume</i>. The figure of a body is what in popular -language is called its <i>shape</i> or <i>form</i>. The volume -of a body is that which is commonly called its <i>size</i>. It -will hence be easily understood, that one body (a globe, -for example) may have ten times the volume of another -(globe), and yet have the same figure; and that two -bodies (as a die and a globe) may have <i>figures</i> altogether<span class="pagenum" id="Page_6">6</span> -different, and yet have equal <i>volumes</i>. What we have -here observed of volumes will also be applicable to lengths -and areas. The arc of a circle and a straight line may -have the same length, although they have different -figures; and, on the other hand, two arcs of different -circles may have the same figure, but very unequal -lengths. The surface of a ball is curved, that of the -table plane; and yet the <i>area</i> of the surface of the ball -may be equal to that of the table.</p> - -<p id="p7">(7.) <i>Atoms—Molecules.</i>—Impenetrability must not -be confounded with inseparability. Every body which -has been brought under human observation is separable -into parts; and these parts, however small, are separable -into others, still more minute. To this process of -division no practical limit has ever been found. Nevertheless, -many of the phenomena which the researches of -those who have successfully examined the laws of nature -have developed, render it highly probable that all bodies -are composed of elementary parts which are indivisible -and unalterable. The component parts, which may be -called <i>atoms</i>, are so minute, as altogether to elude the -senses, even when aided by the most powerful scientific -instruments. The word <i>molecule</i> is often used to signify -component parts of a body so small as to escape sensible -observation, but not ultimate atoms, each molecule -being supposed to be formed of several atoms, arranged -according to some determinate figure. <i>Particle</i> is used -also to express small component parts, but more generally -is applied to those which are not too minute to be -discoverable by observation.</p> - -<p id="p8">(8.) <i>Force.</i>—If the particles of matter were endued -with no property in relation to one another, except their -mutual impenetrability, the universe would be like a -mass of sand, without variety of state or form. Atoms, -when placed in juxtaposition, would neither cohere, -as in solid bodies, nor repel each other, as in aeriform -substances. On the contrary, we find that in some -cases the atoms which compose bodies are not simply -placed together, but a certain effect is manifested in their<span class="pagenum" id="Page_7">7</span> -strong coherence. If they were merely placed in juxtaposition, -their separation would be effected as easily as -any one of them could be removed from one place to another. -Take a piece of iron, and attempt to separate its -parts: the effort will be strongly resisted, and it will -be a matter of much greater facility to move the whole -mass. It appears, therefore, that in such cases the parts -which are in juxtaposition <i>cohere</i> and resist their mutual -separation. This effect is denominated <i>force</i>; and -the constituent atoms are said to cohere with a greater -or less degree of force, according as they oppose a greater -or less resistance to their mutual separation.</p> - -<p>The coherence of particles in juxtaposition is an -effect of the same class as the mutual approach of particles -placed at a distance from each other. It is not -difficult to perceive that the same influence which causes -the bodies A and B to approach each other, when placed -at some distance asunder, will, when they unite, retain -them together, and oppose a resistance to their separation. -Hence this effect of the mutual approximation of bodies -towards each other is also called <i>force</i>.</p> - -<p>Force is generally defined to be “whatever produces -or opposes the production of motion in matter.” In this -sense, it is a name for the unknown cause of a known effect. -It would, however, be more philosophical to give the -name, not to the <i>cause</i>, of which we are ignorant, but -to the <i>effect</i>, of which we have sensible evidence. To -observe and to classify is the whole business of the natural -philosopher. When <i>causes</i> are referred to, it is -implied, that effects of the same class arise from the -agency of the same cause. However probable this assumption -may be, it is altogether unnecessary. All the -objects of science, the enlargement of mind, the extension -and improvement of knowledge, the facility of -its acquisition, are obtained by generalisation alone, and -no good can arise from tainting our conclusions with the -possible errors of hypotheses.</p> - -<p>It may be here, once for all, observed, that the -phraseology of causation and hypotheses has become so<span class="pagenum" id="Page_8">8</span> -interwoven with the language of science, that it is impossible -to avoid the frequent use of it. Thus, we say, -“the magnet <i>attracts</i> iron;” the expression <i>attract</i> -intimating the cause of the observed effect. In such -cases, however, we must be understood to mean the -<i>effect itself</i>, finding it less inconvenient to continue the -use of the received phrases, modifying their signification, -than to introduce new ones.</p> - -<p>Force, when manifested by the mutual approach or -cohesion of bodies, is also called <i>attraction</i>, and it is -variously denominated, according to the circumstances -under which it is observed to act. Thus, the force -which holds together the atoms of solid bodies is called -<i>cohesive attraction</i>. The force which draws bodies to -the surface of the earth, when placed above it, is called -the <i>attraction of gravitation</i>. The force which is exhibited -by the mutual approach, or adhesion, of the loadstone -and iron, is called <i>magnetic attraction</i>, and so on.</p> - -<p>When force is manifested by the motion of bodies from -each other, it is called <i>repulsion</i>. Thus, if a piece of glass, -having been briskly rubbed with a silk handkerchief, touch -successively two feathers, these feathers, if brought near -each other, will move asunder. This effect is called <i>repulsion</i>, -and the feathers are said to <i>repel</i> each other.</p> - -<p id="p9">(9.) The influence which forces have upon the form, -state, arrangement, and motions of material substances -is the principal object of physical science. In its strict -sense, <span class="smcap">Mechanics</span> is a term of very extensive signification. -According to the more popular usage, however, -it has been generally applied to that part of physical -science which includes the investigation of the phenomena -of motion and rest, pressure and other effects developed -by the mutual action of solid masses. The -consideration of similar phenomena, exhibited in bodies -of the liquid form, is consigned to <span class="smcap">Hydrostatics</span>, and -that of aeriform fluids to <span class="smcap">Pneumatics</span>.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_9">9</span></p> - -<h2 class="nobreak" id="CHAP_II">CHAP. II.<br /> - -<span class="title">DIVISIBILITY—POROSITY—DENSITY—COMPRESSIBILITY—ELASTICITY—DILATABILITY.</span></h2> -</div> - -<p id="p10">(10.) <span class="smcap">Besides</span> the qualities of magnitude and impenetrability, -there are several other general properties of -bodies contemplated in mechanical philosophy, and to -which we shall have frequent occasion to refer. Those -which we shall notice in the present chapter are,</p> - -<p class="ml2em"> -1. Divisibility.<br /> -2. Porosity—Density.<br /> -3. Compressibility—Elasticity.<br /> -4. Dilatability.<br /> -</p> - -<p id="p11">(11.) <i>Divisibility.</i>—Observation and experience prove -that all bodies of sensible magnitude, even the most -solid, consist of parts which are separable. To the -practical subdivision of matter there seems to be no -assignable limit. Numerous examples of the division -of matter, to a degree almost exceeding belief, may be -found in experimental enquiries instituted in physical -science; the useful arts furnish many instances not less -striking; but, perhaps, the most conspicuous proofs -which can be produced, of the extreme minuteness of -which the parts of matter are susceptible, arise from the -consideration of certain parts of the organised world.</p> - -<p id="p12">(12.) The relative places of stars in the heavens, as -seen in the field of view of a telescope, are marked by -fine lines of wire placed before the eye-glass, and which -cross each other at right angles. The stars appearing -in the telescope as mere lucid points without sensible -magnitude, it is necessary that the wires which mark -their places should have a corresponding tenuity. But -these wires being magnified by the eye-glass would have -an apparent thickness, which would render them inapplicable -to this purpose, unless their real dimensions -were of a most uncommon degree of minuteness. To -obtain wire for this purpose, Dr. Wollaston invented the<span class="pagenum" id="Page_10">10</span> -following process:—A piece of fine platinum wire, <i>a b</i>, -is extended along the axis of a cylindrical mould, A B, -<i><a href="#i_p058a">fig. 1.</a></i> Into this mould, at A, molten silver is poured. -Since the heat necessary for the fusion of platinum is much -greater than that which retains silver in the liquid form, -the wire <i>a b</i> remains solid, while the mould A B is filled -with the silver. When the metal has become solid by -being cooled, and has been removed from the mould, a -cylindrical bar of silver is obtained, having a platinum -wire in its axis. This bar is then wire-drawn, by forcing -it successively through holes C, D, E, F, G, H, diminishing -in magnitude, the first hole being a little less -than the wire at the beginning of the process. By these -means the platinum <i>a b</i> is wire-drawn at the same time -and in the same proportion with the silver, so that whatever -be the original proportion of the thickness of the -wire <i>a b</i> to that of the mould A B, the same will be the -proportion of the platinum wire to the whole at the -several thicknesses C, D, &c. If we suppose the mould -A B to be ten times the thickness of the wire <i>a b</i>, then -the silver wire, throughout the whole process, will be -ten times the thickness of the platinum wire which it -includes within it. The silver wire may be drawn to a -thickness not exceeding the 300th of an inch. The -platinum will thus not exceed the 3000th of an inch. -The wire is then dipped in nitric acid, which dissolves -the silver, but leaves the platinum solid. By this -method Dr. Wollaston succeeded in obtaining wire, the -diameter of which did not exceed the 18000th of an -inch. A quantity of this wire, equal in bulk to a common -die used in games of chance, would extend from -Paris to Rome.</p> - -<p id="p13">(13.) Newton succeeded in determining the thickness -of very thin laminæ of transparent substances by observing -the colours which they reflect. A soap bubble -is a thin shell of water, and is observed to reflect different -colours from different parts of its surface. Immediately -before the bubble bursts, a black spot may be -observed near the top. At this part the thickness has<span class="pagenum" id="Page_11">11</span> -been proved not to exceed the 2,500,000th of an -inch.</p> - -<p>The transparent wings of certain insects are so attenuated -in their structure that 50,000 of them placed -over each other would not form a pile a quarter of an -inch in height.</p> - -<p id="p14">(14.) In the manufacture of embroidery it is necessary -to obtain very fine gilt silver threads. To accomplish -this, a cylindrical bar of silver, weighing 360 -ounces, is covered with about two ounces of gold. This -gilt bar is then wire-drawn, as in the first example, -until it is reduced to a thread so fine that 3400 feet of -it weigh less than an ounce. The wire is then flattened -by passing it between rollers under a severe pressure, a -process which increases its length, so that about 4000 -feet shall weigh one ounce. Hence, one foot will weigh -the 4000th part of an ounce. The proportion of the gold -to the silver in the original bar was that of 2 to 360, or -1 to 180. Since the same proportion is preserved after -the bar has been wire-drawn, it follows that the quantity -of gold which covers one foot of the fine wire is the -180th part of the 4000th of an ounce; that is the -720,000th part of an ounce.</p> - -<p>The quantity of gold which covers one inch of this -wire will be twelve times less than that which covers -one foot. Hence, this quantity will be the 8,640,000th -part of an ounce. If this inch be again divided into -100 equal parts, every part will be distinctly visible -without the aid of microscopes. The gold which covers -this small but visible portion is the 864,000,000th -part of an ounce. But we may proceed even further; -this portion of the wire may be viewed by a microscope -which magnifies 500 times, so that the 500th part of -it will thus become visible. In this manner, therefore, -an ounce of gold may be divided into 432,000,000,000 -visible parts, each of which will possess all the characters -and qualities found in the largest masses of the -metal. It will retain its solidity, texture, and colour; -it will resist the same agents, and enter into combination -with the same substances. If the gilt wire be dipped<span class="pagenum" id="Page_12">12</span> -in nitric acid, the silver within the coating will be dissolved, -but the hollow tube of gold which surrounded it -will still cohere and remain suspended.</p> - -<p id="p15">(15.) The organised world offers still more remarkable -examples of the inconceivable subtilty of matter.</p> - -<p>The blood which flows in the veins of animals is not, -as it seems, an uniformly red liquid. It consists of -flat discs of a red colour, floating in a transparent fluid -called <i>serum</i>. In different species these discs differ both -in figure and in magnitude. In man and all animals -which suckle their young, they are perfectly circular or -nearly so. In birds, reptiles, and fishes, they are of oval -form. In the human species, the diameter of these -discs is about the 3500th of an inch. Hence it follows, -that in a drop of blood which would remain suspended -from the point of a fine needle, there must be about -3,000,000 of such discs.</p> - -<p>Small as these discs are, the animal kingdom presents -beings whose whole bodies are still more minute. -Animalcules have been discovered, whose magnitude is -such, that a million of them do not exceed the bulk -of a grain of sand; and yet each of these creatures is -composed of members as curiously organised as those of -the largest species; they have life and spontaneous motion, -and are endued with sense and instinct. In the -liquids in which they live, they are observed to move -with astonishing speed and activity; nor are their motions -blind and fortuitous, but evidently governed by -choice, and directed to an end. They use food and -drink, from which they derive nutrition, and are therefore -furnished with a digestive apparatus. They have -great muscular power, and are furnished with limbs and -muscles of strength and flexibility. They are susceptible -of the same appetites, and obnoxious to the same -passions, the gratification of which is attended with -the same results as in our own species. Spallanzani observes, -that certain animalcules devour others so voraciously, -that they fatten and become indolent and sluggish -by over-feeding. After a meal of this kind, if they be<span class="pagenum" id="Page_13">13</span> -confined in distilled water, so as to be deprived of all -food, their condition becomes reduced; they regain -their spirit and activity, and amuse themselves in the -pursuit of the more minute animals, which are supplied -to them; they swallow these without depriving them of -life, for, by the aid of the microscope, the one has been -observed moving within the body of the other. These -singular appearances are not matters of idle and curious -observation. They lead us to enquire what parts are -necessary to produce such results. Must we not conclude -that these creatures have heart, arteries, veins, -muscles, sinews, tendons, nerves, circulating fluids, and -all the concomitant apparatus of a living organised body? -And if so, how inconceivably minute must those parts -be! If a globule of their blood bears the same proportion -to their whole bulk as a globule of our blood bears -to our magnitude, what powers of calculation can give -an adequate notion of its minuteness?</p> - -<p id="p16">(16.) These and many other phenomena observed in the -immediate productions of nature, or developed by mechanical -and chemical processes, prove that the materials -of which bodies are formed are susceptible of minuteness -which infinitely exceeds the powers of sensible observation, -even when those powers have been extended by all -the aids of science. Shall we then conclude that matter -is infinitely divisible, and that there are no original constituent -atoms of determinate magnitude and figure at -which all subdivision must cease? Such an inference -would be unwarranted, even had we no other means of -judging the question, except those of direct observation; -for it would be imposing that limit on the works of -nature which she has placed upon our powers of observing -them. Aided by reason, however, and a due consideration -of certain phenomena which come within our -immediate powers of observation, we are frequently able -to determine other phenomena which are beyond those -powers. The diurnal motion of the earth is not perceived -by us, because all things around us participate in -it, preserve their relative position, and appear to be at<span class="pagenum" id="Page_14">14</span> -rest. But reason tells us that such a motion must produce -the alternations of day and night, and the rising -and setting of all the heavenly bodies; appearances which -are plainly observable, and which betray the cause from -which they arise. Again, we cannot place ourselves at a -distance from the earth, and behold the axis on which it -revolves, and observe its peculiar obliquity to the orbit -in which the earth moves; but we see and feel the -vicissitudes of the seasons, an effect which is the immediate -consequence of that inclination, and by which we -are able to detect it.</p> - -<p id="p17">(17.) So it is in the present case. Although we are unable -by direct observation to prove the existence of constituent -material atoms of determinate figure, yet there are -many observable phenomena which render their existence -in the highest degree probable, if not morally certain. -The most remarkable of this class of effects is observed in -the crystallisation of salts. When salt is dissolved in a -sufficient quantity of pure water, it mixes with the water -in such a manner as wholly to disappear to the sight and -touch, the mixture being one uniform transparent liquid -like the water itself, before its union with the salt. The -presence of the salt in the water may, however, be ascertained -by weighing the mixture, which will be found to -exceed the original weight of the water by the exact -amount of the weight of the salt. It is a well-known -fact, that a certain degree of heat will convert water -into vapour, and that the same degree of heat does not -produce the same effect upon salt. The mixture of -salt and water being exposed to this temperature, the -water will gradually evaporate, disengaging itself from -the salt with which it has been combined. When so -much of the water has evaporated, that what remains is -insufficient to keep in solution the whole of the salt, a -part of the latter thus disengaged from the water will -return to the solid state. The saline constituent will -not in this case collect in irregular solid molecules; but -will exhibit itself in particles of regular figure, terminated -by plane surfaces, the figure being always the same -for the same species of salt, but different for different<span class="pagenum" id="Page_15">15</span> -species. These particles are called <i>crystals</i>. There are -several circumstances in the formation of these <i>crystals</i> -which merit attention.</p> - -<p>If one of them be detached from the others, and the -progress of its formation observed, it will be found gradually -to increase, always preserving its original figure. -Since its increase must be caused by the continued accession -of saline molecules, disengaged by the evaporation -of the water, it follows that these molecules must be so -formed, that by attaching themselves successively to the -crystal, they maintain the regularity of its bounding -planes, and preserve their mutual inclinations unvaried.</p> - -<p>Suppose a crystal to be taken from the liquid during -the process of crystallisation, and a piece broken from it -so as to destroy the regularity of its form: if the crystal -thus broken be restored to the liquid, it will be observed -gradually to resume its regular form, the atoms of salt -successively dismissed by the vaporising water filling up -the irregular cavities produced by the fracture. Hence -it follows, that the saline particles which compose the -surface of the crystal, and those which form the interior -of its mass, are similar, and exert similar attractions on -the atoms disengaged by the water.</p> - -<p>All these details of the process of crystallisation are -very evident indications of a determinate figure in the -ultimate atoms of the substances which are crystallised. -But besides the substances which are thus reduced by art -to the form of crystals, there are larger classes which -naturally exist in that state. There are certain planes, -called <i>planes of cleavage</i>, in the directions of which natural -crystals are easily divided. These planes, in substances -of the same kind, always have the same relative -position, but differ in different substances. The surfaces -of the planes of cleavage are quite invisible before the -crystal is divided; but when the parts are separated, -these surfaces exhibit a most intense polish, which no -effort of art can equal.</p> - -<p>We may conceive crystallised substances to be regular -mechanical structures formed of atoms of a certain<span class="pagenum" id="Page_16">16</span> -figure, on which the figure of the whole structure must -depend. The planes of cleavage are parallel to the -sides of the constituent atoms; and their directions, -therefore, form so many conditions for the determination -of their figure. The shape of the atoms being thus determined, -it is not difficult to assign all the various ways -in which they may be arranged, so as to produce figures -which are accordingly found to correspond with the -various forms of crystals of the same substance.</p> - -<p id="p18">(18.) When these phenomena are duly considered -and compared, little doubt can remain that all substances -susceptible of crystallisation, consist of atoms of determinate -figure. This is the case with all solid bodies -whatever, which have come under scientific observation, -for they have been severally found in or reduced to a -crystallised form. Liquids crystallise in freezing, and -if aëriform fluids could by any means be reduced to the -solid form, they would probably also manifest the same -effect. Hence it appears reasonable to presume, that -all bodies are composed of atoms; that the different -qualities with which we find different substances endued, -depend on the magnitude and figure of these atoms; -that these atoms are indestructible and immutable by -any natural process, for we find the qualities which -depend on them unchangeably the same under all the -influences to which they have been submitted since their -creation; that these atoms are so minute in their magnitude, -that they cannot be observed by any means -which human art has yet contrived; but still that magnitudes -can be assigned which they do not exceed.</p> - -<p>It is proper, however, to observe here, that the various -theorems of mechanical science do not rest upon -any hypothesis concerning these atoms as a basis. These -theorems are not inferred from this or any other supposition, -and therefore their truth would not be in anywise -disturbed, even though it should be established that -matter is physically divisible <i>in infinitum</i>. The basis -of mechanical science is <i>observed facts</i>, and, since the -reasoning is demonstrative, the conclusions have the<span class="pagenum" id="Page_17">17</span> -same degree of certainty as the facts from which they -are deduced.</p> - -<p id="p19">(19.) <i>Porosity.</i>—The <i>volume</i> of a body is the quantity -of space included within its external surfaces. The -<i>mass</i> of a body, is the collection of atoms or material -particles of which it consists. Two atoms or particles -are said to be in contact, when they have approached -each other until arrested by their mutual impenetrability. -If the component particles of a body were in -contact, the <i>volume</i> would be completely occupied by -the <i>mass</i>. But this is not the case. We shall presently -prove, that the component particles of no known -substance are in absolute contact. Hence it follows that -the volume consists partly of material particles, and -partly of interstitial spaces, which spaces are either absolutely -void and empty, or filled by some substance of -a different kind from the body in question. These -interstitial spaces are called <i>pores</i>.</p> - -<p>In bodies which are constituted uniformly throughout -their entire dimensions, the component particles and the -pores are uniformly distributed through the volume; -that is, a given space in one part of the volume will -contain the same quantity of matter and the same -quantity of pores as an equal space in another part.</p> - -<p id="p20">(20.) The proportion of the quantity of matter to -the magnitude is called the <i>density</i>. Thus if of two -substances, one contain in a given space twice as much -matter as the other, it is said to be “twice as dense.” -The density of bodies is, therefore, proportionate to -the closeness or proximity of their particles; and it is -evident, that the greater the density, the less will be the -porosity.</p> - -<p>The pores of a body are frequently filled with another -body of a more subtle nature. If the pores of a body -on the surface of the earth, and exposed to the atmosphere, -be greater than the atoms of air, then the air may -pervade the pores. This is found to be the case with -many sorts of wood which have an open grain. If a piece -of such wood, or of chalk, or of sugar, be pressed to the<span class="pagenum" id="Page_18">18</span> -bottom of a vessel of water, the air which fills the pores -will be observed to escape in bubbles and to rise to the -surface, the water entering the pores, and taking its -place.</p> - -<p>If a tall vessel or tube, having a wooden bottom, be -filled with quicksilver, the liquid metal will be forced -by its own weight through the pores of the wood, and -will be seen escaping in a silver shower from the bottom.</p> - -<p id="p21">(21.) The process of filtration, in the arts, depends -on the presence of pores of such a magnitude as to -allow a passage to the liquid, but to refuse it to those -impurities from which it is to be disengaged. Various -substances are used as filtres; but, whatever be used, this -circumstance should always be remembered, that no -substance can be separated from a liquid by filtration, -except one whose particles are larger than those of the -liquid. In general, filtres are used to separate <i>solid</i> impurities -from a liquid. The most ordinary filtres are -soft stone, paper, and charcoal.</p> - -<p id="p22">(22.) All organised substances in the animal and -vegetable kingdoms are, from their very natures, porous -in a high degree. Minerals are porous in various degrees. -Among the silicious stones is one called <i>hydrophane</i>, -which manifests its porosity in a very remarkable -manner. The stone, in its ordinary state, is semi-transparent. -If, however, it be plunged in water, when it -is withdrawn it is as translucent as glass. The pores, -in this case, previously filled with air, are pervaded by -the water, between which and the stone there subsists a -physical relation, by which the one renders the other -perfectly transparent.</p> - -<p>Larger mineral masses exhibit degrees of porosity not -less striking. Water percolates through the sides and -roofs of caverns and grottoes, and being impregnated -with calcareous and other earths, forms stalactites, or pendant -protuberances, which present a curious appearance.</p> - -<p id="p23">(23.) <i>Compressibility.</i>—That quality, in virtue of -which a body allows its volume to be diminished without -diminishing its mass, is called <i>compressibility</i>. This<span class="pagenum" id="Page_19">19</span> -effect is produced by bringing the constituent particles -more close together, and thereby increasing the density -and diminishing the pores. This effect may be produced -in several ways; but the name “compressibility” -is only applied to it when it is caused by the agency of -mechanical force, as by pressure or percussion.</p> - -<p>All known bodies, whatever be their nature, are capable -of having their dimensions reduced without diminishing -their mass; and this is one of the most conclusive -proofs that all bodies are porous, or that the constituent -atoms are not in contact; for the space by which the -volume may be diminished must, before the diminution, -consist of pores.</p> - -<p id="p24">(24.) <i>Elasticity.</i>—Some bodies, when compressed by -mechanical agency, will resume their former dimensions -with a certain energy when relieved from the operation of -the force which has compressed them. This property is -called <i>elasticity</i>; and it follows, from this definition, -that all elastic bodies must be compressible, although the -converse is not true, compressibility not necessarily implying -elasticity.</p> - -<p id="p25">(25.) <i>Dilatability.</i>—This quality is the opposite of -compressibility. It is the capability observed in bodies -to have their volume enlarged without increasing their -mass. This effect may be produced in several ways. -In ordinary circumstances, a body may exist under the -constant action of a pressure by which its volume and -density are determined. It may happen, that on the occasional -removal of that pressure, the body will <i>dilate</i> -by a quality inherent in its constitution. This is the -case with common air. Dilatation may also be the effect -of heat, as will presently appear.</p> - -<p>The several qualities of bodies which we have noticed -in this chapter, when viewed in relation to each other, -present many circumstances worthy of attention.</p> - -<p id="p26">(26.) It is a physical law, of high generality, that an -increase in the temperature, or degree of heat by which -a body is affected, is accompanied by an increase of -volume; and that a diminution of temperature is ac<span class="pagenum" id="Page_20">20</span>companied -by a diminution of volume. The exceptions -to this law will be noticed and explained in our treatise -on <span class="smcap">Heat</span>. Hence it appears that the reduction of -temperature is an effect which, considered mechanically, -is equivalent to compression or condensation, since it -diminishes the volume without altering the mass; and -since this is an effect of which all bodies whatever -are susceptible, it follows that all bodies whatever have -<i>pores</i>. (<a href="#p23">23</a>.)</p> - -<p>The fact, that the elevation of temperature produces -an increase of volume, is manifested by numerous experiments.</p> - -<p id="p27">(27.) If a flaccid bladder be tied at the mouth, so as -to stop the escape of air, and be then held before a fire, -it will gradually swell, and assume the appearance of -being fully inflated. The small quantity of air contained -in the bladder is, in this case, so much dilated by the heat, -that it occupies a considerably increased space, and fills -the bladder, of which it before only occupied a small -part. When the bladder is removed from the fire, and -allowed to resume its former temperature, the air returns -to its former dimensions, and the bladder becomes again -flaccid.</p> - -<p id="p28">(28.) Let A B, <i><a href="#i_p058a">fig. 2.</a></i> be a glass tube, with a bulb at -the end A; and let the bulb A, and a part of the tube, be -filled with any liquid, coloured so as to be visible. Let -C be the level of the liquid in the tube. If the bulb be -now exposed to heat, by being plunged in hot water, the -level of the liquid C will rapidly rise towards B. This -effect is produced by the dilatation of the liquid in the -bulb, which filling a greater space, a part of it is forced -into the tube. This experiment may easily be made with -a common glass tube and a little port wine.</p> - -<p>Thermometers are constructed on this principle, the -rise of the liquid in the tube being used as an indication -of the degree of heat which causes it. A particular account -of these useful instruments will be found in our -treatise on <span class="smcap">Heat</span>.</p> - -<p id="p29">(29.) The change of dimension of solids produced by<span class="pagenum" id="Page_21">21</span> -changes of temperature being much less than that of -bodies in the liquid or aeriform state, is not so easily -observable. A remarkable instance occurs in the process -of shoeing the wheels of carriages. The rim of iron with -which the wheel is to be bound, is made in the first instance -of a diameter somewhat less than that of the -wheel; but being raised by the application of fire to a -very high temperature, its volume receives such an increase, -that it will be sufficient to embrace and surround -the wheel. When placed upon the wheel it is cooled, -and suddenly contracting its dimensions, binds the parts -of the wheel firmly together, and becomes securely seated -in its place upon the fellies.</p> - -<p id="p30">(30.) It frequently happens that the stopper of a glass -bottle or decanter becomes fixed in its place so firmly, that -the exertion of force sufficient to withdraw it would endanger -the vessel. In this case, if a cloth wetted with -hot-water be applied to the neck of the bottle, the glass -will expand, and the neck will be enlarged, so as to allow -the stopper to be easily withdrawn.</p> - -<p id="p31">(31.) The contraction of metal consequent upon -change of temperature was applied some time ago in -Paris to restore the walls of a tottering building to their -proper position. In the <i>Conservatoire des Arts et Métiers</i>, -the walls of a part of the building were forced out -of the perpendicular by the weight of the roof, so that -each wall was leaning outwards. M. Molard conceived -the notion of applying the irresistible force with which -metals contract in cooling, to draw the walls together. -Bars of iron were placed in parallel directions across the -building, and at right-angles to the direction of the walls. -Being passed through the walls, nuts were screwed on -their ends, outside the building. Every alternate bar -was then heated by lamps, and the nuts screwed close to -the walls. The bars were then cooled, and the lengths -being diminished by contraction, the nuts on their extremities -were drawn together, and with them the walls -were drawn through an equal space. The same process -was repeated with the intermediate bars, and so on alter<span class="pagenum" id="Page_22">22</span>nately -until the walls were brought into a perpendicular -position.</p> - -<p id="p32">(32.) Since there is a continual change of temperature -in all bodies on the surface of the globe, it follows, -that there is also a continual change of magnitude. -The substances which surround us are constantly -swelling and contracting, under the vicissitudes of heat -and cold. They grow smaller in winter, and dilate in -summer. They swell their bulk on a warm day, and -contract it on a cold one. These curious phenomena -are not noticed, only because our ordinary means of observation -are not sufficiently accurate to appreciate them. -Nevertheless, in some familiar instances the effect is -very obvious. In warm weather the flesh swells, the -vessels appear filled, the hand is plump, and the skin -distended. In cold weather, when the body has been -exposed to the open air, the flesh appears to contract, -the vessels shrink, and the skin shrivels.</p> - -<p id="p33">(33.) The phenomena attending change of temperature -are conclusive proofs of the universal porosity -of material substances, but they are not the only proofs. -Many substances admit of compression by the mere -agency of mechanical force.</p> - -<p>Let a small piece of cork be placed floating on the -surface of water in a basin or other vessel, and an empty -glass goblet be inverted over the cork, so that its edge -just meets the water. A portion of air will then be -confined in the goblet, and detached from the remainder -of the atmosphere. If the goblet be now pressed downwards, -so as to be entirely immersed, it will be observed, -that the water will not fill it, being excluded by the -<i>impenetrability</i> of the air inclosed in it. This experiment, -therefore, is decisive of the fact, that air, one of -the most subtle and attenuated substances we know of, -possesses the quality of impenetrability. It absolutely -excludes any other body from the space which it occupies -at any given moment.</p> - -<p>But although the water does not fill the goblet, yet if -the position of the cork which floats upon its surface be<span class="pagenum" id="Page_23">23</span> -noticed, it will be found that the level of the water -within has risen above its edge or rim. In fact, the -water has partially filled the goblet, and the air has been -forced to contract its dimensions. This effect is produced -by the pressure of the incumbent water forcing -the surface in the goblet against the air, which yields -until it is so far compressed that it acquires a force able -to withstand this pressure. Thus it appears that air is -capable of being reduced in its dimensions by mechanical -pressure, independently of the agency of heat. It is -<i>compressible</i>.</p> - -<p>That this effect is the consequence of the pressure of -the liquid will be easily made manifest by showing -that, as the pressure is increased, the air is proportionally -contracted in its dimensions; and as it is diminished, -the dimensions are on the other hand enlarged. If the -depth of the goblet in the water be increased, the cork -will be seen to rise in it, showing that the increased -pressure, at the greater depth, causes the air in the goblet -to be more condensed. If, on the other hand, the -goblet be raised toward the surface, the cork will be -observed to descend toward the edge, showing that as -it is relieved from the pressure of the liquid, the air -gradually approaches to its primitive dimensions.</p> - -<p id="p34">(34.) These phenomena also prove, that air has the -property of <i>elasticity</i>. If it were simply compressible, -and not elastic, it would retain the dimensions to which -it was reduced by the pressure of the liquid; but this is -not found to be the result. As the compressing force is -diminished, so in the same proportion does the air, by -its elastic virtue, exert a force by which it resumes its -former dimensions.</p> - -<p>That it is the air alone which excludes the water from -the goblet, in the preceding experiments, can easily be -proved. When the goblet is sunk deep in the vessel of -water, let it be inclined a little to one side until its mouth -is presented towards the side of the vessel; let this inclination -be so regulated, that the surface of the water -in the goblet shall just reach its edge. Upon a slight<span class="pagenum" id="Page_24">24</span> -increase of inclination, air will be observed to escape -from the goblet, and to rise in bubbles to the surface of -the water. If the goblet be then restored to its position, -it will be found that the cork will rise higher in it than -before the escape of the air. The water in this case -rises and fills the space which the air allowed to escape -has deserted. The same process may be repeated until -all the air has escaped, and then the goblet will be completely -filled by the water.</p> - -<p id="p35">(35.) Liquids are compressible by mechanical force -in so slight a degree, that they are considered in all -hydrostatical treatises as incompressible fluids. They -are, however, not absolutely incompressible, but yield -slightly to very intense pressure. The question of the -compressibility of liquids was raised at a remote period -in the history of science. Nearly two centuries ago, an -experiment was instituted at the Academy <i>del Cimento</i> -in Florence, to ascertain whether water be compressible. -With this view, a hollow ball of gold was filled with the -liquid, and the aperture exactly and firmly closed. The -globe was then submitted to a very severe pressure, by -which its figure was slightly changed. Now it is proved -in geometry, that a globe has this peculiar property, -that any change whatever in its figure must necessarily -diminish its volume or contents. Hence it was inferred, -that if the water did not issue through the pores of the -gold, or burst the globe, its compressibility would be -established. The result of the experiment was, that the -water <i>did</i> ooze through the pores, and covered the surface -of the globe, presenting the appearance of dew, or -of steam cooled by the metal. But this experiment was -inconclusive. It is quite true, that if the water <i>had not</i> -escaped upon the change of figure of the globe, the <i>compressibility</i> -of the liquid would have been established. -The escape of the water does not, however, prove its -<i>incompressibility</i>. To accomplish this, it would be necessary -first to measure accurately the volume of water -which transuded by compression, and next to measure -the diminution of volume which the vessel suffered by<span class="pagenum" id="Page_25">25</span> -its change of figure. If this diminution were greater -than the volume of water which escaped, it would follow -that the water remaining in the globe had been compressed, -notwithstanding the escape of the remainder. -But this could never be accomplished with the delicacy -and exactitude necessary in such an experiment; and, -consequently, as far as the question of the compressibility -of water was concerned, nothing was proved. It forms, -however, a very striking illustration of the porosity of -so dense a substance as gold, and proves that its pores -are larger than the elementary particles of water, since -these are capable of passing through them.</p> - -<p id="p36">(36.) It has since been proved, that water, and -other liquids, are compressible. In the year 1761, -Canton communicated to the Royal Society the results -of some experiments which proved this fact. He provided -a glass tube with a bulb, such as that described -in (<a href="#p28">28</a>), and filled the bulb and a part of the tube with -water well purified from air. He then placed this -in an apparatus called a condenser, by which he was -enabled to submit the surface of the liquid in the tube -to very intense pressure of condensed air. He found -that the level of the liquid in the tube fell in a perceptible -degree upon the application of the pressure. -The same experiment established the fact, that liquids -are <i>elastic</i>; for upon removing the pressure, the liquid -rose to its original level, and therefore resumed its former -dimensions.</p> - -<p id="p37">(37.) Elasticity does not always accompany compressibility. -If lead or iron be submitted to the hammer, -it may be hardened and diminished in its volume; but -it will not resume its former volume after each stroke -of the hammer.</p> - -<p id="p38">(38.) There are some bodies which maintain the state -of density in which they are commonly found by the continual -agency of mechanical pressure; and such bodies -are endued with a quality, in virtue of which they would -enlarge their dimensions without limit, if the pressure -which confines them were removed. Such bodies are<span class="pagenum" id="Page_26">26</span> -called <i>elastic fluids</i> or <i>gases</i>, and always exist in the form -of common air, in whose mechanical properties they participate. -They are hence often called <i>aeriform fluids</i>.</p> - -<p>Those who are provided with an air-pump can easily -establish this property experimentally. Take a flaccid -bladder, such as that already described in (<a href="#p27">27</a>.), and -place it under the glass receiver of an air-pump: by -this instrument we shall be able to remove the air which -surrounds the bladder under the receiver, so as to relieve -the small quantity of air which is inclosed in the bladder -from the pressure of the external air: when this is -accomplished, the bladder will be observed to swell, as if -it were inflated, and will be perfectly distended. The -air contained in it, therefore, has a tendency to dilate, -which takes effect when it ceases to be resisted by the -pressure of surrounding air.</p> - -<p id="p39">(39.) It has been stated that the increase or diminution -of temperature is accompanied by an increase or -diminution of volume. Related to this, there is another -phenomenon too remarkable to pass unnoticed, although -this is not the proper place to dwell upon it: it is the -converse of the former; viz. that an increase or diminution -of bulk is accompanied by a diminution or increase -of temperature. As the application of heat from some -foreign source produces an increase of dimensions, so if -the dimensions be increased from any other cause, a corresponding -portion of the heat which the body had before -the enlargement, will be absorbed in the process, and the -temperature will be thereby diminished. In the same -way, since the abstraction of heat causes a diminution of -volume, so if that diminution be caused by any other -means, the body will <i>give out</i> the heat which in the other -case was abstracted, and will rise in its temperature.</p> - -<p>Numerous and well-known facts illustrate these observations. -A smith by hammering a piece of bar iron, -and thereby compressing it, will render it <i>red hot</i>. -When air is violently compressed, it becomes so hot as -to ignite cotton and other substances. An ingenious -instrument for producing a light for domestic uses has<span class="pagenum" id="Page_27">27</span> -been constructed, consisting of a small cylinder, in which -a solid piston moves air-tight: a little tinder, or dry -sponge, is attached to the bottom of the piston, which is -then violently forced into the cylinder: the air between -the bottom of the cylinder and the piston becomes intensely -compressed, and evolves so much heat as to light -the tinder.</p> - -<p>In all the cases where friction or percussion produces -heat or fire, it is because they are means of compression. -The effects of flints, of pieces of wood rubbed together, -the warmth produced by friction on the flesh, are all to -be attributed to the same cause.</p> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<h2 class="nobreak" id="CHAP_III">CHAP. III.<br /> - -<span class="title">INERTIA.</span></h2> -</div> - - -<p id="p40">(40.) <span class="smcap">The</span> quality of matter which is of all others the -most important in mechanical investigations, is that which -has been called <i>Inertia</i>.</p> - -<p>Matter is incapable of spontaneous change. This is -one of the earliest and most universal results of human -observation: it is equivalent to stating that mere matter -is deprived of life; for spontaneous action is the only -test of the presence of the living principle. If we see a -mass of matter undergo any change, we never seek for -the cause of that change in the body itself; we look for -some external cause producing it. This inability for -voluntary change of state or qualities is a more general -principle than inertia. At any given moment of time a -body must be in one or other of two states, rest or motion. -<i>Inertia</i>, or <i>inactivity</i>, signifies the total absence of power -to change this state. A body endued with inertia cannot -of itself, and independent of all external influence, commence -to move from a state of rest; neither can it when -moving arrest its progress and become quiescent.</p> - -<p id="p41">(41.) The same property by which a body is unable -by any power of its own to pass from a state of rest to<span class="pagenum" id="Page_28">28</span> -one of motion, or <i>vice versâ</i>, also renders it incapable of -increasing or diminishing any motion which it may have -received from an external cause. If a body be moving -in a certain direction at the rate of ten miles per hour, it -cannot by any energy of its own change its rate of motion -to eleven or nine miles an hour. This is a direct -consequence of that manifestation of inertia which has -just been explained. For the same power which would -cause a body moving at ten miles an hour to increase its -rate to eleven miles, would also cause the same body at -rest to commence moving at the rate of one mile an hour; -and the same power which would cause a body moving -at the rate of ten miles an hour to move at the rate of -nine miles in the hour, would cause the same body moving -at the rate of one mile an hour to become quiescent. -It therefore appears, that to increase or diminish the -motion of a body is an effect of the same kind as to -change the state of rest into that of motion, or <i>vice versâ</i>.</p> - -<p id="p42">(42.) The effects and phenomena which hourly fall -under our observation afford unnumbered examples of -the inability of lifeless matter to put itself into motion, -or to increase any motion which may have been communicated -to it. But it does not happen that we have -the same direct and frequent evidence of its inability to -destroy or diminish any motion which it may have received. -And hence it arises, that while no one will -deny to matter the former effect of inertia, few will at -first acknowledge the latter. Indeed, even so late as the -time of <span class="smcap">Kepler</span>, philosophers themselves held it as a -maxim, that “matter is more inclined to rest than to -motion;” we ought not, therefore, to be surprised if in -the present day those who have not been conversant -with physical science are slow to believe that a body -once put in motion would continue for ever to move -with the same velocity, if it were not stopped by some -external cause.</p> - -<p>Reason, assisted by observation, will, however, soon -dispel this illusion. Experience shows us in various -ways, that the same causes which destroy motion in one<span class="pagenum" id="Page_29">29</span> -direction are capable of producing as much motion in -the opposite direction. Thus, if a wheel, spinning on -its axis with a certain velocity, be stopped by a hand -seizing one of the spokes, the effort which accomplishes -this is exactly the same as, had the wheel been previously -at rest, would have put it in motion in the opposite direction -with the same velocity. If a carriage drawn -by horses be in motion, the same exertion of power in -the horses is necessary to stop it, as would be necessary -to <i>back</i> it, if it were at rest. Now, if this be admitted -as a general principle, it must be evident that a body -which can destroy or diminish its own motion must also -be capable of putting itself into motion from a state of -rest, or of increasing any motion which it has received. -But this latter is contrary to all experience, and therefore -we are compelled to admit that a body cannot diminish -or destroy any motion which it has received.</p> - -<p>Let us enquire why we are more disposed to admit -the inability of matter to produce than to destroy motion -in itself. We see most of those motions which take -place around us on the surface of the earth subject to -gradual decay, and if not renewed from time to time, -at length cease. A stone rolled along the ground, a -wheel revolving on its axis, the heaving of the deep -after a storm, and all other motions produced in bodies -by external causes, decay, when the exciting cause is -suspended; and if that cause do not renew its action, -they ultimately cease.</p> - -<p>But is there no exciting cause, on the other hand, -which thus gradually deprives those bodies of their -motion?—and if that cause were removed, or its intensity -diminished, would not the motion continue, or be more -slowly retarded? When a stone is rolled along the -ground, the inequalities of its shape as well as those of -the ground are impediments, which retard and soon -destroy its motion. Render the stone round, and the -ground level, and the motion will be considerably prolonged. -But still small asperities will remain on the -stone, and on the surface over which it rolls: substitute<span class="pagenum" id="Page_30">30</span> -for the stone a ball of highly-polished steel, moving on -a highly-polished steel plane, truly level, and the motion -will continue without sensible diminution for a very -long period; but even here, and in every instance of -motions produced by art, minute asperities must exist -on the surfaces which move in contact with each other, -which must resist, gradually diminish, and ultimately -destroy the motion.</p> - -<p>Independently of the obstructions to the continuation -of motion arising from friction, there is another impediment -to which all motions on the surface of the earth -are liable—the resistance of the air. How much this -may affect the continuation of motion appears by many -familiar effects. On a calm day carry an open umbrella -with its concave side presented in the direction in which -you are moving, and a powerful resistance will be opposed -to your progress, which will increase with every -increase of the speed with which you move.</p> - -<p id="p43">(43.) We are not, however, without direct experience -to prove, that motions when unresisted will for ever continue. -In the heavens we find an apparatus, which -furnishes a sublime verification of this principle. There, -removed from all casual obstructions and resistances, -the vast bodies of the universe roll on in their appointed -paths with unerring regularity, preserving -without diminution all that motion which they received -at their creation from the hand which launched them -into space. This alone, unsupported by other reasons, -would be sufficient to establish the quality of inertia; -but viewed in connection with the other circumstances -previously mentioned, no doubt can remain that this is -an universal law of nature.</p> - -<p id="p44">(44.) It has been proved, that inability to change the -<i>quantity</i> of motion is a consequence of <i>inertia</i>. The -inability to change the <i>direction</i> of motion is another -consequence of this quality. The same cause which increases -or diminishes motion, would also give motion to -a body at rest; and therefore we infer that the same<span class="pagenum" id="Page_31">31</span> -inability which prevents a body from moving itself, will -also prevent it from increasing or diminishing any motion -which it has received. In the same manner we can -show, that any cause which changes the direction of -motion would also give motion to a body at rest; and -therefore if a body change the direction of its own motion, -the same body might move itself from a state of -rest; and therefore the power of changing the direction -of any motion which it may have received is inconsistent -with the quality of inertia.</p> - -<p id="p45">(45.) If a body, moving from A, <i><a href="#i_p058a">fig. 3.</a></i> to B, receive -at B a blow in the direction C B E, it will immediately -change its direction to that of another line B D. The -cause which produces this change of direction would have -put the body in motion in the direction B E, had it been -quiescent at B when it sustained the blow.</p> - -<p id="p46">(46.) Again, suppose G H to be a hard plane surface; -and let the body be supposed to be perfectly inelastic. -When it strikes the surface at B, it will commence to -move along it in the direction B H. This change of -direction is produced by the resistance of the surface. If -the body, instead of meeting the surface in the direction -A B, had moved in the direction E B, perpendicular to -it, all motion would have been destroyed, and the body -reduced to a state of rest.</p> - -<p id="p47">(47.) By the former example it appears that the deflecting -cause would have put a quiescent body in motion, -and by the latter it would have reduced a moving body -to a state of rest. Hence the phenomenon of a change of -direction is to be referred to the same class as the change -from rest to motion, or from motion to rest. The -quality of inertia is, therefore, inconsistent with any -change in the direction of motion which does not arise -from an external cause.</p> - -<p id="p48">(48.) From all that has been here stated, we may -infer generally, that an inanimate parcel of matter is -incapable of changing its state of rest or motion; that, -in whatever state it be, in that state it must for ever<span class="pagenum" id="Page_32">32</span> -continue, unless disturbed by some external cause; that -if it be in motion, that motion must always be <i>uniform</i>, -or must proceed at the same rate, equal spaces being -moved over in the same time: any increase of its rate -must betray some impelling cause; any diminution must -proceed from an impeding cause, and neither of these -causes can exist in the body itself; that such motion -must not only be constantly at the same uniform rate, -but also must be always in the same direction, any deflection -from one uniform direction necessarily arising -from some external influence.</p> - -<p>The language sometimes used to explain the property -of inertia in popular works, is eminently calculated to -mislead the student. The terms resistance and stubbornness -to move are faulty in this respect. Inertia implies -absolute passiveness, a perfect indifference to rest or -motion. It implies as strongly the absence of all resistance -to the reception of motion, as it does the absence -of all power to move itself. The term <i>vis inertiæ</i> or -<i>force of inactivity</i>, so frequently used even by authors -pretending to scientific accuracy, is still more reprehensible. -It is a contradiction in terms; the term <i>inactivity</i> -implying the absence of all force.</p> - -<p class="mt1em" id="p49">(49.) Before we close this chapter, it may be advantageous -to point out some practical and familiar examples -of the general law of inertia. The student must, however, -recollect, that the great object of science is generalisation, -and that his mind is to be elevated to the -contemplation of the <i>laws</i> of nature, and to receive a -habit the very reverse of that which disposes us to enjoy -the descent from generals to particulars. Instances, -taken from the occurrences of ordinary life, may, however, -be useful in verifying the general law, and in impressing -it upon the memory; and for this reason, we -shall occasionally in the present treatise refer to such -examples; always, however, keeping them in subser<span class="pagenum" id="Page_33">33</span>vience -to the general principles of which they are manifestations, -and on which the attention of the student -should never cease to be fixed.</p> - -<p id="p50">(50.) If a carriage, a horse, or a boat, moving with -speed, be suddenly retarded or stopped, by any cause -which does not at the same time affect passengers, riders, -or any loose bodies which are carried, they will be precipitated -in the direction of the motion; because by -reason of their inertia, they persevere in the motion -which they shared in common with that which transported -them, and are not deprived of that motion by the -same cause.</p> - -<p id="p51">(51.) If a passenger leap from a carriage in rapid -motion, he will fall in the direction in which the carriage -is moving at the moment his feet meet the ground; because -his body, on quitting the vehicle, retains, by its -inertia, the motion which it had in common with it. -When he reaches the ground, this motion is destroyed -by the resistance of the ground to the feet, but is retained -in the upper and heavier part of the body; so -that the same effect is produced as if the feet had been -tripped.</p> - -<p id="p52">(52.) When a carriage is once put in motion with a -determinate speed on a level road, the only force necessary -to sustain the motion is that which is sufficient to -overcome the friction of the road; but at starting a -greater expenditure of force is necessary, inasmuch as -not only the friction is to be overcome, but the force with -which the vehicle is intended to move must be communicated -to it. Hence we see that horses make a much -greater exertion at starting than subsequently, when the -carriage is in motion; and we may also infer the inexpediency -of attempting to start at full speed, especially -with heavy carriages.</p> - -<p id="p53">(53.) <i>Coursing</i> owes all its interest to the instinctive -consciousness of the nature of inertia which seems to -govern the measures of the hare. The greyhound is a -comparatively heavy body moving at the same or greater<span class="pagenum" id="Page_34">34</span> -speed in pursuit. The hare <i>doubles</i>, that is, suddenly -changes the direction of her course, and turns back at an -oblique angle with the direction in which she had been -running. The greyhound, unable to resist the tendency -of its body to persevere in the rapid motion it had acquired, -is urged forward many yards before it is able to -check its speed and return to the pursuit. Meanwhile -the hare is gaining ground in the other direction, so that -the animals are at a very considerable distance asunder -when the pursuit is recommenced. In this way a hare, -though much less fleet than a greyhound, will often -escape it.</p> - -<p>In racing, the horses shoot far beyond the winning-post -before their course can be arrested.</p> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<h2 class="nobreak" id="CHAP_IV">CHAP. IV.<br /> - -<span class="title">ACTION AND REACTION.</span></h2> -</div> - - -<p id="p54">(54.) <span class="smcap">The</span> effects of inertia or inactivity, considered -in the last chapter, are such as may be manifested by a -single insulated body, without reference to, or connection -with, any other body whatever. They might all be recognised -if there were but one body existing in the universe. -There are, however, other important results of this law, -to the development of which two bodies at least are -necessary.</p> - -<p id="p55">(55.) If a mass A, <i><a href="#i_p058a">fig. 4.</a></i>, moving towards C, impinge -upon an equal mass, which is quiescent at B, the -two masses will move together towards C after the impact. -But it will be observed, that their speed after the -impact will be only half that of A before it. Thus, -after the impact, A loses half its velocity; and B, which -was before quiescent, receives exactly this amount of motion. -It appears, therefore, in this case, that B receives -exactly as much motion as A loses: so that the real<span class="pagenum" id="Page_35">35</span> -quantity of motion from B to C is the same as the quantity -of motion from A to B.</p> - -<p>Now, suppose that B consisted of two masses, each -equal to A, it would be found that in this case the velocity -of the triple mass after impact would be one-third -of the velocity from A to B. Thus, after impact, A -loses two-thirds of its velocity and, B consisting of two -masses each equal to A, each of these two receives one-third -of A’s motion; so that the whole motion received -by B is two-thirds of the motion of A before impact. -By the impact, therefore, exactly as much motion is -received by B as is lost by A.</p> - -<p>A similar result will be obtained, whatever proportion -may subsist between the masses A and B. Suppose B -to be ten times A; then the whole motion of A must, -after the impact, be distributed among the parts of the -united masses of A and B: but these united masses are, -in this case, eleven times the mass of A. Now, as they -all move with a common motion, it follows that A’s -former motion must be equally distributed among them; -so that each part shall have an eleventh part of it. -Therefore the velocity after impact will be the eleventh -part of the velocity of A before it. Thus A loses by the -impact ten-eleventh parts of its motion, which are precisely -what B receives.</p> - -<p>Again, if the masses of A and B be 5 and 7, then the -united mass after impact will be 12. The motion of A -before impact will be equally distributed between these -twelve parts, so that each part will have a twelfth of it; -but five of these parts belong to the mass A, and seven -to B. Hence B will receive seven-twelfths, while A -retains five-twelfths.</p> - -<p id="p56">(56.) In general, therefore, when a mass A in motion -impinges on a mass B at rest, to find the motion of -the united mass after impact, “divide the whole motion -of A into as many equal parts as there are equal component -masses in A and B together, and then B will receive -by the impact as many parts of this motion as it -has equal component masses.”</p> - -<p><span class="pagenum" id="Page_36">36</span></p> - -<p>This is an immediate consequence of the property of -inertia, explained in the last chapter. If we were to -suppose that by their mutual impact A were to give to B -either more or less motion than that which it (A) loses, it -would necessarily follow, that either A or B must have -a power of producing or of resisting motion, which -would be inconsistent with the quality of inertia already -defined. For if A give to B <i>more</i> motion than it loses, -all the overplus or excess must be excited in B by the -<i>action</i> of A; and, therefore, A is not inactive, but is -capable of exciting motion which it does not possess. On -the other hand, B cannot receive from A <i>less</i> motion than -A loses, because then B must be admitted to have the -power by its resistance of destroying all the deficiency; -a power essentially active, and inconsistent with the quality -of inertia.</p> - -<p id="p57">(57.) If we contemplate the effects of impact, which -we have now described, as facts ascertained by experiment -(which they may be), we may take them as further -verification of the universality of the quality of inertia. -But, on the other hand, we may view them as phenomena -which may certainly be predicted from the previous -knowledge of that quality; and this is one of many -instances of the advantage which science possesses over -knowledge <i>merely</i> practical. Having obtained by observation -or experience a certain number of simple facts, and -thence deduced the general qualities of bodies, we are -enabled, by demonstrative reasoning, to discover <i>other -facts</i> which have never fallen under our observation, or, -if so, may have never excited attention. In this way -philosophers have discovered certain small motions and -slight changes which have taken place among the heavenly -bodies, and have directed the attention of astronomical -observers to them, instructing them with the greatest -precision as to the exact moment of time and the point -of the firmament to which they should direct the telescope, -in order to witness the predicted event.</p> - -<p id="p58">(58.) Since by the quality of inertia a body can<span class="pagenum" id="Page_37">37</span> -neither generate nor destroy motion, it follows that when -two bodies act upon each other in any way whatever, the -total quantity of motion in a given direction, after the -action takes place, must be the same as before it, for -otherwise some motion would be produced by the action -of the bodies, which would contradict the principle that -they are inert. The word “action” is here applied, perhaps -improperly, but according to the usage of mechanical -writers, to express a certain phenomenon or effect. It is, -therefore, not to be understood as implying any active -principle in the bodies to which it is attributed.</p> - -<p id="p59">(59.) In the cases of collision of which we have -spoken, one of the masses B was supposed to be quiescent -before the impact. We shall now suppose it to be moving -in the same direction as A, that is, towards C, but -with a less velocity, so that A shall overtake it, and -impinge upon it. After the impact, the two masses will -move towards C with a common velocity, the amount of -which we now propose to determine.</p> - -<p>If the masses A and B be equal, then their motions -or velocities added together must be the motion of the -united mass after impact, since no motion can either be -created or destroyed by that event. But as A and B -move with a common motion, this sum must be equally -distributed between them, and therefore each will move -with a velocity equal to half the sum of their velocities -before the impact. Thus, if A have the velocity 7, and -B have 5, the velocity of the united mass after impact -is 6, being the half of 12, the sum of 7 and 5.</p> - -<p>If A and B be not equal, suppose them divided into -equal component parts, and let A consist of 8, and B of -6, equal masses: let the velocity of A be 17, so that the -motion of each of the 8 parts being 17, the motion of -the whole will be 136. In the same manner, let the -velocity of B be 10, the motion of each part being 10, -the whole motion of the 6 parts will be 60. The sum -of the two motions, therefore, towards C is 196; and -since none of this can be lost by the impact, nor any<span class="pagenum" id="Page_38">38</span> -motion added to it, this must also be the whole motion -of the united masses after impact. Being equally distributed -among the 14 component parts of which these -united masses consist, each part will have a fourteenth -of the whole motion. Hence, 196 being divided by 14, -we obtain the quotient 14, which is the velocity with -which the whole moves.</p> - -<p id="p60">(60.) In general, therefore, when two masses moving -in the same direction impinge one upon the other, and -after impact move together, their common velocity may -be determined by the following rule: “Express the -masses and velocities by numbers in the usual way, and -multiply the numbers expressing the masses by the numbers -which express the velocities; the two products thus -obtained being added together, and their sum divided by -the sum of the numbers expressing the masses, the quotient -will be the number expressing the required velocity.”</p> - -<p id="p61">(61.) From the preceding details, it appears that -<i>motion</i> is not adequately estimated by <i>speed</i> or <i>velocity</i>. -For example, a certain mass A, moving at a determinate -rate, has a certain quantity of motion. If another equal -mass B be added to A, and a similar velocity be given -to it, as much more motion will evidently be called into -existence. In other words, the <i>two</i> equal masses A and -B united have <i>twice</i> as much motion as the single mass -A had when moving alone, and with the same speed. -The same reasoning will show that <i>three</i> equal masses -will with the same speed have <i>three times</i> the motion of -any one of them. In general, therefore, the velocity -being the same, the quantity of motion will always be -increased or diminished in the same proportion as the -mass moved is increased or diminished.</p> - -<p id="p62">(62.) On the other hand, the quantity of motion does -not depend on the mass <i>only</i>, but also on the speed. If a -certain determinate mass move with a certain determinate -speed, another equal mass which moves with twice the -speed, that is, which moves over twice the space in the -same time, will have twice the quantity of motion. In<span class="pagenum" id="Page_39">39</span> -this manner, the mass being the same, the quantity of -motion will increase or diminish in the same proportion -as the velocity.</p> - -<p id="p63">(63.) The true estimate, then, of the quantity of -motion is found by multiplying together the numbers -which express the mass and the velocity. Thus, in the -example which has been last given of the impact of -masses, the quantities of motion before and after impact -appear to be as follow:</p> - -<div class="center"> -<table width="480" summary=""> -<tr> -<td class="tac" colspan="3"><div>Before Impact.</div></td> -<td class="tac" colspan="3"><div>After Impact.</div></td> -</tr> -<tr> -<td class="tal pl03">Mass of A</td> -<td class="tar"></td> -<td class="tal"> 8</td> -<td class="tal pl03 bl">Mass of A</td> -<td class="tar"></td> -<td class="tal"> 8</td> -</tr> -<tr> -<td class="tal pl03">Velocity of A</td> -<td class="tar"></td> -<td class="tal">17</td> -<td class="tal pl03 bl">Common velocity</td> -<td class="tar"></td> -<td class="tal">14</td> -</tr> -<tr> -<td class="tal plhi">Quantity of<br />motion of A</td> -<td class="tar"><img src="images/25x6br.png" width="6" height="25" alt="" /></td> -<td class="tal btb"><span class="ilb"> 8 × 17* or 136</span></td> -<td class="tal plhi bl">Quantity of<br />motion of A</td> -<td class="tar"><img src="images/25x6br.png" width="6" height="25" alt="" /></td> -<td class="tal btb"><span class="ilb"> 8 × 14 or 112</span></td> -</tr> -<tr> -<td class="tal pl03">Mass of B</td> -<td class="tar"></td> -<td class="tal"> 6</td> -<td class="tal pl03 bl">Mass of B</td> -<td class="tar"></td> -<td class="tal"> 6</td> -</tr> -<tr> -<td class="tal pl03">Velocity of B</td> -<td class="tar"></td> -<td class="tal">10</td> -<td class="tal pl03 bl">Common velocity</td> -<td class="tar"></td> -<td class="tal">14</td> -</tr> -<tr> -<td class="tal plhi">Quantity of<br />motion of B</td> -<td class="tar"><img src="images/25x6br.png" width="6" height="25" alt="" /></td> -<td class="tal btb"><span class="ilb"> 6 × 10 or 60</span></td> -<td class="tal plhi bl">Quantity of<br />motion of B</td> -<td class="tar"><img src="images/25x6br.png" width="6" height="25" alt="" /></td> -<td class="tal btb"><span class="ilb"> 6 × 14 = 84</span></td> -</tr> -</table> -</div> - -<p class="tac fs80">* The sign × placed between two numbers meant that they are to be -multiplied together.</p> - -<p>By this calculation it appears that in the impact A has -lost a quantity of motion expressed by 24, and that B -has received exactly that amount. The effect, therefore, -of the impact is a <i>transfer</i> of motion from A to B; but -no new motion is produced in the direction A C which -did not exist before. This is obviously consistent with -the property of inertia, and indeed an inevitable result -of it.</p> - -<p>These results may be generalised and more clearly -and concisely expressed by the aid of the symbols of -arithmetic.</p> - -<p>Let <i>a</i> express the velocity of A.</p> - -<p>Let <i>b</i> express the velocity of B.</p> - -<p>Let <i>x</i> express the velocity of the united masses of A -and B after impact, each of these velocities being expressed -in feet per second, and the masses of A and B -being expressed by the weight in pounds.</p> - -<p><span class="pagenum" id="Page_40">40</span></p> - -<p>We shall then have the momenta or moving forces of -A and B before impact, expressed by A × <i>a</i> and B × <i>b</i>, -and the moving force of the united mass after impact -will be expressed by (A + B) × <i>x</i>.</p> - -<p>The moving force of A after impact is A × <i>x</i>, and -therefore the force it loses by the collision will be -(A × <i>a</i> - A × <i>x</i>). The force of B after impact will be -B × <i>x</i>, and therefore the force it gains will be B × <i>x</i> -- B × <i>b</i>. But since the force lost by A must be equal to -the force gained by B, we shall have</p> - -<p class="tac">A × <i>a</i> - A × <i>x</i> = B × <i>x</i> - B × <i>b</i></p> - -<p>from which it is easy to infer</p> - -<p class="tac">(A + B) × <i>x</i> = A × <i>a</i> + B × <i>b</i></p> - -<p>and if it be required to express the velocity of the -united masses after impact, we have</p> - -<p class="tac"> -<i>x</i> = <span class="nowrap"><span class="fraction2"><span class="fnum">A × <i>a</i> + B × <i>b</i></span><span class="bar">/</span><span class="fden2">A + B</span></span></span> -</p> - -<p>When it is said that A × <i>a</i> and B × <i>b</i> express the -moving forces of A and B, it must be understood that -the <i>unit</i> of momentum or moving force is in the case -here supposed, the force with which a mass of matter -weighing 1 lb. would move if its velocity were 1 foot per -second, and accordingly the forces with which A and B -move before impact are as many times this as there are -units respectively in the numbers signified by the general -symbols A × <i>a</i> and B × <i>b</i>.</p> - -<p>In like manner, the force of the united masses after -impact is as many times greater than that of 1 lb. moving -through 1 foot per second as there are units in the numbers -expressed by (A + B) × <i>x</i>.</p> - -<p id="p64">(64.) These phenomena present an example of a -law deduced from the property of inertia, and generally -expressed thus—“action and reaction are equal, and -in contrary directions.” The student must, however, be -cautious not to receive these terms in their ordinary<span class="pagenum" id="Page_41">41</span> -acceptation. After the full explanation of inertia given -in the last chapter, it is, perhaps, scarcely necessary -here to repeat, that in the phenomena manifested by the -motion of two bodies, there can be neither “action” nor -“reaction,” properly so called. The bodies are absolutely -incapable either of action or resistance. The sense -in which these words must be received, as used in the -<i>law</i>, is merely an expression of the <i>transfer</i> of a certain -quantity of motion from one body to another, which is -called an <i>action</i> in the body which loses the motion, and -a <i>reaction</i> in the body which receives it. The <i>accession</i> -of motion to the latter is said to proceed from the <i>action</i> -of the former; and the <i>loss</i> of the same motion in the -former is ascribed to the <i>reaction</i> of the latter. The -whole phraseology is, however, most objectionable and -unphilosophical, and is calculated to create wrong notions.</p> - -<p id="p65">(65.) The bodies impinging were, in the last case, -supposed to move in the same direction. We shall now -consider the case in which they move in opposite directions.</p> - -<p>First, let the masses A and B be supposed to be -equal, and moving in opposite directions, with the same -velocity. Let C, <i><a href="#i_p058a">fig. 5.</a></i>, be the point at which they meet. -The equal motions in opposite directions will, in this -case, destroy each other, and both masses will be reduced -to a state of rest. Thus, the mass A loses all -its motion in the direction A C, which it may be supposed -to transfer to B at the moment of impact. But B having -previously had an equal quantity of motion in the direction -B C, will now have two equal motions impressed -upon it, in directions immediately opposite; and these -motions neutralising each other, the mass becomes quiescent. -In this case, therefore, as in all the former -examples, each body transfers to the other all the motion -which it loses, consistently with the principle of “action -and reaction.”</p> - -<p>The masses A and B being still supposed equal, let -them move towards C with different velocities. Let A -move with the velocity 10, and B with the velocity 6.<span class="pagenum" id="Page_42">42</span> -Of the 10 parts of motion with which A is endued, 6 -being transferred to B, will destroy the equal velocity 6, -which B has in the direction B C. The bodies will then -move together in the direction C B, the four remaining -parts of A’s motion being equally distributed between -them. Each body will, therefore, have two parts of A’s -original motion, and 2 therefore will be their common -velocity after impact. In this case, A loses 8 of the 10 -parts of its motion in the direction A C. On the other -hand, B loses the entire of its 6 parts of motion in the -direction B C, and receives 2 parts in the direction A C. -This is equivalent to receiving 8 parts of A’s motion in -the direction A C. Thus, according to the law of -“action and reaction,” B receives exactly what A loses.</p> - -<p>Finally, suppose that both the masses and velocities of -A and B are unequal. Let the mass of A be 8, and its -velocity 9: and let the mass of B be 6, and its velocity -5. The quantity of motion of A will be 72, and that of -B, in the opposite direction, will be 30. Of the 72 -parts of motion, which A has in the direction A C, 30 -being transferred to B, will destroy all its 30 parts of -motion in the direction B C, and the two masses will -move in the direction C B, with the remaining 42 parts -of motion, which will be equally distributed among their -14 component masses. Each component part will, therefore, -receive 3 parts of motion; and accordingly 3 -will be the common velocity of the united mass after -impact.</p> - -<p id="p66">(66.) When two masses moving in opposite directions -impinge and move together, their common velocity -after impact may be found by the following -rule:—“Multiply the numbers expressing the masses -by those which express the velocities respectively, and -subtract the lesser product from the greater; divide the -remainder by the sum of the numbers expressing the -masses, and the quotient will be the common velocity; -the direction will be that of the mass which has the -greater quantity of motion.”</p> - -<p>It may be shown without difficulty, that the example<span class="pagenum" id="Page_43">43</span> -which we have just given obeys the law of “action and -reaction.”</p> - -<div class="center"> -<table width="480" summary=""> -<tr> -<td class="tac" colspan="3"><div>Before Impact.</div></td> -<td class="tac" colspan="3"><div>After Impact.</div></td> -</tr> -<tr> -<td class="tal pl03">Mass of A</td> -<td class="tar"></td> -<td class="tal">8</td> -<td class="tal pl03 bl">Mass of A</td> -<td class="tar"></td> -<td class="tal">8</td> -</tr> -<tr> -<td class="tal pl03">Velocity of A</td> -<td class="tar"></td> -<td class="tal">9</td> -<td class="tal pl03 bl">Common velocity</td> -<td class="tar"></td> -<td class="tal">3</td> -</tr> -<tr> -<td class="tal plhi">Quantity of motion<br />in direction A C</td> -<td class="tar"><img src="images/25x6br.png" width="6" height="25" alt="" /></td> -<td class="tal btb"><span class="ilb">8 × 9 or 72</span></td> -<td class="tal plhi bl">Quantity of motion<br />in direction A C</td> -<td class="tar"><img src="images/25x6br.png" width="6" height="25" alt="" /></td> -<td class="tal btb"><span class="ilb">8 × 3 or 24</span></td> -</tr> -<tr> -<td class="tal pl03">Mass of B</td> -<td class="tar"></td> -<td class="tal">6</td> -<td class="tal pl03 bl">Mass of B</td> -<td class="tar"></td> -<td class="tal">6</td> -</tr> -<tr> -<td class="tal pl03">Velocity of B</td> -<td class="tar"></td> -<td class="tal">5</td> -<td class="tal pl03 bl">Common velocity</td> -<td class="tar"></td> -<td class="tal">3</td> -</tr> -<tr> -<td class="tal plhi">Quantity of motion<br />in direction B C</td> -<td class="tar"><img src="images/25x6br.png" width="6" height="25" alt="" /></td> -<td class="tal btb"><span class="ilb">6 × 5 or 30</span></td> -<td class="tal plhi bl">Quantity of motion<br />in direction A C</td> -<td class="tar"><img src="images/25x6br.png" width="6" height="25" alt="" /></td> -<td class="tal btb"><span class="ilb">6 × 3 = 18</span></td> -</tr> -</table> -</div> - - -<p>Hence it appears that the quantity of motion in the direction -A C of which A has been deprived by the impact -is 48, the difference between 72 and 24. On the -other hand, B loses by the impact the quantity 30 in the -direction B C, which is equivalent to receiving 30 in the -direction A C. But it also acquires a quantity 18 in the -direction A C, which, added to the former 30, gives a -total of 48 received by B in the direction A C. Thus -the same quantity of motion which A loses in the direction -A C, is received by B in the same direction. The -law of “action and reaction” is, therefore, fulfilled.</p> - -<p>This result may in like manner be generalised. -Retaining the former symbols, the moving forces of A -and B before impact will be A × <i>a</i> and B × <i>b</i> and -their forces after impact will be A × <i>x</i> and B × <i>x</i>. -The force lost by A will therefore be A × <i>a</i> - A × <i>x</i>. -The mass B will have lost all the force B × <i>b</i> which -it had in its former direction, and will have received the -force B × <i>x</i> in the opposite direction. Therefore -the actual force imparted to B by the collision will be -B × <i>b</i> + B × <i>x</i>. But since the force lost by A must -be equal to that imparted to B, we shall have</p> - -<p class="tac">A × <i>a</i> - A × <i>x</i> = B × <i>b</i> + B × <i>x</i></p> - -<p>and therefore</p> - -<p class="tac">(A + B) × <i>x</i> = A × <i>a</i> - B × <i>b</i></p> - -<p>and if the common velocity after impact be required, -we have</p> - -<p><span class="pagenum" id="Page_44">44</span></p> - -<p class="tac"><i>x</i> = <span class="nowrap"><span class="fraction2"><span class="fnum">A × <i>a</i> - B × <i>b</i></span><span class="bar">/</span><span class="fden2">A + B</span></span></span></p> - -<p>As a general rule, therefore, to find the common velocity -after impact. Multiply the weights by the previous -velocities and take their sum if the bodies move in -the same direction, and their difference if they move in -opposite directions, and divide the one or the other by -the sum of their weights. The greatest will be the velocity -after impact.</p> - -<p id="p67">(67.) The examples of the equality of action and reaction -in the collision of bodies may be exhibited experimentally -by a very simple apparatus. Let A, <i><a href="#i_p058a">fig. 6.</a></i>, and -B be two balls of soft clay, or any other substance which -is inelastic, or nearly so, and let these be suspended from -C by equal strings, so that they may be in contact; and -let a graduated arc, of which the centre is C, be placed -so that the balls may oscillate over it. One of the balls -being moved from its place of rest along the arc, and -allowed to descend upon the other through a certain number -of degrees, will strike the other with a velocity corresponding -to that number of degrees, and both balls will -then move together with a velocity which may be estimated -by the number of degrees of the arc through -which they rise.</p> - -<p id="p68">(68.) In all these cases in which we have explained -the law of “action and reaction,” the transfer of motion -from one body to the other has been made by impact or -collision. The phenomenon has been selected only because -it is the most ordinary way in which bodies are seen -to affect each other. The law is, however, universal, and -will be fulfilled in whatever manner the bodies may affect -each other. Thus A may be connected with B by a -flexible string, which, at the commencement of A’s motion, -is slack. Until the string becomes stretched, that is, -until A’s distance from B becomes equal to the length of -the string, A will continue to have all the motion first -impressed upon it. But when the string is stretched, a -part of that motion is transferred to B, which is then<span class="pagenum" id="Page_45">45</span> -drawn after A; and whatever motion B in this way -receives, A must lose. All that has been observed of -the effect of motion transferred by impact will be equally -applicable in this case.</p> - -<p>Again, if B, <i><a href="#i_p058a">fig. 4.</a></i>, be a magnet moving in the direction -B C with a certain quantity of motion, and while it -is so moving a mass of iron be placed at rest at A, the -attraction of the magnet will draw the iron after it towards -C, and will thus communicate to the iron a certain -quantity of motion in the direction of C. All the motion -thus communicated to the iron A must be lost by the -magnet B.</p> - -<p>If the magnet and the iron were both placed quiescent -at B and A, the attraction of the magnet would cause the -iron to move from A towards B; but the magnet in this -case not having any motion, cannot be literally said to -<i>transfer</i> a motion to the iron. At the moment, however, -when the iron begins to move from A towards B, -the magnet will be observed to begin also to move from -B towards A; and if the velocities of the two bodies be -expressed by numbers, and respectively multiplied by the -numbers expressing their masses, the quantities of motion -thus obtained will be found to be exactly equal. We -have already explained why a quantity of motion received -in the direction B A, is equivalent to the same -quantity lost in the direction A B. Hence it appears, -that the magnet in receiving as much motion in the -direction B A, as it gives in the direction A B, suffers -an effect which is equivalent to losing as much motion -directed towards C as it has communicated to the iron -in the same direction.</p> - -<p>In the same manner, if the body B had any property -in virtue of which it might <i>repel</i> A, it would itself be -repelled with the same quantity of motion. In a word, -whatever be the manner in which the bodies may affect -each other, whether by collision, traction, attraction, or -repulsion, or by whatever other name the phenomenon -may be designated, still it is an inevitable consequence, -that any motion, in a given direction, which one of the -bodies may receive, must be accompanied by a loss of<span class="pagenum" id="Page_46">46</span> -motion in the same direction, and to the same amount, -by the other body, or the acquisition of as much motion -in the contrary direction; or, finally, by a loss in the -same direction, and an acquisition of motion in the contrary -direction, the combined amount of which is equal -to the motion received by the former.</p> - -<p id="p69">(69.) From the principle, that the force of a body in -motion depends on the mass and the velocity, it follows, -that any body, however small, may be made to move with -the same force as any other body, however great, by giving -to the smaller body a velocity which bears to that of the -greater the same proportion as the mass of the greater -bears to the mass of the smaller. Thus a feather, ten -thousand of which would have the same weight as a -cannon-ball, would move with the same force if it had ten -thousand times the velocity; and in such a case, these -two bodies encountering in opposite directions, would -mutually destroy each other’s motion.</p> - -<p id="p70">(70.) The consequences of the property of inertia, -which have been explained in the present and preceding -chapters, have been given by Newton, in his <span class="smcap">Principia</span>, -and, after him, in most English treatises on mechanics, -under the form of three propositions, which are called -the “laws of motion.” They are as follow:—</p> - - -<p class="tac">I.</p> - -<p class="mrl2em">“Every body must persevere in its state of rest, or of -uniform motion in a straight line, unless it be compelled -to change that state by forces impressed upon it.”</p> - - -<p class="tac">II.</p> - -<p class="mrl2em">“Every change of motion must be proportional to the -impressed force, and must be in the direction of that -straight line in which the force is impressed.”</p> - - -<p class="tac">III.</p> - -<p class="mrl2em">“Action must always be equal and contrary to reaction; -or the actions of two bodies upon each other must -be equal, and directed towards contrary sides.”</p> - -<p><span class="pagenum" id="Page_47">47</span></p> - -<p>When <i>inertia</i> and <i>force</i> are defined, the first law becomes -an identical proposition. The second law cannot -be rendered perfectly intelligible until the student has -read the chapter on the composition and resolution of -forces, for, in fact, it is intended as an expression of the -whole body of results in that chapter. The third law -has been explained in the present chapter, as far as it can -be rendered intelligible in the present stage of our progress.</p> - -<p>We have noticed these formularies more from a respect -for the authorities by which they have been proposed and -adopted, than from any persuasion of their utility. -Their full import cannot be comprehended until nearly -the whole of elementary mechanics has been acquired, -and then all such summaries become useless.</p> - -<p class="mt1em" id="p71">(71.) The consequences deduced from the consideration -of the quality of inertia in this chapter, will account -for many effects which fall under our notice daily, -and with which we have become so familiar, that they -have almost ceased to excite curiosity. One of the facts -of which we have most frequent practical illustration is, -that the quantity of motion or <i>moving force</i>, as it is sometimes -called, is estimated by the velocity of the motion, -and the weight or mass of the thing moved conjointly.</p> - -<p>If the same force impel two balls, one of one pound -weight, and the other of two pounds, it follows, since the -balls can neither give force to themselves, nor resist that -which is impressed upon them, that they will move with -the same force. But the lighter ball will move with -twice the speed of the heavier. The impressed force -which is manifested by giving velocity to a double mass -in the one, is engaged in giving a double velocity to the -other.</p> - -<p>If a cannon-ball were forty times the weight of a -musket-ball, but the musket-ball moved with forty times -the velocity of the cannon-ball, both would strike any -obstacle with the same force, and would overcome the<span class="pagenum" id="Page_48">48</span> -same resistance; for the one would acquire from its -velocity as much force as the other derives from its -weight.</p> - -<p>A very small velocity may be accompanied by enormous -force, if the mass which is moved with that velocity -be proportionally great. A large ship, floating near -the pier wall, may approach it with so small a velocity as -to be scarcely perceptible, and yet the force will be so -great as to crush a small boat.</p> - -<p>A grain of shot flung from the hand, and striking the -person, will occasion no pain, and indeed will scarcely be -felt, while a block of stone having the same velocity -would occasion death.</p> - -<p>If a body in motion strike a body at rest, the striking -body must sustain as great a shock from the collision as -if it had been at rest, and struck by the other body with -the same force. For the loss of force which it sustains -in the one direction, is an effect of the same kind as if, -being at rest, it had received as much force in the opposite -direction. If a man, walking rapidly or running, -encounters another standing still, he suffers as much from -the collision as the man against whom he strikes.</p> - -<p>If a leaden bullet be discharged against a plank of -hard wood, it will be found that the round shape of the -ball is destroyed, and that it has itself suffered a force by -the impact, which is equivalent to the effect which it -produces upon the plank.</p> - -<p>When two bodies moving in opposite directions meet, -each body sustains as great a shock as if, being at rest, it -had been struck by the other body with the united forces -of the two. Thus, if two equal balls, moving at the rate -of ten feet in a second, meet, each will be struck with the -same force as if, being at rest, the other had moved -against it at the rate of twenty feet in a second. In this -case one part of the shock sustained arises from the loss -of force in one direction, and another from the reception -of force in the opposite direction.</p> - -<p>For this reason, two persons walking in opposite -directions receive from their encounter a more violent<span class="pagenum" id="Page_49">49</span> -shock than might be expected. If they be of nearly -equal weight, and one be walking at the rate of three -and the other four miles an hour, each sustains the same -shock as if he had been at rest, and struck by the other -running at the rate of seven miles an hour.</p> - -<p>This principle accounts for the destructive effects -arising from ships running foul of each other at sea. If -two ships of 500 tons burden encounter each other, sailing -at ten knots an hour, each sustains the shock which, -being at rest, it would receive from a vessel of 1000 tons -burden sailing ten knots an hour.</p> - -<p>It is a mistake to suppose, that when a large and small -body encounter, the small body suffers a greater shock -than the large one. The shock which they sustain must -be the same; but the large body may be better able to -bear it.</p> - -<p>When the fist of a pugilist strikes the body of his -antagonist, it sustains as great a shock as it gives; but -the fist being more fitted to endure the blow, the injury -and pain are inflicted on his opponent. This is not the -case, however, when fist meets fist. Then the parts in -collision are equally sensitive and vulnerable, and the -effect is aggravated by both having approached each other -with great force. The effect of the blow is the same as -if one fist, being held at rest, were struck by the other with -the combined force of both.</p> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<h2 class="nobreak" id="CHAP_V">CHAP. V.<br /> - -<span class="title">THE COMPOSITION AND RESOLUTION OF FORCE.</span></h2> -</div> - - -<p id="p72">(72.) <span class="smcap">Motion</span> and pressure are terms too familiar to -need explanation. It may be observed, generally, that -definitions in the first rudiments of a science are seldom, -if ever, comprehended. The force of words is learned by -their application; and it is not until a definition becomes -useless, that we are taught the meaning of the terms in<span class="pagenum" id="Page_50">50</span> -which it is expressed. Moreover, we are perhaps justified -in saying, that in the mathematical sciences the -fundamental notions are of so uncompounded a character, -that definitions, when developed and enlarged -upon, often draw us into metaphysical subtleties and -distinctions, which, whatever be their merit or importance, -would be here altogether misplaced. We shall, -therefore, at once take it for granted, that the words -<i>motion</i> and <i>pressure</i> express phenomena or effects which -are the subjects of constant experience and hourly observation; -and if the scientific use of these words be -more precise than their general and popular application, -that precision will soon be learned by their frequent use -in the present treatise.</p> - -<p id="p73">(73.) <span class="smcap">Force</span> is the name given in mechanics to whatever -produces motion or pressure. This word is also -often used to express the motion or pressure itself; and -when the cause of the motion or pressure is not known, -this is the only correct use of the word. Thus, when a -piece of iron moves toward a magnet, it is usual to say -that the cause of the motion is “the attraction of the -magnet;” but in effect we are ignorant of the <i>cause</i> of -this phenomenon; and the name <i>attraction</i> would be -better applied to the effect of which we have experience. -In like manner the <i>attraction</i> and <i>repulsion</i> of electrified -bodies should be understood, not as names for unknown -causes, but as words expressing observed appearances or -effects.</p> - -<p>When a certain phraseology has, however, gotten into -general use, it is neither easy nor convenient to supersede -it. We shall, therefore, be compelled, in speaking of -motion and pressure, to use the language of causation; -but must advise the student that it is effects and not -causes which will be expressed.</p> - -<p id="p74">(74.) If two forces act upon the same point of a body -in different directions, a single force may be assigned, -which, acting on that point, will produce the same result -as the united effects of the other two.</p> - -<p>Let P, <i><a href="#i_p058a">fig. 7.</a></i>, be the point on which the two forces<span class="pagenum" id="Page_51">51</span> -act, and let their directions be P A and P B. From the -point P, upon the line P A, take a length P <i>a</i>, consisting -of as many inches as there are ounces in the force P A; -and, in like manner, take P <i>b</i>, in the direction P B, consisting -of as many inches as there are ounces in the force -P B. Through <i>a</i> draw a line parallel to P B, and through -<i>b</i> draw a line parallel to P A, and suppose that these lines -meet at <i>c</i>. Then draw P C. A single force, acting in the -direction P C, and consisting of as many ounces as the -line P c consists of inches, will produce upon the point P -the same effect as the two forces P A and P B produce -acting together.</p> - -<p id="p75">(75.) The figure P <i>a c b</i> is called in <span class="lowercase smcap">GEOMETRY</span> a -<i>parallelogram</i>; the lines P <i>a</i>, P <i>b</i>, are called its <i>sides</i>, and -the line P <i>c</i> is called its <i>diagonal</i>. Thus the method of -finding an equivalent for two forces, which we have just -explained, is generally called “the parallelogram of -forces,” and is usually expressed thus: “If two forces -be represented in quantity and direction by the sides of a -parallelogram, an equivalent force will be represented in -quantity and direction by its diagonal.”</p> - -<p id="p76">(76.) A single force, which is thus mechanically -equivalent to two or more other forces, is called their -<i>resultant</i>, and relatively to it they are called its <i>components</i>. -In any mechanical investigation, when the -resultant is used for the components, which it always -may be, the process is called “the composition of force.” -It is, however, frequently expedient to substitute for a -single force two or more forces, to which it is mechanically -equivalent, or of which it is the resultant. This -process is called “the resolution of force.”</p> - -<p id="p77">(77.) To verify experimentally the theorem of the -parallelogram of forces is not difficult. Let two small -wheels, M N, <i><a href="#i_p058a">fig. 8.</a></i>, with grooves in their edges to receive -a thread, be attached to an upright board, or to a wall. -Let a thread be passed over them, having weights A and -B, hooked upon loops at its extremities. From any part -P of the thread between the wheels let a weight C be -suspended: it will draw the thread downwards, so as to<span class="pagenum" id="Page_52">52</span> -form an angle M P N, and the apparatus will settle itself -at rest in some determinate position. In this state it is -evident that since the weight C, acting in the direction -P C, balances the weights A and B, acting in the directions -P M and P N, these two forces must be mechanically -equivalent to a force equal to the weight C, -and acting directly upwards from P. The weight C is -therefore the quantity of the resultant of the forces P M -and P N; and the direction of the resultant is that of a -line drawn directly upwards from P.</p> - -<p>To ascertain how far this is consistent with the -theorem of “the parallelogram of forces,” let a line P O -be drawn upon the upright board to which the wheels -are attached, from the point P upward, in the direction of -the thread C P. Also, let lines be drawn upon the board -immediately under the threads P M and P N. From the -point P, on the line P O, take as many inches as there are -ounces in the weight C. Let the part of P O thus measured -be P <i>c</i>, and from <i>c</i> draw <i>c a</i> parallel to P N, and <i>c b</i> -parallel to P M. If the sides P <i>a</i> and P <i>b</i> of the parallelogram -thus formed be measured, it will be found that -P <i>a</i> will consist of as many inches as there are ounces -in the weight A, and P <i>b</i> of as many inches as there are -ounces in the weight B.</p> - -<p>In this illustration, <i>ounces</i> and <i>inches</i> have been used -as the subdivisions of <i>weight</i> and <i>length</i>. It is scarcely -necessary to state, that any other measures of these -quantities would serve as well, only observing that the -same denominations must be preserved in all parts of the -same investigation.</p> - -<p id="p78">(78.) Among the philosophical apparatus of the -University of London, is a very simple and convenient -instrument which I constructed for the experimental -illustration of this important theorem. The wheels -M N are attached to the tops of two tall stands, the -heights of which may be varied at pleasure by an adjusting -screw. A jointed parallelogram, A B C D, <i><a href="#i_p058a">fig. 9.</a></i>, -is formed, whose sides are divided into inches, and the -joints at A and B are moveable, so as to vary the lengths<span class="pagenum" id="Page_53">53</span> -of the sides at pleasure. The joint C is fixed at the -extremity of a ruler, also divided into inches, while the -opposite joint A is attached to a brass loop, which surrounds -the diagonal ruler loosely, so as to slide freely -along it. An adjusting screw is provided in this loop so -as to clamp it in any required position.</p> - -<p>In making the experiment, the sides A B and A D, C B -and C D are adjusted by the joints B and A to the same -number of inches respectively as there are ounces in the -weights A and B, <i><a href="#i_p058a">fig. 8.</a></i> Then the diagonal A C is adjusted -by the loop and screw at A, to as many inches as -there are ounces in the weight C. This done, the point -A is placed behind P, <i><a href="#i_p058a">fig. 8.</a></i>, and the parallelogram is held -upright, so that the diagonal A C shall be in the direction -of the vertical thread P C. The sides A B and A D will -then be found to take the direction of the threads P M -and P N. By changing the weights and the lengths of -the diagonal and sides of the parallelogram, the experiment -may be easily varied at pleasure.</p> - -<p id="p79">(79.) In the examples of the composition of forces -which we have here given, the effects of the forces are -the production of pressures, or, to speak more correctly, -the theorem which we have illustrated, is “the composition -of pressures.” For the point P is supposed to be -at rest, and to be drawn or pressed in the directions -P M and P N. In the definition which has been given -of the word force, it is declared to include motions as -well as pressures. In fact, if motion be resisted, the -effect is converted into pressure. The same cause acting -upon a body, will either produce motion or pressure, -according as the body is free or restrained. If the body -be free, motion ensues; if restrained, pressure, or both -these effects together. It is therefore consistent with -analogy to expect that the same theorems which regulate -pressures, will also be applicable to motions; and we find -accordingly a most exact correspondence.</p> - -<p id="p80">(80.) If a body have a motion in the direction A B, -and at the point P it receive another motion, such as -would carry it in the direction P C, <i><a href="#i_p058a">fig. 10.</a></i>, were it pre<span class="pagenum" id="Page_54">54</span>viously -quiescent at P, it is required to determine the -direction which the body will take, and the speed with -which it will move, under these circumstances.</p> - -<p>Let the velocity with which the body is moving from -A to B be such, that it would move through a certain -space, suppose P N, in one second of time, and let the -velocity of the motion impressed upon it at P be such, -that if it had no previous motion it would move from P -to M in one second. From the point M draw a line -parallel to P B, and from N draw a line parallel to P C, -and suppose these lines to meet at some point, as O. -Then draw the line P O. In consequence of the two -motions, which are at the same time impressed upon the -body at P, it will move in the straight line from P to O.</p> - -<p>Thus the two motions, which are expressed in quantity -and direction by the sides of a parallelogram, will, -when given to the same body, produce a single motion, -expressed in quantity and direction by its diagonal; a -theorem which is to motions exactly what the former -was to pressures.</p> - -<p>There are various methods of illustrating experimentally -the composition of motion. An ivory ball, being -placed upon a perfectly level square table, at one of the -corners, and receiving two equal impulses, in the directions -of the sides of the table, will move along the diagonal. -Apparatus for this experiment differ from each -other only in the way of communicating the impulses to -the ball.</p> - -<p id="p81">(81.) As two motions simultaneously communicated -to a body are equivalent to a single motion in an intermediate -direction, so also a single motion may be mechanically -replaced, by two motions in directions expressed -by the sides of any parallelogram, whose diagonal -represents the single motion. This process is -“the resolution of motion,” and gives considerable clearness -and facility to many mechanical investigations.</p> - -<p id="p82">(82.) It is frequently necessary to express the portion -of a given force, which acts in some given direction different -from the immediate direction of the force itself.<span class="pagenum" id="Page_55">55</span> -Thus, if a force act from A, <i><a href="#i_p058a">fig. 11.</a></i>, in the direction A C, -we may require to estimate what part of that force acts -in the direction A B. If the force be a pressure, take -as many inches A P from A, on the line A C, as there -are ounces in the force, and from P draw P M perpendicular -to A B; then the part of the force which acts -along A B will be as many ounces as there are inches in -A M. The force A B is mechanically equivalent to two -forces, expressed by the sides A M and A N of the parallelogram; -but A N, being perpendicular to A B, can -have no effect on a body at A, in the direction of A B, -and therefore the effective part of the force A P in the -direction A B is expressed by A M.</p> - -<p id="p83">(83.) Any number of forces acting on the same point -of a body may be replaced by a single force, which is -mechanically equivalent to them, and which is, therefore, -their resultant. This composition may be effected -by the successive application of the parallelogram of -forces. Let the several forces be called A, B, C, D, E, -&c. Draw the parallelogram whose sides express the -forces A and B, and let its diagonal be <span class="ilb">A′</span>. The force -expressed by <span class="ilb">A′</span> will be equivalent to A and B. Then -draw the parallelogram whose sides express the forces -<span class="ilb">A′</span> and C, and let its diagonal be <span class="ilb">B′</span>. This diagonal -will express a force mechanically equivalent to <span class="ilb">A′</span> and C. -But <span class="ilb">A′</span> is mechanically equivalent to A and B, and -therefore <span class="ilb">B′</span> is mechanically equivalent to A, B, and C. -Next construct a parallelogram, whose sides express the -forces <span class="ilb">B′</span> and D, and let its diagonal be <span class="ilb">C′</span>. The force -expressed by <span class="ilb">C′</span> will be mechanically equivalent to the -forces <span class="ilb">B′</span> and D; but the force <span class="ilb">B′</span> is equivalent to A, B, -C, and therefore <span class="ilb">C′</span> is equivalent to A, B, C, and D. -By continuing this process it is evident, that a single -force may be found, which will be equivalent to, and -may be always substituted for, any number of forces -which act upon the same point.</p> - -<p>If the forces which act upon the point neutralise -each other, so that no motion can ensue, they are said -to be in equilibrium.</p> - -<p><span class="pagenum" id="Page_56">56</span></p> - -<p id="p84">(84.) Examples of the composition of motion and -pressure are continually presenting themselves. They -occur in almost every instance of motion or force which -falls under our observation. The difficulty is to find -an example which, strictly speaking, is a simple motion.</p> - -<p>When a boat is rowed across a river, in which there -is a current, it will not move in the direction in which -it is impelled by the oars. Neither will it take the direction -of the stream, but will proceed exactly in that -intermediate direction which is determined by the composition -of force.</p> - -<p>Let A, <i><a href="#i_p058a">fig. 12.</a></i>, be the place of the boat at starting; -and suppose that the oars are so worked as to impel the -boat towards B with a force which would carry it to B -in one hour, if there were no current in the river. But, -on the other hand, suppose the rapidity of the current -is such, that without any exertion of the rowers the boat -would float down the stream in one hour to C. From -C draw C D parallel to A B, and draw the straight line -A D diagonally. The combined effect of the oars and -the current will be, that the boat will be carried along -A D, and will arrive at the opposite bank in one hour, at -the point D.</p> - -<p>If the object be, therefore, to reach the point B, -starting from A, the rowers must calculate, as nearly as -possible, the velocity of the current. They must imagine -a certain point E at such a distance above B that the -boat would be floated by the stream from E to B in the -time taken in crossing the river in the direction A E, -if there were no current. If they row towards the -point E, the boat will arrive at the point B, moving in -the line A B.</p> - -<p>In this case the boat is impelled by two forces, that -of the oars in the direction A E, and that of the current -in the direction A C. The result will be, according -to the parallelogram of forces, a motion in the diagonal -A B.</p> - -<p><span class="pagenum" id="Page_57">57</span></p> - -<p>The wind and tide acting upon a vessel is a case of a -similar kind. Suppose that the wind is made to impel -the vessel in the direction of the keel; while the tide -may be acting in any direction oblique to that of the -keel. The course of the vessel is determined exactly in -the same manner as that of the boat in the last example.</p> - -<p>The action of the oars themselves, in impelling the -boat, is an example of the composition of force. Let -A, <i><a href="#i_p058a">fig. 13.</a></i>, be the head, and B the stern of the boat. -The boatman presents his face towards B, and places -the oars so that their blades press against the water in -the directions C E, D F. The resistance of the water -produces forces on the side of the boat, in the directions -G L and H L, which, by the composition of force, are -equivalent to die diagonal force K L, in the direction of -the keel.</p> - -<p>Similar observations will apply to almost every body -impelled by instruments projecting from its sides, and -acting against a fluid. The motions of fishes, the act of -swimming, the flight of birds, are all instances of the -same kind.</p> - -<p id="p85">(85.) The action of wind upon the sails of a vessel, -and the force thereby transmitted to the keel, modified -by the rudder, is a problem which is solved by the principles -of the composition and resolution of force; but it -is of too complicated and difficult a nature to be introduced -with all its necessary conditions and limitations in this -place. The question may, however, be simplified, if we -consider the canvass of the sails to be stretched so completely -as to form a plane surface. Let A B, <i><a href="#i_p058a">fig. 14.</a></i>, be -the position of the sail, and let the wind blow in the -direction C D. If the line C D be taken to express the -force of the wind, let D E C F be a parallelogram, of -which it is the diagonal. The force C D is equivalent -to two forces, one in the direction F D of the plane of -the canvass, and the other E D perpendicular to the sail. -The effect, therefore, is the same as if there were <i>two -winds</i>, one blowing in the direction of F D or B A, that -is against the edge of the sail, and the other, E D, blow<span class="pagenum" id="Page_58">58</span>ing -full against its face. It is evident that the former -will produce no effect whatever upon the sail, and that -the latter will urge the vessel in the direction D G.</p> - -<p>Let us now consider this force D G as acting in the -diagonal of the parallelogram D H G I. It will be equivalent -to two forces, D H and D I, acting along the sides. -One of these forces, D H, is in the direction of the keel, -and the other, D I, at right angles to the length of the -vessel, so as to urge it <i>sideways</i>. The form of the vessel -is evidently such as to offer a great resistance to the -latter force, and very little to the former. It consequently -proceeds with considerable velocity in the direction -D H of its keel, and makes way very slowly in -the sideward direction D I. The latter effect is called -<i>lee-way</i>.</p> - -<p>From this explanation it will be easily understood, -how a wind which is nearly opposed to the course of -a vessel may, nevertheless, be made to impel it by the -effect of sails. The angle B D V, formed by the sail -and the direction of the keel, may be very oblique, as -may also be the angle C D B formed by the direction of -the wind and that of the sail. Therefore the angle -C D V, made up of these two, and which is that formed -by the direction of the wind and that of the keel, may -be very oblique. In <i><a href="#i_p058a">fig. 15.</a></i> the wind is nearly contrary -to the direction of the keel, and yet there is an impelling -force expressed by the line D H, the line C D expressing, -as before, the whole force of the wind.</p> - -<p>In this example there are two successive decompositions -of force. First, the original force of the wind C D -is resolved into two, E D and F D; and next the element -E D, or its equal D G, is resolved into D I and D H; so -that the original force is resolved into three, viz. F D, -D I, D H, which, taken together, are mechanically equivalent -to it. The part F D is entirely ineffectual; it -glides off on the surface of the canvass without producing -any effect upon the vessel. The part D I produces -<i>lee-way</i>, and the part D H impels.</p> - -<div class="figcenter" id="i_p058a" style="max-width: 31.25em;"> - <img src="images/i_p058a.jpg" alt="" /> - <div class="caption"> - -<p class="tar"><i>H. Adlard, sc.</i></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<p id="p86">(86.) If the wind, however, be directly contrary to<span class="pagenum" id="Page_59">59</span> -the course which it is required that the vessel should -take, there is no position which can be given to the sails -which will impel the vessel. In this case the required -course itself is resolved into two, in which the vessel -sails alternately, a process which is called <i>tacking</i>. Thus, -suppose the vessel is required to move from A to E, <i><a href="#i_p058a">fig. 16.</a></i>, -the wind setting from E to A. The motion A B being -resolved into two, by being assumed as the diagonal of -a parallelogram, the sides A <i>a</i>, <i>a</i> B of the parallelogram -are successively sailed over, and the vessel by this -means arrives at B, instead of moving along the diagonal -A B. In the same manner she moves along B <i>b</i>, <i>b</i> C, -C <i>c</i>, <i>c</i> D, D <i>d</i>, <i>d</i> E, and arrives at E. She thus sails -continually at a sufficient angle with the wind to obtain -an impelling force, yet at a sufficiently small angle to -make way in her proposed course.</p> - -<p>The consideration of the effect of the rudder, which -we have omitted in the preceding illustration, affords -another instance of the resolution of force. We shall -not, however, pursue this example further.</p> - -<p id="p87">(87.) A body falling from the top of the mast when -the vessel is in full sail, is an example of the composition -of motion. It might be expected, that during the -descent of the body, the vessel having sailed forward, -would leave it behind, and that, therefore, it would fall -in the water behind the stern, or at least on the deck, -considerably behind the mast. On the other hand, it is -found to fall at the foot of the mast, exactly as it would -if the vessel were not in motion. To account for this, -let A B, <i><a href="#i_p058a">fig. 17.</a></i>, be the position of the mast when the -body at the top is disengaged. The mast is moving -onwards with the vessel in the direction A C, so that in -the time which the body would take to fall to the deck, -the top of the mast would move from A to C. But the -body being on the mast at the moment it is disengaged, -has this motion A C in common with the mast; and -therefore in its descent it is affected by two motions, -viz. that of the vessel expressed by A C, and its descending -motion expressed by A B. Hence, by the com<span class="pagenum" id="Page_60">60</span>position -of motion, it will be found at the opposite angle -D of the parallelogram, at the end of the fall. During -the fall, however, the mast has moved with the vessel, -and has advanced to C D, so that the body falls at the -foot of the mast.</p> - -<p id="p88">(88.) An instance of the composition of motion, which -is worthy of some attention, as it affords a proof of the -diurnal motion of the earth, is derived from observing -the descent of a body from a very high tower. To render -the explanation of this more simple, we shall suppose -the tower to be on the equator of the earth. Let -E P Q, <i><a href="#i_p104a">fig. 18.</a></i>, be a section of the earth through the equator, -and let P T be the tower. Let us suppose that the -earth moves on its axis in the direction E P Q. The -foot P of the tower will, therefore, in one day move -over the circle E P Q, while the top T moves over the -greater circle T <span class="ilb">T′</span> R. Hence it is evident, that the top -of the tower moves with greater speed than the foot, -and therefore in the same time moves through a greater -space. Now suppose a body placed at the top; it participates -in the motion which the top of the tower has -in common with the earth. If it be disengaged, it also -receives the descending motion T P. Let us suppose -that the body would take five seconds to fall from T to -P, and that in the same time the top T is moved by the -rotation of the earth from T to <span class="ilb">T′</span>, the foot being moved -from P to <span class="ilb">P′</span>. The falling body is therefore endued -with two motions, one expressed by T <span class="ilb">T′</span>, and the other -by T P. The combined effect of these will be found in -the usual way by the parallelogram. Take T <i>p</i> equal to -T <span class="ilb">T′</span>; the body will move from T to <i>p</i> in the time -of the fall, and will meet the ground at <i>p</i>. But since -T <span class="ilb">T′</span> is greater than P <span class="ilb">P′</span>, it follows that the point <i>p</i> must -be at a distance from <span class="ilb">P′</span> equal to the excess of T <span class="ilb">T′</span> -above P <span class="ilb">P′</span>. Hence the body will not fall exactly at the -foot of the tower, but at a certain distance from it, in -the direction of the earth’s motion, that is, eastward. -This is found, by experiment, to be actually the case; -and the distance from the foot of the tower, at which<span class="pagenum" id="Page_61">61</span> -the body is observed to fall, agrees with that which is -computed from the motion of the earth, to as great a -degree of exactness as could be expected from the nature -of the experiment.</p> - -<p id="p89">(89.) The properties of compounded motions cause -some of the equestrian feats exhibited at public spectacles -to be performed by a kind of exertion very different -from that which the spectators generally attribute -to the performer. For example, the horseman standing -on the saddle leaps over a garter extended over the -horse at right angles to his motion; the horse passing -under the garter, the rider lights upon the saddle at -the opposite side. The exertion of the performer, in -this case, is not that which he would use were he to -leap from the ground over a garter at the same height. -In the latter case, he would make an exertion to rise, -and, at the same time, to project his body forward. In -the case, however, of the horseman, he merely makes -that exertion which is necessary to rise directly upwards -to a sufficient height to clear the garter. The -motion which he has in common with the horse, compounded -with the elevation acquired by his muscular -power, accomplishes the leap.</p> - -<p>To explain this more fully, let A B C, <i><a href="#i_p104a">fig. 19.</a></i>, be the -direction in which the horse moves, A being the point -at which the rider quits the saddle, and C the point at -which he returns to it. Let D be the highest point -which is to be cleared in the leap. At A the rider makes -a leap towards the point E, and this must be done at -such a distance from B, that he would rise from B to E -in the time in which the horse moves from A to B. On -departing from A, the rider has, therefore, two motions, -represented by the lines A E and A B, by which he will -move from the point A to the opposite angle D of the -parallelogram. At D, the exertion of the leap being -overcome by the weight of his body, he begins to return -downward, and would fall from D to B in the time in -which the horse moves from B to C. But at D he still -retains the motion which he had in common with the -horse; and therefore, in leaving the point D, he has<span class="pagenum" id="Page_62">62</span> -two motions, expressed by the lines D F and D B. The -compounded effects of these motions carry him from D -to C. Strictly speaking, his motion from A to D, and -from D to C, is not in straight lines, but in a curve. It -is not necessary here, however, to attend to this circumstance.</p> - -<p id="p90">(90.) If a billiard-ball strike the cushion of the table -obliquely, it will be reflected from it in a certain direction, -forming an angle with the direction in which it struck it. -This affords an example of the resolution and composition -of motion. We shall first consider the effect which -would ensue if the ball struck the cushion perpendicularly.</p> - -<p>Let A B, <i><a href="#i_p104a">fig. 20.</a></i>, be the cushion, and C D the direction -in which the ball moves towards it. If the ball and -the cushion were perfectly inelastic, the resistance of the -cushion would destroy the motion of the ball, and it -would be reduced to a state of rest at D. If, on the -other hand, the ball were perfectly elastic, it would be -reflected from the cushion, and would receive as much -motion from D to C after the impact, as it had from -C to D before it. Perfect elasticity, however, is a -quality which is never found in these bodies. They -are always elastic, but imperfectly so. Consequently the -ball after the impact will be reflected from D towards C, -but with a less motion than that with which it approached -from C to D.</p> - -<p>Now let us suppose that the ball, instead of moving -from C to D, moves from E to D. The force with which -it strikes D being expressed by D <span class="ilb">E′</span>, equal to E D, may -be resolved into two, D F and D <span class="ilb">C′</span>. The resistance of -the cushion destroys D <span class="ilb">C′</span>, and the elasticity produces a -contrary force in the direction D C, but less than D C or -D <span class="ilb">C′</span>, because that elasticity is imperfect. The line D C -expressing the force in the direction C D, let D G (less -than D C) express the reflective force in the direction -D C. The other element D F, into which the force D <span class="ilb">E′</span> -is resolved by the impact, is not destroyed or modified by -the cushion, and therefore, on leaving the cushion at D,<span class="pagenum" id="Page_63">63</span> -the ball is influenced by two forces, D F (which is equal -to C E) and D G. Consequently it will move in the diagonal -D H.</p> - -<p id="p91">(91.) The angle E D C is in this case called the “angle -of incidence,” and C D H is called “the angle of reflection.” -It is evident, from what has been just inferred, -that the ball, being imperfectly elastic, the angle of incidence -must always be less than the angle of reflection, -and with the same obliquity of incidence, the more imperfect -the elasticity is, the less will be the angle of reflection.</p> - -<p>In the impact of a perfectly elastic body, the angle of -reflection would be equal to the angle of incidence. For -then the line D G, expressing the reflective force, would -be taken equal to C D, and the angle C D H would be -equal to C D E. This is found by experiment to be the -case when light is reflected from a polished surface of -glass or metal.</p> - -<p>Motion is sometimes distinguished into <i>absolute</i> and -<i>relative</i>. What “relative motion” means is easily explained. -If a man walk upon the deck of a ship from -stem to stern, he has a relative motion which is measured -by the space upon the deck over which he walks in a given -time. But while he is thus walking from stem to stern, -the ship and its contents, including himself, are impelled -through the deep in the opposite direction. If it so -happen that the motion of the man, from stem to stern, -be exactly equal to the motion of the ship in the contrary -way, the man will be, relatively to the surface of the sea -and that of the earth, at rest. Thus, relatively to the -ship, he is in motion, while, relatively to the surface -of the earth, he is at rest. But still this is not absolute -rest. The surface itself is moving by the diurnal -rotation of the earth upon its axis, as well as by the -annual motion in its orbit round the sun. These motions, -and others to which the earth is subject, must be all -compounded by the theorem of the parallelogram of -forces before we can obtain the <i>absolute state</i> of the body -with respect to motion or rest.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_64">64</span></p> - -<h2 class="nobreak" id="CHAP_VI">CHAP. VI.<br /> - -<span class="title">ATTRACTION.</span></h2> -</div> - - -<p id="p92">(92.) <span class="smcap">Whatever</span> produces, or tends to produce, a -change in the state of a particle or mass of matter with -respect to motion or rest, is a force. Rest, or uniform -rectilinear motion, are therefore the only states in which -any body can exist which is not subject to the present -action of some force. We are not, however, entitled to -conclude, that because a body is observed in one or other -of these states, it is therefore uninfluenced by any forces. -It may be under the immediate action of forces which -neutralise each other: thus two forces may be acting -upon it which are equal, and in opposite directions. In -such a case, its state of rest, or of uniform rectilinear -motion, will be undisturbed. The state of uniform rectilinear -motion declares more with respect to the body -than the state of rest; for the former betrays the action -of a force upon the body at some antecedent period; this -action having been suspended, while its effect continues -to be observed in the motion which it has produced.</p> - -<p id="p93">(93.) When the state of a body is changed from rest -to uniform rectilinear motion, the action of the force -is only momentary, in which case it is called an <i>impulse</i>. -If a body in uniform rectilinear motion receive an impulse -in the direction in which it is moving, the effect -will be, that it will continue to move uniformly in the -same direction, but its velocity will be increased by the -amount of speed which the impulse would have given it -had it been previously quiescent. Thus, if the previous -motion be at the rate of ten feet in a second, and the impulse -be such as would move it from a state of rest at five -feet in a second, the velocity, after the impulse, will be -fifteen feet in a second.</p> - -<p>But if the impulse be received in a direction immediately -opposed to the previous motion, then it will diminish -the speed by that amount of velocity which it would<span class="pagenum" id="Page_65">65</span> -give to the body had it been previously at rest. In the -example already given, if the impulse were opposed to -the previous motion, the velocity of the body after the -impulse would be five feet in a second. If the impulse -received in the direction opposed to the motion be such -as would give to the body at rest a velocity equal to that -with which it is moving, then the effect will be, that -after the impulse no motion will exist; and if the impulse -would give it a still greater velocity, the body will -be moved in the opposite direction with an uniform velocity -equal to the excess of that due to the impulse over -that which the body previously had.</p> - -<p>When a body in a state of uniform motion receives -an impulse in a direction not coinciding with that of its -motion, it will move uniformly after the impulse in an -intermediate direction, which may be determined by the -principles established for the composition of motion in -the last chapter.</p> - -<p>Thus it appears, that whenever the state of a body is -changed either from rest to uniform rectilinear motion -or <i>vice versa</i>, or from one state of uniform rectilinear -motion to another, differing from that either in velocity -or direction, or in both, the phenomenon is produced by -that peculiar modification of force whose action continues -but for a single instant, and which has been called <i>an -impulse</i>.</p> - -<p id="p94">(94.) In most cases, however, the mechanical state of -a body is observed to be subject to a continual change or -tendency to change. We are surrounded by innumerable -examples of this. A body is placed on the table. -A continual pressure is excited on the surface of the -table. This pressure is only the consequence of the -continual tendency of the body to move downwards. If -the body were excited by a force of the nature of an -impulse, the effect upon the table would be instantaneous, -and would immediately cease. It would, in fact, be <i>a -blow</i>. But the continuation of the pressure proves the -continuation of the action of the force.</p> - -<p>If the table be removed from beneath the body, the<span class="pagenum" id="Page_66">66</span> -force which excites it being no longer resisted, will produce -motion; it is manifested, not as before, by a tendency -to produce motion, but by the actual exhibition of -that phenomenon. Now if the exciting force were an -impulse, the body would descend to the ground with an -uniform velocity. On the other hand, as will hereafter -appear, every moment of its fall increases its speed, and -that speed is greatest at the instant it meets the ground.</p> - -<p>A piece of iron placed at a distance from a magnet -approaches it, but not with an uniform velocity. The -force of the magnet continues to act during the approach -of the iron, and each moment gives it increased motion.</p> - -<p id="p95">(95.) The forces which are thus in constant operation, -proceed from secret agencies which the human -mind has never been able to detect. All the analogies of -nature prove that they are not the immediate results of -the divine will, but are secondary causes, that is, effects -of some more remote principles. To ascend to these -secondary causes, and thus as it were approach one step -nearer to the Creator, is the great business of philosophy; -and the most certain means for accomplishing -this, is diligently to observe, to compare, and to classify -the phenomena, and to avoid assuming the existence of -any thing which has not either been directly observed, -or which cannot be inferred demonstratively from natural -phenomena. Philosophy should follow nature, and -not lead her.</p> - -<p>While the law of inertia, established by observation -and reason, declares the inability of matter, from any -principle resident in it, to change its state, all the phenomena -of the universe prove that state to be in constant -but regular fluctuation. There is not in existence a -single instance of the phenomenon of absolute rest, or of -motion which is absolutely uniform and rectilinear. In -bodies, or the parts of bodies, there is no known instance -of simple passive juxtaposition unaccompanied by pressure -or tension, or some other “tendency to motion.” -Innumerable secret powers are ever at work, compensating, -as it were, for inertia, and supplying the material<span class="pagenum" id="Page_67">67</span> -world with a substitute for the principles of action and -will, which give such immeasurable superiority to the -character of life.</p> - -<p id="p96">(96.) The forces which are thus in continual operation, -whose existence is demonstrated by their observed -effects, but whose nature, seat, and mode of operation -are unknown to us, are called by the general name <i>attractions</i>. -These forces are classified according to the -analogies which prevail among their effects, in the same -manner, and according to the same principles, as organised -beings are grouped in natural history. In that -department of natural science, when individuals are distributed -in classes, the object is merely to generalise, -and thereby promote the enlargement of knowledge; but -nothing is or ought to be thus assumed respecting -the essence, or real internal constitution of the individuals. -According to their external and observable characters -and qualities they are classed; and this classification -should never be adduced as an evidence of any -thing except that similitude of qualities to which it owed -its origin.</p> - -<p>Phenomena are to the natural philosopher what organised -beings are to the naturalist. He groups and -classifies them on the same principles, and with a like -object. And as the naturalist gives to each species a -name applicable to the individual beings which exhibit -corresponding qualities, so the philosopher gives to each -force or attraction a name corresponding to the phenomena -of which it is the cause. The naturalist is ignorant -of the real essence or internal constitution of the -thing which he nominates, and of the manner in which -it comes to possess or exhibit those qualities which form -the basis of his classification; and the natural philosopher -is equally ignorant of the nature, seat, and mode of operation -of the force which he assigns as the cause of an -observed class of effects.</p> - -<p>These observations respecting the true import of the -term “attraction” seem the more necessary to be premised, -because the general phraseology of physical science,<span class="pagenum" id="Page_68">68</span> -taken as language is commonly received, will seem to -convey something more. The names of the several attractions -which we shall have to notice, frequently refer -the seat of the cause to specific objects, and seem to -imply something respecting its mode of operation. Thus, -when we say “the magnet attracts a piece of iron,” the -true philosophical import of the words is, “that a piece -of iron placed in the vicinity of the magnet, will move -towards it, or placed in contact, will adhere to it, so that -some force is necessary to separate them.” In the ordinary -sense, however, something more than this simple -fact is implied. It is insinuated that the magnet is the -seat of the force which gives motion to the iron; that -in the production of the phenomenon, the magnet is an -<i>agent</i> exerting a certain influence, of which the iron is -the <i>subject</i>. Of all this, however, there is no proof; -on the contrary, since the magnet must move towards -the iron with just as much force as the iron moves towards -the magnet, there is as much reason to place the -seat of the force in the iron, and consider it as an agent -affecting the magnet. But, in fact, the influence which -produces this phenomenon may not be resident in either -the one body or the other. It may be imagined to be a -property of a medium in which both are placed, or to -arise from some third body, the presence of which is -not immediately observed. However attractive these -and like speculations may be, they cannot be allowed a -place in physical investigations, nor should consequences -drawn from such hypotheses be allowed to taint our -conclusions with their uncertainty.</p> - -<p>The student ought, therefore, to be aware, that whatever -may seem to be implied by the language used in -this science in relation to attractions, nothing is permitted -to form the basis of reasoning respecting them -except <i>their effects</i>; and whatever be the common signification -of the terms used, it is to these effects, and to -these alone, they should be referred.</p> - -<p id="p97">(97.) Attractions may be primarily distributed into -two classes; one consisting of those which exist between<span class="pagenum" id="Page_69">69</span> -the molecules or constituent parts of bodies, and the -other between bodies themselves. The former are sometimes -called, for distinction, <i>molecular</i> or <i>atomic</i> attractions.</p> - -<p>Without the agency of molecular forces, the whole -face of nature would be deprived of variety and beauty; -the universe would be a confused heap of material atoms -dispersed through space, without form, shape, coherence, -or motion. Bodies would neither have the forms of -solid, liquid, or air; heat and light would no longer -produce their wonted effects; organised beings could -not exist; life itself, as connected with body, would be -extinct. Atoms of matter, whether distant or in juxtaposition, -would have no tendency to change their places, -and all would be eternal stillness and rest. If, then, -we are asked for a proof of the existence of molecular -forces, we may point to the earth and to the heavens; -we may name every object which can be seen or felt. -The whole material world is one great result of the -influence of these powerful agents.</p> - -<p id="p98">(98.) It has been proved (<a href="#p11">11</a>. <i>et seq.</i>) that the constituent -particles of bodies are of inconceivable minuteness, -and that they are not in immediate contact (<a href="#p23">23</a>), -but separated from each other by interstitial spaces, -which, like the atoms themselves, although too small to -be directly observed, yet are incontestably proved to -exist, by observable phenomena, from which their existence -demonstratively follows. The resistance which -every body opposes to compression, proves that a repulsive -influence prevails between the particles, and that -this repulsion is the cause which keeps the atoms separate, -and maintains the interstitial spaces just mentioned. -Although this repulsion is found to exist between the -molecules of all substances whatever, yet it has different -degrees of energy in different bodies. This is proved -by the fact, that some substances admit of easy compression, -while in others, the exertion of considerable -force is necessary to produce the smallest diminution in -bulk.</p> - -<p><span class="pagenum" id="Page_70">70</span></p> - -<p>The space around each atom of a body, through which -this repulsive influence extends, is generally limited, and -immediately beyond it, a force of the opposite kind is -manifested, viz. attraction. Thus, in solid bodies, the -particles resist separation as well as compression, and the -application of force is as necessary to break the body, or -divide it into separate parts, as to force its particles into -closer aggregation. It is by virtue of this attraction that -solid bodies maintain their figure, and that their parts -are not separated and scattered like those of fluids, merely -by their own weight. This force is called the <i>attraction -of cohesion</i>.</p> - -<p>The cohesive force acts in different substances with -different degrees of energy: in some its intensity is very -great; but the sphere of its influence apparently very -limited. This is the case with all bodies which are hard, -strong, and brittle, which no force can extend or stretch -in any perceptible degree, and which require a great -force to break or tear them asunder. Such, for example, -is cast iron, certain stones, and various other substances. -In some bodies the cohesive force is weak, but the sphere -of its action considerable. Bodies which are easily -extended, without being broken or torn asunder, furnish -examples of this. Such are Indian-rubber, or caoutchouc, -several animal and vegetable products, and, in general, all -solids of a soft and viscid kind.</p> - -<p>Between these extremes, the cohesive force may be -observed in various degrees. In lead and other soft -metals, its sphere of action is greater, and its energy -less, than in the former examples; but its sphere less, -and energy greater, than in the latter ones. It is from -the influence of this force, and that of the repulsion, -whose sphere of action is still closer to the component -atoms, that all the varieties of texture which we denominate -hard, soft, tough, brittle, ductile, pliant, &c. arise.</p> - -<p>After having been broken, or otherwise separated, the -parts of a solid may be again united by their cohesion, -provided any considerable number of points be brought -into sufficiently close contact. When this is done by me<span class="pagenum" id="Page_71">71</span>chanical -means, however, the cohesion is not so strong as -before their separation, and a comparatively small force -will be sufficient again to disunite them. Two pieces of -lead freshly cut, with smooth surfaces, will adhere when -pressed together, and will require a considerable force to -separate them. In the same manner if a piece of Indian-rubber -be torn, the parts separated will again cohere, by -being brought together with a slight pressure. The -union of the parts in such instances is easy, because the -sphere through which the influence of cohesion extends -is considerable; but even in bodies in which this influence -extends through a more limited space, the cohesion -of separate pieces will be manifested, provided their surfaces -be highly polished, so as to insure the near approach -of a great number of their particles. Thus, two polished -surfaces of glass, metal, or stone, will adhere when -brought into contact.</p> - -<p>In all these cases, if the bodies be disunited by mechanical -force, they will separate at exactly the parts at -which they had been united, so that after their separation -no part of the one will adhere to the other; proving -that the force of cohesion of the surfaces brought into -contact is less than that which naturally held the particles -of each together.</p> - -<p id="p99">(99.) When a body is in the liquid form, the weight -of its particles greatly predominates over their mutual -cohesion, and consequently if such a body be unconfined -it will be scattered by its own weight; if it be placed in -any vessel, it will settle itself, by the force of its weight, -into the lowest parts, so that no space in the vessel below -the upper surface of the liquid will be unoccupied. The -particles of a solid body placed in the vessel have exactly -the same tendency, by reason of their weight; but this -tendency is resisted and prevented from taking effect by -their strong cohesion.</p> - -<p>Although this cohesion in solids is much greater than -in liquids, and productive of more obvious effects, yet the -principle is not altogether unobserved in liquids. Water -converted into vapour by heat, is divided into incon<span class="pagenum" id="Page_72">72</span>ceivably -minute particles, which ascend in the atmosphere. -When it is there deprived of a part of that heat -which gave it the vaporous form, the particles, in virtue -of their cohesive force, collect into round drops, in which -form they descend to the earth.</p> - -<p>In the same manner, if a liquid be allowed to fall -gradually from the lip of a vessel, it will not be dismissed -in particles indefinitely small, as if its mass were incoherent, -like sand or powder, but will fall in drops of -considerable magnitude. In proportion as the cohesive -force is greater, these drops affect a greater size. Thus, -oil and viscid liquids fall in large drops; ether, alcohol, -and others in small ones.</p> - -<p>Two drops of rain trickling down a window pane will -coalesce when they approach each other; and the same -phenomenon is still more remarkable, if a few drops of -quicksilver be scattered on an horizontal plate of glass.</p> - -<p>It is the cohesive principle which gives rotundity to -grains of shot: the liquid metal is allowed to fall like -rain from a great elevation. In its descent the drops -become truly globular, and before they reach the end of -their fall they are hardened by cooling, so that they -retain their shape.</p> - -<p>It is also, probably, to the cohesive attraction that we -should assign the globular forms of all the great bodies -of the universe; the sun, planets, satellites, &c., which -originally may have been in the liquid state.</p> - -<p id="p100">(100.) Molecular attraction is also exhibited between -the particles of liquids and solids. A drop of water will -not descend freely when it is in contact with a perpendicular -glass plane: it will adhere to the glass; its descent -will be retarded; and if its weight be insufficient to overcome -the adhesive force, it will remain suspended.</p> - -<p>If a plate of glass be placed upon the surface of water -without being permitted to sink, it will require more -force to raise it from the water than is sufficient merely -to balance the weight of the glass. This shows the -adhesion of the water and glass, and also the cohesive force -with which the particles of the water resist separation.</p> - -<p><span class="pagenum" id="Page_73">73</span></p> - -<p>If a needle be dipped in certain liquids, a drop will -remain suspended at its point when withdrawn from -them: and, in general, when a solid body has been -immersed in a liquid and withdrawn, it is <i>wet</i>; that is, -some of the liquid has adhered to its surfaces. If no -attraction existed between the solid and liquid, the -solid would be in the same state after immersion as -before. This is proved by liquids and solids between -which no attraction exists. If a piece of glass be immersed -in mercury, it will be in the same state when -withdrawn as before it was immersed. No mercury -will adhere to it; it will not be <i>wet</i>.</p> - -<p>When it rains, the person and vesture are affected -only because this attraction exists between them and -water. If it rained mercury, none would adhere to them.</p> - -<p id="p101">(101.) When molecular attraction is exhibited by -liquids pervading the interstices of porous bodies, ascending -in crevices or in the bores of small tubes, it is called -<i>capillary attraction</i>. Instances of this are innumerable. -Liquids are thus drawn into the pores of sponge, sugar, -lamp-wick, &c. The animal and vegetable kingdom -furnish numerous examples of this class of effects.</p> - -<p>A weight being suspended by a dry rope, will be -drawn upwards through a considerable height, if the rope -be moistened with a wet sponge. The attraction of the -particles composing the rope for the water is in this -case so powerful, that the tension produced by several -hundred weight cannot expel them.</p> - -<p>A glass tube, of small bore, being dipped in water -tinged by mixture with a little ink, will retain a quantity -of the liquid suspended when withdrawn. The -height of the liquid in the tube will be seen by looking -through it. It is found that the less the bore of the tube -is, the greater will be the height of the column sustained. -A series of such tubes fixed in the same frame, -with their lower orifices at the same level, and with bores -gradually decreasing, being dipped in the liquid, will -exhibit columns gradually increasing.</p> - -<p>A <i>capillary syphon</i> is formed of a hank of cotton<span class="pagenum" id="Page_74">74</span> -threads, one end of which is immersed in the vessel containing -the liquid, and the other is carried into the vessel -into which the liquid is to be transferred. The liquid -may be thus drawn from the one vessel into the other. -The same effect may be produced by a glass syphon with -a small bore.</p> - -<p id="p102">(102.) It frequently happens that a <i>molecular repulsion</i> -is exhibited between a solid and a liquid. If a piece -of wood be immersed in quicksilver, the liquid will be -depressed at that part of the surface which is near the -wood; and in like manner, if it be contained in a glass -vessel, it will be depressed at the edges. In a barometer -tube, the surface of the mercury is convex, owing partly -to the repulsion between the glass and mercury.</p> - -<p>All solids, however, do not repel mercury. If any -golden trinket be dipped in that liquid, or even be exposed -for a moment to contact with it, the gold will be -instantly intermingled with particles of quicksilver, the -metal changes its colour, and becomes white like silver, -and the mercury can only be extricated by a difficult -process. Chains, seals, rings, &c. should always be laid -aside by those engaged in experiments or other processes -in which mercury is used.</p> - -<p id="p103">(103.) Of all the forms under which molecular force -is exhibited, that in which it takes the name of <i>affinity</i> -is attended with the most conspicuous effects. Affinity -is in chemistry what inertia is in mechanics, the basis -of the science. The present treatise is not the proper -place for any detailed account of this important class of -natural phenomena. Those who seek such knowledge -are referred to our treatise on <span class="smcap">Chemistry</span>. Since, however, -affinity sometimes influences the mechanical state -of bodies, and affects their mechanical properties, it will -be necessary here to state so much respecting it as to -render intelligible those references which we may have -occasion to make to such effects.</p> - -<p>When the particles of different bodies are brought -into close contact, and more especially when, being in a -fluid state, they are mixed together, their union is fre<span class="pagenum" id="Page_75">75</span>quently -observed to produce a compound body, differing -in its qualities from either of the component bodies. -Thus the bulk of the compound is often greater or less -than the united volumes of the component bodies. The -component bodies may be of the ordinary temperature of -the atmosphere, and yet the compound may be of a much -higher or lower temperature. The components may be -liquid, and the compound solid. The colour of the -compound may bear no resemblance whatever to that of -the components. The species of molecular action between -the components, which produce these and similar, -effects, is called <i>affinity</i>.</p> - -<p id="p104">(104.) We shall limit ourselves here to the statement -of a few examples of these phenomena.</p> - -<p>If a pint of water and a pint of sulphuric acid be -mixed, the compound will be considerably less than a -quart. The density of the mixture is, therefore, greater -than that which would result from the mere diffusion -of the particles of the one fluid through those of the -other. The particles have assumed a greater proximity, -and therefore exhibit a mutual attraction.</p> - -<p>In this experiment, although the liquids before being -mixed be of the temperature of the surrounding air, -the mixture will be so intensely hot, that the vessel -which contains it cannot be touched without pain.</p> - -<p>If the two aeriform fluids, called oxygen and hydrogen, -be mixed together in a certain proportion, the compound -will be water. In this case, the components are -different from the compound, not merely in the one being -<i>air</i> and the other <i>liquid</i>, but in other respects not -less striking. The compound water extinguishes fire, -and yet of the components, hydrogen is one of the most -inflammable substances in nature, and the presence of -oxygen is indispensably necessary to sustain the phenomenon -of combustion.</p> - -<p>Oxygen gas, united with quicksilver, produces a compound -of a black colour, the quicksilver being white and -the gas colourless. When these substances are combined -in another proportion, they give a red compound.</p> - -<p><span class="pagenum" id="Page_76">76</span></p> - -<p id="p105">(105.) Having noticed the principal molecular forces, -we shall now proceed to the consideration of those attractions -which are exhibited between bodies existing in -masses. The influence of molecular attractions is limited -to insensible distances. On the contrary, the forces -which are now to be noticed act at considerable distances, -and to the influence of some there is no limit, the effect, -however, decreasing as the distance increases.</p> - -<p>The effect of the loadstone on iron is well known, -and is one of this class of forces. For a detailed account -of this force, and the various phenomena of which -it is the cause, the reader is referred to our treatise on -<span class="smcap">Magnetism</span>.</p> - -<p>When glass, wax, amber, and other substances are -submitted to friction with silken or woollen cloth, they -are observed to attract feathers, and other light bodies -placed near them. A like effect is produced in several -other ways, and is attended with other phenomena, the -discussion of which forms a principal part of physical -science. The force thus exhibited is called electricity. -For details respecting it, and for its connection with -magnetism, the reader is referred to our treatises on -<span class="smcap">Electricity</span> and <span class="smcap">Electro-magnetism</span>.</p> - -<p id="p106">(106.) These attractions exist either between bodies -of particular kinds, or are developed by reducing the -bodies which manifest them to a certain state by friction, -or some other means. There is, however, an attraction, -which is manifested between bodies of all -species, and under all circumstances whatever; an attraction, -the intensity of which is wholly independent -of the nature of the bodies, and only depends on their -masses and mutual distances. Thus, if a mass of metal -and a mass of clay be placed in the vast abyss of space, -at a mile asunder, they will instantly commence to approach -each other with certain velocities. Again, if a -mass of stone and of wood respectively equal to the -former, be placed at a like distance, they will also commence -to approach each other with the same velocities -as the former. This universal attraction, which only<span class="pagenum" id="Page_77">77</span> -depends on the quantity of the masses and their mutual -distances, is called the “attraction of gravitation.” We -shall first explain the “law” of this attraction, and -shall then point out some of the principal phenomena -by which its existence and its laws are known.</p> - -<p id="p107">(107.) The “law of gravitation” sometimes from -its universality called the “law of nature,” may be -explained as follows:</p> - -<p>Let us suppose two masses, A and B, placed beyond -the influence or attraction of any other bodies, -in a state of rest, and at any proposed distance from -each other. By their mutual attraction they will approach -each other, but not with the same velocity. The -velocity of A will be greater than that of B, in the same -proportion as its mass is less than that of B. Thus, if -the mass of B be twice that of A, while A approaches -B through a space of two feet, B will approach A -through a space of one foot. Hence it follows, that -the force with which A moves towards B is equal to -the force with which B moves towards A (<a href="#p68">68</a>). This -is only a consequence of the property of inertia, and is -an example of the equality of action and reaction, as -explained in Chapter <a href="#CHAP_IV">IV</a>. The velocity with which A -and B approach each other is estimated by the diminution -of their distance, A B, by their mutual approach -in a given time. Thus, if in one second A move -towards B through a space of two feet, and in the same -time B moves towards A through the space of one foot, -they will approach each other through a space of three -feet in a second, which will be their relative velocity -(<a href="#p91">91</a>).</p> - -<p>If the mass of B be doubled, it will attract A with -double the former force, or, what is the same, will cause -A to approach B with double the former velocity. If -the mass of B be trebled, it will attract A with treble -the first force, and, in general, while the distance A B -remains the same, the attractive force of B upon A will -increase or diminish in exactly the same proportion as -the mass of B is increased or diminished.</p> - -<p><span class="pagenum" id="Page_78">78</span></p> - -<p>In the same manner, if the mass A be doubled, it will -be attracted by B with a double force, because B exerts -the same degree of attraction on every part of the mass -A, and any addition which it may receive will not diminish -or otherwise affect the influence of B on its -former mass.</p> - -<p>To express this in general arithmetical symbols let -<i>a</i> and <i>b</i> express the space through which A and B respectively -would be moved towards each other by their -mutual attraction. We would then have</p> - -<p class="tac"> -A × <i>a</i> = B × <i>b</i>. -</p> - -<p>Thus, it is a general law of gravitation, that so long -as the distance between two bodies remains the same, -each will attract and be attracted by the other, in proportion -to its mass; and any increase or decrease of the -mass will cause a corresponding increase or decrease in -the amount of the attraction.</p> - -<p id="p108">(108.) We shall now explain the law, according to -which the attraction is changed, by changing the distance -between the bodies. At the distance of one mile -the body B attracts A with a certain force. At the -distance of two miles, the masses not being changed, the -attraction of B upon A will be one-fourth of its amount -at the distance of one mile. At the distance of three -miles, it will be one-ninth of its original amount; at -four miles, it is reduced to a sixteenth, and so on. The -following table exhibits the diminution of the attraction -corresponding to the successive increase of distance:</p> - -<div class="center"> -<table width="350" border="1" cellpadding="4" summary=""> -<tr> -<td class="tal">Distance</td> -<td class="tac"><div>1</div></td> -<td class="tac"><div>2</div></td> -<td class="tac"><div>3</div></td> -<td class="tac"><div>4</div></td> -<td class="tac"><div>5</div></td> -<td class="tac"><div>6</div></td> -<td class="tac"><div>7</div></td> -<td class="tac"><div>8</div></td> -<td class="tac"><div>&c.</div></td> -</tr> -<tr> -<td class="tal">Attraction</td> -<td class="tac">1</td> -<td class="tac"><span class="nowrap"> <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">4</span></span></span></td> -<td class="tac"><span class="nowrap"> <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">9</span></span></span></td> -<td class="tac"><span class="nowrap"> <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">16</span></span></span></td> -<td class="tac"><span class="nowrap"> <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">25</span></span></span></td> -<td class="tac"><span class="nowrap"> <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">36</span></span></span></td> -<td class="tac"><span class="nowrap"> <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">49</span></span></span></td> -<td class="tac"><span class="nowrap"> <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">64</span></span></span></td> -<td class="tac">&c.</td> -</tr> -</table> -</div> - -<p>In <span class="lowercase smcap">ARITHMETIC</span>, that number which is found by multiplying -any proposed number by itself, is called its -<i>square</i>. Thus 4, that is, 2 multiplied by 2, is the -square of 2; 9 that is, 3 times 3, is the square of 3, -and so on. On inspecting the above table, it will be -apparent, therefore, that the attraction of gravitation -decreases in the same proportion as the square of the -distance from the attracting body increases, the mass of<span class="pagenum" id="Page_79">79</span> -both bodies in this case being supposed to remain the -same; but if the mass of either be increased or diminished, -the attraction will be increased or diminished in -the same proportion.</p> - -<p id="p109">(109.) Hence the <i>law of gravitation</i> may be thus expressed: -“The mutual attraction of two bodies increases -in the same proportion as their masses are increased, and -as the square of their distance is decreased; and it decreases -in proportion as their masses are decreased, and -as the square of their distance is increased.”</p> - -<p>This law may be more clearly expressed by means of -general symbols. Let <i>f</i> express the force with which a -mass weighing 1 lb. will attract another mass weighing -1 lb., at the distance of 1 foot. The force with which -they will mutually attract, when removed to the distance -expressed in feet by D, will be</p> - -<p class="tac"><span class="nowrap"><span class="fraction2"><span class="fnum"><i>f</i></span><span class="bar">/</span><span class="fden2">D<sup>2</sup></span></span></span></p> - -<p>that is, the force <i>f</i> divided by the square of the number -D.</p> - -<p>If one of the bodies, instead of weighing 1 lb., weigh -the number of pounds expressed by A, their mutual -attraction will be increased A times, and will therefore -be expressed by</p> - -<p class="tac"><span class="nowrap"><span class="fraction2"><span class="fnum">A × <i>f</i></span><span class="bar">/</span><span class="fden2">D<sup>2</sup></span></span></span></p> - -<p>In fine, if the other be also the number of pounds -expressed by B, their mutual attraction will be</p> - -<p class="tac"><span class="nowrap"><span class="fraction2"><span class="fnum">A × B × <i>f</i></span><span class="bar">/</span><span class="fden2">D<sup>2</sup></span></span></span></p> - -<p id="p110">(110.) Having explained the law of gravitation, we -shall now proceed to show how the existence of this -force is proved, and its law discovered.</p> - -<p>The earth is known to be a globular mass of matter, -incomparably greater than any of the detached bodies -which are found upon its surface. If one of these bodies -suspended at any proposed height above the surface of -the earth be disengaged, it will be observed to descend<span class="pagenum" id="Page_80">80</span> -perpendicularly to the earth, that is, in the direction of -the earth’s centre. The force with which it descends -will also be found to be in proportion to the mass, without -any regard to the species of the body. These circumstances -are consistent with the account which we -have given of gravitation. But by that account we -should expect, that as the falling body is attracted with -a certain force towards the earth, the earth itself should -be attracted towards it by the same force; and instead -of the falling body moving towards the earth, which is -the phenomenon observed, the earth and it should move -towards each other, and meet at some intermediate point. -This, in fact, is the case, although it is impossible to -render the motion of the earth observable, for reasons -which will easily be understood.</p> - -<p>Since all the bodies around us participate in this motion, -it would not be directly observable, even though its -quantity were sufficiently great to be perceived under -other circumstances. But setting aside this consideration, -the space through which the earth moves in such a case -is too minute to be the subject of sensible observation. -It has been stated (<a href="#p107">107</a>), that when two bodies attract -each other, the space through which the greater approaches -the lesser, bears to that through which the lesser -approaches the greater, the same proportion as the mass -of the lesser bears to the mass of the greater. Now the -mass of the earth is more than 1000,000,000,000,000 -times the mass of any body which is observed to -fall on its surface; and therefore if even the largest -body which can come under observation were to fall -through an height of 500 feet, the corresponding motion -of the earth would be through a space less than the -1000,000,000,000,000th part of 500 feet, which is -less than the 100,000,000,000th part of an inch.</p> - -<p>The attraction between the earth and detached bodies -on its surface is not only exhibited by the descent of -these bodies when unsupported, but by their pressure -when supported. This pressure is what is called <i>weight</i>. -The phenomena of weight, and the descent of heavy -bodies, will be fully investigated in the next chapter.</p> - -<p><span class="pagenum" id="Page_81">81</span></p> - -<p id="p111">(111.) It is not alone by the direct fall of bodies -that the gravitation of the earth is manifested. The -curvilinear motion of bodies projected in directions different -from the perpendicular, is a combination of the -effects of the uniform velocity which has been given to the -projectile by the impulse which it has received, and the -accelerated velocity which it receives from the earth’s attraction. -Suppose a body placed at any point P, <i><a href="#i_p104a">fig. 21.</a></i>, -above the surface of the earth, and let P C be the direction -of the earth’s centre. If the body were allowed to move -without receiving any impulse, it would descend to the -earth in the direction P A, with an accelerated motion. -But suppose that at the moment of its departure from -P, it receives an impulse in the direction P B, which -would carry it to B in the time the body would fall from -P to A, then, by the composition of motion, the body -must at the end of that time be found in the line B D, -parallel to P A. If the motion in the direction of P A -were uniform, the body P would in this case move in -the straight line from P to D. But this is not the case. -The velocity of the body in the direction P A is at first -so small as to produce very little deflection of its motion -from the line P B. As the velocity, however, increases, -this deflection increases, so that it moves from P to D -in a curve, which is convex, towards P B.</p> - -<p>The greater the velocity of the projectile in the direction -P A, the greater sweep the curve will take. Thus -it will successively take the forms P D, P E, P F, &c., -and that velocity can be computed, which (setting aside -the resistance of the air) would cause the projectile to -go completely round the earth, and return to the point -P from which it departed. In this case, the body P -would continue to revolve round the earth like the moon. -Hence it is obvious, that the phenomenon of the revolution -of the moon round the earth, is nothing more than -the combined effects of the earth’s attraction, and the -impulse which it received when launched into space by -the hand of its Creator.</p> - -<p id="p112">(112.) This is a great step in the analysis of the<span class="pagenum" id="Page_82">82</span> -phenomenon of gravitation. We have thus reduced to -the same class two effects apparently very dissimilar, the -rectilinear descent of a heavy body, and the nearly circular -revolution of the moon round the earth. Hence -we are conducted to a generalisation still more extensive.</p> - -<p>As the moon’s revolution round the earth, in an orbit -nearly circular, is caused by the combination of the -earth’s attraction, and an original projectile impulse, so -also the singular phenomena of the planets’ revolution -round the sun in orbits nearly circular, must be considered -an effect of the same class, as well as the revolution -of the satellites of those planets which are attended by -such bodies. Although the orbits in which the comets -move deviate very much from circles, yet this does not -hinder the application of the same principle to them, -their deviation from circles not depending on the sun’s -attraction, but only on the direction and force of the -original impulse which put them in motion.</p> - -<p id="p113">(113.) We therefore conclude that gravitation is the -principle which, as it were, animates the universe. All -the great changes and revolutions of the bodies which -compose our system, can be traced to or derived from -this principle. It still remains to show how that remarkable -law, by which this force is declared to increase -or decrease in the same proportion as the square -of the distance from the attracting body is decreased or -increased, may be verified and established.</p> - -<p>It has been shown, that the curvilinear path of a projectile -depends on, and can be derived, by mathematical -reasoning, from the consideration of the intensity of the -earth’s attraction, and the force of the original impulse, -or the velocity of projection. In the same manner, by -a reverse process, when we know the curve in which a -projectile moves, we can infer the amount of the attracting -force which gives the curvature to its path. In this -way, from our knowledge of the curvature of the moon’s -orbit, and the velocity with which she moves, the intensity -of the attraction which the earth exerts upon her -can be exactly ascertained. Upon comparing this with -the force of gravitation at the earth’s surface, it is found<span class="pagenum" id="Page_83">83</span> -that the latter is as many times greater than the former, -as the square of the moon’s distance is greater than the -square of the distance of a body on the surface of the -earth from its centre.</p> - -<p id="p114">(114.) If this were the only fact which could be -brought to establish the law of gravitation, it might be -thought to be an accidental relation, not necessarily characterising -the attraction of gravitation. Upon examining -the orbits and velocities of the several planets, the -same result is, however, obtained. It is found that the -forces with which they are severally attracted by the -sun are great, in exactly the same proportion as the -squares of the several numbers expressing their distances -are small. The mutual gravitation of bodies on the -surface of the earth towards each other is lost in the -predominating force exerted by the earth upon all of -them. Nevertheless, in some cases, this effect has not -only been observed, but actually measured.</p> - -<p>A plumb-line, under ordinary circumstances, hangs in -a direction truly vertical; but if it be near a large mass -of matter, as a mountain, it has been observed to be -deflected from the true vertical, towards the mountain. -This effect was observed by Dr. Maskeline near the -mountain called Skehallien, in Scotland, and by French -astronomers near Chimboraco. For particulars of these -observations, see our treatise on <span class="smcap">Geodæsy</span>.</p> - -<p>Cavendish succeeded in exhibiting the effects of the -mutual gravitation of metallic spheres. Two globes of -lead A, B, each about a foot in diameter, were placed at -a certain distance asunder. A light rod, to the ends of -which were attached small metallic balls C, D, was suspended -at its centre E from a fine wire, and the rod -was placed as in <i><a href="#i_p104a">fig. 22.</a></i>, so that the attractions of -each of the leaden globes had a tendency to turn the -rod round the centre E in the same direction. A manifest -effect was produced upon the balls C, D, by the -gravitation of the spheres. In this experiment, care -must be taken that no magnetic substance is intermixed -with the materials of the balls.</p> - -<p><span class="pagenum" id="Page_84">84</span></p> - -<p>Having so far stated the principles on which the law -of gravitation is established, we shall dismiss this subject -without further details, since it more properly belongs -to the subject of <span class="smcap">Physical Astronomy</span>; to which we -refer the reader for a complete demonstration of the law, -and for the detailed development of its various and important -consequences.</p> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<h2 class="nobreak" id="CHAP_VII">CHAP. VII.<br /> - -<span class="title">TERRESTRIAL GRAVITY.</span></h2> -</div> - - -<p id="p115">(115.) <span class="smcap">Gravitation</span> is the general name given to -this attraction, by whatever masses of matter it may be -manifested. As exhibited in the effects produced by the -earth upon surrounding bodies, it is called “terrestrial -gravity.”</p> - -<p>As the attraction of the earth is directed towards its -centre, it might be expected that two plumb-lines should -appear not to be parallel, but so inclined to each other -as to converge to a point under the surface of the earth. -Thus, if A B and C D, <i><a href="#i_p104a">fig. 23.</a></i>, be two plumb-lines, each -will be directed to the centre O, where, if their directions -were continued, they would meet. In like manner, if -two bodies were allowed to fall from A and C, they would -descend in the directions A B and C D, which converge -to O. Observation, on the contrary, shows, that plumb-lines -suspended in places not far distant from each other -are truly parallel; and that bodies allowed to fall descend -in parallel lines. This apparent parallelism of the direction -of terrestrial gravity is accounted for by the -enormous proportion which the magnitude of the earth -bears to the distance between the two plumb-lines or the -two falling bodies which are compared. If the distance -between the places B, D, were 1200 feet, the inclination -of the lines A B and C D would not amount to a quarter -of a minute, or the 240th part of a degree. But the distance, -in cases where the parallelism is assumed, is never -greater than, and seldom so great as, a few yards; and -hence the inclination of the directions A B and C D is<span class="pagenum" id="Page_85">85</span> -too small to be appreciated by any practical measure. -In the investigation of the phenomena of falling bodies, -we shall, therefore, assume, that all the particles of the -same body are attracted in parallel directions, perpendicular -to an horizontal plane.</p> - -<p id="p116">(116.) Since the intensity of terrestrial gravity increases -as the square of the distance decreases, it might -be expected that, as a falling body approaches the earth, -the force which accelerates it should be continually increasing, -and, strictly speaking, it is so. But any height -through which we observe falling bodies to descend -bears so very small a proportion to the whole distance -from the centre, that the change of intensity of the -force of gravity is quite beyond any practical means of -estimating it. The radius, or the distance from the -surface of the earth to its centre, is 4000 miles. Now, -suppose a body descended through the height of half a -mile, a distance very much beyond those used in experimental -enquiries, the distances from the centre, at -the beginning and end of the fall, are then in the proportion -of 8000 to 8001, and therefore the proportion -of the force of attraction at the commencement to the -force at the end, being that of the squares of these -numbers, is 64,000,000 to 64,016,001, which, in the -whole descent, is an increase of about one part in 4000; -a quantity practically insignificant. We shall, therefore, -in explaining the laws of falling bodies, assume -that, in the entire descent, the body is urged by a force -of uniform intensity.</p> - -<p>Although the force which attracts all parts of the -same body during its descent in a given place is the -same, yet the force of gravity, at different parts of the -earth’s surface, has different intensities. The intensity -diminishes with the latitude, so that it is greater -towards the poles, and lesser towards the equator. The -causes of this variation, its law, and the experimental -proofs of it, will be explained, when we shall treat of -centrifugal force, and the motion of pendulums. It is -sufficient merely to advert to it in this place.</p> - -<p><span class="pagenum" id="Page_86">86</span></p> - -<p id="p117">(117.) Since the earth’s attraction acts separately -and equally on every particle of matter, without regard -to the nature or species of the body, it follows that all -bodies, of whatever kind, or whatever be their masses, -must be moved with the same velocity. If two equal -particles of matter be placed at a certain distance above -the surface of the earth, they will fall in parallel lines, -and with exactly the same speed, because the earth attracts -them equally. In the same manner, a thousand -particles would fall with equal velocities. Now, these -circumstances will in no wise be changed if those 1000 -particles, instead of existing separately, be aggregated -into two solid masses, one consisting of 990 particles, -and the other of 10. We shall thus have a heavy body -and a light one, and, according to our reasoning, they -must fall to the earth with the same speed.</p> - -<p>Common experience, however, is not always consistent -with this doctrine. What are called light substances, -as feathers, gold-leaf, paper, &c., are observed -to fall slowly and irregularly, while heavier masses, as -solid pieces of metal, stones, &c., fall rapidly. Nay, -there are not a few instances in which the earth, instead -of attracting bodies, seems to repel them, as in the case -of smoke, vapours, balloons, and other substances which -actually ascend. We are to consider that the mass of -the earth is not the only agent engaged in these phenomena. -The earth is surrounded by an atmosphere -composed of an elastic or aeriform fluid. This atmosphere -has certain properties, which will be explained in -our treatise on <span class="smcap">Pneumatics</span>, and which are the causes -of the anomalous circumstances alluded to. Light -bodies rise in the atmosphere, for the same reason that -a piece of cork rises from the bottom of a vessel of water; -and other light bodies fall more slowly than heavy ones, -for the same reason that an egg in water falls to the -bottom more slowly than a leaden bullet. This treatise -is not the place to give a direct explanation of these -phenomena. It will be sufficient for our present purpose -to show, that if there were no atmosphere, all bodies, -heavy and light, would fall at the same rate. This may<span class="pagenum" id="Page_87">87</span> -easily be accomplished by the aid of an air-pump. -Having by that instrument abstracted the air from a -tall glass vessel, we are enabled, by means of a wire -passing air-tight through a hole in the top, to let fall -several bodies from the top of the vessel to the bottom. -These, whether they be feathers, paper, gold-leaf, pieces -of money, &c. all descend with the same speed, and strike -the bottom at the same moment.</p> - -<p id="p118">(118.) Every one who has seen a heavy body fall -from a height, has witnessed the fact, that its velocity -increases as it approaches the ground. But if this were -not observable by the eye, it would be betrayed by the -effects. It is well known, that the force with which a -body strikes the ground increases with the height from -whence it has fallen. This force, however, is proportional -to the velocity which it has at the moment it meets -the ground, and therefore this velocity increases with the -height.</p> - -<p>When the observations on attraction in the last -chapter are well understood, it will be evident that -the velocity which a body has acquired in falling -from any height, is the accumulated effects of the -attraction of terrestrial gravity during the whole time of -the fall. Each instant of the fall a new impulse is given -to the body, from which it receives additional velocity; -and its final velocity is composed of the aggregation of -all the small increments of velocity which are thus communicated. -As we are at present to suppose the intensity -of the attraction invariable, it will follow that the -velocity communicated to the body in each instant of -time will be the same, and therefore that the whole quantity -of velocity produced or accumulated at the end of -any time is proportional to the length of that time. -Thus, if a certain velocity be produced in a body having -fallen for one second, twice that velocity will be produced -when it has fallen for two seconds, thrice that -velocity in three seconds, and so on. Such is the -fundamental principle or characteristic of <i>uniformly -accelerated motion</i>.</p> - -<p><span class="pagenum" id="Page_88">88</span></p> - -<p id="p119">(119.) In examining the circumstances of the descent -of a body, the time of the fall and the velocity at each -instant of that time are not the only things to be attended -to. The spaces through which it falls in given intervals -of time, counted either from the commencement of its -fall, or from any proposed epoch of the descent, are -equally important objects of enquiry. To estimate the -space in reference to the time and the final velocity, we -must consider that this space has been moved through -with varying speed. From a state of rest at the beginning -of the fall, the speed gradually increases with the -time, and the final velocity is greater still than that which -the body had at any preceding instant during its descent. -We cannot, therefore, <i>directly</i> appreciate the space moved -through in this case by the time and final velocity. But -as the velocity increases uniformly with the time, we -shall obtain the average speed, by finding that which the -body had in the middle of the interval which elapsed -between the beginning and end of the fall, and thus -the space through which the body has actually fallen is -that through which it would move in the same time with -this average velocity uniformly continued.</p> - -<p>But since the velocity which the body receives in any -time, counted from the beginning of its descent, is in the -proportion of that time, it follows that the velocity of the -body after half the whole time of descent is half the final -velocity. From whence it appears, that the height from -which a body falls in any proposed time is equal to the -space through which a body would move in the same time -with half the final velocity, and it is therefore equal to -half the space which would be moved through in the -same time with the final velocity.</p> - -<p id="p120">(120.) It follows from this reasoning, that between -the three quantities, the height, the time, and the final -velocity, which enter into the investigation of the phenomena -of falling bodies, there are two fixed relations: -<i>First</i>, the time, counted from the beginning of the fall -and the final velocity, are proportional the one to the -other; so that as one increases, the other increases in the -same proportion. <i>Secondly</i>, the height being equal to<span class="pagenum" id="Page_89">89</span> -half the space which would be moved through in the <i>time</i> -of the fall, with the <i>final velocity</i>, must have a fixed -proportion to these two quantities, viz. the <i>time</i> and the -<i>final velocity</i>, or must be proportional to the product of -the two numbers which express them.</p> - -<p>But since the time is always proportional to the final -velocity, they may be expressed by equal numbers, and -the product of equal numbers is the square of either of -them. Hence, the product of the numbers expressing -the time and final velocity is equivalent to the square of -the number expressing the time, or to the square of the -number expressing the final velocity. Hence we infer, -that the height is always proportional to the square of -the time of the fall, or to the square of the final -velocity.</p> - -<p id="p121">(121.) The use of a few mathematical characters will -render these results more distinct, even to students not -conversant with mathematical science.</p> - -<p>Let S = the height from which the body falls, expressed -in feet.</p> - -<p class="ml17em">V = the velocity at the end of the fall in feet per -second.</p> - -<p class="ml17em">T = the number of seconds in the time of the -fall.</p> - -<p class="ml17em"><i>g</i> = the number of feet through which a body -would fall in one second.</p> - -<p>It will therefore follow that the velocity acquired in -one second will be 2<i>g</i>, and the velocity acquired in T -seconds will therefore be 2<i>g</i> × T; so that</p> - -<p class="tac">V = 2<i>g</i> × T  [1]</p> - -<p>Since the space which a body falls through in T seconds -is found by multiplying the space it falls through -in one second by T<sup>2</sup>, we shall have</p> - -<p class="tac">S = <i>g</i> × T<sup>2</sup>  [2]</p> - -<p>from which, combined with [1] we deduce</p> - -<p><span class="pagenum" id="Page_90">90</span></p> - -<p class="tac"><span class="nowrap"><span class="fraction2"><span class="fnum">S = V<sup>2</sup></span><span class="bar">/</span><span class="fden2">4<i>g</i></span></span></span>     [3]</p> - -<p class="tac">S = <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>V × T  [4]</p> - -<p>By these formularies, if the height through which a -body falls freely in one second be known, the height -through which it will fall in any proposed time may be -computed. For since the height is proportional to the -square of the time, the height through which it will fall -in <i>two</i> seconds will be <i>four</i> times that which it falls -through in <i>one</i> second. In <i>three</i> seconds it will fall -through <i>nine</i> times that space; in <i>four</i> seconds, <i>sixteen</i> -times; in <i>five</i> seconds, <i>twenty-five</i> times, and so on. The -following, therefore, is a general rule to find the height -through which a body will fall in any given time: -“Reduce the given time to seconds, take the square -of the number of seconds in it, and multiply the height -through which a body falls in one second by that number; -the result will be the height sought.”</p> - -<p>The following table exhibits the heights and corresponding -times as far as 10 seconds:</p> - -<div class="center"> -<table width="350" border="1" cellpadding="4" summary=""> -<tr> -<td class="tal">Time</td> -<td class="tac"><div>1</div></td> -<td class="tac"><div>2</div></td> -<td class="tac"><div>3</div></td> -<td class="tac"><div>4</div></td> -<td class="tac"><div>5</div></td> -<td class="tac"><div>6</div></td> -<td class="tac"><div>7</div></td> -<td class="tac"><div>8</div></td> -<td class="tac"><div>9</div></td> -<td class="tac"><div>10</div></td> -</tr> -<tr> -<td class="tal">Height</td> -<td class="tac"><div>1</div></td> -<td class="tac"><div>4</div></td> -<td class="tac"><div>9</div></td> -<td class="tac"><div>16</div></td> -<td class="tac"><div>25</div></td> -<td class="tac"><div>36</div></td> -<td class="tac"><div>49</div></td> -<td class="tac"><div>64</div></td> -<td class="tac"><div>81</div></td> -<td class="tac"><div>100</div></td> -</tr> -</table> -</div> - -<p>Each unit in the numbers of the first row expresses a -second of time, and each unit in those of the second row -expresses the height through which a body falls freely -in a second.</p> - -<p id="p122">(122.) If a body fall continually for several successive -seconds, the spaces which it falls through in each succeeding -second have a remarkable relation among each -other, which may be easily deduced from the preceding -table. Taking the space moved through in the first -second still as our unit, four times that space will be -moved through in the first two seconds. Subtract from -this 1, the space moved through in the first second, and -the remainder 3 is the space through which the body falls -in the <i>second</i> second. In like manner if 4, the height -fallen through in the first two seconds, be subtracted<span class="pagenum" id="Page_91">91</span> -from 9, the height fallen through in the first three seconds, -the remainder 5 will be the space fallen through in -the third second. To find the space fallen through in the -fourth second, subtract 9, the space fallen through in the -first three seconds, from 16, the space fallen through in -the first four seconds, and the result is 7, and so on. It -thus appears that if the space fallen through in the first -second be called 1, the spaces described in the second, -third, fourth, fifth, &c. seconds, will be expressed by -the odd numbers respectively, 3, 5, 7, 9, &c. This -places in a striking point of view the accelerated motion -of a falling body, the spaces moved through in each -succeeding second being continually increased.</p> - -<p id="p123">(123.) If velocity be estimated by the space through -which the body would move uniformly in one second, -then the final velocity of a body falling for one second -will be 2; for with that final velocity the body would in -one second move through twice the height through which -it has fallen.</p> - -<p id="p124">(124.) Since the final velocity increases in the same -proportion as the time, it follows that after two seconds -it is twice its amount after one, and after three seconds -thrice that, and so on. Thus, the following table exhibits -the final velocities corresponding to the times of -descent:</p> - -<div class="center"> -<table width="350" border="1" cellpadding="4" summary=""> -<tr> -<td class="tal">Time</td> -<td class="tac"><div>1</div></td> -<td class="tac"><div>2</div></td> -<td class="tac"><div>3</div></td> -<td class="tac"><div>4</div></td> -<td class="tac"><div>5</div></td> -<td class="tac"><div>6</div></td> -<td class="tac"><div>7</div></td> -<td class="tac"><div>8</div></td> -<td class="tac"><div>9</div></td> -<td class="tac"><div>10</div></td> -</tr> -<tr> -<td class="tal">Final velocity</td> -<td class="tac"><div>2</div></td> -<td class="tac"><div>4</div></td> -<td class="tac"><div>6</div></td> -<td class="tac"><div>8</div></td> -<td class="tac"><div>10</div></td> -<td class="tac"><div>12</div></td> -<td class="tac"><div>14</div></td> -<td class="tac"><div>16</div></td> -<td class="tac"><div>18</div></td> -<td class="tac"><div>20</div></td> -</tr> -</table> -</div> - -<p>The numbers in the second row express the spaces -through which a body with the final velocity would move -in one second, the unit being, as usual, the space through -which a body falls freely in one second.</p> - -<p id="p125">(125.) Having thus developed theoretically the laws -which characterise the descent of bodies, falling freely -by the force of gravity, or by any other uniform force -of the same kind, it is necessary that we should show -how these laws can be exhibited by actual experiment. -There are some circumstances attending the fall of heavy -bodies which would render it difficult, if not impossible, -to illustrate, by the direct observation of this pheno<span class="pagenum" id="Page_92">92</span>menon, -the properties which have been explained in -this chapter. A body falling freely by the force of -gravity, as we shall hereafter prove, descends in one -second of time through a height of about 16 <span class="nowrap">feet<a id="FNanchor_1" href="#Footnote_1" class="fnanchor">1</a></span>; in -two seconds, it would, therefore, fall through four times -that space, or 64 feet; in three seconds, through 9 times -the height, or 144 feet; and in four seconds, through 256 -feet. In order, therefore, to be enabled to observe the -phenomena for only four seconds, we should command -an height of at least 256 feet. But further; the velocity -at the end of the first second would be at the rate of 32 -feet per second; at the end of the second second, it -would be 64 feet per second; and towards the end of the -fall it would be about 120 feet per second. It is evident -that this great degree of rapidity would be a serious impediment -to accurate observation, even though we should -be able to command the requisite height. It appears -therefore that the number expressed by <i>g</i> in the preceding -formulæ is 16·083.</p> - -<p>It occurred to Mr. George Attwood, a mathematician -and natural philosopher of the last century, that all the -phenomena of falling bodies might be experimentally -exhibited and accurately observed, if a force of the same -kind as gravity, viz. an uniformly accelerating force, be -used, but of a much less intensity; so that while the -motion continues to be governed by the same laws, its -quantity may be so much diminished, that the final velocity, -even after a descent of many seconds, shall be -so moderated as to admit of most deliberate and exact -observation. This being once accomplished, nothing -more would remain but to find the height through which -a body would fall in one second, or, what is the same, the -proportion of the force of gravity to the mitigated but -uniform accelerating force thus substituted for it.</p> - -<p id="p126">(126.) To realise this notion, Attwood constructed a -wheel turning on its axle with very little friction, and -having a groove on its edge to receive a string. Over -this wheel, and in the groove, he placed a fine silken cord, -to the ends of which were attached equal cylindrical<span class="pagenum" id="Page_93">93</span> -weights. Thus placed, the weights perfectly balance each -other, and no motion ensues. To one of the weights he -then added a small quantity, so as to give it a slight -preponderance. The loaded weight now began to descend, -drawing up on the other side the unloaded weight. The -descent of the loaded weight, under these circumstances, -is a motion exactly of the <i>same kind</i> as the descent of a -heavy body falling freely by the force of gravity; that is, -it increases according to the same laws, though at a very -diminished rate. To explain this, suppose that the -loaded weight descends from a state of rest through one -inch in a second, it will descend through 4 inches in two -seconds, through 9 in three, through 16 in four, and so -on. Thus in 20 seconds, it would descend through 400 -inches, or 33 feet 4 inches, a height which, if it were -necessary, could easily be commanded.</p> - -<p>It might, perhaps, be thought, that since the weights -suspended at the ends of the thread are in equilibrium, -and therefore have no tendency either to move or to -resist motion, the additional weight placed upon one of -them ought to descend as rapidly as it would if it were -allowed to fall freely and unconnected with them. It -is very true that this weight will receive from the attraction -of the earth the same force when placed upon -one of the suspended weights, as it would if it were -disengaged from them; but in the consequences which -ensue, there is this difference. If it were unconnected -with the suspended weights, the whole force impressed -upon it would be expended in accelerating its descent; -but being connected with the equal weights which sustain -each other in equilibrium, by the silken cord passing -over the wheel, the force which is impressed upon the -added weight is expended, not as before, in giving velocity -to the added weight alone, but to it together with -the two equal weights appended to the string, one of -which descends with the added weight, and the other -rises on the opposite side of the wheel. Hence, setting -aside any effect which the wheel itself produces, the -velocity of the descent must be lessened just in proportion -as the mass among which the impressed force is to be<span class="pagenum" id="Page_94">94</span> -distributed is increased; and therefore the <i>rate</i> of the -fall bears to that of a body falling freely the same proportion -as the added weight bears to the sum of the -masses of the equal suspended weights and the added -weight. Thus the smaller the added weight is, and the -greater the equal suspended weights are, the slower will -the rate of descent be.</p> - -<p>To render the circumstances of the fall conveniently -observable, a vertical shaft (see <i><a href="#i_p104a">fig. 24.</a></i>) is usually provided, -which is placed behind the descending weight. -This pillar is divided to inches and halves, and of course -may be still more minutely graduated, if necessary. A -stage to receive the falling weight is moveable on this -pillar, and capable of being fixed in any proposed position -by an adjusting screw. A pendulum vibrating -seconds, the beat of which ought to be very audible, is -placed near the observer. The loaded weight being thus -allowed to descend for any proposed time, or from any -required height, all the circumstances of the descent may -be accurately observed, and the several laws already explained -in this chapter may be experimentally verified.</p> - -<p id="p127">(127.) The laws which govern the descent of bodies -by gravity, being reversed, will be applicable to the -ascent of bodies projected upwards. If a body be -projected directly upwards with any given velocity, it -will rise to the height from which it should have fallen -to acquire that velocity. The earth’s attraction will, in -this case, gradually deprive the body of the velocity -which is communicated to it at the moment at which it -is projected. Consequently, the phenomenon will be -that of <i>retarded motion</i>. At each part of its ascent it -will have the same velocity which it would have if it -descended to the same place from the highest point to -which it rises. Hence it is clear, that all the particulars -relative to the ascent of bodies may be immediately -inferred from those of their descent, and therefore this -subject demands no further notice.</p> - -<p>To complete the investigation of the phenomena of -falling bodies, it would now only remain to explain the<span class="pagenum" id="Page_95">95</span> -method of ascertaining the exact height through which -a body would descend in one second, if unresisted by -the atmosphere, or any other disturbing cause. As the -solution of this problem, however, requires the aid of -principles not yet explained, it must for the present be -postponed.</p> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<h2 class="nobreak" id="CHAP_VIII">CHAP. VIII.<br /> - -<span class="title">OF THE MOTION OF BODIES ON INCLINED PLANES AND CURVES.</span></h2> -</div> - - -<p id="p128">(128.) <span class="smcap">In</span> the last chapter, we investigated the phenomena -of bodies descending freely in the vertical direction, -and determined the laws which govern, not their -motion alone, but that of bodies urged by any uniformly -accelerating force whatever. We shall now consider -some of the most ordinary cases in which the free descent -of bodies is impeded, and the effects of their gravitation -modified.</p> - -<p id="p129">(129.) If a body, urged by any forces whatever, be -placed upon a hard unyielding surface, it will evidently -remain at rest, if the resultant <a href="#p76">(76)</a> of all the forces -which are applied to it be directed perpendicularly against -the surface. In this case, the effect produced is pressure, -but no motion ensues. If only one force act upon -the body, it will remain at rest, provided the direction -of that force be perpendicular to the surface.</p> - -<p>But the effect will be different, if the resultant of the -forces which are applied to the body be oblique to the -surface. In that case this resultant, which, for simplicity, -may be taken as a single force, may be considered -as mechanically equivalent to two forces <a href="#p76">(76)</a>, one in -the direction of the surface, and the other perpendicular -to it. The latter element will be resisted, and will produce -a pressure; the former will cause the body to -move. This will perhaps be more clearly apprehended -by the aid of a diagram.</p> - -<p>Let A B, <i><a href="#i_p104a">fig. 25.</a></i>, be the surface, and let P be a particle<span class="pagenum" id="Page_96">96</span> -of matter placed upon it, and urged by a force in the direction -P D, perpendicular to A B. It is manifest, that -this force can only press the particle P against A B, but -cannot give it any motion.</p> - -<p>But let us suppose, that the force which urges P is -in a direction P F, oblique to A B. Taking P F as the -diagonal of a parallelogram, whose sides are P D and -P C <a href="#p74">(74)</a>, the force P F is mechanically equivalent to -two forces, expressed by the lines P D and P C. But -P D, being perpendicular to A B, produces pressure without -motion, and P C, being in the direction of A B, produces -motion without pressure. Thus the effect of the -force P F is distributed between motion and pressure in -a certain proportion, which depends on the obliquity of -its direction to that of the surface. The two extreme -cases are, 1. When it is in the direction of the surface; -it then produces motion without pressure: and, 2. When -it is perpendicular to the surface; it then produces pressure -without motion. In all intermediate directions, -however, it will produce both these effects.</p> - -<p id="p130">(130.) It will be very apparent, that the more oblique -the direction of the force P F is to A B, the greater -will be that part of it which produces motion, and -the less will be that which produces pressure. This -will be evident by inspecting <i><a href="#i_p104a">fig. 26.</a></i> In this figure the -line P F, which represents the force, is equal to P F in -<i><a href="#i_p104a">fig. 25.</a></i> But P D, which expresses the pressure, is less -in <i><a href="#i_p104a">fig. 26.</a></i> than in <i><a href="#i_p104a">fig. 25.</a></i>, while P C, which expresses the -motion, is greater. So long, then, as the obliquity of the -directions of the surface and the force remain unchanged, -so long will the distribution of the force between motion -and pressure remain the same; and therefore, if the force -itself remain the same, the parts of it which produce -motion and pressure will be respectively equal.</p> - -<p id="p131">(131.) These general principles being understood, -no difficulty can arise in applying them to the motion of -bodies urged on inclined planes or curves by the force -of gravity. If a body be placed on an unyielding horizontal -plane, it will remain at rest, producing a pressure -on the plane equal to the total amount of its weight. -For in this case the force which urges the body, being -that of terrestrial gravity, its direction is vertical, and -therefore perpendicular to the horizontal plane.</p> - -<p>But if the body P, <i><a href="#i_p104a">fig. 25.</a></i>, be placed upon a plane -A B, oblique to the direction of the force of gravity, -then, according to what has been proved <a href="#p129">(129)</a>, the -weight of the body will be distributed into two parts, -P C and P D; one, P D, producing a pressure on the -plane A B, and the other, P C, producing motion down -the plane. Since the obliquity of the perpendicular direction -P F of the weight to that of the plane A B -must be the same on whatever part of the plane the -weight may be placed, it follows <a href="#p130">(130)</a>, that the proportion -P C of the weight which urges the body down -the plane must be the same throughout its whole descent.</p> - -<p id="p132">(132.) Hence it may easily be inferred, that the force -down the plane is uniform; for since the weight of the -body P is always the same, and since its proportion to -that part which urges it down the plane is the same, it -follows that the quantity of this part cannot vary. The -motion of a heavy body down an inclined plane is -therefore an uniformly-accelerated motion, and is characterised -by all the properties of uniformly-accelerated -motion, explained in the last chapter.</p> - -<p>Since P F represents the force of gravity, that is, the -force with which the body would descend freely in the vertical -direction, and P C the force with which it moves -down the plane, it follows that a body would fall freely -in the vertical direction from P to F in the same time as -on the plane it would move from P to C. In this manner, -therefore, when the height through which a body would -fall vertically is known, the space through which it would -descend in the same time down any given inclined plane -may be immediately determined. For let A B, <i><a href="#i_p104a">fig. 25.</a></i>, be -the given inclined plane, and let P F be the space through -which the body would fall in one second. From F draw -F C perpendicular to the plane, and the space P C is that -through which the body P will fall in one second on -the plane.</p> - -<p id="p133">(133.) As the angle B A H, which measures the elevation -of the plane, is increased, the obliquity of the -vertical direction P F with the plane is also increased. -Consequently, according to what has been proved <a href="#p130">(130)</a>, -it follows, that as the elevation of the plane is increased, -the force which urges the body down the plane is also -increased, and as the elevation is diminished, the force -suffers a corresponding diminution. The two extreme -cases are, 1. When the plane is raised until it becomes -perpendicular, in which case the weight is permitted to -fall freely, without exerting any pressure upon the plane; -and, 2. When the plane is depressed until it becomes -horizontal, in which case the whole weight is supported, -and there is no motion.</p> - -<p>From these circumstances it follows, that by means of -an inclined plane we can obtain an uniformly-accelerating -force of any magnitude less than that of gravity.</p> - -<p>We have here omitted, and shall for the present in -every instance omit, the effects of <i>friction</i>, by which the -motion down the plane is retarded. Having first investigated -the mechanical properties of bodies supposed to be -free from friction, we shall consider friction separately, -and show how the present results are modified by it.</p> - -<p id="p134">(134.) The accelerating forces on different inclined -planes may be compared by the principle explained in -(<a href="#p131">131</a>). Let <i><a href="#i_p104a">figs. 25.</a></i> and <i>26.</i> be two inclined planes, and -take the lines P F in each figure equal, both expressing -the force of gravity, then P C will be the force which in -each case urges the body down the plane.</p> - -<p>As the force down an inclined plane is less than that -which urges a body falling freely in the vertical direction, -the space through which the body must fall to -attain a certain final velocity must be just so much -greater as the accelerating force is less. On this principle -we shall be able to determine the final velocity in -descending through any space on a plane, compared with -the final velocity attained in falling freely in the vertical -direction. Suppose the body P, <i><a href="#i_p104a">fig. 27.</a></i>, placed at the top -of the plane, and from H draw the perpendicular H C. If -B H represent the force of gravity, B C will represent the -force down the plane <a href="#p131">(131)</a>. In order that the body<span class="pagenum" id="Page_97">97</span> -moving down the plane shall have a final velocity equal -to that of one which has fallen freely from B to H, it -will be necessary that it should move from B down the -plane, through a space which bears the same proportion -to B H as B H does to B C. But since the triangle -A B H is in all respects similar to H B C, only made -upon a larger scale, the line A B bears the same proportion -to B H as B H bears to B C. Hence, in falling on -the inclined plane from B to A, the final velocity is the -same as in falling freely from B to H.</p> - -<p>It is evident that the same will be true at whatever -level an horizontal line be drawn. Thus, if I K be horizontal, -the final velocity in falling on the plane from B -to I will be the same as the final velocity in falling -freely from B to K.</p> - -<p id="p135">(135.) The motion of a heavy body down a curve -differs in an important respect from the motion down -an inclined plane. Every part of the plane being -equally inclined to the vertical direction, the effect of -gravity in the direction of the plane is uniform; and, -consequently, the phenomena obey all the established -laws of uniformly-accelerated motion. If, however, we -suppose the line B A, on which the body P descends, to -be curved as in <i><a href="#i_p104a">fig. 28.</a></i>, the obliquity of its direction -at different parts, to the direction P F of gravity, will -evidently vary. In the present instance, this obliquity -is greater towards B and less towards A, and hence the -part of the force of gravity which gives motion to the -body is greater towards B than towards A <a href="#p130">(130)</a>. The -force, therefore, which urges the body, instead of being -uniform as in the inclined plane, is here gradually diminished. -The rate of this diminution depends entirely -on the nature of the curve, and can be deduced -from the properties of the curve by mathematical reasoning. -The details of such an investigation are not, -however, of a sufficiently elementary character to allow -of being introduced with advantage into this treatise. -We must therefore limit ourselves to explain such of the -results as may be necessary for the development of the -other parts of the science.</p> - -<p><span class="pagenum" id="Page_98">98</span></p> - -<p id="p136">(136.) When a heavy body is moved down an inclined -plane by the force of gravity, the plane has been proved -to sustain a pressure, arising from a certain part of the -weight P D, <i><a href="#i_p104a">fig. 25.</a></i>, which acts perpendicularly to the -plane. This is also the case in moving down a curve such -as B A, <i><a href="#i_p104a">fig. 28.</a></i> In this case, also, the whole weight is -distributed between that part which is directed down -the curve, and that which, being perpendicular to the -curve, produces a pressure upon it. There is, however, -another cause which produces pressure upon the curve, and -which has no operation in the case of the inclined plane. -By the property of inertia, when a body is put in motion -in any direction, it must persevere in that direction, -unless it be deflected from it by an efficient force. In -the motion down an inclined plane the direction is never -changed, and therefore by its inertia the falling body -retains all the motion impressed upon it continually in -the same direction; but when it descends upon a curve, -its direction is constantly varying, and the resistance of -the curve being the deflecting cause, the curve must -sustain a pressure equal to that force, which would thus -be capable of continually deflecting the body from -the rectilinear path in which it would move in virtue -of its inertia. This pressure entirely depends on the -curvature of the path in which the body is constrained -to move, and on its inertia, and is therefore altogether -independent of the weight, and would, in fact, exist if -the weight were without effect.</p> - -<p id="p137">(137.) This pressure has been denominated <i>centrifugal -force</i>, because it evinces a tendency of the moving -body to <i>fly from</i> the centre of the curve in which it -is moved. Its quantity depends conjointly on the velocity -of the motion and the curvature of the path -through which the body is moved. As circles may be -described with every degree of curvature, according to -the length of the radius, or the distance from their circumference -to their centre, it follows that, whatever be -the curve in which the body moves, a circle can always -be assigned which has the same curvature as is found at<span class="pagenum" id="Page_99">99</span> -any proposed point of the given curve. Such a circle -is called “the circle of curvature” at that point of the -curve; and as all curves, except the circle, vary their -degrees of curvature at different points, it follows that -different parts of the same curve will have different -circles of curvature. It is evident that the greater the -radius of a circle is, the less is its curvature: thus the -circle with the radius A B, <i><a href="#i_p104a">fig. 29.</a></i>, is more curved than -that whose radius is C D, and that in the exact proportion -of the radius C D to the radius A B. The radius -of the circle of curvature for any part of a curve is -called “the radius of curvature” of that part.</p> - -<p id="p138">(138.) The centrifugal pressure increases as the radius -of curvature increases; but it also has a dependence -on the velocity with which the moving body swings -round the centre of the circle of curvature. This velocity -is estimated either by the actual space through which -the body moves, or by the <i>angular velocity</i> of a line -drawn from the centre of the circle to the moving body. -That body carries one end of this line with it, while -the other remains fixed at the centre. As this angular -swing round the centre increases, the centrifugal pressure -increases. To estimate the rate at which this pressure -in general varies, it is necessary to multiply the square -of the number expressing the angular velocity by that -which expresses the radius of curvature, and the force -increases in the same proportion as the product thus -obtained.</p> - -<p id="p139">(139.) We have observed that the same causes which -produce pressure on a body restrained, will produce motion -if the body be free. Accordingly, if a body be -moved by any efficient cause in a curve, it will, by reason -of the centrifugal force, <i>fly off</i>, and the moving force -with which it will thus retreat from the centre round -which it is whirled will be a measure of the centrifugal -force. Upon this principle an apparatus called a <i>whirling -table</i> has been constructed, for the purpose of -exhibiting experimental illustrations of the laws of centrifugal -force. By this machine we are enabled to place -any proposed weights at any given distances from cen<span class="pagenum" id="Page_100">100</span>tres -round which they are whirled, either with the same -angular velocity, or with velocities having a certain proportion. -Threads attached to the whirling weights are -carried to the centres round which they respectively -revolve, and there, passing over pulleys, are connected -with weights which may be varied at pleasure. When -the whirling weights fly from their respective centres, -by reason of the centrifugal force, they draw up the -weights attached to the other ends of the threads, and -the amount of the centrifugal force is estimated by the -weight which it is capable of raising.</p> - -<p>With this instrument the following experiments may -be exhibited:—</p> - -<p>Exp. 1. Equal weights whirled with the same velocity -at equal distances from the centre raise the same -weight, and therefore have the same centrifugal force.</p> - -<p>Exp. 2. Equal weights whirled with the same angular -velocity at distances from the centre in the proportion -of one to two, will raise weights in the same -proportion. Therefore the centrifugal forces are in that -proportion.</p> - -<p>Exp. 3. Equal weights whirled at equal distances -with angular velocities which are as one to two, will -raise weights as one to four, that is, as the squares of -the angular velocities. Therefore the centrifugal forces -are in that proportion.</p> - -<p>Exp. 4. Equal weights whirled at distances which are -as two to three, with angular velocities which are as one -to two, will raise weights which are as two to twelve; -that is, as the products of the distances two and three, -and the squares one and four, of the angular velocities. -Hence, the centrifugal forces are in this proportion.</p> - -<p>The centrifugal force must also increase as the mass -of the body moved increases; for, like attraction, each -particle of the moving body is separately and equally -affected by it. Hence a double mass, moving at the same -distance, and with the same velocity, will have a double -force. The following experiment verifies this:—</p> - -<p>Exp. 5. If weights, which are as one to two, be -whirled at equal distances with the same velocity, they -will raise weights which are as one to two.</p> - -<p><span class="pagenum" id="Page_101">101</span></p> - -<p>The law which governs centrifugal force may then be -expressed in general symbols briefly thus:—</p> - -<p>Let <i>c</i> = the centrifugal force with which a weight of -one lb. revolving in a circle in one second, the radius of -which is one foot, would act on a string connecting it -with the centre. The force with which it would act on -a string, the length of which is R feet, would be <i>c</i> × R; -and if instead of revolving in one second it revolved in -T seconds, the force would be</p> - -<p class="tac"><span class="nowrap"><span class="fraction2"><span class="fnum"><i>c</i> × R</span><span class="bar">/</span><span class="fden2">T<sup>2</sup></span></span></span>;</p> - -<p>and if the revolving mass were W lbs. the force would be</p> - -<p class="tac">C = <span class="nowrap"><span class="fraction2"><span class="fnum"><i>c</i> × W × R</span><span class="bar">/</span><span class="fden2">T<sup>2</sup></span></span></span>.</p> - -<p>This formula includes the entire theory of centrifugal -force.</p> - -<p>But it can be shown that the number expressed by <i>c</i> -is 1·226, and consequently</p> - -<p class="tac">C = <span class="nowrap"><span class="fraction2"><span class="fnum">1·226 × W × R</span><span class="bar">/</span><span class="fden2">T<sup>2</sup></span></span></span>.</p> - -<p>It is often more convenient to use the number of revolutions -made in a given time than the time of one -revolution. Let N then express the number of revolutions, -or fraction of a revolution, made in one second, -and we shall have</p> - -<p class="tac">T = <span class="nowrap"><span class="fraction2"><span class="fnum">1</span><span class="bar">/</span><span class="fden2">N</span></span></span>.</p> - -<p>Therefore</p> - -<p class="tac">C = 1·226 × W × R × N<sup>2</sup>.</p> - -<p id="p140">(140.) The consideration of centrifugal force proves, -that if a body be observed to move in a curvilinear path, -some efficient cause must exist which prevents it from -flying off, and which compels it to revolve round the -centre. If the body be connected with the centre by a -thread, cord, or rod, then the effect of the centrifugal -force is to give tension to the thread, cord, or rod. If -an unyielding curved surface be placed on the convex side -of the path, then the force will produce pressure on this<span class="pagenum" id="Page_102">102</span> -surface. But if a body is observed to move in a curve -without any visible material connection with its centre, -and without any obstruction on the convex side of its path -to resist its retreat, as is the case with the motions of -the planets round the sun, and the satellites round the -planets, it is usual to assign the cause to the attraction -of the body which occupies the centre: in the present -instance the sun is that body, and it is customary to say -that the <i>attraction</i> of the sun, neutralising the effects -of the centrifugal force of the planets, <i>retains them</i> in -their orbits. We have elsewhere animadverted on the -inaccurate and unphilosophical style of this phraseology, -in which terms are admitted which intimate not -only an unknown cause, but assign its seat, and intimate -something of its nature. All that we are entitled to declare -in this case is, that a motion is continually impressed -upon the planet; that this motion is directed towards -the sun; that it counteracts the centrifugal force; but -from whence this motion proceeds, whether it be a virtue -resident in the sun, or a property of the medium or space -in which both sun and planets are placed, or whatever -other influence may be its proximate cause, we are altogether -ignorant.</p> - -<p class="mt1em" id="p141">(141.) Numerous examples of the effects of centrifugal -force may be produced.</p> - -<p>If a stone or other weight be placed in a sling, which -is whirled round by the hand in a direction perpendicular -to the ground, the stone will not fall out of the sling, even -when it is at the top of its circuit, and, consequently, -has no support beneath it. The centrifugal force, in this -case, acting from the hand, which is the centre of rotation, -is greater than the weight of the body, and therefore -prevents its fall.</p> - -<p>In like manner, a glass of water may be whirled so -rapidly that even when the mouth of the glass is presented -downwards, the water will still be retained in it -by the centrifugal force.</p> - -<p>If a bucket of water be suspended by a number of -threads, and these threads be twisted by turning round<span class="pagenum" id="Page_103">103</span> -the bucket many times in the same direction, on allowing -the cords to untwist, the bucket will be whirled rapidly -round, and the water will be observed to rise on its sides -and sink at its centre, owing to the centrifugal force with -which it is driven from the centre. This effect might -be carried so far, that all the water would flow over and -leave the bucket nearly empty.</p> - -<p id="p142">(142.) A carriage, or horseman, or pedestrian, passing -a corner moves in a curve, and suffers a centrifugal force, -which increases with the velocity, and which impresses -on the body a force directed from the corner. An animal -causes its weight to resist this force, by voluntarily -inclining its body towards the corner. In this case, let -A B, <i><a href="#i_p104a">fig. 30.</a></i>, be the body; C D is the direction of the -weight perpendicular to the ground, and C F is the direction -of the centrifugal force parallel to the ground and -<i>from</i> the corner. The body A B is inclined to the corner, -so that the diagonal force <a href="#p74">(74)</a>, which is mechanically -equivalent to the weight and centrifugal force, shall be in -the direction C A, and shall therefore produce the pressure -of the feet upon the ground.</p> - -<p>As the velocity is increased, the centrifugal force is -also increased, and therefore a greater inclination of the -body is necessary to resist it. We accordingly find that -the more rapidly a corner is turned, the more the animal -inclines his body towards it.</p> - -<p>A carriage, however, not having voluntary motion, -cannot make this compensation for the disturbing force -which is called into existence by the gradual change of -direction of the motion; consequently it will, under -certain circumstances, be overturned, falling of course -outwards, or <i>from</i> the corner. If A B be the carriage, -and C, <i><a href="#i_p104a">fig. 31.</a></i>, the place at which the weight is principally -collected, this point C will be under the influence of -two forces: the weight, which may be represented by the -perpendicular C D, and the centrifugal force, which will -be represented by a line C F, which shall have the same -proportion to C D as the centrifugal force has to the -weight. Now the combined effect of these two forces will -be the same as the effect of a single force, represented by<span class="pagenum" id="Page_104">104</span> -C G. Thus, the pressure of the carriage on the road is -brought nearer to the outer wheel B. If the centrifugal -force bear the same proportion to the weight as C F (or -D B), <i><a href="#i_p104a">fig. 32.</a></i>, bears to C D, the whole pressure is thrown -upon the wheel B.</p> - -<p>If the centrifugal force bear to the weight a greater -proportion than D B has to C D, then the line C F, which -represents it, <i><a href="#i_p104a">fig. 33.</a></i>, will be greater than D B. The -diagonal C G, which represents the combined effects of -the weight and centrifugal force, will in this case pass -outside the wheel B, and therefore this resultant will be -unresisted. To perceive how far it will tend to overturn -the carriage, let the force C G be resolved into two, one -in the direction of C B, and the other C K, perpendicular -to C B. The former C B will be resisted by the road, -but the latter C K will tend to lift the carriage over the -external wheel. If the velocity and the curvature of the -course be continued for a sufficient time to enable this -force C K to elevate the weight, so that the line of direction -shall fall on B, the carriage will be overthrown.</p> - -<p>It is evident from what has been now stated, that the -chances of overthrow under these circumstances depend -on the proportion of B D to C D, or what is to the -same purpose, of the distance between the wheels to -the height of the principal seat of the load. It will be -shown in the next chapter, that there is a certain point, -called the centre of gravity, at which the entire weight -of the vehicle and its load may be conceived to be concentrated. -This is the point which in the present investigation -we have marked C. The security of the carriage, -therefore, depends on the greatness of the distance -between the wheels and the smallness of the elevation of -the centre of gravity above the road; for either or both -of these circumstances will increase the proportion of -B D to C D.</p> - -<p id="p143">(143.) In the equestrian feat exhibited in the ring -at the amphitheatre, when the horse moves round with -the performer standing on the saddle, both the horse and -rider incline continually towards the centre of the ring, -and the inclination increases with the velocity of the<span class="pagenum" id="Page_105">105</span> -motion: by this inclination their weights counteract the -effect of the centrifugal force, exactly as in the case -already mentioned (<a href="#p142">142</a>.)</p> - -<div class="figcenter" id="i_p104a" style="max-width: 31.25em;"> - <img src="images/i_p104a.jpg" alt="" /> - <div class="caption"> - -<p class="tar"><i>H. Adlard, sc.</i></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<p id="p144">(144.) If a body be allowed to fall by its weight down -a convex surface, such as A B, <i><a href="#i_p120a">fig. 34.</a></i>, it would continue -upon the surface until it arrive at B but for the effect of -the centrifugal force: this, giving it a motion from the -centre of the curve, will cause it to quit the curve at -a certain point C, which can be easily found by mathematical -computation.</p> - -<p id="p145">(145.) The most remarkable and important manifestation -of centrifugal force is observed in the effects -produced by the rotation of the earth upon its axis. -Let the circle in <i><a href="#i_p120a">fig. 35.</a></i> represent a section of the -earth, A B being the axis on which it revolves. This -rotation causes the matter which composes the mass of -the earth to revolve in circles round the different points -of the axis as centres at the various distances at which -the component parts of this mass are placed. As they -all revolve with the same angular velocity, they will be -affected by centrifugal forces, which will be greater or -less in proportion as their distances from the centre are -greater or less. Consequently the parts of the earth which -are situated about the equator, D, will be more strongly -affected by centrifugal force than those about the poles, -A B. The effect of this difference has been that the -component matter about the equator has actually been -driven farther from the centre than that about the poles, -so that the figure of the earth has swelled out at the sides, -and appears proportionally depressed at the top and bottom, -resembling the shape of an orange. An exaggerated -representation of this figure is given in <i><a href="#i_p120a">fig. 36.</a></i>; the -real difference between the distances of the poles and -equator from the centre being too small to be perceptible -in a diagram. The exact proportion of C A to C D has -never yet been certainly ascertained. Some observations -make C D exceed C A by <span class="nowrap"> <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">277,</span></span></span> and others by only <span class="nowrap"> <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">333</span></span></span>. -The latter, however, seems the more probable. It may -be considered to be included between these limits.</p> - -<p><span class="pagenum" id="Page_106">106</span></p> - -<p>The same cause operates more powerfully in other -planets which revolve more rapidly on their axes. Jupiter -and Saturn have forms which are considerably more -elliptical.</p> - -<p id="p146">(146.) The centrifugal force of the earth’s rotation -also affects detached bodies on its surface. If such -bodies were not held upon the surface by the earth’s -attraction, they would be immediately flung off by the -whirling motion in which they participate. The centrifugal -force, however, really diminishes the effects of the -earth’s attraction on those bodies, or, what is the same, -diminishes their weights. If the earth did not revolve -on its axis, the weight of bodies in all places equally -distant from the centre would be the same; but this is -not so when the bodies, as they do, move round with the -earth. They acquire from the centrifugal force a tendency -to fly from the axis, which increases with their -distance from that axis, and is therefore greater the -nearer they are to the equator, and less as they approach -the pole. But there is another reason why the centrifugal -force is more efficient, in the opposition which it -gives to gravity near the equator than near the poles. -This force does not act from the centre of the earth, but -is directed from the earth’s axis. It is, therefore, not -directly opposed to gravity, except on the equator itself. -On leaving the equator, and proceeding towards the poles, -it is less and less opposed to gravity, as will be plain on -inspecting <i><a href="#i_p120a">fig. 35.</a></i>, where the lines P C all represent the -direction of gravity, and the lines P F represent the direction -of the centrifugal force.</p> - -<p>Since, then, as we proceed from the equator towards -the poles, not only the amount of the centrifugal force -is continually diminished, but also it acts less and less in -opposition to gravity, it follows that the weights of bodies -are most diminished by it at the equator, and less so -towards the poles.</p> - -<p>Since bodies are commonly weighed by balancing -them against other bodies of known weight, it may be -asked, how the phenomena we have been just describing -can be ascertained as a matter of fact? for whatever be<span class="pagenum" id="Page_107">107</span> -the body against which it may be balanced, that body -must suffer just as much diminution of weight as every -other, and consequently, all being diminished in the same -proportion, the balance will be preserved though the -weights be changed.</p> - -<p>To render this effect observable, it will be necessary -to compare the effects of gravity with some phenomenon -which is not affected by the centrifugal force of the -earth’s rotation, and which will be the same at every -part of the earth. The means of accomplishing this will -be explained in a subsequent chapter.</p> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<h2 class="nobreak" id="CHAP_IX">CHAP. IX.<br /> - -<span class="title">THE CENTRE OF GRAVITY.</span></h2> -</div> - - -<p id="p147">(147.) <span class="smcap">By</span> the earth’s attraction, all the particles -which compose the mass of a body are solicited by equal -forces in parallel directions downwards. If these component -particles were placed in mere juxtaposition, -without any mechanical connection, the force impressed -on any one of them could in nowise affect the others, -and the mass would, in such a case, be contemplated as -an aggregation of small particles of matter, each urged -by an independent force. But the bodies which are the -subjects of investigation in mechanical science are not -found in this state. Solid bodies are coherent masses, -the particles of which are firmly bound together, so that -any force which affects one, being modified according -to circumstances, will be transmitted through the whole -body. Liquids accommodate themselves to the shape of -the surfaces on which they rest, and forces affecting any -one part are transmitted to others, in a manner depending -on the peculiar properties of this class of bodies.</p> - -<p>As all bodies, which are subjects of mechanical enquiry, -on the surface of the earth, must be continually<span class="pagenum" id="Page_108">108</span> -influenced by terrestrial gravity, it is desirable to obtain -some easy and summary method of estimating the effect -of this force. To consider it, as is unavoidable in the -first instance, the combined action of an infinite number -of equal and parallel forces soliciting the elementary -molecules downwards, would be attended with manifest -inconvenience. An infinite number of forces, and an -infinite subdivision of the mass, would form parts of -every mechanical problem.</p> - -<p>To overcome this difficulty, and to obtain all the ease -and simplicity which can be desired in elementary investigations, -it is only necessary to determine some force, -whose single effect shall be equivalent to the combined -effects of the gravitation of all the molecules of the -body. If this can be accomplished, that single force -might be introduced into all problems to represent the -whole effect of the earth’s attraction, and no regard need -be had to any particles of the body, except that on which -this force acts.</p> - -<p id="p148">(148.) To discover such a force, if it exist, we shall -first enquire what properties must necessarily characterise -it. Let A B, <i><a href="#i_p120a">fig. 37.</a></i>, be a solid body placed near -the surface of the earth. Its particles are all solicited -downwards, in the directions represented by the arrows. -Now, if there be any single force equivalent to these -combined effects, two properties may be at once assigned -to it: 1. It must be presented downwards, in the common -direction of those forces to which it is mechanically -equivalent; and, 2. it must be equal in intensity -to their sum, or, what is the same, to the force with -which the whole mass would descend. We shall then -suppose it to have this intensity, and to have the direction -of the arrow D E. Now, if the single force, in the -direction D E, be equivalent to all the separate attractions -which affect the particles, we may suppose all these -attractions removed, and the body A B influenced only -by a single attraction, acting in the direction D E. This -being admitted, it follows that if the body be placed -upon a prop, immediately under the direction of the line<span class="pagenum" id="Page_109">109</span> -D E, or be suspended from a fixed point immediately -above its direction, it will remain motionless. For the -whole attracting force in the direction D E will, in the -one case, press the body on the prop, and, in the other -case, will give tension to the cord, rod, or whatever -other means of suspension be used.</p> - -<p id="p149">(149.) But suppose the body were suspended from -some point P, not in the direction of the line D E. Let -P C be the direction of the thread by which the body is -suspended. Its whole weight, according to the supposition -which we have adopted, must then act in the -direction C E. Taking C F to represent the weight; it -may be considered as mechanically equivalent to two -forces <a href="#p74">(74)</a>, C I and C H. Of these C H, acting directly -from the point P, merely produces pressure upon -it, and gives tension to the cord P C; but C I, acting at -right angles to C P, produces motion round P as a centre, -and in the direction C I, towards a vertical line P G, -drawn through the point P. If the body A B had been -on the other side of the line P G, it would have moved -in like manner towards it, and therefore in the direction -contrary to its present motion.</p> - -<p>Hence we must infer, that when the body is suspended -from a fixed point, it cannot remain at rest, if -that fixed point be not placed in the direction of the line -D E; and, on the other hand, that if the fixed point <i>be</i> -in the direction of that line, it cannot move. A practical -test is thus suggested, by which the line D E may be at -once discovered. Let a thread be attached to any point -of the body, and let it be suspended by this thread from -a hook or other fixed point. The direction of the -thread, when the body becomes quiescent, will be that -of a single force equivalent to the gravitation of all the -component parts of the mass.</p> - -<p id="p150">(150.) An enquiry is here suggested: does the direction -of the equivalent force thus determined depend -on the position of the body with respect to the surface -of the earth, and how is the direction of the equivalent -force affected by a change in that position? This ques<span class="pagenum" id="Page_110">110</span>tion -may be at once solved if the body be suspended by -different points, and the directions which the suspending -thread takes in each case relatively to the figure and dimensions -of the body examined.</p> - -<p>The body being suspended in this manner from any -point, let a small hole be bored through it, in the exact -direction of the thread, so that if the thread were continued -below the point where it is attached to the body, -it would pass through this hole. The body being successively -suspended by several different points on its -surface, let as many small holes be bored through it in the -same manner. If the body be then cut through, so as to -discover the directions which the several holes have taken, -they will be all found to cross each other at one point -within the body; or the same fact may be discovered -thus: a thin wire, which nearly fills the holes being -passed through any one of them, it will be found -to intercept the passage of a similar wire through any -other.</p> - -<p>This singular fact teaches us, what indeed can be -proved by mathematical reasoning without experiment, -that there is <i>one</i> point in every body through which the -single force, which is equivalent to the gravitation of all -its particles, must pass, in whatever position the body be -placed. This point is called <i>the centre of gravity</i>.</p> - -<p id="p151">(151.) In whatever situation a body may be placed, -the centre of gravity will have a tendency to descend in -the direction of a line perpendicular to the horizon, and -which is called the <i>line of direction</i> of the weight. If -the body be altogether free and unrestricted by any resistance -or impediment, the centre of gravity will actually -descend in this direction, and all the other points -of the body will move with the same velocity in parallel -directions, so that during its fall the position of the -parts of the body, with respect to the ground, will be -unaltered. But if the body, as is most usual, be subject -to some resistance or restraint, it will either remain -unmoved, its weight being expended in exciting pressure -on the restraining points or surfaces, or it will move in<span class="pagenum" id="Page_111">111</span> -a direction and with a velocity depending on the circumstances -which restrain it.</p> - -<p>In order to determine these effects, to predict the -pressure produced by the weight if the body be quiescent, -or the mixed effects of motion and pressure, if it -be not so, it is necessary in all cases to be able to assign -the place of the centre of gravity. When the magnitude -and figure of the body, and the density of the -matter which occupies its dimensions, are known, the -place of the centre of gravity can be determined with -the greatest precision by mathematical calculation. The -process by which this is accomplished, however, is not -of a sufficiently elementary nature to be properly introduced -into this treatise. To render it intelligible would -require the aid of some of the most advanced analytical -principles; and even to express the position of the point -in question, except in very particular instances, would -be impossible, without the aid of peculiar symbols.</p> - -<p id="p152">(152.) There are certain particular forms of body in -which, when they are uniformly dense, the place of the -centre of gravity can be easily assigned, and proved by -reasoning, which is generally intelligible; but in all -cases whatever, this point may be easily determined by -experiment.</p> - -<p id="p153">(153.) If a body uniformly dense have such a shape -that a point may be found on either side of which in -all directions around it the materials of the body are -similarly distributed, that point will obviously be the -centre of gravity. For if it be supported, the gravitation -of the particles on one side drawing them downwards, -is resisted by an effect of exactly the same kind -and of equal amount on the opposite side, and so the -body remains balanced on the point.</p> - -<p>The most remarkable body of this kind is a globe, -the centre of which is evidently its centre of gravity.</p> - -<p>A figure, such as <i><a href="#i_p120a">fig. 38.</a></i>, called an <i>oblate spheroid</i>, has -its centre of gravity at its centre, C. Such is the figure -of the earth. The same may be observed of the elliptical -solid, <i><a href="#i_p120a">fig. 39.</a></i>, which is called a prolate spheroid.</p> - -<p><span class="pagenum" id="Page_112">112</span></p> - -<p>A cube, and some other regular solids, bounded by -plane surfaces, have a point within them, such as above -described, and which is, therefore, their centre of gravity. -Such are <i><a href="#i_p120a">fig. 40.</a></i></p> - -<p>A straight wand of uniform thickness has its centre -of gravity at the centre of its length; and a cylindrical -body has its centre of gravity in its centre, at the middle -of its length or axis. Such is the point C, <i><a href="#i_p120a">fig. 41.</a></i></p> - -<p>A flat plate of any uniform substance, and which has -in every part an equal thickness, has its centre of gravity -at the middle of its thickness, and under a point of its -surface, which is to be determined by its shape. If it -be circular or elliptical, this point is its centre. If it -have any regular form, bounded by straight edges, it is -that point which is equally distant from its several angles, -as C in <i><a href="#i_p120a">fig. 42.</a></i></p> - -<p id="p154">(154.) There are some cases in which, although the -place of the centre of gravity is not so obvious as in the -examples just given, still it may be discovered without -any mathematical process, which is not easily understood. -Suppose A B C, <i><a href="#i_p120a">fig. 43.</a></i>, to be a flat triangular plate of -uniform thickness and density. Let it be imagined to be -divided into narrow bars, by lines parallel to the side -A C, as represented in the figure. Draw B D from the -angle B to the middle point D of the side A C. It is not -difficult to perceive, that B D will divide equally all the -bars into which the triangle is conceived to be divided. -Now if the flat triangular plate A B C be placed in -a horizontal position on a straight edge coinciding with -the line B D, it will be balanced: for the bars parallel -to A C will be severally balanced by the edge immediately -under their middle point; since that middle -point is the centre of gravity of each bar. Since, then, -the triangle is balanced on the edge, the centre of gravity -must be somewhere immediately over it, and must, -therefore, be within the plate at some point under the -line B D.</p> - -<p>The same reasoning will prove that the centre of -gravity of the plate is under the line A E, drawn from<span class="pagenum" id="Page_113">113</span> -the angle A to the middle point E of the side B C. To -perceive this, it is only necessary to consider the triangle -divided into bars parallel to B C, and thence to show -that it will be balanced on an edge placed under A E. -Since then the centre of gravity of the plate is under -the line B D, and also under A E, it must be under the -point G, at which these lines cross each other; and it is -accordingly at a depth beneath G, equal to half the -thickness of the plate.</p> - -<p>This may be experimentally verified by taking a piece -of tin or card, and cutting it into a triangular form. -The point G being found by drawing B D and A E, -which divide two sides equally, it will be balanced if placed -upon the point of a pin at G.</p> - -<p>The centre of gravity of a triangle being thus determined, -we shall be able to find the position of the centre -of gravity of any plate of uniform thickness and density -which is bounded by straight edges, as will be shown -hereafter. (<a href="#p173">173</a>.)</p> - -<p id="p155">(155.) The centre of gravity is not always included -within the volume of the body, that is, it is not enclosed -by its surfaces. Numerous examples of this can be produced. -If a piece of wire be bent into any form, the -centre of gravity will rarely be in the wire. Suppose -it be brought to the form of a ring. In that case, -the centre of gravity of the wire will be the centre of -the circle, a point not forming any part of the wire itself: -nevertheless this point may be proved to have the -characteristic property of the centre of gravity; for if -the ring be suspended by any point, the centre of the ring -must always settle itself under the point of suspension. -If this centre could be supposed to be connected with -the ring by very fine threads, whose weight would be -insignificant, and which might be united by a knot or -otherwise at the centre, the ring would be balanced upon -a point placed under the knot.</p> - -<p>In like manner, if the wire be formed into an ellipse, -or any other curve similarly arranged round a centre -point, that point will be its centre of gravity.</p> - -<p><span class="pagenum" id="Page_114">114</span></p> - -<p id="p156">(156.) To find the centre of gravity experimentally, -the method described in (<a href="#p149">149</a>, 150) may be used. In -this case two points of suspension will be sufficient to determine -it; for the directions of the suspending cord being -continued through the body, will cross each other at the -centre of gravity. These directions may also be found -by placing the body on a sharp point, and adjusting it -so as to be balanced upon it. In this case a line drawn -through the body directly upwards from the point will -pass through the centre of gravity, and therefore two -such lines must cross at that point.</p> - -<p id="p157">(157.) If the body have two flat parallel surfaces -like sheet metal, stiff paper, card, board, &c., the centre -of gravity may be found by balancing the body in -two positions on an horizontal straight edge. The -point where the lines marked by the edge cross each -other will be immediately under the centre of gravity. -This may be verified by showing that the body will be -balanced on a point thus placed, or that if it be suspended, -the point thus determined will always come -under the point of suspension.</p> - -<p>The position of the centre of gravity of such bodies -may also be found by placing the body on an horizontal -table having a straight edge. The body being moved -beyond the edge until it is in that position in which the -slightest disturbance will cause it to fall, the centre of -gravity will then be immediately over the edge. This -being done in two positions, the centre of gravity will -be determined as before.</p> - -<p id="p158">(158.) It has been already stated, that when the -body is perfectly free, the centre of gravity must necessarily -move downwards, in a direction perpendicular -to an horizontal plane. When the body is not free, the -circumstances which restrain it generally permit the -centre of gravity to move in certain directions, but obstruct -its motion in others. Thus if a body be suspended -from a fixed point by a flexible cord, the centre -of gravity is free to move in every direction except those -which would carry it farther from the point of suspen<span class="pagenum" id="Page_115">115</span>sion -than the length of the cord. Hence if we conceive -a globe or sphere to surround the point of suspension -on every side to a distance equal to that of the centre of -gravity from the point of suspension, when the cord is -fully stretched, the centre of gravity will be at liberty -to move in every direction within this sphere.</p> - -<p>There are an infinite variety of circumstances under -which the motion of a body may be restrained, and in -which a most important and useful class of mechanical -problems originate. Before we notice others, we shall, -however, examine that which has just been described -more particularly.</p> - -<p>Let P, <i><a href="#i_p120a">fig. 44.</a></i>, be the point of suspension, and C the -centre of gravity, and suppose the body so placed that C -shall be within the sphere already described. The cord -will therefore be slackened, and in this state the body will -be free. The centre of gravity will therefore descend -in the perpendicular direction until the cord becomes -fully extended; the tension will then prevent its further -motion in the perpendicular direction. The downward -force must now be considered as the diagonal of a parallelogram, -and equivalent to two forces C D and C E, in -the directions of the sides, as already explained in <a href="#p149">(149)</a>. -The force C D will bring the centre of gravity into the -direction P F, perpendicularly under the point of suspension. -Since the force of gravity acts continually on -C in its approach to P F, it will move towards that line -with accelerated speed, and when it has arrived there it -will have acquired a force to which no obstruction is -immediately opposed, and consequently by its inertia it -retains this force, and moves beyond P F on the other -side. But when the point C gets into the line P F, it -is in the lowest possible position; for it is at the lowest -point of the sphere which limits its motion. When it -passes to the other side of P F, it must therefore begin -to ascend, and the force of gravity, which, in the former -case, accelerated its descent, will now for the same reason, -and with equal energy, oppose its ascent. This -will be easily understood. Let <span class="ilb">C′</span> be any point which it<span class="pagenum" id="Page_116">116</span> -may have attained in ascending; <span class="ilb">C′</span> <span class="ilb">G′</span>, the force of -gravity, is now equivalent to <span class="ilb">C′</span> <span class="ilb">D′</span> and <span class="ilb">C′</span> <span class="ilb">E′</span>. The -latter as before produces tension; but the former <span class="ilb">C′</span> <span class="ilb">D′</span> -is in a direction immediately opposed to the motion, and -therefore retards it. This retardation will continue -until all the motion acquired by the body in its descent -from the first position has been destroyed, and then it -will begin to return to P F, and so it will continue to -vibrate from the one side to the other until the friction -on the point P, and the resistance of the air, gradually -deprive it of its motion, and bring it to a state of rest -in the direction P F.</p> - -<p>But for the effects of friction and atmospheric resistance, -the body would continue for ever to oscillate equally -from side to side of the line P F.</p> - -<p id="p159">(159.) The phenomenon just developed is only an -example of an extensive class. Whenever the circumstances -which restrain the body are of such a nature -that the centre of gravity is prevented from descending -below a certain level, but not, on the other hand, restrained -from rising above it, the body will remain at -rest if the centre of gravity be placed at the lowest limit -of its level; any disturbance will cause it to oscillate -around this state, and it cannot return to a state of rest -until friction or some other cause have deprived it of -the motion communicated by the disturbing force.</p> - -<p id="p160">(160.) Under the circumstances which we have just -described, the body could not maintain itself in a state -of rest in any position except that in which the centre -of gravity is, at the lowest point of the space in which -it is free to move. This, however, is not always the -case. Suppose it were suspended by an inflexible rod -instead of a flexible string; the centre of gravity would -then not only be prevented from receding from the point -of suspension, but also from approaching it; in fact, it -would be always kept at the same distance from it. -Thus, instead of being capable of moving anywhere -within the sphere, it is now capable of moving on its -surface only. The reasoning used in the last case may<span class="pagenum" id="Page_117">117</span> -also be applied here, to prove that when the centre of -gravity is on either side of the perpendicular P F, it will -fall towards P F and oscillate, and that if it be placed in -the line P F, it will remain in equilibrium. But in this -case there is another position, in which the centre of -gravity may be placed so as to produce equilibrium. If -it be placed at the highest point of the sphere in which -it moves, the whole force acting on it will then be directed -on the point of suspension, perpendicularly downwards, -and will be entirely expended in producing -pressure on that point; consequently, the body will -in this case be in equilibrium. But this state of equilibrium -is of a character very different from that in -which the centre of gravity was at the lowest part of -the sphere. In the present case any displacement, however -slight, of the centre of gravity, will carry it to a -lower level, and the force of gravity will then prevent -its return to its former state, and will impel it downwards -until it attain the lowest point of the sphere, and -round that point it will oscillate.</p> - -<p id="p161">(161.) The two states of equilibrium which have -been just noticed, are called stable and instable equilibrium. -The character of the former is, that any disturbance -of the state produces oscillation about it; but -any disturbance of the latter state produces a total overthrow, -and finally causes oscillation around the state of -stable equilibrium.</p> - -<p>Let A B, <i><a href="#i_p120a">fig. 45.</a></i>, be an elliptical board resting on its -edge on an horizontal plane. In the position here represented, -the extremity P of the lesser axis being the -point of support, the board is in stable equilibrium; -for any motion on either side must cause the centre of -gravity C to ascend in the directions C O, and oscillation -will ensue. If, however, it rest upon the smaller end, as -in <i><a href="#i_p120a">fig. 46.</a></i>, the position would still be a state of equilibrium, -because the centre of gravity is directly above -the point of support; but it would be instable equilibrium, -because the slightest displacement of the centre of gravity -would cause it to descend.</p> - -<p><span class="pagenum" id="Page_118">118</span></p> - -<p>Thus an egg or a lemon may be balanced on the end, -but the least disturbance will overthrow it. On the -contrary, it will easily rest on the side, and any disturbance -will produce oscillation.</p> - -<p id="p162">(162.) When the circumstances under which the -body is placed allow the centre of gravity to move only -in an horizontal line, the body is in a state which may -be called <i>neutral equilibrium</i>. The slightest force will -move the centre of gravity, but will neither produce -oscillation nor overthrow the body, as in the last two -cases.</p> - -<p>An example of this state is furnished by a cylinder -placed upon an horizontal plane. As the cylinder is -rolled upon the plane, the centre of gravity C, <i><a href="#i_p120a">fig. 47.</a></i>, -moves in a line parallel to the plane A B, and distant -from it by the radius of the cylinder. The body will -thus rest indifferently in any position, because the line -of direction always falls upon a point P at which the -body rests upon the plane.</p> - -<p>If the plane were inclined, as in <i><a href="#i_p120a">fig. 48.</a></i>, a body might -be so shaped, that while it would roll the centre of gravity -would move horizontally. In this case the body -would rest indifferently on any part of the plane, as if -it were horizontal, provided the friction be sufficient to -prevent the body from sliding down the plane.</p> - -<p>If the centre of gravity of a cylinder happen not to -coincide with its centre by reason of the want of uniformity -in the materials of which it is composed, it will -not be in a state of neutral equilibrium on an horizontal -plane, as in <i><a href="#i_p120a">fig. 47.</a></i> In this case let G, <i><a href="#i_p120a">fig. 49.</a></i>, be the -centre of gravity. In the position here represented, -where the centre of gravity is immediately <i>below</i> the -centre C, the state will be stable equilibrium, because a -motion on either side would cause the centre of gravity -to ascend; but in <i><a href="#i_p120a">fig. 50.</a></i>, where G is immediately above -C, the state is instable equilibrium, because a motion on -either side would cause G to descend, and the body -would turn into the position <i><a href="#i_p120a">fig. 49.</a></i></p> - -<p id="p163">(163.) A cylinder of this kind will, under certain<span class="pagenum" id="Page_119">119</span> -circumstances, roll up an inclined plane. Let A B, -<i><a href="#i_p120a">fig. 51.</a></i>, be the inclined plane, and let the cylinder be so -placed that the line of direction from G shall be <i>above</i> -the point P at which the cylinder rests upon the plane. -The whole weight of the body acting in the direction -G D will obviously cause the cylinder to roll towards A, -provided the friction be sufficient to prevent sliding; -but although the cylinder in this case ascends, the centre -of gravity G really descends.</p> - -<p>When G is so placed that the line of direction G D -shall fall on the point P, the cylinder will be in equilibrium, -because its weight acts upon the point on which -it rests. There are two cases represented in <i><a href="#i_p128a">fig. 52.</a></i> -and <i><a href="#i_p128a">fig. 53.</a></i>, in which G takes this position. <i>Fig. 52.</i> -represents the state of stable, and <i><a href="#i_p128a">fig. 53.</a></i> of instable -equilibrium.</p> - -<p id="p164">(164.) When a body is placed upon a base, its stability -depends upon the position of the line of direction -and the height of the centre of gravity above the base. -If the line of direction fall within the base, the body -will stand firm; if it fall on the edge of the base, it will -be in a state in which the slightest force will overthrow -it on that side at which the line of direction falls; and -if the line of direction fall without the base, the body -must turn over that edge which is nearest to the line of -direction.</p> - -<p>In <i><a href="#i_p128a">fig. 54.</a></i> and <i><a href="#i_p128a">fig. 55.</a></i>, the line of direction G P falls -within the base, and it is obvious that the body will -stand firm; for any attempt to turn it over either edge -would cause the centre of gravity to ascend. But in -<i><a href="#i_p128a">fig. 56.</a></i> the line of direction falls upon the edge, and if -the body be turned over, the centre of gravity immediately -commences to descend. Until it be turned -over, however, the centre of gravity is supported by the -edge.</p> - -<p>In <i><a href="#i_p128a">fig. 57.</a></i> the line of direction falls outside the base, -the centre of gravity has a tendency to descend from G -towards A, and the body will accordingly fall in that direction.</p> - -<p><span class="pagenum" id="Page_120">120</span></p> - -<p id="p165">(165.) When the line of direction falls within the -base, bodies will always stand firm, but not with the -same degree of stability. In general, the stability depends -on the height through which the centre of gravity -must be elevated before the body can be overthrown. -The greater this height is, the greater in the same proportion -will be the stability.</p> - -<p>Let B A C, <i><a href="#i_p128a">fig. 58.</a></i>, be a pyramid, the centre of gravity -being at G. To turn this over the edge B, the -centre of gravity; must be carried over the arch G E, and -must therefore be raised through the height H E. If, -however, the pyramid were taller relatively to its base, as -in <i><a href="#i_p128a">fig. 59.</a></i>, the height H E would be proportionally less; -and if the base were very small in reference to the height, -as in <i><a href="#i_p128a">fig. 60.</a></i>, the height H E would be very small, and a -slight force would throw it over the edge B.</p> - -<p>It is obvious that the same observations may be applied -to all figures whatever, the conclusions just deduced -depending only on the distance of the line of direction -from the edge of the base, and the height of the centre of -gravity above it.</p> - -<p id="p166">(166.) Hence we may perceive the principle on which -the stability of loaded carriages depends. When the -load is placed at a considerable elevation above the wheels, -the centre of gravity is elevated, and the carriage becomes -proportionally insecure. In coaches for the conveyance -of passengers, the luggage is therefore sometimes placed -below the body of the coach; light parcels of large bulk -may be placed on the top with impunity.</p> - -<p>When the centre of gravity of a carriage is much -elevated, there is considerable danger of overthrow, if -a corner be turned sharply and with a rapid pace; for the -centrifugal force then acting on the centre of gravity will -easily raise it through the small height which is necessary -to turn the carriage over the external wheels <a href="#p142">(142)</a>.</p> - -<p id="p167">(167.) The same waggon will have greater stability -when loaded with a heavy substance which occupies a -small space, such as metal, than when it carries the -same weight of a lighter substance, such as hay; because<span class="pagenum" id="Page_121">121</span> -the centre of gravity in the latter case will be much more -elevated.</p> - -<div class="figcenter" id="i_p120a" style="max-width: 31.25em;"> - <img src="images/i_p120a.jpg" alt="" /> - <div class="caption"> - -<p class="tar"><i>H. Adlard, sc.</i></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<p>If a large table be placed upon a single leg in its centre, -it will be impracticable to make it stand firm; but if the -pillar on which it rests terminate in a tripod, it will have -the same stability as if it had three legs attached to the -points directly over the places where the feet of the tripod -rest.</p> - -<p id="p168">(168.) When a solid body is supported by more points -than one, it is not necessary for its stability that the line -of direction should fall on one of those points. If there -be only two points of support, the line of direction must -fall between them. The body is in this case supported -as effectually as if it rested on an edge coinciding with a -straight line drawn from one point of support to the -other. If there be three points of support, which are -not ranged in the same straight line, the body will be -supported in the same manner as it would be by a base -coinciding with the triangle formed by straight lines joining -the three points of support. In the same manner, -whatever be the number of points on which the body -may rest, its virtual base will be found by supposing -straight lines drawn, joining the several points successively. -When the line of direction falls within this base, -the body will always stand firm, and otherwise not. -The degree of stability is determined in the same manner -as if the base were a continued surface.</p> - -<p id="p169">(169.) Necessity and experience teach an animal to -adapt its postures and motions to the position of the -centre of gravity of his body. When a man stands, the -line of direction of his weight must fall within the base -formed by his feet. If A B, C D, <i><a href="#i_p128a">fig. 61.</a></i>, be the feet, this -base is the space A B D C. It is evident, that the more -his toes are turned outwards, the more contracted the -base will be in the direction E F, and the more liable he -will be to fall backwards or forwards. Also, the closer -his feet are together, the more contracted the base will be -in the direction G H, and the more liable he will be to -fall towards either side.</p> - -<p><span class="pagenum" id="Page_122">122</span></p> - -<p>When a man walks, the legs are alternately lifted -from the ground, and the centre of gravity is either unsupported -or thrown from the one side to the other. -The body is also thrown a little forward, in order that -the tendency of the centre of gravity to fall in the direction -of the toes may assist the muscular action in propelling -the body. This forward inclination of the body -increases with the speed of the motion.</p> - -<p>But for the flexibility of the knee-joint the labour of -walking would be much greater than it is; for the centre -of gravity would be more elevated by each step. The -line of motion of the centre of gravity in walking is represented -by <i><a href="#i_p128a">fig. 62.</a></i>, and deviates but little from a regular -horizontal line, so that the elevation of the centre of -gravity is subject to very slight variation. But if there -were no knee-joint, as when a man has wooden legs, the -centre of gravity would move as in <i><a href="#i_p128a">fig. 63.</a></i>, so that at each -step the weight of the body would be lifted through a -considerable height, and therefore the labour of walking -would be much increased.</p> - -<p>If a man stand on one leg, the line of direction of his -weight must fall within the space on which his foot -treads. The smallness of this space, compared with the -height of the centre of gravity, accounts for the difficulty -of this feat.</p> - -<p>The position of the centre of gravity of the body -changes with the posture and position of the limbs. If -the arm be extended from one side, the centre of gravity -is brought nearer to that side than it was when the arm -hung perpendicularly. When dancers, standing on one -leg, extend the other at right angles to it, they must -incline the body in the direction opposite to that in -which the leg is extended, in order to bring the centre -of gravity over the foot which supports them.</p> - -<p>When a porter carries a load, his position must be -regulated by the centre of gravity of his body and the -load taken together. If he bore the load on his back, -the line of direction would pass beyond his heels, and he<span class="pagenum" id="Page_123">123</span> -would fall backwards. To bring the centre of gravity -over his feet he accordingly leans forward, <i><a href="#i_p128a">fig. 64.</a></i></p> - -<p>If a nurse carry a child in her arms, she leans back -for a like reason.</p> - -<p>When a load is carried on the head, the bearer stands -upright, that the centre of gravity may be over his feet.</p> - -<p>In ascending a hill, we appear to incline forward; and -in descending, to lean backward, but in truth, we are -standing upright with respect to a level plane. This is -necessary to keep the line of direction between the feet, -as is evident from <i><a href="#i_p128a">fig. 65.</a></i></p> - -<p>A person sitting on a chair which has no back cannot -rise from it without either stooping forward to bring the -centre of gravity over the feet, or drawing back the feet -to bring them under the centre of gravity.</p> - -<p>A quadruped never raises both feet on the same side -simultaneously, for the centre of gravity would then be -unsupported. Let A B C D, <i><a href="#i_p128a">fig. 66.</a></i>, be the feet. The -base on which it stands is A B C D, and the centre of -gravity is nearly over the point O, where the diagonals -cross each other. The legs A and C being raised together, -the centre of gravity is supported by the legs B and -D, since it falls between them; and when B and D are -raised it is, in like manner, supported by the feet A and -C. The centre of gravity, however, is often unsupported -for a moment; for the leg B is raised from the -ground before A comes to it, as is plain from observing -the track of a horse’s feet, the mark of A being upon or -before that of B. In the more rapid paces of all animals -the centre of gravity is at intervals unsupported.</p> - -<p>The feats of rope-dancers are experiments on the -management of the centre of gravity. The evolutions -of the performer are found to be facilitated by holding -in his hand a heavy pole. His security in this case depends, -not on the centre of gravity of his body, but on -that of his body and the pole taken together. This -point is near the centre of the pole, so that, in fact, he -may be said to hold in his hands the point on the position -of which the facility of his feats depends. Without<span class="pagenum" id="Page_124">124</span> -the aid of the pole the centre of gravity would be within -the trunk of the body, and its position could not be -adapted to circumstances with the same ease and rapidity.</p> - -<p id="p170">(170.) The centre of gravity of a mass of fluid is -that point which would have the properties which have -been proved to belong to the centre of gravity of a solid, -if the fluid were solidified without changing in any respect -the quantity or arrangement of its parts. This -point also possesses other properties, in reference to -fluids, which will be investigated in <span class="smcap">Hydrostatics</span> and -<span class="smcap">Pneumatics</span>.</p> - -<p id="p171">(171.) The centre of gravity of two bodies separated -from one another, is that point which would possess -the properties ascribed to the centre of gravity, if the -two bodies were united by an inflexible line, the weight -of which might be neglected. To find this point mathematically -is a very simple problem. Let A and B, -<i><a href="#i_p128a">fig. 67.</a></i>, be the two bodies, and <i>a</i> and <i>b</i> their centres of -gravity. Draw the right line <i>a b</i>, and divide it at C, in -such a manner that <i>a</i> C shall have the same proportion -to <i>b</i> C as the mass of the body B has to the mass of the -body A.</p> - -<p>This may easily be verified experimentally. Let A -and B be two bodies, whose weight is considerable, in -comparison with that of the rod <i>a b</i>, which joins them. -Let a fine silken string, with its ends attached to them, be -hung upon a pin; and on the same pin let a plumb-line -be suspended. In whatever position the bodies may be -hung, it will be observed that the plumb-line will cross -the rod <i>a b</i> at the same point, and that point will divide -the line <i>a b</i> into parts <i>a</i> C and <i>b</i> C, which are in the proportion -of the mass of B to the mass of A.</p> - -<p id="p172">(172.) The centre of gravity of three separate bodies -is defined in the same manner as that of two, and -may be found by first determining the centre of gravity -of two; and then supposing their masses concentrated -at that point, so as to form one body, and finding the -centre of gravity of that and the third.</p> - -<p><span class="pagenum" id="Page_125">125</span></p> - -<p>In the same manner the centre of gravity of any -number of bodies may be determined.</p> - -<p id="p173">(173.) If a plate of uniform thickness be bounded by -straight edges, its centre of gravity may be found by -dividing it into triangles by diagonal lines, as in <i><a href="#i_p176a">fig. 68.</a></i>, -and having determined by <a href="#p154">(154)</a> the centres of gravity of -the several triangles, the centre of gravity of the whole -plate will be their common centre of gravity, found as -above.</p> - -<p id="p174">(174.) Although the centre of gravity takes its name -from the familiar properties which it has in reference -to detached bodies of inconsiderable magnitude, -placed on or near the surface of the earth, yet it possesses -properties of a much more general and not less important -nature. One of the most remarkable of these is, that -the centre of gravity of any number of separate bodies is -never affected by the mutual attraction, impact, or other -influence which the bodies may transmit from one to -another. This is a necessary consequence of the equality -of action and reaction explained in Chapter <a href="#CHAP_IV">IV</a>. For if -A and B, <i><a href="#i_p128a">fig. 67.</a></i>, attract each other, and change their -places to <span class="ilb">A′</span> and <span class="ilb">B′</span>, the space a a′ will have to <i>b b′</i> the -same proportion as B has to A, and therefore by what -has just been proved <a href="#p171">(171)</a> the same proportion as <i>a</i> C -has to <i>b</i> C. It follows, that the remainders <i><span class="ilb">a′</span></i> C and <i><span class="ilb">b′</span></i> C -will be in the proportion of B to A, and that C will -continue to be the centre of gravity of the bodies after -they have approached by their mutual attraction.</p> - -<p>Suppose, for example, that A and B were 12lbs. and -8lbs. respectively, and that <i>a b</i> were 40 feet. The point -C must <a href="#p171">(171)</a> divide <i>a b</i> into two parts, in the proportion -of 8 to 12, or of 2 to 3. Hence it is obvious that <i>a</i> C -will be 16 feet, and <i>b</i> C 24 feet. Now, suppose that A -and B attract each other, and that A approaches B -through two feet. Then B must approach A through -three feet. Their distances from C will now be 14 -feet and 21 feet, which, being in the proportion of B -to A, the point C will still be their centre of gravity.</p> - -<p>Hence it follows, that if a system of bodies, placed at<span class="pagenum" id="Page_126">126</span> -rest, be permitted to obey their mutual attractions, although -the bodies will thereby be severally moved, yet -their common centre of gravity must remain quiescent.</p> - -<p id="p175">(175.) When one of two bodies is moving in a straight -line, the other being at rest, their common centre of -gravity must move in a parallel straight line. Let A -and B, <i><a href="#i_p176a">fig. 69.</a></i>, be the centres of gravity of the bodies, -and let A move from A to <i>a</i>, B remaining at rest. -Draw the lines A B and <i>a</i> B. In every position which -the body B assumes during its motion, the centre of -gravity C divides the line joining them into parts A C, -B C, which are in the proportion of the mass B to the -mass A. Now, suppose any number of lines drawn from -B to the line A <i>a</i>; a parallel C <i>c</i> to A <i>a</i> through C divides -all these lines in the same proportion; and therefore, -while the body A moves from A to <i>a</i>, the common -centre of gravity moves from C to <i>c</i>.</p> - -<p>If both the bodies A and B moved uniformly in -straight lines, the centre of gravity would have a motion -compounded <a href="#p74">(74)</a> of the two motions with which it -would be affected, if each moved while the other remained -at rest. In the same manner, if there were three -bodies, each moving uniformly in a straight line, their -common centre of gravity would have a motion compounded -of that motion which it would have if one remained -at rest while the other two moved, and that -which the motion of the first would give it if the last -two remained at rest; and in the same manner it may -be proved, that when any number of bodies move each -in a straight line, their common centre of gravity will -have a motion compounded of the motions which it receives -from the bodies severally.</p> - -<p>It may happen that the several motions which the -centre of gravity receives from the bodies of the system -will neutralise each other; and this does, in fact, take -place for such motions as are the consequences of the -mutual action of the bodies upon one another.</p> - -<p id="p176">(176.) If a system of bodies be not under the immediate -influence of any forces, and their mutual attrac<span class="pagenum" id="Page_127">127</span>tion -be conceived to be suspended, they must severally -be either at rest or in uniform rectilinear motion in -virtue of their inertia. Hence, their common centre of -gravity must also be either at rest or in uniform rectilinear -motion. Now, if we suppose their mutual attractions -to take effect, the state of their common centre of -gravity will not be changed, but the bodies will severally -receive motions compounded of their previous -uniform rectilinear motions and those which result from -their mutual attractions. The combined effects will -cause each body to revolve in an orbit round the common -centre of gravity, or will precipitate it towards -that point. But still that point will maintain its former -state undisturbed.</p> - -<p>This constitutes one of the general laws of mechanical -science, and is of great importance in physical -astronomy. It is known by the title “the conservation -of the motion of the centre of gravity.”</p> - -<p id="p177">(177.) The solar system is an instance of the class -of phenomena to which we have just referred. All the -motions of the bodies which compose it can be traced -to certain uniform rectilinear motions, received from -some former impulse, or from a force whose action has -been suspended, and those motions which necessarily -result from the principle of gravitation. But we shall -not here insist further on this subject, which more properly -belongs to another department of the science.</p> - -<p id="p178">(178.) If a solid body suffer an impact in the direction -of a line passing through its centre of gravity, all -the particles of the body will be driven forward with -the same velocity in lines parallel to the direction of -the impact, and the whole force of the motion will be -equal to that of the impact. The common velocity of -the parts of the body will in this case be determined by -the principles explained in Chapter <a href="#CHAP_IV">IV</a>. The impelling -force being equally distributed among all the parts, the -velocity will be found by dividing the numerical value -of that force by the number expressing the mass.</p> - -<p>If any number of impacts be given simultaneously to<span class="pagenum" id="Page_128">128</span> -different points of a body, a certain complex motion will -generally ensue. The mass will have a relative motion -round the centre of gravity as if it were fixed, while that -point will move forward uniformly in a straight line, -carrying the body with it. The relative motion of the -mass round the centre of gravity may be found by considering -the centre of gravity as a fixed point, round -which the mass is free to move, and then determining the -motion which the applied forces would produce. This -motion being supposed to continue uninterrupted, let all -the forces be imagined to be applied in their proper -directions and quantities to the centre of gravity. By -the principles for the composition of force they will be -mechanically equivalent to a single force through that -point. In the direction of this single force the centre of -gravity will move and have the same velocity as if the -whole mass were there concentrated and received the -impelling forces.</p> - -<p id="p179">(179.) These general properties, which are entirely -independent of gravity, render the “centre of gravity” -an inadequate title for this important point. Some physical -writers have, consequently, called it the “centre of -inertia.” The “centre of gravity,” however, is the -name by which it is still generally designated.</p> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<h2 class="nobreak" id="CHAP_X">CHAP. X.<br /> - -<span class="title">THE MECHANICAL PROPERTIES OF AN AXIS.</span></h2> -</div> - - -<p id="p180">(180.) <span class="smcap">When</span> a body has a motion of rotation, the line -round which it revolves is called an <i>axis</i>. Every point -of the body must in this case move in a circle, whose -centre lies in the axis, and whose radius is the distance -of the point from the axis. Sometimes while the body -revolves, the axis itself is moveable, and not unfrequently -in a state of actual motion. The motions of the<span class="pagenum" id="Page_129">129</span> -earth and planets, or that of a common spinning-top, are -examples of this. The cases, however, which will be -considered in the present chapter, are chiefly those in -which the axis is immovable, or at least where its motion -has no relation to the phenomena under investigation. -Instances of this are so frequent and obvious, that it -seems scarcely necessary to particularise them. Wheel-work -of every description, the moving parts of watches -and clocks, turning lathes, mill-work, doors and lids on -hinges, are all obvious examples. In tools or other instruments -which work on joints or pivots, such as scissors, -shears, pincers, although the joint or pivot be not absolutely -fixed, it is to be considered so in reference to the -mechanical effect.</p> - -<div class="figcenter" id="i_p128a" style="max-width: 31.25em;"> - <img src="images/i_p128a.jpg" alt="" /> - <div class="caption"> -<p class="tar"><i>H. Adlard, sc.</i></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<p>In some cases, as in most of the wheels of watches and -clocks, fly-wheels and chucks of the turning lathe, and -the arms of wind-mills, the body turns continually in the -same direction, and each of its points traverses a complete -circle during every revolution of the body round its axis. -In other instances the motion is alternate or reciprocating, -its direction being at intervals reversed. Such is -the case in pendulums of clocks, balance-wheels of chronometers, -the treddle of the lathe, doors and lids on -hinges, scissors, shears, pincers, &c. When the alternation -is constant and regular, it is called <i>oscillation</i> or -<i>vibration</i>, as in pendulums and balance-wheels.</p> - -<p id="p181">(181.) To explain the properties of an axis of rotation -it will be necessary to consider the different kinds of -forces to the action of which a body moveable on such an -axis may be submitted, to show how this action depends -on their several quantities and directions, to distinguish -the cases in which the forces neutralise each other and -mutually equilibrate from those in which motion ensues, -to determine the effect which the axis suffers, and, in the -cases where motion is produced, to estimate the effects of -those centrifugal forces (<a href="#p137">137</a>.) which are created by the -mass of the body whirling round the axis.</p> - -<p>Forces in general have been distinguished by the duration -of their action into instantaneous and continued<span class="pagenum" id="Page_130">130</span> -forces. The effect of an instantaneous force is produced -in an infinitely short time. If the body which sustains -such an action be previously quiescent and free, it will -move with a uniform velocity in the direction of the impressed -force. (<a href="#p93">93</a>.) If, on the other hand, the body be -not free, but so restrained that the impulse cannot put it in -motion, then the fixed points or lines which resist the -motion sustain a corresponding shock at the moment of -the impulse. This effect, which is called <i>percussion</i>, is, -like the force which causes it, instantaneous.</p> - -<p>A continued force produces a continued effect. If -the body be free and previously quiescent, this effect is a -continual increase of velocity. If the body be so restrained -that the applied force cannot put it in motion, -the effect is a continued pressure on the points or lines -which sustain it. (<a href="#p94">94</a>.)</p> - -<p>It may happen, however, that although the body be -not absolutely free to move in obedience to the force applied -to it, yet still it may not be altogether so restrained -as to resist the effect of that force and remain at rest. If -the point at which a force is applied be free to move in -a certain direction not coinciding with that of the applied -force, that force will be resolved into two elements; one -of which is in the direction in which the point is free to -move, and the other at right angles to that direction. -The point will move in obedience to the former element, -and the latter will produce percussion or pressure on the -points or lines which restrain the body. In fact, in such -cases the resistance offered by the circumstances which confine -the motion of the body modifies the motion which it -receives, and as every change of motion must be the consequence -of a force applied (<a href="#p44">44</a>.), the fixed points or lines -which offer the resistance must suffer a corresponding -effect.</p> - -<p>It may happen that the forces impressed on the body, -whether they be continued or instantaneous, are such as, -were it free, would communicate to it a motion which -the circumstances which restrain it do not forbid it to -receive. In such a case the fixed points or lines which<span class="pagenum" id="Page_131">131</span> -restrain the body sustain no force, and the phenomena -will be the same in all respects as if these points or lines -were not fixed.</p> - -<p>It will be easy to apply these general reflections to the -case in which a solid body is moveable on a fixed axis. -Such a body is susceptible of no motion except one of -rotation on that axis. If it be submitted to the action -of instantaneous forces, one or other of the following -effects must ensue. 1. The axis may resist the forces, -and prevent any motion. 2. The axis may modify the -effect of the forces sustaining a corresponding percussion, -and the body receiving a motion of rotation. 3. The -forces applied may be such as would cause the body to -spin round the axis even were it not fixed, in which case -the body will receive a motion of rotation, but the axis -will suffer no percussion.</p> - -<p>What has been just observed of the effect of instantaneous -forces is likewise applicable to continued ones. 1. -The axis may entirely resist the effect of such forces, in -which case it will suffer a pressure which may be estimated -by the rules for the composition of force. 2. It -may modify the effect of the applied forces, in which case -it must also sustain a pressure, and the body must receive -a motion of rotation which is subject to constant variation, -owing to the incessant action of the forces. 3. The -forces may be such as would communicate to the body -the same rotatory motion if the axis were not fixed. -In this case the forces will produce no pressure on the -axis.</p> - -<p>The impressed forces are not the only causes which -affect the axis of a body during the phenomenon of rotation. -This species of motion calls into action other forces -depending on the inertia of the mass, which produce effects -upon the axis, and which play a prominent part in the theory -of rotation. While the body revolves on its axis, the -component particles of its mass move in circles, the centres -of which are placed in the axis. The radius of the circle -in which each particle moves is the line drawn from that -particle perpendicular to the axis. It has been already<span class="pagenum" id="Page_132">132</span> -proved that a particle of matter, moving round a centre, is -attended with a centrifugal force proportionate to the radius -of the circle in which it moves and to the square of its -angular velocity. When a solid body revolves on its axis, -all its parts are whirled round together, each performing -a complete revolution in the same time. The angular velocity -is consequently the same for all, and the difference -of the centrifugal forces of different particles must entirely -depend upon their distances from the axis. The tendency -of each particle to fly from the axis, arising from the centrifugal -force, is resisted by the cohesion of the parts of -the mass, and in general this tendency is expended in exciting -a pressure or strain upon the axis. It ought to be -recollected, however, that this pressure or strain is altogether -different from that already mentioned, and produced -by the forces which give motion to the body. The latter -depends entirely upon the quantity and directions of the -applied forces in relation to the axis: the former depends -on the figure and density of the body, and the velocity -of its motion.</p> - -<p>These very complex effects render a simple and elementary -exposition of the mechanical properties of a fixed -axis a matter of considerable difficulty. Indeed, the -complete mathematical development of this theory long -eluded the skill of the most acute geometers, and it was -only at a comparatively late period that it yielded to the -searching analysis of modern science.</p> - -<p id="p182">(182.) To commence with the most simple case, we -shall consider the body as submitted to the action of a -single force. The effect of this force will vary according -to the relation of its direction to that of the axis. There -are two ways in which a body may be conceived to be -moveable around an axis. 1. By having pivots at two -points which rest in sockets, so that when the body is -moved it must revolve round the right line joining the -pivots as an axis. 2. A thin cylindrical rod may pass -through the body, on which it may turn in the same -manner as a wheel upon its axle.</p> - -<p>If the force be applied to the body in the direction of<span class="pagenum" id="Page_133">133</span> -the axis, it is evident that no motion can ensue, and the -effect produced will be a pressure on that pivot towards -which the force is directed. If in this case the body -revolved on a cylindrical rod, the tendency of the force -would be to make it slide along the rod without revolving -round it.</p> - -<p>Let us next suppose the force to be applied not in the -direction of the axis itself, but parallel to it. Let A B, -<i><a href="#i_p176a">fig. 70.</a></i>, be the axis, and let C D be the direction of the -force applied. The pivots being supposed to be at A and -B, draw A G and B F perpendicular to A B. The force -C D will be equivalent to three forces, one acting from B -towards A, equal in quantity to the force C D. This -force will evidently produce a corresponding pressure on -the pivot A. The other two forces will act in the directions -A G and B F, and will have respectively to the -force C D the same proportion as A E has to A B. Such -will be the mechanical effect of a force C D parallel to the -axis. And as these effects are all directed on the pivots, -no motion can ensue.</p> - -<p>If the body revolve on a cylindrical rod, the forces A G -and B F would produce a strain upon the axis, while the -third force in the direction B A would have a tendency -to make the body slide along it.</p> - -<p id="p183">(183.) If the force applied to the body be directed -upon the axis, and at right angles to it, no motion can be -produced. In this case, if the body be supported by pivots -at A and B, the force K L, perpendicular to the line -A B, will be distributed between the pivots, producing -a pressure on each proportional to its distance from the -other. The pressure on A having to the pressure on B -the same proportion as L B has to L A.</p> - -<p>If the force K H be directed obliquely to the axis, -it will be equivalent to two forces (<a href="#p76">76</a>.), one K L perpendicular -to the axis, and the other K M parallel to it. -The effect of each of these may be investigated as in the -preceding cases.</p> - -<p>In all these observations the body has been supposed -to be submitted to the action of one force only. If<span class="pagenum" id="Page_134">134</span> -several forces act upon it, the direction of each of them -crossing the axis either perpendicularly or obliquely, or -taking the direction of the axis or any parallel direction, -their effects may be similarly investigated. In the same -manner we may determine the effects of any number of -forces whose combined results are mechanically equivalent -to forces which either intersect the axis or are parallel -to it.</p> - -<p id="p184">(184.) If any force be applied whose direction lies in a -plane oblique to the axis, it can always be resolved into -two elements (<a href="#p76">76</a>.), one of which is parallel to the axis, -and the other in a plane perpendicular to it. The effect -of the former has been already determined, and therefore -we shall at present confine our attention to the latter.</p> - -<p>Suppose the axis to be perpendicular to the paper, and -to pass through the point G, <i><a href="#i_p176a">fig. 71.</a></i> and let A B C be -a section of the body. It will be convenient to consider -the section vertical and the axis horizontal, omitting, -however, any notice of the effect of the weight of the -body.</p> - -<p>Let a weight W be suspended by a cord Q W from -any point Q. This weight will evidently have a tendency -to turn the body round in the direction A B C. -Let another cord be attached to any other point P, and, -being carried over a wheel R, let a dish S be attached to -it, and let fine sand be poured into this dish until the -tendency of S to turn the body round the axis in the -direction of C B A balances the opposite tendency of W. -Let the weights of W and S be then exactly ascertained, -and also let the distances G I and G H of the cords -from the axis be exactly measured. It will be found -that, if the number of ounces in the weight S be multiplied -by the number of inches in G H, and also the -number of ounces in W by the number of inches in G I, -equal products will be obtained. This experiment may -be varied by varying the position of the wheel R, and -thereby changing the direction of the string P R, in -which cases it will be always found necessary to vary -the weight of S in such a manner, that when the num<span class="pagenum" id="Page_135">135</span>ber -of ounces in it is multiplied by the number of inches -in the distance of the string from the axis, the product -obtained shall be equal to that of the weight W by the -distance G I. We have here used ounces and inches as -the measures of weight and distance; but it is obvious -that any other measures would be equally applicable.</p> - -<p>From what has been just stated it follows, that the -energy of the weight of S to move the body on its axis, -does not depend alone upon the actual amount of that -weight, but also upon the distance of the string from -the axis. If, while the position of the string remains -unaltered, the weight of S be increased or diminished, -the resisting weight W must be increased or diminished -in the same proportion. But if, while the weight of S -remains unaltered, the distance of the string P R from -the axis G be increased or diminished, it will be found -necessary to increase or diminish the resisting weight W -in exactly the same proportion. It therefore appears -that the increase or diminution of the distance of the -direction of a force from the axis has the same effect -upon its power to give rotation as a similar increase or -diminution of the force itself. The power of a force to -produce rotation is, therefore, accurately estimated, not -by the force alone, but by the product found by multiplying -the force by the distance of its direction from the -axis. It is frequently necessary in mechanical science -to refer to this power of a force, and, accordingly, the -product just mentioned has received a particular denomination. -It is called the <i>moment</i> of the force round the -axis.</p> - -<p id="p185">(185.) The distance of the direction of a force from -the axis is sometimes called the <i>leverage</i> of the force. -The <i>moment</i> of a force is therefore found by multiplying -the force by its leverage, and the energy of a given -force to turn a body round an axis is proportional to the -leverage of that force.</p> - -<p>From all that has been observed it may easily be inferred -that, if several forces affect a body moveable on -an axis, having tendencies to turn it in different direc<span class="pagenum" id="Page_136">136</span>tions, -they will mutually neutralise each other and produce -equilibrium, if the sum of the moments of those -forces which tend to turn the body in one direction be -equal to the sum of the moments of those which tend to -turn it in the opposite direction. Thus, if the forces -A, B, C, . . . tend to turn the body from right to left, and -the distances of their directions from the axis be <i>a</i>, <i>b</i>, <i>c</i>, . . . -and the forces <span class="ilb">A′</span>, <span class="ilb">B′</span>, <span class="ilb">C′</span>, . . . tend to move it from left to -right, and the distances of their directions from the axis -be <i><span class="ilb">a′</span></i>, <i><span class="ilb">b′</span></i>, <i><span class="ilb">c′</span></i>, . . .; then these forces will produce equilibrium, -if the products found by multiplying the ounces -in A, B, C, . . . respectively by the inches in <i>a</i>, <i>b</i>, <i>c</i>, . . . when -added together be equal to the products found by multiplying -the ounces in <span class="ilb">A′</span>, <span class="ilb">B′</span>, <span class="ilb">C′</span>, . . . by the inches in -<i><span class="ilb">a′</span></i>, <i><span class="ilb">b′</span></i>, <i><span class="ilb">c′</span></i>, . . . respectively when added together. But if -either of these sets of products when added together exceed -the other, the corresponding set of forces will prevail, -and the body will revolve on its axis.</p> - -<p id="p186">(186.) When a body receives an impulse in a direction -perpendicular to the axis, but not crossing it, a uniform -rotatory motion is produced. The velocity of this motion -depends on the force of the impulse, the distance of the -direction of the impulse from the axis, and the manner -in which the mass of the body is distributed round the -axis. It is to be considered that the whole force of the -impulse is shared amongst the various parts of the -mass, and is transmitted to them from the point where -the impulse is applied by reason of the cohesion and -tenacity of the parts, and the impossibility of one part -yielding to a force without carrying all the other parts -with it. The force applied acts upon those particles -nearer to the axis than its own direction under advantageous -circumstances; for, according to what has been -already explained, their power to resist the effect of the -applied force is small in the same proportion with their -distance. On the other hand, the applied force acts -upon particles of the mass, at a greater distance than its -own direction, under circumstances proportionably disadvantageous; -for their resistance to the applied force -is great in proportion to their distances from the axis.</p> - -<p><span class="pagenum" id="Page_137">137</span></p> - -<p>Let C D, <i><a href="#i_p176a">fig. 72.</a></i>, be a section of the body made by a -plane passing through the axis A B. Suppose the impulse -to be applied at P, perpendicular to this plane, and -at the distance P O from the axis. The effect of the impulse -being distributed through the mass will cause the -body to revolve on A B, with a uniform velocity. There is -a certain point G, at which, if the whole mass were concentrated, -it would receive from the impulse the same -velocity round the axis. The distance O G is called the -<i>radius of gyration</i> of the axis A B, and the point G is -called the <i>centre of gyration</i> relatively to that axis. The -effect of the impulse upon the mass concentrated at G is -great in exactly the same proportion as O G is small. -This easily follows from the property of moments which -has been already explained; from whence it may be -inferred, that the greater the radius of gyration is, the -less will be the velocity which the body will receive from -a given impulse.</p> - -<p id="p187">(187.) Since the radius of gyration depends on the -manner in which the mass is arranged round the axis, it -follows that for different axes in the same body there -will be different radii of gyration. Of all axes taken in -the same body parallel to each other, that which passes -through the centre of gravity has the least radius of -gyration. If the radius of gyration of any axis passing -through the centre of gravity be given, that of any -parallel axis can be found; for the square of the -radius of gyration of any axis is equal to the square of -the distance of that axis from the centre of gravity added -to the square of the radius of gyration of the parallel -axis through the centre of gravity.</p> - -<p id="p188">(188.) The product of the numerical expressions for -the mass of the body and the square of the radius of -gyration is a quantity much used in mechanical science, -and has been called the <i>moment of inertia</i>. The moments -of inertia, therefore, for different axes in the same body -are proportional to the squares of the corresponding radii -of gyration; and consequently increase as the distances -of the axes from the centre of gravity increase. (<a href="#p187">187</a>.)</p> - -<p><span class="pagenum" id="Page_138">138</span></p> - -<p id="p189">(189.) From what has been explained in (<a href="#p187">187</a>.), it -follows, that the moment of inertia of any axis may be -computed by common arithmetic, if the moment of inertia -of a parallel axis through the centre of gravity be -previously known. To determine this last, however, -would require analytical processes altogether unsuitable -to the nature and objects of the present treatise.</p> - -<p>The velocity of rotation which a body receives from -a given impulse is great in exactly the same proportion -as the moment of inertia is small. Thus the moment -of inertia may be considered in rotatory motion analogous -to the mass of the body in rectilinear motion.</p> - -<p>From what has been explained in (<a href="#p187">187</a>.) it follows -that a given impulse at a given distance from the axis -will communicate the greatest angular velocity when -the axis passes through the centre of gravity, and that -the velocity which it will communicate round other -axes will be diminished in the same proportion as the -squares of their distances from the centre of gravity -added to the square of the radius of gyration for a -parallel axis through the centre of gravity are augmented.</p> - -<p id="p190">(190.) If any point whatever be assumed in a body, -and right lines be conceived to diverge in all directions -from that point, there are generally two of these lines, -which being taken as axes of rotation, one has a greater -and the other a less moment of inertia than any of the -others. It is a remarkable circumstance, that, whatever -be the nature of the body, whatever be its shape, and -whatever be the position of the point assumed, these -two axes of greatest and least moment will always be -at right angles to each other.</p> - -<p>These axes and a third through the same point, and -at right angles to both of them, are called the <i>principal -axes</i> of that point from which they diverge. To form -a distinct notion of their relative position, let the axis -of greatest moment be imagined to lie horizontally from -north to south, and the axis of least moment from east -to west; then the third principal axis will be presented<span class="pagenum" id="Page_139">139</span> -perpendicularly upwards and downwards. The first -two being called the principal axes of greatest and least -moment, the third may be called the <i>intermediate principal -axis</i>.</p> - -<p id="p191">(191.) Although the moments of the three principal -axes be in general unequal, yet bodies may be found -having certain axes for which these moments may be -equal. In some cases the moment of the intermediate -axis is equal to that of the principal axis of greatest -moment: in others it is equal to that of the principal -axis of least moment, and in others the moments of all -the three principal axes are equal to each other.</p> - -<p>If the moments of any two of three principal axes be -equal, the moments of all axes through the same point -and in their plane will also be equal; and if the moments -of the three principal axes through a point be -equal, the moments of all axes whatever, through the -same point, will be equal.</p> - -<p id="p192">(192.) If the moments of the principal axes through -the centre of gravity be known, the moments for all -other axes through that point may be easily computed. -To effect this it is only necessary to multiply the moments -of the principal axes by the squares of the co-sines -of the angles formed by them respectively with the -axis whose moment is sought. The products being -added together will give the required moment.</p> - -<p id="p193">(193.) By combining this result with that of (<a href="#p189">189</a>.), -it will be evident that the moment of all axes whatever -may be determined, if those of the principal axes -through the centre of gravity be known.</p> - -<p id="p194">(194.) It is obvious that the principal axis of least -moment through the centre of gravity has a less moment -of inertia than any other axis whatever. For it -has, by its definition (<a href="#p190">190</a>.) a less moment of inertia -than any other axis through the centre of gravity, and -every other axis through the centre of gravity has a less -moment of inertia than a parallel axis through any -other point (<a href="#p187">187</a>.) and (<a href="#p189">189</a>.)</p> - -<p id="p195">(195.) If two of the principal axes through the -centre of gravity have equal moments of inertia, all axes<span class="pagenum" id="Page_140">140</span> -in any plane parallel to the plane of these axes, and -passing through the point where a perpendicular from -the centre of gravity meets that plane, must have equal -moments of inertia. For by (<a href="#p191">191</a>.) all axes in the -plane of those two have equal moments, and by (<a href="#p189">189</a>.) -the axes in the parallel plane have moments which -exceed these by the same quantity, being equally distant -from them. (<a href="#p187">187</a>.)</p> - -<p>Hence it is obvious that if the three principal axes -through the centre of gravity have equal moments, all -axes situated in any given plane, and passing through -the point where the perpendicular from the centre of -gravity meets that plane, will have equal moments, -being equally distant from parallel axes through the -centre of gravity.</p> - -<p id="p196">(196.) If the three principal axes through the -centre of gravity have unequal moments, there is no -point whatever for which all axes will have equal -moments; but if the principal axis of least moment -and the intermediate principal axis through the centre -of gravity have equal moments, then there will be two -points on the principal axis of greatest moment, equally -distant at opposite sides of the centre of gravity, at -which all axes will have equal moments. If the three -principal axes through the centre of gravity have equal -moments, no other point of the body can have principal -axes of equal moment.</p> - -<p id="p197">(197.) When a body revolves on a fixed axis, the -parts of its mass are whirled in circles round the -axis; and since they move with a common angular -velocity, they will have centrifugal forces proportional to -their distances from the axis. If the component parts -of the mass were not united together by cohesive forces -of energies greater than these centrifugal forces, they -would be separated, and would fly off from the axis; -but their cohesion prevents this, and causes the effects -of the different centrifugal forces, which affect the -different parts of the mass, to be transmitted so as to -modify each other, and finally to produce one or more -forces mechanically equivalent to the whole, and which<span class="pagenum" id="Page_141">141</span> -are exerted upon the axis and resisted by it. We -propose now to explain these effects, as far as it is -possible to render them intelligible without the aid of -mathematical language.</p> - -<p>It is obvious that any number of equal parts of the -mass, which are uniformly arranged in a circle round -the axis, have equal centrifugal forces acting from the -centre of the circle in every direction. These mutually -neutralise each other, and therefore exert no force on -the axis. The same may be said of all parts of the -mass which are regularly and equally distributed on -every side of the axis.</p> - -<p>Also if equal masses be placed at equal distances on -opposite sides of the axis, their centrifugal forces will -destroy each other. Hence it appears that the pressure -which the axis of rotation sustains from the centrifugal -forces of the revolving mass, arises from the unequal -distribution of the matter around it.</p> - -<p>From this reasoning it will be easily perceived that -in the following examples the axis of rotation will -sustain no pressure.</p> - -<p>A globe revolving on any of its diameters, the density -being the same at equal distances from the centre.</p> - -<p>A spheroid or a cylinder revolving on its axis, the -density being equal at equal distances from the axis.</p> - -<p>A cube revolving on an axis which passes through -the centre of two opposite bases, being of uniform -density.</p> - -<p>A circular plate of uniform thickness and density -revolving on one of its diameters as an axis.</p> - -<p id="p198">(198.) In all these examples it will be observed that -the axis of rotation passes through the centre of gravity. -The general theorem, of which they are only particular -instances, is, “if a body revolve on a principal axis, passing -through the centre of gravity, the axis will sustain -no pressure from the centrifugal force of the revolving -mass.” This is a property in which the principal axes -through the centre of gravity are unique. There is no -other axis on which a body could revolve without -pressure.</p> - -<p><span class="pagenum" id="Page_142">142</span></p> - -<p>If two of the principal axes through the centre of -gravity have equal moments, every axis in their plane -has the same moment, and is to be considered equally -as a principal axis. In this case the body would revolve -on any of these axes without pressure.</p> - -<p>A homogeneous spheroid furnishes an example of -this. If any of the diameters of the earth’s equator -were a fixed axis, the earth would revolve on it without -producing pressure.</p> - -<p>If the three principal axes through the centre of -gravity have equal moments, all axes through the -centre of gravity are to be considered as principal -axes. In this case the body would revolve without -pressure on any axis through the centre of gravity.</p> - -<p>A globe, in which the density of the mass at equal -distances from the centre is the same, is an example -of this. Such a body would revolve without pressure -on any axis through its centre.</p> - -<p id="p199">(199.) Since no pressure is excited on the axis in -these cases, the state of the body will not be changed, -if during its rotation the axis cease to be fixed. The -body will notwithstanding continue to revolve round -the axis, and the axis will maintain its position.</p> - -<p>Thus a spinning-top of homogeneous material and -symmetrical form will revolve steadily in the same -position, until the friction of its point with the surface -on which it rests deprives it of motion. This is a -phenomenon which can only be exhibited when the -axis of rotation is a principal axis through the centre -of gravity.</p> - -<p id="p200">(200.) If the body revolve round any axis through -the centre of gravity, which is not a principal axis, -the centrifugal pressure is represented by two forces, -which are equal and parallel, but which act in opposite -directions on different points of the axis. The effect of -these forces is to produce a strain upon the axis, and -give the body a tendency to move round another axis -at right angles to the former.</p> - -<p id="p201">(201.) If the fixed axis on which a body revolves<span class="pagenum" id="Page_143">143</span> -be a principal axis through any point different from -the centre of gravity, then a pressure will be produced -by the centrifugal force of the revolving mass, and this -pressure will act at right angles to the axis on the point -to which it is a principal axis, and in the plane through -that axis and the centre of gravity. The amount of -the pressure will be proportional to the mass of the -body, the distance of the centre of gravity from the -axis, and the square of the velocity of rotation.</p> - -<p id="p202">(202.) Since the whole pressure is in this case excited -on a single point, the stability of the axis will not -be disturbed, provided that point alone be fixed. So -that even though the axis should be free to turn on that -point, no motion will ensue as long as no external -forces act upon the body.</p> - -<p id="p203">(203.) If the axis of rotation be not a principal axis, -the centrifugal forces will produce an effect which -cannot be represented by a single force. The effect -may be understood by conceiving two forces to act on -<i>different points</i> of the axis at right angles to it and to -each other. The quantities of these pressures and -their directions depend on the figure and density of -the mass and the position of the axis, in a manner -which cannot be explained without the aid of mathematical -language and principles.</p> - -<p id="p204">(204.) The effects upon the axis which have been -now explained are those which arise from the motion -of rotation, from whatever cause that motion may have -arisen. The forces which produce that motion, however, -are attended with effects on the axis which still -remain to be noticed. When these forces, whether -they be of the nature of instantaneous actions or continued -forces, are entirely resisted by the axis, their -directions must severally be in a plane passing through -the axis, or they must, by the principles of the composition -of force [(<a href="#p74">74</a>.) et seq.], be mechanically equivalent -to forces in that plane. In every other case the -impressed forces <i>must</i> produce motion, and, except in -certain cases, must also produce effects upon the axis.</p> - -<p><span class="pagenum" id="Page_144">144</span></p> - -<p>By the rules for the composition of force it is possible -in all cases to resolve the impressed forces into -others which are either in planes through the axis, or -in planes perpendicular to it, or, finally, some in planes -through it, and others in planes perpendicular to it. -The effect of those which are in planes through the -axis has been already explained; and we shall now -confine our attention to those impelling forces which -act at right angles to the axis, and which produce -motion.</p> - -<p>It will be sufficient to consider the effect of a single -force at right angles to the axis; for whatever be the -number of forces which act either simultaneously or -successively, the effect of the whole will be decided by -combining their separate effects. The effect which a -single force produces depends on two circumstances, -1. The position of the axis with respect to the figure -and mass of the body, and 2. The quantity and direction -of the force itself.</p> - -<p>In general the shock which the axis sustains from -the impact may be represented by two impacts applied -to it at different points, one parallel to the impressed -force, and the other perpendicular to it, but both perpendicular -to the axis. There are certain circumstances, -however, under which this effect will be -modified.</p> - -<p>If the impulse which the body receives be in a -direction perpendicular to a plane through the axis and -the centre of gravity, and at a distance from the axis -which bears to the radius of gyration (<a href="#p186">186</a>.) the same -proportion as that line bears to the distance of the -centre of gravity from the axis, there are certain cases -in which the impulse will produce no percussion. To -characterise these cases generally would require analytical -formulæ which cannot conveniently be translated -into ordinary language. That point of the plane, however, -where the direction of the impressed force meets -it, when no percussion on the axis is produced, is -called the <i>centre of percussion</i>.</p> - -<p><span class="pagenum" id="Page_145">145</span></p> - -<p>If the axis of rotation be a principal axis, the centre -of percussion must be in the right line drawn through -the centre of gravity, intersecting the axis at right angles, -and at the distance from the axis already explained.</p> - -<p>If the axis of rotation be parallel to a principal axis -through the centre of gravity, the centre of percussion -will be determined in the same manner.</p> - -<p id="p205">(205.) There are many positions which the axis may -have in which there will be no centre of percussion; that -is, there will be no direction in which an impulse could -be applied without producing a shock upon the axis. One -of these positions is when it is a principal axis through -the centre of gravity. This is the only case of rotation -round an axis in which no effect arises from the centrifugal -force; and therefore it follows that the only case -in which the axis sustains no effect from the motion -produced, is one in which it must necessarily suffer an -effect from that which produces the motion.</p> - -<p>If the body be acted upon by continued forces, their -effect is at each instant determined by the general principles -for the composition of force.</p> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<h2 class="nobreak" id="CHAP_XI">CHAP. XI.<br /> - -<span class="title">ON THE PENDULUM.</span></h2> -</div> - - -<p id="p206">(206.) <span class="smcap">When</span> a body is placed on a horizontal axis -which does not pass through its centre of gravity, it will -remain in permanent equilibrium only when the centre -of gravity is immediately below the axis. If this point -be placed in any other situation, the body will oscillate -from side to side, until the atmospherical resistance and -the friction of the axis destroy its motion. (<a href="#p159">159</a>, 160.) -Such a body is called a <i>pendulum</i>. The swinging motion -which it receives is called <i>oscillation</i> or <i>vibration</i>.</p> - -<p id="p207">(207.) The use of the pendulum, not only for philosophical -purposes, but in the ordinary economy of life,<span class="pagenum" id="Page_146">146</span> -renders it a subject of considerable importance. It furnishes -the most exact means of measuring time, and of -determining with precision various natural phenomena. -By its means the variation of the force of gravity in -different latitudes is discovered, and the law of that -variation experimentally exhibited. In the present -chapter, we propose to explain the general principles -which regulate the oscillation of pendulums. Minute -details concerning their construction will be given in the -twenty-first chapter of this volume.</p> - -<p id="p208">(208.) A simple pendulum is composed of a heavy -molecule attached to the end of a flexible thread, and -suspended by a fixed point O, <i><a href="#i_p176a">fig. 73.</a></i> When the pendulum -is placed in the position O C, the molecule being -vertically below the point of suspension, it will remain -in equilibrium; but if it be drawn into the position O A -and there liberated, it will descend towards C, moving -through the arc A C with accelerated motion. Having -arrived at C and acquired a certain velocity, it will, by -reason of its inertia, continue to move in the same -direction. It will therefore commence to ascend the arc -C <span class="ilb">A′</span> with the velocity so acquired. During its ascent, -the weight of the molecule retards its motion in exactly -the same manner as it had accelerated it in descending -from A to C; and when the molecule has ascended -through the arc C <span class="ilb">A′</span> equal to C A, its entire velocity -will be destroyed, and it will cease to move in that direction. -It will thus be placed at <span class="ilb">A′</span> in the same manner -as in the first instance it had been placed at A, and consequently -it will descend from <span class="ilb">A′</span> to C with accelerated -motion, in the same manner as it first moved from A to -C. It will then ascend from C to A, and so on, continually. -In this case the thread, by which the molecule -is suspended, is supposed to be perfectly flexible, inextensible, -and of inconsiderable weight. The point of -suspension is supposed to be without friction, and the -atmosphere to offer no resistance to the motion.</p> - -<p>It is evident from what has been stated, that the -times of moving from A to <span class="ilb">A′</span> and from <span class="ilb">A′</span> to A are<span class="pagenum" id="Page_147">147</span> -equal, and will continue to be equal so long as the pendulum -continues to vibrate. If the number of vibrations -performed by the pendulum were registered, and the -time of each vibration known, this instrument would -become a chronometer.</p> - -<p>The rate at which the motion of the pendulum is -accelerated in its descent towards its lowest position is -not uniform, because the force which impels it is -continually decreasing, and altogether disappears at the -point C. The impelling force arises from the effect of -gravity on the suspended molecule, and this effect is always -produced in the vertical direction A V. The greater the -angle O A V is, the less efficient the force of gravity will -be in accelerating the molecule: this angle evidently -increases as the molecule approaches C, which will -appear by inspecting <i><a href="#i_p176a">fig. 73.</a></i> At C, the force of gravity -acting in the direction C B is totally expended in giving -tension to the thread, and is inefficient in moving the -molecule. It follows, therefore, that the impelling force -is greatest at A, and continually diminishes from A to C, -where it altogether vanishes. The same observations -will be applicable to the retarding force from C to <span class="ilb">A′</span>, and -to the accelerating force from <span class="ilb">A′</span> to C, and so on.</p> - -<p>When the length of the thread and the intensity of -the force of gravity are given, the time of vibration -depends on the length of the arc A C, or on the magnitude -of the angle A O C. If, however, this angle do not -exceed a certain limit of magnitude, the time of vibration -will be subject to no sensible variation, however -that angle may vary. Thus the time of oscillation will -be the same, whether the angle A O C be 2°, or 1° 30′, or -1°, or any lesser magnitude. This property of a pendulum -is expressed by the word <i>isochronism</i>. The -strict demonstration of this property depends on mathematical -principles, the details of which would not be -suitable to the present treatise. It is not difficult, -however, to explain generally how it happens that the -same pendulum will swing through greater and smaller -arcs of vibration in the same time. If it swing from A,<span class="pagenum" id="Page_148">148</span> -the force of gravity at the commencement of its motion -impels it with an effect depending on the obliquity of -the lines O A and A V. If it commence its motion -from <i>a</i>, the impelling effect from the force of gravity -will be considerably less than at A; consequently, the -pendulum begins to move at a slower rate, when it -swings from <i>a</i> than when it moves from A: the greater -magnitude of the swing is therefore compensated by the -increased velocity, so that the greater and the smaller arcs -of vibration are moved through in the same time.</p> - -<p id="p209">(209.) To establish this property experimentally, it is -only necessary to suspend a small ball of metal, or other -heavy substance, by a flexible thread, and to put it in a -state of vibration, the entire arc of vibration not exceeding -4° or 5°, the friction on the point of suspension and -other causes will gradually diminish the arc of vibration, -so that after the lapse of some hours it will be so small, -that the motion will scarcely be discerned without -microscopic aid. If the vibration of this pendulum be -observed in reference to a correct timekeeper, at the -commencement, at the middle, and towards the end -of its motion, the rate will be found to suffer no sensible -change.</p> - -<p>This remarkable law of isochronism was one of the -earliest discoveries of Galileo. It is said, that when very -young, he observed a chandelier suspended from the roof -of a church in Pisa swinging with a pendulous motion, -and was struck with the uniformity of the rate even when -the extent of the swing was subject to evident variation.</p> - -<p id="p210">(210.) It has been stated in (<a href="#p117">117</a>.) that the attraction -of gravity affects all bodies equally, and moves them with -the same velocity, whatever be the nature or quantity of -the materials of which they are composed. Since it is -the force of gravity which moves the pendulum, we should -therefore expect that the circumstances of that motion -should not be affected either by the quantity or quality -of the pendulous body. And we find this, in fact, to be -the case; for if small pieces of different heavy substances -such as lead, brass, ivory, &c., be suspended by fine<span class="pagenum" id="Page_149">149</span> -threads of equal length, they will vibrate in the same -time, provided their weights bear a considerable proportion -to the atmospherical resistance, or that they be suspended -<i>in vacuo</i>.</p> - -<p id="p211">(211.) Since the time of vibration of a pendulum, -which oscillates in small arcs, depends neither on the magnitude -of the arc of vibration nor on the quality or -weight of the pendulous body, it will be necessary to explain -the circumstances on which the variation of this -time depends.</p> - -<p>The first and most striking of these circumstances is -the length of the suspending thread. The rudest experiments -will demonstrate the fact, that every increase -in the length of this thread will produce a corresponding -increase in the time of vibration; but according to what -law does this increase proceed? If the length of the thread -be doubled or trebled, will the time of vibration also be increased -in a double or treble proportion? This problem -is capable of exact mathematical solution, and the result -shows that the time of vibration increases not in the -proportion of the increased length of the thread, but as -the square root of that length; that is to say, if the -length of the thread be increased in a four-fold proportion, -the time of vibration will be augmented in a two-fold -proportion. If the thread be increased to nine times its -length, the time of vibration will be trebled, and so on. -This relation is exactly the same as that which was proved -to subsist between the spaces through which a body falls -freely, and the times of fall. In the table, page 89, if the -figures representing the height be understood to express -the length of different pendulums, the figures immediately -above them will express the corresponding times of vibration.</p> - -<p>This law of the proportion of the lengths of pendulums -to the squares of the time of vibration may be experimentally -established in the following manner:—</p> - -<p>Let A, B, C, <i><a href="#i_p176a">fig. 74.</a></i>, be three small pieces of metal -each attached by threads to two points of suspension, and -let them be placed in the same vertical line under the<span class="pagenum" id="Page_150">150</span> -point O; suppose them so adjusted that the distances -O A, O B, and O C shall be in the proportion of the -numbers 1, 4, and 9. Let them be removed from the -vertical in a direction at right angles to the plane of the -paper, so that the threads shall be in the same plane, and -therefore the three pendulums will have the same angle -of vibration. Being now liberated, the pendulum A will -immediately gain upon B, and B upon C, so that A will -have completed one vibration before B or C. At the end -of the second vibration of A, the pendulum B will have -arrived at the end of its first vibration, so that the suspending -threads of A and B will then be separated by -the whole angle of vibration; at the end of the fourth vibration -of A the suspending threads of A and B will return -to their first position, B having completed two vibrations; -thus the proportion of the times of vibration of B and A -will be 2 to 1, the proportion of their lengths being -4 to 1. At the end of the third vibration of A, C will -have completed one vibration, and the suspending strings -will coincide in the position distant by the whole angle -of vibration from their first position. So that three vibrations -of A are performed in the same time as one of -C: the proportion of the time of vibration of C and A -are, therefore, 3 to 1, the proportion of their lengths being -9 to 1, conformably to the law already explained.</p> - -<p id="p212">(212.) In all the preceding observations we have assumed -that the material of the pendulous body is of inconsiderable -magnitude, its whole weight being conceived -to be collected in a physical point. This is generally -called a simple pendulum; but since the conditions of a -suspending thread without weight, and a heavy molecule -without magnitude, cannot have practical existence, the -simple pendulum must be considered as imaginary, and -merely used to establish hypothetical theorems, which, -though inapplicable in practice, are nevertheless the means -of investigating the laws which govern the real phenomena -of pendulous bodies.</p> - -<p>A pendulous body being of determinate magnitude, its -several parts will be situated at different distances from<span class="pagenum" id="Page_151">151</span> -the axis of suspension. If each component part of such -a body were separately connected with the axis of suspension -by a fine thread, it would, being unconnected with -the other particles, be an independent simple pendulum, -and would oscillate according to the laws already explained. -It therefore follows that those particles of the -body which are nearest to the axis of suspension would, -if liberated from their connection with the others, vibrate -more rapidly than those which are more remote. The -connection, however, which the particles of the body -have, by reason of their solidity, compels them all to -vibrate in the same time. Consequently, those particles -which are nearer the axis are retarded by the slower -motion of those which are more remote; while the more -remote particles, on the other hand, are urged forward by -the greater tendency of the nearer particles to rapid vibration. -This will be more readily comprehended, if we -conceive two particles of matter A and B, <i><a href="#i_p176a">fig. 75.</a></i>, to be -connected with the same axis O by an inflexible wire O C, -the weight of which may be neglected. If B were removed, -A would vibrate in a certain time depending upon -the distance O A. If A were removed, and B placed upon -the wire at a distance B O equal to four times A O, B -would vibrate in twice the former time. Now if both be -placed on the wire at the distances just mentioned, the tendency -of A to vibrate more rapidly will be transmitted to -B by means of the wire, and will urge B forward more -quickly than if A were not present: on the other hand, -the tendency of B to vibrate more slowly will be transmitted -by the wire to A, and will cause it to move more -slowly than if B were not present. The inflexible quality -of the connecting wire will in this case compel A and B to -vibrate simultaneously, the time of vibration being greater -than that of A, and less than that of B, if each vibrated -unconnected with the other.</p> - -<p>If, instead of supposing two particles of matter placed -on the wire, a greater number were supposed to be -placed at various distances from O, it is evident the -same reasoning would be applicable. They would mu<span class="pagenum" id="Page_152">152</span>tually -affect each other’s motion; those placed nearest to -point O accelerating the motion of those more remote, -and being themselves retarded by the latter. Among -these particles one would be found in which all these -effects would be mutually neutralised, all the particles -nearer O being retarded in reference to that motion -which they would have if unconnected with the rest, -and those more remote being in the same respect accelerated. -The point at which such a particle is placed is -called <i>the centre of oscillation</i>.</p> - -<p>What has been here observed of the effects of -particles of matter placed upon rigid wire will be -equally applicable to the particles of a solid body. -Those which are nearer to the axis are urged forward -by those which are more remote, and are in their turn -retarded by them; and as with the particles placed upon -the wire, there is a certain particle of the body at which -the effects are mutually neutralised, and which vibrates -in the same time as it would if it were unconnected -with the other parts of the body, and simply connected -by a fine thread to the axis. By this centre of oscillation -the calculations respecting the vibration of a solid -body are rendered as simple as those of a molecule of -inconsiderable magnitude. All the properties which have -been explained as belonging to a simple pendulum may -thus be transferred to a vibrating body of any magnitude -and figure, by considering it as equivalent to a single -particle of matter vibrating at its centre of oscillation.</p> - -<p id="p213">(213.) It follows from this reasoning, that the virtual -length of a pendulum is to be estimated by the distance -of its centre of oscillation from the axis of suspension, -and therefore that the times of vibration of -different pendulums are in the same proportion as the -square roots of the distances of their centres of oscillation -from their axes.</p> - -<p>The investigation of the position of the centre of -oscillation is, in most cases, a subject of intricate mathematical -calculation. It depends on the magnitude and -figure of the pendulous body, the manner in which the<span class="pagenum" id="Page_153">153</span> -mass is distributed through its volume, or the density -of its several parts, and the position of the axis on which -it swings.</p> - -<p>The place of the centre of oscillation may be determined -when the position of the centre of gravity and -the centre of gyration are known; for the distance of -the centre of oscillation from the axis will always be -obtained by dividing the square of the radius of gyration -(<a href="#p186">186</a>.) by the distance of the centre of gravity from the -axis. Thus if 6 be the radius of gyration, and 9 the -distance of gravity from the axis, 36 divided by 9, -which is 4, will be the distance of the centre of oscillation -from the axis. Hence it may be inferred generally, -that the greater the proportion which the radius of gyration -bears to the distance of the centre of gravity from -the axis, the greater will be the distance of the centre of -oscillation.</p> - -<p>It follows from this reasoning, that the length of a -pendulum is not limited by the dimensions of its volume. -If the axis be so placed that the centre of gravity is -near it, and the centre of gyration comparatively removed -from it, the centre of oscillation may be placed -far beyond the limits of the pendulous body. Suppose -the centre of gravity is at a distance of one inch from -the axis, and the centre of gyration 12 inches, the centre -of oscillation will then be at the distance of 144 inches, -or 12 feet. Such a pendulum may not in its greatest -dimensions exceed one foot, and yet its time of vibration -would be equal to that of a simple pendulum whose -length is 12 feet.</p> - -<p>By these means pendulums of small dimensions may -be made to vibrate as slowly as may be desired. The -instruments called <i>metronomes</i>, used for marking the -time of musical performances, are constructed on this -principle.</p> - -<p id="p214">(214.) The centre of oscillation is distinguished by -a very remarkable property in relation to the axis of -suspension. If A, <i><a href="#i_p176a">fig. 76.</a></i>, be the point of suspension, -and O the corresponding centre of oscillation, the time<span class="pagenum" id="Page_154">154</span> -of vibration of the pendulum will not be changed if -it be raised from its support, inverted, and suspended -from the point O. It follows, therefore, that if O be -taken as the point of suspension, A will be the corresponding -centre of oscillation. These two points are, therefore, -convertible. This property may be verified experimentally -in the following manner. A pendulum being put into -a state of vibration, let a small heavy body be suspended -by a fine thread, the length of which is so adjusted that -it vibrates simultaneously with the pendulum. Let the -distance from the point of suspension to the centre of -the vibrating body be measured, and take this distance -on the pendulum from the axis of suspension downwards; -the place of the centre of oscillation will thus -be obtained, since the distance so measured from the -axis is the length of the equivalent simple pendulum. -If the pendulum be now raised from its support, inverted, -and suspended from the centre of oscillation thus obtained, -it will be found to vibrate simultaneously with -the body suspended by the thread.</p> - -<p id="p215">(215.) This property of the interchangeable nature -of the centres of oscillation and suspension has been, -at a late period, adopted by Captain Kater, as an accurate -means of determining the length of a pendulum. -Having ascertained with great accuracy two points of -suspension at which the same body will vibrate in the -same time, the distance between these points being -accurately measured, is the length of the equivalent -simple pendulum. See Chapter <a href="#CHAP_XXI">XXI</a>.</p> - -<p id="p216">(216.) The manner in which the time of vibration of -a pendulum depends on its length being explained, we -are next to consider how this time is affected by the -attraction of gravity. It is obvious that, since the pendulum -is moved by this attraction, the rapidity of its -motion will be increased, if the impelling force receive -any augmentation; but it still is to be decided, in what -exact proportion the time of oscillation will be diminished -by any proposed increase in the intensity of the -earth’s attraction. It can be demonstrated mathema<span class="pagenum" id="Page_155">155</span>tically, -that the time of one vibration of a pendulum has -the same proportion to the time of falling freely in the -perpendicular direction, through a height equal to half -the length of the pendulum, as the circumference of a -circle has to its diameter. Since, therefore, the times -of vibration of pendulums are in a fixed proportion to -the times of falling freely through spaces equal to the -halves of their lengths, it follows that these times have -the same relation to the force of attraction as the times -of falling freely through their lengths have to that force. -If the intensity of the force of gravity were increased in -a four-fold proportion, the time of falling through a -given height would be diminished in a two-fold proportion; -if the intensity were increased to a nine-fold proportion, -the time of falling through a given space would -be diminished in a three-fold proportion, and so on; the -rate of diminution of the time being always as the -square root of the increased force. By what has been -just stated this law will also be applicable to the -vibration of pendulums. Any increase in the intensity -of the force of gravity would cause a given pendulum -to vibrate more rapidly, and the increased rapidity of -the vibration would be in the same proportion as the -square root of the increased intensity of the force of -gravity.</p> - -<p id="p217">(217.) The laws which regulate the times of vibration -of pendulums in relation to one another being well -understood, the whole theory of these instruments will -be completed, when the method of ascertaining the actual -time of vibration of any pendulum, in reference to its -length, has been explained. In such an investigation, the -two elements to be determined are, 1. the exact time of a -single vibration, and, 2. the exact distance of the centre -of oscillation from the point of suspension.</p> - -<p>The former is ascertained by putting a pendulum in -motion in the presence of a good chronometer, and -observing precisely the number of oscillations which are -made in any proposed number of hours. The entire -time during which the pendulum swings, being divided<span class="pagenum" id="Page_156">156</span> -by the number of oscillations made during that time, the -exact time of one oscillation will be obtained.</p> - -<p>The distance of the centre of oscillation from the point -of suspension may be rendered a matter of easy calculation, -by giving a certain uniform figure and material to -the pendulous body.</p> - -<p id="p218">(218.) The time of vibration of one pendulum of -known length being thus obtained, we shall be enabled -immediately to solve either of the following problems.</p> - -<p>“To find the length of a pendulum which shall -vibrate in a given time.”</p> - -<p>“To find the time of vibration of a pendulum of a -given length.”</p> - -<p>The former is solved as follows: the time of vibration -of the known pendulum is to the time of vibration -of the required pendulum, as the square root of the -length of the known pendulum is to the square root of -the length of the required pendulum. This length is -therefore found by the ordinary rules of arithmetic.</p> - -<p>The latter may be solved as follows: the length of the -known pendulum is to the length of the proposed pendulum, -as the square of the time of vibration of the -known pendulum is to the square of the time of vibration -of the proposed pendulum. The latter time may therefore -be found by arithmetic.</p> - -<p id="p219">(219.) Since the rate of a pendulum has a known -relation to the intensity of the earth’s attraction, we are -enabled, by this instrument, not only to detect certain -variations in that attraction in various parts of the earth, -but also to discover the actual amount of the attraction -at any given place.</p> - -<p>The actual amount of the earth’s attraction at any given -place is estimated by the height through which a body -would fall freely at that place in any given time, as -in one second. To determine this, let the length of a -pendulum which would vibrate in one second at that -place be found. As the circumference of a circle is to -its <span class="nowrap">diameter<a id="FNanchor_2" href="#Footnote_2" class="fnanchor">2</a></span> (a known proportion), so will one second -be to the time of falling through a height equal to<span class="pagenum" id="Page_157">157</span> -half the length of this pendulum. This time is therefore -a matter of arithmetical calculation. It has been -proved in (<a href="#p120">120</a>.), that the heights, through which a body -falls freely, are in the same proportion as the squares of -the times; from whence it follows, that the square of -the time of falling through a height equal to half the -length of the pendulum is to one second as half the -length of that pendulum is to the height through which -a body would fall in one second. This height, therefore, -may be immediately computed, and thus the actual -amount of the force of gravity at any given place may -be ascertained.</p> - -<p id="p220">(220.) To compare the force of gravity in different -parts of the earth, it is only necessary to swing the same -pendulum in the places under consideration, and to -observe the rapidity of its vibrations. The proportion -of the force of gravity in the several places will be that -of the squares of the velocity of the vibration. Observations -to this effect have been made at several places, -by Biot, Kater, Sabine, and others.</p> - -<p>The earth being a mass of matter of a form nearly -spherical, revolving with considerable velocity on an -axis, its component parts are affected by a centrifugal -force; in virtue of which, they have a tendency to fly off -in a direction perpendicular to the axis. This tendency -increases in the same proportion as the distance of any -part from the axis increases, and consequently those parts -of the earth which are near the equator, are more strongly -affected by this influence than those near the pole. It -has been already explained (<a href="#p145">145</a>.) that the figure of -the earth is affected by this cause, and that it has -acquired a spheroidal form. The centrifugal force, -acting in opposition to the earth’s attraction, diminishes -its effects; and consequently, where this force is more -efficient, a pendulum will vibrate more slowly. By -these means the rate of vibration of a pendulum becomes -an indication of the amount of the centrifugal force. -But this latter varies in proportion to the distance of the -place from the earth’s axis; and thus the rate of a<span class="pagenum" id="Page_158">158</span> -pendulum indicates the relation of the distances of -different parts of the earth’s surface from its axis. The -figure of the earth may be thus ascertained, and that -which theory assigns to it, it may be practically proved to -have.</p> - -<p>This, however, is not the only method by which the -figure of the earth may be determined. The meridians -being sections of the earth through its axis, if their -figure were exactly determined, that of the earth would -be known. Measurements of arcs of meridians on a -large scale have been executed, and are still being made -in various parts of the earth, with a view to determine -the curvature of a meridian at different latitudes. This -method is independent of every hypothesis concerning -the density and internal structure of the earth, and is -considered by some to be susceptible of more accuracy than -that which depends on the observations of pendulums.</p> - -<p id="p221">(221.) It has been stated that, when the arc of -vibration of a pendulum is not very small, a variation in -its length will produce a sensible effect on the time of -vibration. To construct a pendulum such that the time -of vibration may be independent of the extent of the -swing, was a favourite speculation of geometers. This -problem was solved by Huygens, who showed that the -curve called a <i>cycloid</i>, previously discovered and described -by Galileo, possessed the isochronal property; that is, -that a body moving in it by the force of gravity, would -vibrate in the same time, whatever be the length of the -arc described.</p> - -<p>Let O A, <i><a href="#i_p176a">fig. 77.</a></i>, be a horizontal line, and let O B be -a circle placed below this line, and in contact with it. -If this circle be rolled upon the line from O towards A, -a point upon its circumference, which at the beginning of -the motion is placed at O, will during the motion trace -the curve O C A. This curve is called a <i>cycloid</i>. If -the circle be supposed to roll in the opposite direction -towards <span class="ilb">A′</span>, the same point will trace another cycloid -O <span class="ilb">C′</span> <span class="ilb">A′</span>. The points C and <span class="ilb">C′</span> being the lowest points -of the curves, if the perpendiculars C D and <span class="ilb">C′</span> <span class="ilb">D′</span> be<span class="pagenum" id="Page_159">159</span> -drawn, they will respectively be equal to the diameter of -the circle. By a known property of this curve, the arcs -O C and O <span class="ilb">C′</span> are equal to twice the diameter of the -circle. From the point O suppose a flexible thread to -be suspended, whose length is twice the diameter of the -circle, and which sustains a pendulous body P at its -extremity. If the curves O C and O <span class="ilb">C′</span>, from the plane -of the paper, be raised so as to form surfaces to which -the thread may be applied, the extremity P will extend -to the points C and <span class="ilb">C′</span>, when the entire thread has been -applied to either of the curves. As the thread is deflected -on either side of its vertical position, it is applied to a -greater or lesser portion of either curve, according to the -quantity of its deflection from the vertical. If it be -deflected on each side until the point P reaches the -points C and <span class="ilb">C′</span>, the extremity would trace a cycloid -C P <span class="ilb">C′</span> precisely equal and similar to those already mentioned. -Availing himself of this property of the curve, -Huygens constructed his cycloidal pendulum. The time -of vibration was subject to no variation, however the arc -of vibration might change, provided only that the length -of the string O P continued the same. If small arcs -of the cycloid be taken on either side of the point P, they -will not sensibly differ from arcs of a circle described -with the centre O and the radius O P; for, in slight -deflections from the vertical position, the effect of the -curves O C and O <span class="ilb">C′</span> on the thread O P is altogether -inconsiderable. It is for this reason that when the arcs -of vibration of a circular pendulum are small, they partake -of the property of isochronism peculiar to those of -a cycloid. But when the deflection of P from the -vertical is great, the effect of the curves O C and O <span class="ilb">C′</span> -on the thread produces a considerable deviation of the -point P from the arc of the circle whose centre is O and -whose radius is O P, and consequently the property of -isochronism will no longer be observed in the circular -pendulum.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_160">160</span></p> - -<h2 class="nobreak" id="CHAP_XII">CHAP. XII.<br /> - -OF SIMPLE MACHINES.</h2> -</div> - - -<p id="p222">(222.) A <span class="lowercase smcap">MACHINE</span> is an instrument by which force or -motion may be transmitted and modified as to its quantity -and direction. There are two ways in which a machine -may be applied, and which give rise to a division of mechanical -science into parts denominated <span class="lowercase smcap">STATICS</span> and <span class="lowercase smcap">DYNAMICS</span>; -the one including the theory of equilibrium, and -the other the theory of motion. When a machine is considered -statically, it is viewed as an instrument by which -forces of determinate quantities and direction are made to -balance other forces of other quantities and other directions. -If it be viewed dynamically, it is considered -as a means by which certain motions of determinate -quantity and direction may be made to produce other motions -in other directions and quantities. It will not be -convenient, however, in the present treatise, to follow this -division of the subject. We shall, on the other hand, as -hitherto, consider the phenomena of equilibrium and motion -together.</p> - -<p>The effects of machinery are too frequently described -in such a manner as to invest them with the appearance of -paradox, and to excite astonishment at what appears to -contradict the results of the most common experience. -It will be our object here to take a different course, and to -attempt to show that those effects which have been held -up as matters of astonishment are the necessary, natural, -and obvious results of causes adapted to produce them -in a manner analogous to the objects of most familiar -experience.</p> - -<p id="p223">(223.) In the application of a machine there are three -things to be considered. 1. The force or resistance which -is required to be sustained, opposed, or overcome. 2. -The force which is used to sustain, support, or overcome -that resistance. 3. The machine itself by which the<span class="pagenum" id="Page_161">161</span> -effect of this latter force is transmitted to the former. Of -whatever nature be the force or the resistance which is to -be sustained or overcome, it is technically called the <i>weight</i>, -since, whatever it be, a weight of equivalent effect may -always be found. The force which is employed to sustain -or overcome it is technically called the <i>power</i>.</p> - -<p id="p224">(224.) In expressing the effect of machinery it is -usual to say that the power sustains the weight; but this, -in fact, is not the case, and hence arises that appearance of -paradox which has already been alluded to. If, for example, -it is said that a power of one ounce sustains the -weight of one ton, astonishment is not unnaturally excited, -because the fact, as thus stated, if the terms be literally -interpreted, is physically impossible. No power less -than a ton can, in the ordinary acceptation of the word, support -the weight of a ton. It will, however, be asked how it -happens that a machine <i>appears</i> to do this? how it happens -that by holding a silken thread, which an ounce weight -would snap, many hundred weight may be sustained? -To explain this it will only be necessary to consider the -effect of a machine, when the power and weight are in -equilibrium.</p> - -<p id="p225">(225.) In every machine there are some fixed points or -props; and the arrangement of the parts is always such, -that the pressure, excited by the power or weight, or both, -is distributed among these props. If the weight amount -to twenty hundred, it is possible so to distribute it, that -any proportion, however great, of it may be thrown on the -fixed points or props of the machine; the remaining part -only can properly be said to be supported by the power, -and this part can never be greater than the power. Considering -the effect in this way, it appears that the power -supports just so much of the weight and no more as is -equal to its own force, and that all the remaining part of -the weight is sustained by the machine. The force of -these observations will be more apparent when the nature -and properties of the mechanic powers and other machines -have been explained.</p> - -<p id="p226">(226.) When a machine is considered dynamically,<span class="pagenum" id="Page_162">162</span> -its effects are explained on different principles. It is true -that, in this case, a very small power may elevate a very -great weight; but nevertheless, in so doing, whatever be -the machine used, the total expenditure of power, in raising -the weight through any height, is never less than that -which would be expended if the power were immediately -applied to the weight without the intervention of any -machine. This circumstance arises from an universal -property of machines by which the velocity of the weight -is always less than that of the power, in exactly the same -proportion as the power itself is less than the weight; so -that when a certain power is applied to elevate a weight, the -rate at which the elevation is effected is always slow in the -same proportion as the weight is great. From a due -consideration of this remarkable law, it will easily be understood, -that a machine can never diminish the total expenditure -of power necessary to raise any weight or to -overcome any resistance. In such cases, all that a machine -ever does or ever can do, is to enable the power to be expended -at a slow rate, and in a more advantageous direction -than if it were immediately applied to the weight or the -resistance.</p> - -<p>Let us suppose that P is a power amounting to an ounce, -and that W is a weight amounting to 50 ounces, and that -P elevates W by means of a machine. In virtue of the -property already stated, it follows, that while P moves -through 50 feet, W will be moved through 1 foot; but -in moving P through 50 feet, 50 distinct efforts are -made, by each of which 1 ounce is moved through 1 foot, -and by which collectively 50 distinct ounces might be -successively raised through 1 foot. But the weight W -is 50 ounces, and has been raised through 1 foot; from -whence it appears, that the expenditure of power is equal -to that which would be necessary to raise the weight without -the intervention of any machine.</p> - -<p>This important principle may be presented under another -aspect, which will perhaps render it more apparent. -Suppose the weight W were actually divided into -50 equal parts, or suppose it were a vessel of liquid<span class="pagenum" id="Page_163">163</span> -weighing 50 ounces, and containing 50 equal measures; -if these 50 measures were successively lifted through a -height of 1 foot; the efforts necessary to accomplish this -would be the same as those used to move the power P -through 50 feet, and it is obvious, that the total expenditure -of force would be the same as that which would -be necessary to lift the entire contents of the vessel through -1 foot.</p> - -<p>When the nature and properties of the mechanic powers -and other machines have been explained, the force of these -observations will be more distinctly perceived. The effects -of props and fixed points in sustaining a part of the -weight, and sometimes the whole, both of the weight -and power, will then be manifest, and every machine will -furnish a verification of the remarkable proportion between -the velocities of the weight and power, which has enabled -us to explain what might otherwise be paradoxical -and difficult of comprehension.</p> - -<p id="p227">(227.) The most simple species of machines are those -which are commonly denominated the <span class="lowercase smcap">MECHANIC POWERS</span>. -These have been differently enumerated by different writers. -If, however, the object be to arrange in distinct -classes, and in the smallest possible number of them, those -machines which are alike in principle, the mechanic -powers may be reduced to three.</p> - -<p class="ml2em"> -1. The lever.<br /> -2. The cord.<br /> -3. The inclined plane.<br /> -</p> - -<p>To one or other of these classes all simple machines -whatever may be reduced, and all complex machines -may be resolved into simple elements which come under -them.</p> - -<p id="p228">(228.) The first class includes every machine which is -composed of a solid body revolving on a fixed axis, although -the name lever has been commonly confined to -cases where the machine affects certain particular forms. -This is by far the most useful class of machines, and will -require in subsequent chapters very detailed development.<span class="pagenum" id="Page_164">164</span> -The general principle, upon which equilibrium is established -between the power and weight in machines of -this class has been already explained in (<a href="#p183">183</a>.) The -power and weight are always supposed to be applied in -directions at right angles to the axis. If lines be drawn -from the axis perpendicular to the directions of power -and weight, equilibrium will subsist, provided the power -multiplied by the perpendicular distance of its direction -from the axis, be equal to the weight multiplied by the -perpendicular distance of its direction from the axis. -This is a principle to which we shall have occasion to -refer in explaining the various machines of this class.</p> - -<p id="p229">(229.) If the moment of the power (<a href="#p184">184</a>.) be greater -than that of the weight, the effect of the power will prevail -over that of the weight, and elevate it; but if, on the -other hand, the moment of the power be less than that of -the weight, the power will be insufficient to support the -weight, and will allow it to fall.</p> - -<p id="p230">(230.) The second class of simple machines includes -all those cases in which force is transmitted by means of -flexible threads, ropes, or chains. The principle, by which -the effects of these machines are estimated, is, that the -tension throughout the whole length of the same cord, -provided it be perfectly flexible, and free from the effects -of friction, must be the same. Thus, if a force acting at -one end be balanced by a force acting at the other -end, however the cord may be bent, or whatever -course it may be compelled to take, by any causes which -may affect it between its ends, these forces must be equal, -provided the cord be free to move over any obstacles which -may deflect it.</p> - -<p>Within this class of machines are included all the various -forms of <i>pulleys</i>.</p> - -<p id="p231">(231.) The third class of simple machines includes -all those cases in which the weight or resistance is supported -or moved on a hard surface inclined to the vertical -direction.</p> - -<p>The effects of such machines are estimated by resolving -the whole weight of the body into two elements by the -parallelogram of forces. One of these elements is perpen<span class="pagenum" id="Page_165">165</span>dicular -to the surface, and supported by its resistance; -the other is parallel to the surface, and supported by the -power. The proportion, therefore, of the power to the -weight will always depend on the obliquity of the surface -to the direction of the weight. This will be easily understood -by referring to what has been already explained -in Chapter <span class="lowercase smcap">VIII</span>.</p> - -<p>Under this class of machines come the inclined plane, -commonly so called, the wedge, the screw, and various -others.</p> - -<p id="p232">(232.) In order to simplify the development of the -elementary theory of machines, it is expedient to omit -the consideration of many circumstances, of which, however, -a strict account must be taken before any practically -useful application of that theory can be attempted. A -machine, as we must for the present contemplate it, is a -thing which can have no real or practical existence. Its -various parts are considered to be free from friction: all -surfaces which move in contact are supposed to be infinitely -smooth and polished. The solid parts are conceived -to be absolutely inflexible. The weight and inertia -of the machine itself are wholly neglected, and we -reason upon it as if it were divested of these qualities. Cords -and ropes are supposed to have no stiffness, to be infinitely -flexible. The machine, when it moves, is supposed -to suffer no resistance from the atmosphere, and to be in -all respects circumstanced as if it were <i>in vacuo</i>.</p> - -<p>It is scarcely necessary to state, that, all these suppositions -being false, none of the consequences deduced -from them can be true. Nevertheless, as it is the -business of art to bring machines as near to this state of -ideal perfection as possible, the conclusions which are -thus obtained, though false in a strict sense, yet deviate -from the truth in but a small degree. Like the first -outline of a picture, they resemble in their general -features that truth to which, after many subsequent -corrections, they must finally approximate.</p> - -<p>After a first approximation has been made on the -several false suppositions which have been mentioned,<span class="pagenum" id="Page_166">166</span> -various effects, which have been previously neglected, -are successively taken into account. Roughness, rigidity, -imperfect flexibility, the resistance of air and other -fluids, the effects of the weight and inertia of the -machine, are severally examined, and their laws and -properties detected. The modifications and corrections, -thus suggested as necessary to be introduced into our -former conclusions, are applied, and a second approximation, -but still <i>only</i> an approximation, to truth is made. -For, in investigating the laws which regulate the several -effects just mentioned, we are compelled to proceed upon -a new group of false suppositions. To determine the -laws which regulate the friction of surfaces, it is necessary -to assume that every part of the surfaces of contact -are uniformly rough; that the solid parts which are -imperfectly rigid, and the cords which are imperfectly -flexible, are constituted throughout their entire dimensions -of a uniform material; so that the imperfection -does not prevail more in one part than another. Thus, -all irregularity is left out of account, and a general -average of the effects taken. It is obvious, therefore, -that by these means we have still failed in obtaining a -result exactly conformable to the real state of things; but -it is equally obvious, that we have obtained one much -more conformable to that state than had been previously -accomplished, and sufficiently near it for most practical -purposes.</p> - -<p>This apparent imperfection in our instruments and -powers of investigation is not peculiar to mechanics: -it pervades all departments of natural science. In -astronomy, the motions of the celestial bodies, and their -various changes and appearances as developed by theory, -assisted by observation and experience, are only approximations -to the real motions and appearances which -take place in nature. It is true that these approximations -are susceptible of almost unlimited accuracy; but still they -are, and ever will continue to be, only approximations. -Optics and all other branches of natural science are liable -to the same observations.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_167">167</span></p> - -<h2 class="nobreak" id="CHAP_XIII">CHAP. XIII.<br /> - -<span class="title">OF THE LEVER.</span></h2> -</div> - - -<p id="p233">(233.) <span class="smcap">An</span> inflexible, straight bar, turning on an axis, -is commonly called a <i>lever</i>. The <i>arms</i> of the lever are -those parts of the bar which extend on each side of the -axis.</p> - -<p>The axis is called the <i>fulcrum</i> or <i>prop</i>.</p> - -<p id="p234">(234.) Levers are commonly divided into three kinds, -according to the relative positions of the power, the -weight, and the fulcrum.</p> - -<p>In a lever of the first kind, as in <i><a href="#i_p176a">fig. 78.</a></i>, the fulcrum -is between the power and weight.</p> - -<p>In a lever of the second kind, as in <i><a href="#i_p176a">fig. 79.</a></i>, the weight -is between the fulcrum and power.</p> - -<p>In a lever of the third kind, as in <i><a href="#i_p176a">fig. 80.</a></i>, the power -is between the fulcrum and weight.</p> - -<p id="p235">(235.) In all these cases, the power will sustain the -weight in equilibrium, provided its moment be equal to -that of the weight. (<a href="#p184">184</a>.) But the moment of the -power is, in this case, equal to the product obtained by -multiplying the power by its distance from the fulcrum; -and the moment of the weight by multiplying the weight -by its distance from the fulcrum. Thus, if the number -of ounces in P, being multiplied by the number of inches -in P F, be equal to the number of ounces in W, multiplied -by the number of inches in W F, equilibrium will -be established. It is evident from this, that as the -distance of the power from the fulcrum increases in -comparison to the distance of the weight from the -fulcrum, in the same degree exactly will the proportion -of the power to the weight diminish. In other words, -the proportion of the power to the weight will be always -the same as that of their distances from the fulcrum -taken in a reverse order.</p> - -<p>In cases where a small power is required to sustain or<span class="pagenum" id="Page_168">168</span> -elevate a great weight, it will therefore be necessary -either to remove the power to a great distance from the -fulcrum, or to bring the weight very near it.</p> - -<p id="p236">(236.) Numerous examples of levers of the first -kind may be given. A crow-bar, applied to elevate a -stone or other weight, is an instance. The fulcrum is -another stone placed near that which is to be raised, and -the power is the hand placed at the other end of the -bar.</p> - -<p>A handspike is a similar example.</p> - -<p>A poker applied to raise fuel is a lever of the first -kind, the fulcrum being the bar of the grate.</p> - -<p>Scissors, shears, nippers, pincers, and other similar -instruments are composed of two levers of the first -kind; the fulcrum being the joint or pivot, and the -weight the resistance of the substance to be cut or -seized; the power being the fingers applied at the other -end of the levers.</p> - -<p>The brake of a pump is a lever of the first kind; -the pump-rods and piston being the weight to be -raised.</p> - -<p id="p237">(237.) Examples of levers of the second kind, -though not so frequent as those just mentioned, are -not uncommon.</p> - -<p>An oar is a lever of the second kind. The reaction -of the water against the blade is the fulcrum. The -boat is the weight, and the hand of the boatman the -power.</p> - -<p>The rudder of a ship or boat is an example of this -kind of lever, and explained in a similar way.</p> - -<p>The chipping knife is a lever of the second kind. -The end attached to the bench is the fulcrum, and the -weight the resistance of the substance to be cut, placed -beneath it.</p> - -<p>A door moved upon its hinges is another example.</p> - -<p>Nut-crackers are two levers of the second kind; the -hinge which unites them being the fulcrum, the resistance -of the shell placed between them being the weight, -and the hand applied to the extremity being the power.</p> - -<p><span class="pagenum" id="Page_169">169</span></p> - -<p>A wheelbarrow is a lever of the second kind; the -fulcrum being the point at which the wheel presses on -the ground, and the weight being that of the barrow -and its load, collected at their centre of gravity.</p> - -<p>The same observation may be applied to all two-wheeled -carriages, which are partly sustained by the -animal which draws them.</p> - -<p id="p238">(238.) In a lever of the third kind, the weight, being -more distant from the fulcrum than the power, must be -proportionably less than it. In this instrument, therefore, -the power acts upon the weight to a mechanical -disadvantage, inasmuch as a greater power is necessary -to support or move the weight than would be required -if the power were immediately applied to the weight, -without the intervention of a machine. We shall, -however, hereafter show that the advantage which is -lost in force is gained in despatch, and that in proportion -as the weight is less than the power which moves -it, so will the speed of its motion be greater than that -of the power.</p> - -<p>Hence a lever of the third kind is only used in cases -where the exertion of great power is a consideration -subordinate to those of rapidity and despatch.</p> - -<p>The most striking example of levers of the third -kind is found in the animal economy. The limbs of -animals are generally levers of this description. The -socket of the bone is the fulcrum; a strong muscle attached -to the bone near the socket is the power; and -the weight of the limb, together with whatever resistance -is opposed to its motion, is the weight. A slight -contraction of the muscle in this case gives a considerable -motion to the limb: this effect is particularly conspicuous -in the motion of the arms and legs in the -human body; a very inconsiderable contraction of the -muscles at the shoulders and hips giving the sweep to -the limbs from which the body derives so much activity.</p> - -<p>The treddle of the turning lathe is a lever of the -third kind. The hinge which attaches it to the floor is<span class="pagenum" id="Page_170">170</span> -the fulcrum, the foot applied to it near the hinge is the -power, and the crank upon the axis of the fly-wheel, -with which its extremity is connected, is the weight.</p> - -<p>Tongs are levers of this kind, as also the shears -used in shearing sheep. In these cases the power is the -hand placed immediately below the fulcrum or point -where the two levers are connected.</p> - -<p id="p239">(239.) When the power is said to support the -weight by means of a lever or any other machine, it -is only meant that the power keeps the machine in -equilibrium, and thereby enables it to sustain the weight. -It is necessary to attend to this distinction, to remove -the difficulty which may arise from the paradox of a -small power sustaining a great weight.</p> - -<p>In a lever of the first kind, the fulcrum F, <i><a href="#i_p176a">fig. 78.</a></i>, -or axis, sustains the united forces of the power and -weight.</p> - -<p>In a lever of the second kind, if the power be supposed -to act over a wheel R, <i><a href="#i_p176a">fig. 79.</a></i>, the fulcrum F -sustains a pressure equal to the difference between the -power and weight, and the axis of the wheel R sustains -a pressure equal to twice the power; so that the total -pressures on F and R are equivalent to the united forces -of the power and weight.</p> - -<p>In a lever of the third kind similar observations are -applicable. The wheel R, <i><a href="#i_p176a">fig. 80.</a></i>, sustains a pressure -equal to twice the power, and the fulcrum F sustains a -pressure equal to the difference between the power and -weight.</p> - -<p>These facts may be experimentally established by -attaching a string to the lever immediately over the fulcrum, -and suspending the lever by that string from the -arm of a balance. The counterpoising weight, when -the fulcrum is removed, will, in the first case, be equal -to the sum of the weight and power, and in the last -two cases equal to their difference.</p> - -<p id="p240">(240.) We have hitherto omitted the consideration -of the effect of the weight of the lever itself. If the -centre of gravity of the lever be in the vertical line<span class="pagenum" id="Page_171">171</span> -through the axis, the weight of the instrument will have -no other effect than to increase the pressure on the axis -by its own amount. But if the centre of gravity be on -the same side of the axis with the weight, as at G, it -will oppose the effect of the power, a certain part of -which must therefore be allowed to support it. To -ascertain what part of the power is thus expended, it is -to be considered that the moment of the weight of the -lever collected at G, is found by multiplying that weight -by the distance G F. The moment of that part of the -power which supports this must be equal to it; therefore, -it is only necessary to find how much of the power -multiplied by P F will be equal to the weight of the -lever multiplied by G F. This is a question in common -arithmetic.</p> - -<p>If the centre of gravity of the lever be at a different -side of the axis from the weight, as at <span class="ilb">G′</span>, the weight of -the instrument will co-operate with the power in sustaining -the weight W. To determine what portion of -the weight W is thus sustained by the weight of the -lever, it is only necessary to find how much of W, multiplied -by the distance W F, is equal to the weight of the -lever multiplied by <span class="ilb">G′</span> F.</p> - -<p>In these cases the pressure on the fulcrum, as already -estimated, will always be increased by the weight of the -lever.</p> - -<p id="p241">(241.) The sense in which a small power is said to -sustain a great weight, and the manner of accomplishing -this, being explained, we shall now consider how the -power is applied in moving the weight. Let P W, -<i><a href="#i_p176a">fig. 81.</a></i>, be the places of the power and weight, and F -that of the fulcrum, and let the power be depressed to -<span class="ilb">P′</span> while the weight is raised to <span class="ilb">W′</span>. The space P <span class="ilb">P′</span> -evidently bears the same proportion to W <span class="ilb">W′</span>, as the -arm P F to W F. Thus if P F be ten times W F, P <span class="ilb">P′</span> -will be ten times W <span class="ilb">W′</span>. A power of one pound at P -being moved from P to <span class="ilb">P′</span>, will carry a weight of ten -pounds from W to <span class="ilb">W′</span>. But in this case it ought not -to be said, that a lesser weight moves a greater, for it is<span class="pagenum" id="Page_172">172</span> -not difficult to show, that the total expenditure of force -in the motion of one pound from P to <span class="ilb">P′</span> is exactly the -same as in the motion of ten pounds from W to <span class="ilb">W′</span>. If -the space P <span class="ilb">P′</span> be ten inches, the space W <span class="ilb">W′</span> will be -one inch. A weight of one pound is therefore moved -through ten successive inches, and in each inch the -force expended is that which would be sufficient to move -one pound through one inch. The total expenditure of -force from P to <span class="ilb">P′</span> is ten times the force necessary to -move one pound through one inch, or what is the same, -it is that which would be necessary to move ten pounds -through one inch. But this is exactly what is accomplished -by the opposite end W of the lever; for the -weight W is ten pounds, and the space W <span class="ilb">W′</span> is one -inch.</p> - -<p>If the weight W of ten pounds could be conveniently -divided into ten equal parts of one pound each, each -part might be separately raised through one inch, without -the intervention of the lever or any other machine. -In this case, the same quantity of power would be expended, -and expended in the same manner as in the case -just mentioned.</p> - -<p>It is evident, therefore, that when a machine is applied -to raise a weight or to overcome resistance, as much force -must be really used as if the power were immediately -applied to the weight or resistance. All that is accomplished -by the machine is to enable the power to do -that by a succession of distinct efforts which should be -otherwise performed by a single effort. These observations -will be found to be applicable to all machines -whatever.</p> - -<p id="p242">(242.) Weighing machines of almost every kind, -whether used for commercial or philosophical purposes, -are varieties of the lever. The common balance, which, -of all weighing machines, is the most perfect and best -adapted for ordinary use, whether in commerce or experimental -philosophy, is a lever with equal arms. In the -steel-yard one weight serves as a counterpoise and measure -of others of different amount, by receiving a leverage<span class="pagenum" id="Page_173">173</span> -variable according to the varying amount of the weight -against which it acts. A detailed account of such instruments -will be found in Chapter <a href="#CHAP_XXI">XXI</a>.</p> - -<p id="p243">(243.) We have hitherto considered the power and -weight as acting on the lever, in directions perpendicular -to its length and parallel to each other. This does not -always happen. Let A B, <i><a href="#i_p176a">fig. 83.</a></i>, be a lever whose fulcrum -is F, and let A R be the direction of the power, and B S -the direction of the weight. If the lines R A and S B be -continued, and perpendiculars F C and F D drawn from -the fulcrum to those lines, the moment of the power will -be found by multiplying the power by the line F C, and -the moment of the weight by multiplying the weight -by F D. If these moments be equal, the power will -sustain the weight in equilibrium. (<a href="#p185">185)</a>.</p> - -<p>It is evident, that the same reasoning will be applicable -when the arms of the lever are not in the same -direction. These arms may be of any figure or shape, -and may be placed relatively to each other in any -position.</p> - -<p id="p244">(244.) In the rectangular lever the arms are perpendicular -to each other, and the fulcrum F, <i><a href="#i_p182a">fig. 84.</a></i>, is at -the right angle. The moment of the power, in this case, -is P multiplied by A F, and that of the weight W -multiplied by B F. When the instrument is in equilibrium -these moments must be equal.</p> - -<p>When the hammer is used for drawing a nail, it is a -lever of this kind: the claw of the hammer is the shorter -arm; the resistance of the nail is the weight; and the -hand applied to the handle the power.</p> - -<p id="p245">(245.) When a beam rests on two props A B, <i><a href="#i_p182a">fig. 85.</a></i>, -and supports, at some intermediate place C, a weight W, -this weight is distributed between the props in a manner -which may be determined by the principles already -explained. If the pressure on the prop B be considered -as a power sustaining the weight W, by means of the -lever of the second kind B A, then this power multiplied -by B A must be equal to the weight multiplied by C A. -Hence the pressure on B will be the same fraction of the<span class="pagenum" id="Page_174">174</span> -weight as the part A C is of A B. In the same manner -it may be proved, that the pressure on A is the same -fraction of the weight as B C is of B A. Thus, if A C -be one third, and therefore B C two thirds of B A, the -pressure on B will be one third of the weight, and the -pressure on A two thirds of the weight.</p> - -<p>It follows from this reasoning, that if the weight be -in the middle, equally distant from B and A, each prop -will sustain half the weight. The effect of the weight -of the beam itself may be determined by considering it -to be collected at its centre of gravity. If this point, -therefore, be equally distant from the props, the weight -of the beam will be equally distributed between them.</p> - -<p>According to these principles, the manner in which a -load borne on poles between two bearers is distributed -between them may be ascertained. As the efforts of the -bearers and the direction of the weight are always -parallel; the position of the poles relatively to the -horizon makes no difference in the distribution of the -weights between the bearers. Whether they ascend or -descend, or move on a level plane, the weight will be -similarly shared between them.</p> - -<p>If the beam extend beyond the prop, as in <i><a href="#i_p182a">fig. 86.</a></i>, -and the weight be suspended at a point not placed between -them, the props must be applied at different sides -of the beam. The pressures which they sustain may be -calculated in the same manner as in the former case. -The pressure of the prop B may be considered as a -power sustaining the weight W by means of the lever -B C. Hence, the pressure of B, multiplied by B A, -must be equal to the weight W multiplied by A C. -Therefore, the pressure on B bears the same proportion -to the weight as A C does to A B. In the same manner, -considering B as a fulcrum, and the pressure of the -prop A as the power, it may be proved that the pressure -of A bears the same proportion to the weight as the line -B C does to A B. It therefore appears, that the pressure -on the prop A is greater than the weight.</p> - -<p id="p246">(246.) When great power is required, and it is inconve<span class="pagenum" id="Page_175">175</span>nient -to construct a long lever, a combination of levers -may be used. In <i><a href="#i_p182a">fig. 87.</a></i> such a system of levers is -represented, consisting of three levers of the first kind. -The manner in which the effect of the power is transmitted -to the weight may be investigated by considering -the effect of each lever successively. The power at P -produces an upward force at <span class="ilb">P′</span>, which bears to P the -same proportion as <span class="ilb">P′</span> F to P F. Therefore, the effect -at <span class="ilb">P′</span> is as many times the power as the line P F is of -<span class="ilb">P′</span> F. Thus, if P F be ten times <span class="ilb">P′</span> F, the upward force -at <span class="ilb">P′</span> is ten times the power. The arm <span class="ilb">P′</span> <span class="ilb">F′</span> of the -second lever is pressed upwards by a force equal to ten -times the power at P. In the same manner this may be -shown to produce an effect at <span class="ilb">P″</span> as many times greater -than <span class="ilb">P′</span> as <span class="ilb">P′</span> <span class="ilb">F′</span> is greater than <span class="ilb">P″</span> <span class="ilb">F′</span>. Thus, if <span class="ilb">P′</span> <span class="ilb">F′</span> be -twelve times <span class="ilb">P″</span> <span class="ilb">F′</span>, the effect at <span class="ilb">P″</span> will be twelve times -that of <span class="ilb">P′</span>. But this last was ten times the power, and -therefore the <span class="ilb">P″</span> will be one hundred and twenty times -the power. In the same manner it may be shown that -the weight is as many times greater than the effect at <span class="ilb">P″</span> -as <span class="ilb">P″</span> <span class="ilb">F″</span> is greater than W <span class="ilb">F″</span>. If <span class="ilb">P″</span> <span class="ilb">F″</span> be five times -W <span class="ilb">F″</span>, the weight will be five times the effect at <span class="ilb">P″</span>. But -this effect is one hundred and twenty times the power, -and therefore the weight would be six hundred times -the power.</p> - -<p>In the same manner the effect of any compound -system of levers may be ascertained by taking the proportion -of the weight to the power in each lever separately, -and multiplying these numbers together. In the example -given, these proportions are 10, 12, and 5, which -multiplied together give 600. In <i><a href="#i_p182a">fig. 87.</a></i> the levers -composing the system are of the first kind; but the -principles of the calculation will not be altered if they -be of the second or third kind, or some of one kind and -some of another.</p> - -<p id="p247">(247.) That number which expresses the proportion -of the weight to the equilibrating power in any machine, -we shall call the <i>power of the machine</i>. Thus, if, in a -lever, a power of one pound support a weight of ten<span class="pagenum" id="Page_176">176</span> -pounds, the power of the machine is <i>ten</i>. If a power -of 2lbs. support a weight of 11lbs., the power of the -machine is <span class="nowrap">5<span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>, 2 being contained in 11 <span class="nowrap">5<span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> times.</p> - -<p id="p248">(248.) As the distances of the power and weight -from the fulcrum of a lever may be varied at pleasure, -and any assigned proportion given to them, a lever may -always be conceived having a power equal to that of any -given machine. Such a lever may be called, in relation -to that machine, the <i>equivalent lever</i>.</p> - -<p>As every complex machine consists of a number of -simple machines acting one upon another, and as each -simple machine may be represented by an equivalent -lever, the complex machine will be represented by a -compound system of equivalent levers. From what has -been proved in (<a href="#p246">246</a>.), it therefore follows that the power -of a complex machine may be calculated by multiplying -together the powers of the several simple machines of -which it is composed.</p> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<h2 class="nobreak" id="CHAP_XIV">CHAP. XIV.<br /> - -<span class="title">OF WHEEL-WORK.</span></h2> -</div> - - -<p id="p249">(249.) <span class="smcap">When</span> a lever is applied to raise a weight, or -overcome a resistance, the space through which it acts at -any one time is small, and the work must be accomplished -by a succession of short and intermitting efforts. -In <i><a href="#i_p176a">fig. 81.</a></i>, after the weight has been raised from W to -<span class="ilb">W′</span>, the lever must again return to its first position, to -repeat the action. During this return the motion of the -weight is suspended, and it will fall downwards unless -some provision be made to sustain it. The common lever is, -therefore, only used in cases where weights are required -to be raised through small spaces, and under these -circumstances its great simplicity strongly recommends -it. But where a continuous motion is to be produced, as -in raising ore from the mine, or in weighing the anchor of -a vessel, some contrivance must be adopted to remove<span class="pagenum" id="Page_177">177</span> -the intermitting action of the lever, and render it continual. -The various forms given to the lever, with a -view to accomplish this, are generally denominated the -<i>wheel and axle</i>.</p> - -<div class="figcenter" id="i_p176a" style="max-width: 31.25em;"> - <img src="images/i_p176a.jpg" alt="" /> - <div class="caption"> -<p class="tar"><i>H. Adlard, sc.</i></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<p>In <i><a href="#i_p182a">fig. 88.</a></i>, A B is a horizontal axle, which rests in -pivots at its extremities, or is supported in gudgeons, and -capable of revolving. Round this axis a rope is coiled, -which sustains the weight W. On the same axis a -wheel C is fixed, round which a rope is coiled in a contrary -direction, to which is appended the power P. The -moment of the power is found by multiplying it by the -radius of a wheel, and the moment of the weight, by multiplying -it by the radius of its axle. If these moments -be equal (<a href="#p185">185</a>.), the machine will be in equilibrium. -Whence it appears that the power of the machine (<a href="#p247">247</a>.) -is expressed by the proportion which the radius of the -wheel bears to the radius of the axle; or, what is the same, -of the diameter of the wheel to the diameter of the axle. -This principle is applicable to the wheel and axle in -every variety of form under which it can be presented.</p> - -<p id="p250">(250.) It is evident that as the power descends continually, -and the rope is uncoiled from the wheel, the -weight will be raised continually, the rope by which it is -suspended being at the same time coiled upon the axle.</p> - -<p>When the machine is in equilibrium, the forces of -both the weight and power are sustained by the axle, and -distributed between its props, in the manner explained -in (<a href="#p245">245</a>.)</p> - -<p>When the machine is applied to raise a weight, the -velocity with which the power moves is as many times -greater than that with which the weight rises, as the -weight itself is greater than the power. This is a principle -which has already been noticed, and which is common -to all machines whatsoever. It may hence be -proved, that in the elevation of the weight a quantity of -power is expended equal to that which would be necessary -to elevate the weight if the power were immediately -applied to it, without the intervention of any machine. -This has been explained in the case of the lever in (<a href="#p241">241</a>.),<span class="pagenum" id="Page_178">178</span> -and may be explained in the present instance in nearly -the same words.</p> - -<p>In one revolution of the machine the length of rope -uncoiled from the wheel is equal to the circumference of -the wheel, and through this space the power must therefore -move. At the same time the length of rope coiled -upon the axle is equal to the circumference of the axle, -and through this space the weight must be raised. The -spaces, therefore, through which the power and weight -move in the same time, are in the proportion of the circumferences -of the wheel and axle; but these circumferences -are in the same proportion as their diameters. Therefore -the velocity of the power will bear to the velocity of -the weight the same proportion as the diameter of the -wheel bears to the diameter of the axle, or, what is the -same, as the weight bears to the power (<a href="#p249">249)</a>.</p> - -<p id="p251">(251.) We have here omitted the consideration of -the thickness of the rope. When this is considered, -the force must be conceived as acting in the direction of -the centre of the rope, and therefore the thickness of the -rope which supports the power ought to be added to the -diameter of the wheel, and the thickness of the rope -which supports the weight to the diameter of the axle. -It is the more necessary to attend to this circumstance, -as the strength of the rope necessary to support the -weight causes its thickness to bear a considerable proportion -to the diameter of the axle; while the rope which -sustains the power not requiring the same strength, and -being applied to a larger circle, bears a very inconsiderable -proportion to its diameter.</p> - -<p id="p252">(252.) In numerous forms of the wheel and axle, -the weight or resistance is applied by a rope coiled -upon the axle; but the manner in which the power is -applied is very various, and not often by means of a -rope. The circumference of a wheel sometimes carries -projecting pins, as represented in <i><a href="#i_p182a">fig. 88.</a></i>, to which the -hand is applied to turn the machine. An instance of -this occurs in the wheel used in the steerage of a vessel.</p> - -<p>In the common <i>windlass</i>, the power is applied by<span class="pagenum" id="Page_179">179</span> -means of a <i>winch</i>, which is a rectangular lever, as represented -in <i><a href="#i_p182a">fig. 89.</a></i> The arm B C of the winch represents -the radius of the wheel, and the power is applied to C D -at right angles to B C.</p> - -<p>In some cases no wheel is attached to the axle; but -it is pierced with holes directed towards its centre, in -which long levers are incessantly inserted, and a continuous -action produced by several men working at the -same time; so that while some are transferring the levers -from hole to hole, others are working the windlass.</p> - -<p>The axle is sometimes placed in a vertical position, -the wheel or levers being moved horizontally. The <i>capstan</i> -is an example of this: a vertical axis is fixed in the -deck of the ship; the circumference is pierced with holes -presented towards its centre. These holes receive long -levers, as represented in <i><a href="#i_p182a">fig. 90.</a></i> The men who work -the capstan walk continually round the axle, pressing -forward the levers near their extremities.</p> - -<p>In some cases the wheel is turned by the weight of -animals placed at its circumference, who move forward -as fast as the wheel descends, so as to maintain -their position continually at the extremity of the horizontal -diameter. The <i>treadmill</i>, <i><a href="#i_p182a">fig. 91.</a></i>, and certain -<i>cranes</i>, such as <i><a href="#i_p182a">fig. 92.</a></i>, are examples of this.</p> - -<p>In water-wheels, the power is the weight of water -contained in buckets at the circumference, as in <i><a href="#i_p182a">fig. 93.</a></i>, -which is called an over-shot wheel: and sometimes by -the impulse of water against float-boards at the circumference, -as in the under-shot wheel, <i><a href="#i_p188a">fig. 94.</a></i> Both these -principles act in the breast-wheel, <i><a href="#i_p188a">fig. 95.</a></i></p> - -<p>In the paddle-wheel of a steam-boat, the power is the -resistance which the water offers to the motion of the -paddle-boards.</p> - -<p>In windmills, the power is the force of the wind acting -on various parts of the arms, and may be considered -as different powers simultaneously acting on different -wheels having the same axle.</p> - -<p id="p253">(253.) In most cases in which the wheel and axle is -used, the action of the power is liable to occasional sus<span class="pagenum" id="Page_180">180</span>pension -or intermission, in which case some contrivance -is necessary to prevent the recoil of the weight. A -ratchet wheel R, <i><a href="#i_p182a">fig. 88.</a></i>, is provided for this purpose, -which is a contrivance which permits the wheel to turn -in one direction; but a catch which falls between the -teeth of a fixed wheel prevents its motion in the other -direction. The effect of the power or weight is sometimes -transmitted to the wheel or axle by means of a -straight bar, on the edge of which teeth are raised, which -engage themselves in corresponding teeth on the wheel -or axle. Such a bar is called a rack; and an instance of -its use may be observed in the manner of working the -pistons of an air-pump.</p> - -<p id="p254">(254.) The power of the wheel and axle being expressed -by the number of times the diameter of the axle -is contained in that of the wheel, there are obviously only -two ways by which this power may be increased; viz. -either by increasing the diameter of the wheel, or diminishing -that of the axle. In cases where great power -is required, each of these methods is attended with practical -inconvenience and difficulty. If the diameter of -the wheel be considerably enlarged, the machine will -become unwieldy, and the power will work through an -unmanageable space. If, on the other hand, the power -of the machine be increased by reducing the thickness of -the axle, the strength of the axle will become insufficient -for the support of that weight, the magnitude of which -had rendered the increase of the power of the machine -necessary. To combine the requisite strength with moderate -dimensions and great mechanical power is, therefore, -impracticable in the ordinary form of the wheel and -axle. This has, however, been accomplished by giving -different thicknesses to different parts of the axle, and -carrying a rope, which is coiled on the thinner part, -through a wheel attached to the weight, and coiling it -in the opposite direction on the thicker part, as in <i><a href="#i_p188a">fig. 96.</a></i> -To investigate the proportion of the power to the weight -in this case, let <i><a href="#i_p188a">fig. 97.</a></i> represent a section of the apparatus -at right angles to the axis. The weight is equally<span class="pagenum" id="Page_181">181</span> -suspended by the two parts of the rope, S and <span class="ilb">S′</span>, and -therefore each part is stretched by a force equal to half -the weight. The moment of the force, which stretches -the rope S, is half the weight multiplied by the radius -of the thinner part of the axle. This force being at the -same side of the centre with the power, co-operates with -it in supporting the force which stretches <span class="ilb">S′</span>, and which -acts at the other side of the centre. By the principle -established in (<a href="#p185">185</a>.), the moments of P and S must be -equal to that of <span class="ilb">S′</span>; and therefore if P be multiplied by -the radius of the wheel, and added to half the weight -multiplied by the radius of the thinner part of the axle, -we must obtain a sum equal to half the weight multiplied -by the radius of the thicker part of the axle. -Hence it is easy to perceive, that the power multiplied -by the radius of the wheel is equal to half the -weight multiplied by the difference of the radii of the -thicker and thinner parts of the axle; or, what is the -same, the power multiplied by the diameter of the wheel, -is equal to the weight multiplied by half the difference of -the diameters of the thinner and thicker parts of the axle.</p> - -<p>A wheel and axle constructed in this manner is equivalent -to an ordinary one, in which the wheel has -the same diameter, and whose axle has a diameter -equal to half the difference of the diameters of the -thicker and thinner parts. The power of the machine -is expressed by the proportion which the diameter of the -wheel bears to half the difference of these diameters; -and therefore this power, when the diameter of the wheel -is given, does not, as in the ordinary wheel and axle, -depend on the smallness of the axle, but on the smallness -of the difference of the thinner and thicker parts of it. -The axle may, therefore, be constructed of such a thickness -as to give it all the requisite strength, and yet the -difference of the diameters of its different parts may be -so small as to give it all the requisite power.</p> - -<p id="p255">(255.) It often happens that a varying weight is to -be raised, or resistance overcome by a uniform power. -If, in such a case, the weight be raised by a rope coiled<span class="pagenum" id="Page_182">182</span> -upon a uniform axle, the action of the power would not -be uniform, but would vary with the weight. It is, -however, in most cases desirable or necessary that the -weight or resistance, even though it vary, shall be moved -uniformly. This will be accomplished if by any means -the leverage of the weight is made to increase in the -same proportion as the weight diminishes, and to diminish -in the same proportion as the weight increases: for in -that case the moment of the weight will never vary, -whatever it gains by the increase of weight being lost -by the diminished leverage, and whatever it loses by the -diminished weight being gained by the increased leverage. -An axle, the surface of which is curved in such a -manner, that the thickness on which the rope is coiled -continually increased or diminishes in the same proportion -as the weight or resistance diminishes or increases, -will produce this effect.</p> - -<p>It is obvious that all that has been said respecting a variable -weight or resistance, is also applicable to a variable -power, which, therefore, may, by the same means, be made -to produce a uniform effect. An instance of this occurs -in a watch, which is moved by a spiral spring. When -the watch has been wound up, this spring acts with its -greatest intensity, and as the watch goes down, the elastic -force of the spring gradually loses its energy. This -spring is connected by a chain with an axle of varying -thickness, called a <i>fusee</i>. When the spring is at its -greatest intensity, the chain acts upon the thinnest part -of the fusee, and as it is uncoiled it acts upon a part of -the fusee which is continually increasing in thickness, -the spring at the same time losing its elastic power in -exactly the same proportion. A representation of the -fusee, and the cylindrical box which contains the spring, -is given in <i><a href="#i_p188a">fig. 98.</a></i>, and of the spring itself in <i><a href="#i_p188a">fig. 99.</a></i></p> - -<p id="p256">(256.) When great power is required, wheels and -axles may be combined in a manner analogous to a compound -system of levers, explained in (<a href="#p246">246</a>.) In this -case the power acts on the circumference of the first -wheel, and its effect is transmitted to the circumference<span class="pagenum" id="Page_183">183</span> -of the first axle. That circumference is placed in connection -with the circumference of the second wheel, and -the effect is thereby transmitted to the circumference of -the second axle, and so on. It is obvious from what was -proved in (<a href="#p248">248</a>.), that the power of such a combination -of wheels and axles will be found by multiplying together -the powers of the several wheels of which it is composed. -It is sometimes convenient to compute this power by -numbers expressing the proportions of the circumferences -or diameters of the several wheels, to the circumferences -or diameters of the several axles respectively. This -computation is made by first multiplying the numbers -together which express the circumferences or diameters -of the wheels, and then multiplying together the numbers -which express the circumferences or diameters of -the several axles. The proportion of the two products -will express the power of the machine. Thus, -if the circumferences or diameters be as the numbers -10, 14, and 15, their product will be 2100; and if the -circumferences or diameters of the axles be expressed by -the numbers 3, 4, and 5, their product will be 60, and -the power of the machine will be expressed by the proportion -of 2100 and 60, or 35 to 1.</p> - -<div class="figcenter" id="i_p182a" style="max-width: 31.25em;"> - <img src="images/i_p182a.jpg" alt="" /> - <div class="caption"> -<p class="tar"><i>H. Adlard, sc.</i></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<p id="p257">(257.) The manner in which the circumferences of -the axles act upon the circumferences of the wheels in -compound wheel-work is various. Sometimes a strap -or cord is applied to a groove in the circumference of the -axle, and carried round a similar groove in the circumference -of the succeeding wheel. The friction of this -cord or strap with the groove is sufficient to prevent its -sliding and to communicate the force from the axle to -the wheel, or <i>vice versa</i>. This method of connecting -wheel-work is represented in <i><a href="#i_p188a">fig. 100.</a></i></p> - -<p>Numerous examples of wheels and axles driven by -straps or cords occur in machinery applied to almost -every department of the arts and manufactures. In the -turning lathe, the wheel worked by the treddle is connected -with the mandrel by a catgut cord passing through -grooves in the wheel and axle. In all great factories,<span class="pagenum" id="Page_184">184</span> -revolving shafts are carried along the apartments, on which, -at certain intervals, straps are attached passing round -their circumferences and carried round the wheels which -give motion to the several machines. If the wheels, connected -by straps or cords, are required to revolve in the -same direction, these cords are arranged as in <i><a href="#i_p188a">fig. 100.</a></i>; -but if they are required to revolve in contrary directions, -they are applied as in <i><a href="#i_p188a">fig. 101.</a></i></p> - -<p>One of the chief advantages of the method of transmitting -motion between wheels and axles by straps or cords, -is that the wheel and axle may be placed at any distance -from each other which may be found convenient, and -may be made to turn either in the same or contrary -directions.</p> - -<p id="p258">(258.) When the circumference of the wheel acts -immediately on the circumference of the succeeding axle, -some means must necessarily be adopted to prevent the -wheel from moving in contact with the axle without compelling -the latter to turn. If the surfaces of both were -perfectly smooth, so that all friction were removed, it is -obvious that either would slide over the surface of the -other, without communicating motion to it. But, on -the other hand, if there were any asperities, however -small, upon these surfaces, they would become mutually -inserted among each other, and neither the wheel nor -axle could move without causing the asperities with -which its edge is studded to encounter those asperities -which project from the surface of the other; and thus, -until these projections should be broken off, both wheel -and axle must be moved at the same time. It is on this -account that if the surfaces of the wheels and axles are -by any means rendered rough, and pressed together with -sufficient force, the motion of either will turn the other, -provided the load or resistance be not greater than the -force necessary to break off these small projections which -produce the friction.</p> - -<p>In cases where great power is not required, motion -is communicated in this way through a train of wheel-work, -by rendering the surface of the wheel and axle<span class="pagenum" id="Page_185">185</span> -rough, either by facing them with buff leather, or with -wood cut across the grain. This method is sometimes -used in spinning machinery, where one large buffed -wheel, placed in a horizontal position, revolves in contact -with several small buffed rollers, each roller communicating -motion to a spindle. The position of the -wheel W, and the rollers R R, &c., are represented in -<i><a href="#i_p188a">fig. 102.</a></i> Each roller can be thrown out of contact with -the wheel, and restored to it at pleasure.</p> - -<p>The communication of motion between wheels and -axles by friction has the advantage of great smoothness -and evenness, and of proceeding with little noise; but this -method can only be used in cases where the resistance -is not very considerable, and therefore is seldom adopted -in works on a large scale. Dr. Gregory mentions an instance -of a saw mill at Southampton, where the wheels -act upon each other by the contact of the end grain of -wood. The machinery makes very little noise, and -wears very well, having been used not less than 20 -years.</p> - -<p id="p259">(259.) The most usual method of transmitting motion -through a train of wheel-work is by the formation -of teeth upon their circumferences, so that these indentures -of each wheel fall between the corresponding -ones of that in which it works, and ensure the action -so long as the strain is not so great as to fracture the -tooth.</p> - -<p>In the formation of teeth very minute attention must -be given to their figure, in order that the motion may -be communicated from wheel to wheel with smoothness -and uniformity. This can only be accomplished by -shaping the teeth according to curves of a peculiar kind, -which mathematicians have invented, and assigned rules -for drawing. The ill consequences of neglecting this -will be very apparent, by considering the nature of the -action which would be produced if the teeth were formed -of square projecting pins, as in <i><a href="#i_p188a">fig. 103.</a></i> When the -tooth A comes into contact with B, it acts obliquely -upon it, and, as it moves, the corner of B slides upon the<span class="pagenum" id="Page_186">186</span> -plane surface of A in such a manner as to produce much -friction, and to grind away the side of A and the end of -B. As they approach the position C D, they sustain a -jolt the moment their surfaces come into full contact; -and after passing the position of C D, the same scraping -and grinding effect is produced in the opposite direction, -until by the revolution of the wheels the teeth become -disengaged. These effects are avoided by giving to the -teeth the curved forms represented in <i><a href="#i_p188a">fig. 104.</a></i> By such -means the surfaces of the teeth roll upon each other with -very inconsiderable friction, and the direction in which -the pressure is excited is always that of a line M N, -touching the two wheels, and at right angles to the -radii. Thus the pressure being always the same, and -acting with the same leverage, produces a uniform -effect.</p> - -<p id="p260">(260.) When wheels work together, their teeth must -necessarily be of the same size, and therefore the proportion -of their circumferences may always be estimated by -the number of teeth which they carry. Hence it follows, -that in computing the power of compound wheel-work, -the number of teeth may always be used to express the -circumferences respectively, or the diameters which are -proportional to these circumferences. When teeth are -raised upon an axle, it is generally called a <i>pinion</i>, and -in that case the teeth are called <i>leaves</i>. The rule for -computing the train of wheel-work given in (<a href="#p256">256</a>.) will -be expressed as follows: when the wheel and axle carry -teeth, multiply together the number of teeth in each of -the wheels, and next the number of leaves in each of -the pinions; the proportion of the two products will -express the power of the machine. If some of the -wheels and axles carry teeth, and others not, this computation -may be made by using for those circumferences -which do not bear teeth the number of teeth which -would fill them. <i><a href="#i_p188a">Fig. 105.</a></i> represents a train of three -wheels and pinions. The wheel F which bears the -power, and the axle which bears the weight, have no -teeth; but it is easy to find the number of teeth which -they would carry.</p> - -<p><span class="pagenum" id="Page_187">187</span></p> - -<p id="p261">(261.) It is evident that each pinion revolves much -more frequently in a given time than the wheel which -it drives. Thus, if the pinion C be furnished with -ten teeth, and the wheel E, which it drives, have sixty -teeth, the pinion C must turn six times, in order to turn -the wheel E once round. The velocities of revolution -of every wheel and pinion which work in one another -will therefore have the same proportion as their number -of teeth taken in a reverse order, and by this means the -relative velocity of wheels and pinions may be determined -according to any proposed rate.</p> - -<p>Wheel-work, like all other machinery, is used to transmit -and modify force in every department of the arts -and manufactures; but it is also used in cases where motion -alone, and not force, is the object to be attained. -The most remarkable example of this occurs in watch -and clock-work, where the object is merely to produce -uniform motions of rotation, having certain proportions, -and without any regard to the elevation of weights, or -the overcoming of resistances.</p> - -<p id="p262">(262.) A <i>crane</i> is an example of combination of -wheel-work used for the purpose of raising or lowering -great weights. <i><a href="#i_p196a">Fig. 106.</a></i> represents a machine of this -kind. A B is a strong vertical beam, resting on a pivot, -and secured in its position by beams in the floor. It -is capable, however, of turning on its axis, being confined -between rollers attached to the beams and fixed -in the floor. C D is a projecting arm called a <i>gib</i>, -formed of beams which are mortised into A B. The -wheel-work is mounted in two cast-iron crosses, bolted -on each side of the beams, one of which appears at -E F G H. The winch at which the power is applied is -at I. This carries a pinion immediately behind H. -This pinion works in a wheel K, which carries another -pinion upon its axle. This last pinion works in a larger -wheel L, which carries upon its axis a barrel M, on -which a chain or rope is coiled. The chain passes over -a pulley D at the top of the gib. At the end of the -chain a hook O is attached, to support the weight W. -During the elevation of the weight it is convenient that<span class="pagenum" id="Page_188">188</span> -its recoil should be hindered in case of any occasional -suspension of the power. This is accomplished by a -ratchet wheel attached to the barrel M, as explained in -(<a href="#p253">253</a>.); but when the weight W is to be lowered, the -catch must be removed from this ratchet wheel. In this -case the too rapid descent of the weight is in some cases -checked by pressure excited on some part of the wheel-work, -so as to produce sufficient friction to retard the -descent in any required degree, or even to suspend it, if -necessary. The vertical beam at B resting on a pivot, -and being fixed between rollers, allows the gib to be -turned round in any direction; so that a weight raised -from one side of the crane may be carried round, and -deposited on another side, at any distance within the -range of the gib. Thus, if a crane be placed upon a -wharf near a vessel, weights may be raised, and when -elevated, the gib may be turned round so as to let them -descend into the hold.</p> - -<p>The power of this machine may be computed upon -the principles already explained. The magnitude of the -circle, in which the power at I moves, may be determined -by the radius of the winch, and therefore the -number of teeth which a wheel of that size would carry -may be found. In like manner we may determine the -number of leaves in a pinion whose magnitude would be -equal to the barrel M. Let the first number be multiplied -by the number of teeth in the wheel K, and that -product by the number of teeth in the wheel L. Next let -the number of leaves in the pinion H be multiplied by -the number of leaves in the pinion attached to the axle -of the wheel K, and let that product be multiplied by -the number of leaves in a pinion, whose diameter is -equal to that of the barrel M. These two products will -express the power of the machine.</p> - -<p id="p263">(263.) Toothed wheels are of three kinds, distinguished -by the position which the teeth bear with respect -to the axis of the wheel. When they are raised upon -the edge of the wheel as in <i><a href="#i_p188a">fig. 105.</a></i>, they are called <i>spur -wheels</i>, or <i>spur gear</i>. When they are raised parallel to the -axis, as in <i><a href="#i_p196a">fig. 107.</a></i>, it is called a <i>crown wheel</i>. When<span class="pagenum" id="Page_189">189</span> -the teeth are raised on a surface inclined to the plane of -the wheel, as in <i><a href="#i_p196a">fig. 108.</a></i>, they are called <i>bevelled wheels</i>.</p> - -<div class="figcenter" id="i_p188a" style="max-width: 31.25em;"> - <img src="images/i_p188a.jpg" alt="" /> - <div class="caption"> -<p class="tar"><i>H. Adlard, sc.</i></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<p>If a motion round one axis is to be communicated to -another axis parallel to it, spur gear is generally used. -Thus, in <i><a href="#i_p188a">fig. 105.</a></i>, the three axes are parallel to each -other. If a motion round one axis is to be communicated -to another at right angles to it, a crown wheel, -working in a spur pinion, as in <i><a href="#i_p196a">fig. 107.</a></i>, will serve. Or -the same object may be obtained by two bevelled wheels, -as in <i><a href="#i_p196a">fig. 108.</a></i></p> - -<p>If a motion round one axis is required to be communicated -to another inclined to it at any proposed angle, -two bevelled wheels can always be used. In <i><a href="#i_p196a">fig. 109.</a></i> let -A B and A C be the two axles; two bevelled wheels, -such as D E and E F, on these axles will transmit the -motion or rotation from one to the other, and the relative -velocity may, as usual, be regulated by the proportional -magnitude of the wheels.</p> - -<p id="p264">(264.) In order to equalise the wear of the teeth of -a wheel and pinion, which work in one another, it is -necessary that every leaf of the pinion should work in -succession through every tooth of the wheel, and not -continually act upon the same set of teeth. If the teeth -could be accurately shaped according to mathematical -principles, and the materials of which they are formed -be perfectly uniform, this precaution would be less necessary; -but as slight inequalities, both of material and -form, must necessarily exist, the effects of these should -be as far as possible equalised, by distributing them -through every part of the wheel. For this purpose it is -usual, especially in mill-work, where considerable force -is used, so to regulate the proportion of the number of -teeth in the wheel and pinion, that the same leaf of the -pinion shall not be engaged twice with any one tooth of -the wheel, until after the action of a number of teeth, -expressed by the product of the number of teeth in the -wheel and pinion. Let us suppose that the pinion contains -ten leaves, which we shall denominate by the numbers -1, 2, 3, &c., and that the wheel contains 60 teeth<span class="pagenum" id="Page_190">190</span> -similarly denominated. At the commencement of the -motion suppose the leaf 1 of the pinion engages the -tooth 1 of the wheel; then after one revolution the leaf -1 of the pinion will engage the tooth 11 of the wheel, -and after two revolutions the leaf 1 of the pinion will -engage the tooth 21 of the wheel; and in like manner, -after 3, 4, and 5 revolutions of the pinion, the leaf 1 -will engage successively the teeth 31, 41, and 51 of the -wheel. After the sixth revolution, the leaf 1 of the -pinion will again engage the tooth 1 of the wheel. Thus -it is evident, that in the case here supposed the leaf 1 of -the pinion will continually be engaged with the teeth 1, -11, 21, 31, 41, and 51 of the wheel, and no others. -The like may be said of every leaf of the pinion. Thus -the leaf 2 of the pinion will be successively engaged -with the teeth 2, 12, 22, 32, 42, and 52 of the wheel, -and no others. Any accidental inequalities of these -teeth will therefore continually act upon each other, -until the circumference of the wheel be divided into -parts of ten teeth each, unequally worn. This effect -would be avoided by giving either the wheel or pinion -one tooth more or one tooth less. Thus, suppose the -wheel, instead of having sixty teeth, had sixty-one, then -after six revolutions of the pinion the leaf 1 of the pinion -would be engaged with the tooth 61 of the wheel; and -after one revolution of the wheel, the leaf 2 of the pinion -would be engaged with the tooth 1 of the wheel. Thus, -during the first revolution of the wheel the leaf 1 of the -pinion would be successively engaged with the teeth 1, -11, 21, 31, 41, 51, and 61 of the wheel: at the commencement -of the second revolution of the wheel the -leaf 2 of the pinion would be engaged with the tooth 1 -of the wheel; and during the second revolution of the -wheel the leaf 1 of the pinion would be successively -engaged with the teeth 10, 20, 30, 40, 50, and 60 of -the wheel. In the same manner it may be shown, that -in the third revolution of the wheel the leaf 1 of the -pinion would be successively engaged with the teeth 9, -19, 29, 39, 49, and 59 of the wheel: during the fourth<span class="pagenum" id="Page_191">191</span> -revolution of the wheel the leaf 1 of the pinion would -be successively engaged with the teeth 8, 18, 28, 38, -48, and 58 of the wheel. By continuing this reasoning -it will appear, that during the tenth revolution of the -wheel the leaf 1 of the pinion will be engaged successively -with the teeth 2, 12, 22, 32, 42, and 52 of the -wheel. At the commencement of the eleventh revolution -of the wheel the leaf 1 of the pinion will be engaged -with the tooth 1 of the wheel, as at the beginning -of the motion. It is evident, therefore, that during the -first ten revolutions of the wheel each leaf of the pinion -has been successively engaged with every tooth of the -wheel, and that during these ten revolutions the pinion -has revolved sixty-one times. Thus the leaves of the -pinion have acted six hundred and ten times upon the -teeth of the wheel, before two teeth can have acted twice -upon each other.</p> - -<p>The odd tooth which produces this effect is called by -millwrights the <i>hunting cog</i>.</p> - -<p id="p265">(265.) The most familiar case in which wheel-work -is used to produce and regulate motion merely, without -any reference to weights to be raised or resistances to be -overcome, is that of chronometers. In watch and clock -work the object is to cause a wheel to revolve with a -uniform velocity, and at a certain rate. The motion of -this wheel is indicated by an index or hand placed upon -its axis, and carried round with it. In proportion to the -length of the hand the circle over which its extremity -plays is enlarged, and its motion becomes more perceptible. -This circle is divided, so that very small fractions -of a revolution of the hand may be accurately observed. -In most chronometers it is required to give motion to -two hands, and sometimes to three. These motions -proceed at different rates, according to the subdivisions -of time generally adopted. One wheel revolves in a -minute, bearing a hand which plays round a circle divided -into sixty equal parts; the motion of the hand -over each part indicating one second, and a complete -revolution of the hand being performed in one minute.<span class="pagenum" id="Page_192">192</span> -Another wheel revolves once, while the former revolves -sixty times; consequently the hand carried by this wheel -revolves once in sixty minutes, or one hour. The circle -on which it plays is, like the former, divided into sixty -equal parts, and the motion of the hand over each division -is performed in one minute. This is generally -called the <i>minute hand</i>, and the former the <i>second -hand</i>.</p> - -<p>A third wheel revolves once, while that which carries -the minute hand revolves twelve times; consequently -this last wheel, which carries the <i>hour hand</i>, revolves at -a rate twelve times less than that of the minute hand, -and therefore seven hundred and twenty times less than -the second hand. We shall now endeavour to explain -the manner in which these motions are produced and -regulated. Let A, B, C, D, E, <i><a href="#i_p196a">fig. 110.</a></i>, represent a -train of wheels, and <i>a</i>, <i>b</i>, <i>c</i>, <i>d</i> represent their pinions, <i>e</i> -being a cylinder on the axis of the wheel E, round which -a rope is coiled, sustaining a weight W. Let the effect -of this weight transmitted through the train of wheels be -opposed by a power P acting upon the wheel A, and let -this power be supposed to be of such a nature as to cause -the weight W to descend with a uniform velocity, and -at any proposed rate. The wheel E carries on its circumference -eighty-four teeth. The wheel D carries -eighty teeth; the wheel C is also furnished with eighty -teeth, and the wheel B with seventy-five. The pinions -<i>d</i> and <i>c</i> are each furnished with twelve leaves, and the -pinions <i>b</i> and <i>a</i> with ten.</p> - -<p>If the power at P be so regulated as to allow the -wheel A to revolve once in a minute, with a uniform velocity, -a hand attached to the axis of this wheel will -serve as the <i>second hand</i>. The pinion <i>a</i> carrying ten -teeth must revolve seven times and a half to produce one -revolution of B, consequently fifteen revolutions of the -wheel A will produce two revolutions of the wheel B; -the wheel B, therefore, revolves twice in fifteen minutes. -The pinion <i>b</i> must revolve eight times to produce one -revolution of the wheel C, and therefore the wheel C<span class="pagenum" id="Page_193">193</span> -must revolve once in four quarters of an hour, or in one -hour. If a hand be attached to the axis of this wheel, -it will have the motion necessary for the minute hand. -The pinion <i>c</i> must revolve six times and two thirds to -produce one revolution of the wheel D, and therefore -this wheel must revolve once in six hours and two -thirds. The pinion <i>d</i> revolves seven times for one revolution -of the wheel E, and therefore the wheel E will -revolve once in forty-six hours and two thirds.</p> - -<p>On the axis of the wheel C a second pinion may be -placed, furnished with seven leaves, which may lead a -wheel of eighty-four teeth, so that this wheel shall turn -once during twelve turns of the wheel C. If a hand be -fixed upon the axis, this hand will revolve once for -twelve revolutions of the minute hand fixed upon the -axis of the wheel C; that is, it will revolve once in -twelve hours. If it play upon a dial divided into twelve -equal parts, it will move over each part in an hour, and will -serve the purpose of the hour hand of the chronometer.</p> - -<p>We have here supposed that the second hand, the -minute hand, and the hour hand move on separate dials. -This, however, is not necessary. The axis of the hour -hand is commonly a tube, inclosing within it that of the -minute hand, so that the same dial serves for both. The -second hand, however, is generally furnished with a separate -dial.</p> - -<p id="p266">(266.) We shall now explain the manner in which a -power is applied to the wheel A, so as to regulate and -equalise the effect of the weight W. Suppose the wheel -A furnished with thirty teeth, as in <i><a href="#i_p196a">fig. 111.</a></i>; if nothing -check the motion, the weight W would descend with an -accelerated velocity, and would communicate an accelerated -motion to the wheel A. This effect, however, is -interrupted by the following contrivance:—L M is a pendulum -vibrating on the centre L, and so regulated that -the time of its oscillation is one second. The pallets -I and K are connected with the pendulum, so as to oscillate -with it. In the position of the pendulum represented -in the figure, the pallet I stops the motion of the<span class="pagenum" id="Page_194">194</span> -wheel A, and entirely suspends the action of the weight -W, <i><a href="#i_p196a">fig. 110.</a></i>, so that for a moment the entire machine is -motionless. The weight M, however, falls by its gravity -towards the lowest position, and disengages the pallet -I from the tooth of the wheel. The weight W begins -then to take effect, and the wheel A turns from A -towards B. Meanwhile the pendulum M oscillates -to the other side, and the pallet K falls under a tooth -of the wheel A, and checks for a moment its further -motion. On the returning vibration the pallet K becomes -again disengaged, and allows the tooth of the -wheel to escape, and by the influence of the weight W -another tooth passes before the motion of the wheel A is -again checked by the interposition of the pallet I.</p> - -<p>From this explanation it will appear that, in two vibrations -of the pendulum, one tooth of the wheel A -passes the pallet I, and therefore, if the wheel A be -furnished with 30 teeth, it will be allowed to make one -revolution during 60 vibrations of the pendulum. If, -therefore, the pendulum be regulated so as to vibrate -seconds, this wheel will revolve once in a minute. From -the action of the pallets in checking the motion of the -wheel A, and allowing its teeth alternately to <i>escape</i>, -this has been called the <i>escapement</i> wheel; and the wheel -and pallets together are generally called the <i>escapement</i>, -or <i>’scapement</i>.</p> - -<p>We have already explained, that by reason of the -friction on the points of support, and other causes, the -swing of the pendulum would gradually diminish, and -its vibration at length cease. This, however, is prevented -by the action of the teeth of the scapement wheel -upon the pallets, which is just sufficient to communicate -that quantity of force to the pendulum which is necessary -to counteract the retarding effects, and to maintain -its motion. It thus appears, that although the effect of -the gravity of the weight W in giving motion to the machine -is at intervals suspended, yet this part of the force -is not lost, being, during these intervals, employed in -giving to the pendulum all that motion which it would -lose by the resistances to which it is inevitably exposed.</p> - -<p><span class="pagenum" id="Page_195">195</span></p> - -<p>In stationary clocks, and in other cases in which the -bulk of the machine is not an objection, a descending -weight is used as the moving power. But in watches -and portable chronometers, this would be attended with -evident inconvenience. In such cases, a spiral spring, -called the <i>mainspring</i>, is the moving power. The manner -in which this spring communicates rotation to an -axis, and the ingenious method of equalising the effect -of its variable elasticity by giving to it a leverage, which -increases as the elastic force diminishes, have been already -explained. (<a href="#p255">255</a>.)</p> - -<p>A similar objection lies against the use of a pendulum -in portable chronometers. A spiral spring of a similar -kind, but infinitely more delicate, called a <i>hair spring</i>, -is substituted in its place. This spring is connected -with a nicely-balanced wheel, called <i>the balance wheel</i>, -which plays in pivots. When this wheel is turned to -a certain extent in one direction, the hair spring is coiled -up, and its elasticity causes the wheel to recoil, and -return to a position in which the energy of the spring -acts in the opposite direction. The balance wheel then -returns, and continually vibrates in the same manner. -The axis of this wheel is furnished with pallets similar -to those of the pendulum, which are alternately engaged -with the teeth of a crown wheel, which takes the place -of the scapement wheel already described.</p> - -<p>A general view of the work of a common watch is -represented in <i><a href="#i_p196a">fig. 111.</a></i> <i>bis.</i> A is the balance wheel bearing -pallets <i>p</i> <i>p</i> upon its axis; C is the crown wheel, whose -teeth are suffered to escape alternately by those pallets in -the manner already described in the scapement of a -clock. On the axis of the crown wheel is placed a -pinion <i>d</i>, which drives another crown wheel K. On the -axis of this is placed the pinion <i>c</i>, which plays in the -teeth of the third wheel L. The pinion <i>b</i> on the axis -of L is engaged with the wheel M, called the centre -wheel. The axle of this wheel is carried up through -the centre of the dial. A pinion <i>a</i> is placed upon it, -which works in the great wheel N. On this wheel the<span class="pagenum" id="Page_196">196</span> -mainspring immediately acts. O P is the mainspring -stripped of its barrel. The axis of the wheel M passing -through the centre of the dial is squared at the end to -receive the minute hand. A second pinion Q is placed -upon this axle which drives a wheel T. On the axle of -this wheel a pinion <i>g</i> is placed, which drives the hour -wheel V. This wheel is placed upon a tubular axis, -which incloses within it the axis of the wheel M. This -tubular axis passing through the centre of the dial, carries -the hour hand. The wheels A, B, C, D, E, <i>fig. -110.</i>, correspond to the wheels C, K, L, M, N, <i><a href="#i_p204a">fig. 112.</a></i>; -and the pinions <i>a</i>, <i>b</i>, <i>c</i>, <i>d</i>, <i>e</i>, <i><a href="#i_p196a">fig. 109.</a></i>, correspond to the -pinions <i>d</i>, <i>c</i>, <i>b</i>, <i>a</i>, <i>fig. 111</i>. From what has already been -explained of these wheels, it will be obvious that the -wheel M, <i><a href="#i_p196a">fig. 111.</a></i>, revolves once in an hour, causing the -minute hand to move round the dial once in that time. -This wheel at the same time turns the pinion Q which -leads the wheel T. This wheel again turns the pinion -<i>g</i> which leads the hour wheel V. The leaves and teeth -of these pinions and wheels are proportioned, as already -explained, so that the wheel V revolves once during -twelve revolutions of the wheel M. The hour hand, -therefore, which is carried by the tubular axle of the -wheel V, moves once round the dial in twelve hours.</p> - -<p>Our object here has not been to give a detailed account -of watch and clock work, a subject for which we -must refer the reader to the proper department of this -work. Such a general account has only been attempted -as may explain how tooth and pinion work may be applied -to regulate motion.</p> - -<div class="figcenter" id="i_p196a" style="max-width: 31.25em;"> - <img src="images/i_p196a.jpg" alt="" /> - <div class="caption"> -<p class="tar"><i>H. Adlard, sc.</i></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_197">197</span></p> - -<h2 class="nobreak" id="CHAP_XV">CHAP. XV.<br /> - -<span class="title">OF THE PULLEY.</span></h2> -</div> - - -<p id="p267">(267.) <span class="smcap">The</span> next class of simple machines, which present -themselves to our attention, is that which we have -called the <i>cord</i>. If a rope were perfectly flexible, and -were capable of being bent over a sharp edge, and of -moving upon it without friction, we should be enabled -by its means to make a force in any one direction overcome -resistance, or communicate motion in any other -direction. Thus if P, <i><a href="#i_p204a">fig. 112.</a></i>, be such an edge, a perfectly -flexible rope passing over it would be capable of -transmitting a force S F to a resistance Q R, so as to -support or overcome R, or by a motion in the direction -of S F to produce another motion in the direction R Q. -But as no materials of which ropes can be constructed -can give them perfect flexibility, and as in proportion -to the strength by which they are enabled to transmit -force their rigidity increases, it is necessary, in practice, -to adopt means to remove or mitigate those effects -which attend imperfect flexibility, and which would -otherwise render cords practically inapplicable as machines.</p> - -<p>When a cord is used to transmit a force from one -direction to another, its stiffness renders some force necessary -in bending it over the angle P, which the two -directions form; and if the angle be sharp, the exertion -of such a force may be attended with the rupture of the -cord. If, instead of bending the rope at one point over -a single angle, the change of direction were produced by -successively deflecting it over several angles, each of -which would be less sharp than a single one could be, -the force requisite for the deflection, as well as the -liability of rupturing the cord, would be considerably -diminished. But this end will be still more perfectly -attained if the deflection of the cord be produced by -bending it over the surface of a curve.</p> - -<p><span class="pagenum" id="Page_198">198</span></p> - -<p>If a rope were applied only to sustain, and not to -move a weight, this would be sufficient to remove the -inconveniences arising from its rigidity. But when motion -is to be produced, the rope, in passing over the -curved surface, would be subject to excessive friction, -and consequently to rapid wear. This inconvenience -is removed by causing the surface on which the rope -runs to move with it, so that no more friction is produced -than would arise from the curved surface rolling -upon the rope.</p> - -<p id="p268">(268.) All these ends are attained by the common -pulley, which consists of a wheel called a <i>sheave</i>, fixed -in a block and turning on a pivot. A groove is formed -in the edge of the wheel in which the rope runs, the -wheel revolving with it. Such an apparatus is represented -in <i><a href="#i_p204a">fig. 113.</a></i></p> - -<p>We shall, for the present, omit the consideration of -that part of the effects of the stiffness and friction of -the machine which is not removed by the contrivance -just explained, and shall consider the rope as perfectly -flexible and moving without friction.</p> - -<p>From the definition of a flexible cord, it follows, that -its tension, or the force by which it is stretched throughout -its entire length, must be uniform. From this principle, -and this alone, all the mechanical properties of -pulleys may be derived.</p> - -<p>Although, as already explained, the whole mechanical -efficacy of this machine depends on the qualities of the -cord, and not on those of the block and sheave, which -are only introduced to remove the accidental effects of -stiffness and friction; yet it has been usual to give the -name pulley to the block and sheave, and a combination -of blocks, sheaves, and ropes is called a <i>tackle</i>.</p> - -<p id="p269">(269.) When the rope passes over a single wheel, -which is fixed in its position, as in <i><a href="#i_p204a">fig. 113.</a></i>, the machine -is called a <i>fixed pulley</i>. Since the tension of the cord is -uniform throughout its length, it follows, that in this -machine the power and weight are equal. For the -weight stretches that part of the cord which is between<span class="pagenum" id="Page_199">199</span> -the weight and pulley, and the power stretches that part -between the power and the pulley. And since the tension -throughout the whole length is the same, the weight -must be equal to the power.</p> - -<p>Hence it appears that no mechanical advantage is -gained by this machine. Nevertheless, there is scarcely -any engine, simple or complex, attended with more convenience. -In the application of power, whether of men -or animals, or arising from natural forces, there are always -some directions in which it may be exerted to -much greater convenience and advantage than others, -and in many cases the exertion of these powers is limited -to a single direction. A machine, therefore, which enables -us to give the most advantageous direction to the -moving power, whatever be the direction of the resistance -opposed to it, contributes as much practical convenience -as one which enables a small power to balance -or overcome a great weight. In directing the power -against the resistance, it is often necessary to use two -fixed pulleys. Thus, in elevating a weight A, <i><a href="#i_p204a">fig. 114.</a></i>, -to the summit of a building, by the strength of a horse -moving below, two fixed pulleys B and C may be used. -The rope is carried from A over the pulley B; and, -passing downwards, is brought under C, and finally -drawn by the animal on the horizontal plane. In -the same manner sails are spread, and flags hoisted on -the yards and masts of a ship, by sailors pulling a rope -on the deck.</p> - -<p>By means of the fixed pulley a man may raise himself -to a considerable height, or descend to any proposed -depth. If he be placed in a chair or bucket attached to -one end of a rope which is carried over a fixed pulley, -by laying hold of this rope on the other side, as represented -in <i><a href="#i_p204a">fig. 115.</a></i>, he may, at will, descend to a depth -equal to half of the entire length of the rope, by continually -yielding rope on the one side, and depressing -the bucket or chair by his weight on the other. Fire-escapes -have been constructed on this principle, the -fixed pulley being attached to some part of the building.</p> - -<p><span class="pagenum" id="Page_200">200</span></p> - -<p id="p270">(270.) A <i>single moveable pulley</i> is represented in -<i><a href="#i_p204a">fig. 116.</a></i> A cord is carried from a fixed point F, and -passing through a block B, attached to a weight W, -passes over a fixed pulley C, the power being applied at -P. We shall first suppose the parts of the cord on each -side the wheel B to be parallel; in this case, the whole -weight W being sustained by the parts of the cords B C -and B F, and these parts being equally stretched (<a href="#p268">268</a>.), -each must sustain half the weight, which is therefore the -tension of the cord. This tension is resisted by the -power at P, which must, therefore, be equal to half the -weight. In this machine, therefore, the weight is twice -the power.</p> - -<p id="p271">(271.) If the parts of the cord B C and B F be not -parallel, as in <i><a href="#i_p204a">fig. 117.</a></i>, a greater power than half the -weight is therefore necessary to sustain it. To determine -the power necessary to support a given weight, -in this case take the line B A in the vertical direction, -consisting of as many inches as the weight consists of -ounces; from A draw A D parallel to B C, and A E -parallel to B F; the force of the weight represented by -A B will be equivalent to two forces represented by B D -and B E. (<a href="#p74">74</a>.) The number of inches in these lines -respectively will represent the number of ounces which -are equivalent to the tensions of the parts B F and B C -of the cord. But as these tensions are equal, B D and -B E must be equal, and each will express the amount of -the power P, which stretches the cord at P C.</p> - -<p>It is evident that the four lines, A E, E B, B D, and -D A, are equal. And as each of them represents the -power, the weight which is represented by A B must -be less than twice the power which is represented by -A E and E B taken together. It follows, therefore, that -as parts of the ropes which support the weight depart -from parallelism the machine becomes less and less -efficacious; and there are certain obliquities at which -the equilibrating power would be much greater than the -weight.</p> - -<p id="p272">(272.) The mechanical power of pulleys admits of<span class="pagenum" id="Page_201">201</span> -being almost indefinitely increased by combination. Systems -of pulleys may be divided into two classes; those -in which a single rope is used, and those which consist -of several distinct ropes. <i><a href="#p204">Fig. 118.</a></i> and <i>119.</i> represent -two systems of pulleys, each having a single rope. -The weight is in each case attached to a moveable block, -B, in which are fixed two or more wheels; A is a fixed -block, and the rope is successively passed over the wheels -above and below, and, after passing over the last wheel -above, is attached to the power. The tension of that -part of the cord to which the power is attached is produced -by the power, and therefore equivalent to it, and -the same tension must extend throughout its whole -length. The weight is sustained by all those parts of -the cord which pass from the lower block, and as the -force which stretches them all is the same, viz. that of -the power, the effect of the weight must be equally distributed -among them, their directions being supposed to -be parallel. It will be evident, from this reasoning, that -the weight will be as many times greater than the power -as the number of cords which support the lower block. -Thus, if there be six cords, each cord will support a -sixth part of the weight, that is, the weight will be six -times the tension of the cord, or six times the power. -In <i><a href="#i_p204a">fig. 118.</a></i> the cord is represented as being finally attached -to a hook on the upper block. But it may be carried -over an additional wheel fixed in that block, and finally -attached to a hook in the lower block, as in <i><a href="#i_p204a">fig. 119.</a></i>, by -which one will be added to the power of the machine, the -number of cords at the lower block being increased by -one. In the system represented in <i><a href="#i_p204a">fig. 118.</a></i> the wheels are -placed in the blocks one above the other; in <i><a href="#i_p204a">fig. 119.</a></i> they -are placed side by side. In all systems of pulleys of this -class, the weight of the lower block is to be considered -as a part of the weight to be raised, and in estimating -the power of the machine, this should always be attended -to.</p> - -<p id="p273">(273.) When the power of the machine, and therefore -the number of wheels, is considerable, some diffi<span class="pagenum" id="Page_202">202</span>culty -arises in the arrangement of the wheels and cords. -The celebrated Smeaton contrived a tackle, which takes -its name from him, in which there are ten wheels in -each block: five large wheels placed side by side, and -five smaller ones similarly placed above them in the -lower block, and below them in the upper. <i><a href="#i_p204a">Fig. 120.</a></i> -represents Smeaton’s blocks without the rope. The -wheels are marked with the numbers 1, 2, 3, &c., in the -order in which the rope is to be passed over them. As -in this pulley 20 distinct parts of the rope support the -lower block, the weight, including the lower block, will -be 20 times the equilibrating power.</p> - -<p id="p274">(274.) In all these systems of pulleys, every wheel -has a separate axle, and there is a distinct wheel for -every turn of the rope at each block. Each wheel is -attended with friction on its axle, and also with friction -between the sheave and block. The machine is by this -means robbed of a great part of its efficacy, since, to -overcome the friction alone, a considerable power is in -most cases necessary.</p> - -<p>An ingenious contrivance has been suggested, by -which all the advantage of a large number of wheels -may be obtained without the multiplied friction of -distinct sheaves and axles. To comprehend the excellence -of this contrivance, it will be necessary to consider -the rate at which the rope passes over the several -wheels of such a system, as <i><a href="#i_p204a">fig. 118.</a></i> If one foot of the -rope G F pass over the pulley F, two feet must pass over -the pulley E, because the distance between F and E -being shortened one foot, the total length of the rope -G F E must be shortened two feet. These two feet of -rope must pass in the direction E D, and the wheel D, -rising one foot, three feet of rope must consequently pass -over it. These three feet of rope passing in the direction -D C, and the rope D C being also shortened one foot -by the ascent of the lower block, four feet of rope must -pass over the wheel C. In the same way it may be -shown that five feet must pass over B, and six feet over -A. Thus, whatever be the number of wheels in the<span class="pagenum" id="Page_203">203</span> -upper and lower blocks, the parts of the rope which pass -in the same time over the wheels in the lower block are -in the proportion of the odd numbers 1, 3, 5, &c.; and -those which pass over the wheels in the upper block in -the same time, are as the even numbers 2, 4, 6, &c. If -the wheels were all of equal size, as in <i><a href="#i_p204a">fig. 119.</a></i>, they -would revolve with velocities proportional to the rate at -which the rope passes over them. So that, while the -first wheel below revolves once, the first wheel above -will revolve twice; the second wheel below three times; -the second wheel above, four times, and so on. If, -however, the wheels differed in size in proportion to the -quantity of rope which must pass over them, they would -evidently revolve in the same time. Thus, if the first -wheel above were twice the size of the first wheel below, -one revolution would throw off twice the quantity of -rope. Again, if the second wheel below were thrice -the size of the first wheel below, it would throw off in -one revolution thrice the quantity of rope, and so on. -Wheels thus proportioned, revolving in exactly the -same time, might be all placed on one axle, and would -partake of one common motion, or, what is to the same -effect, several grooves might be cut upon the face of one -solid wheel, with diameters in the proportion of the odd -numbers 1, 3, and 5, &c., for the lower pulley, and corresponding -grooves on the face of another solid wheel -represented by the even numbers 2, 4, 6, &c., for the -upper pulley. The rope being passed successively over -the grooves of such wheels, would be thrown off exactly -in the same manner as if every groove were upon a separate -wheel, and every wheel revolved independently of -the others. Such is White’s pulley, represented in -<i><a href="#i_p204a">fig. 121.</a></i></p> - -<p>The advantage of this machine, when accurately constructed, -is very considerable. The friction, even when -great resistances are to be opposed, is very trifling; but, -on the other hand, it has corresponding disadvantages -which greatly circumscribe its practical utility. In the -workmanship of the grooves great difficulty is found in<span class="pagenum" id="Page_204">204</span> -giving them the exact proportions. In doing which, the -thickness of the rope must be accurately allowed for; and -consequently it follows, that the same pulley can never act -except with a rope of a particular diameter. A very -slight deviation from the true proportion of the grooves -will cause the rope to be unequally stretched, and will -throw on some parts of it an undue proportion of the -weight, while other parts become nearly, and sometimes -altogether slack. Besides these defects, the rope is so -liable to derangement by being thrown out of the grooves, -that the pulley can scarcely be considered portable.</p> - -<p>For these and other reasons, this machine, ingenious -as it unquestionably is, has never been extensively used.</p> - -<p id="p275">(275.) In the several systems of pulleys just explained, -the hook to which the fixed block is attached supports -the entire of both the power and weight. When the -machine is in equilibrium, the power only supports so -much of the weight as is equal to the tension of the -cord, all the remainder of the weight being thrown on -the fixed point, according to what was observed in (<a href="#p225">225</a>.)</p> - -<p>If the power be moved so as to raise the weight, -it will move with a velocity as many times greater -than that of the weight as the weight itself is greater -than the power. Thus in <i><a href="#i_p204a">fig. 118.</a></i> if the weight -attached to the lower block ascend one foot, six feet of -line will pass over the pulley A, according to what has been -already proved. Thus, the power will descend through six -feet, while the weight rises one foot. But, in this case, -the weight is six times the power. All the observations -in (<a href="#p226">226</a>.) will therefore be applicable to the cases of great -weights raised by small powers by means of the system -of pulleys just described.</p> - -<p id="p276">(276.) When two or more ropes are used, pulleys may -be combined in various ways so as to produce any degree -of mechanical effect. If to any of the systems already -described a single moveable pulley be added, the power -of the machine would be doubled. In this case, the -second rope is attached to the hook of the lower block, -as in <i><a href="#i_p214a">fig. 122.</a></i>, and being carried through a moveable<span class="pagenum" id="Page_205">205</span> -pulley attached to the weight, it is finally brought up to -a fixed point. The tension of the second cord is equal -to half the weight (<a href="#p270">270</a>.); and therefore the power P, by -means of the first cord, will have only half the tension -which it would have if the weight were attached to the -lower block. A moveable pulley thus applied is called a -<i>runner</i>.</p> - -<div class="figcenter" id="i_p204a" style="max-width: 31.25em;"> - <img src="images/i_p204a.jpg" alt="" /> - <div class="caption"><p> -<span class="l-align"><i>C. Varley, del.</i></span> - -<span class="r-align"><i>H. Adlard, sc.</i></span></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<p id="p277">(277.) Two systems of pulleys, called <i>Spanish bartons</i>, -having each two ropes, are represented in <i><a href="#i_p214a">fig. 123.</a></i> The -tension of the rope P A B C in the first system is equal -to the power; and therefore the parts B A and B C -support a portion of the weight equal to twice the power. -The rope E A supports the tensions of A P and A B; -and therefore the tension of A E D is twice the power. -Thus, the united tensions of the ropes which support -the pulley B is four times the power, which is therefore -the amount of the weight. In the second system, the -rope P A D is stretched by the power. The rope A E B C -acts against the united tensions A P and A D; and -therefore the tension of A E or E B is twice the power. -Thus, the weight acts against three tensions; two of -which are equal to twice the power, and the remaining -one is equal to the power. The weight is therefore -equal to five times the power.</p> - -<p>A single rope may be so arranged with one moveable -pulley as to support a weight equal to three times the -power. In <i><a href="#i_p214a">fig. 124.</a></i> this arrangement is represented, -where the numbers sufficiently indicate the tension of -the rope, and the proportion of the weight and power. -In <i><a href="#i_p214a">fig. 125.</a></i> another method of producing the same effect -with two ropes is represented.</p> - -<p id="p278">(278.) If several single moveable pulleys be made -successively to act upon each other, the effect is doubled -by every additional pulley: such a system as this is -represented in <i><a href="#i_p214a">fig. 126.</a></i> The tension of the first rope is -equal to the power; the second rope acts against twice -the tension of the first, and therefore it is stretched -with a force equal to twice the power: the third rope -acts against twice this tension, and therefore it is stretched<span class="pagenum" id="Page_206">206</span> -with a force equal to four times the power, and so on. -In the system represented in <i><a href="#i_p214a">fig. 126.</a></i> there are three -ropes, and the weight is eight times the power. Another -rope would render it sixteen times the power, and so on.</p> - -<p>In this system, it is obvious that the ropes will require -to have different degrees of strength, since the tension to -which they are subject increases in a double proportion -from the power to the weight.</p> - -<p id="p279">(279.) If each of the ropes, instead of being attached -to fixed points at the top, are carried over fixed pulleys, -and attached to the several moveable pulleys respectively, -as in <i><a href="#i_p214a">fig. 127.</a></i>, the power of the machine will be greatly -increased; for in that case the forces which stretch the -successive ropes increase in a treble instead of a double -proportion, as will be evident by attending to the numbers -which express the tensions in the figure. One rope -would render the weight three times the power, two -ropes nine times, three ropes twenty-seven times, and -so on. An arrangement of pulleys is represented in <i><a href="#i_p214a">fig. -128.</a></i>, by which each rope, instead of being finally -attached to a fixed point, as in <i><a href="#i_p214a">fig. 126.</a></i>, is attached to the -weight. The weight is in this case supported by three -ropes; one stretched with a force equal to the power; -another with a force equal to twice the power; and a -third with a force equal to four times the power. The -weight is therefore, in this case, seven times the power.</p> - -<p id="p280">(280.) If the ropes, instead of being attached to the -weight, pass through wheels, as in <i><a href="#i_p214a">fig. 129.</a></i>, and are -finally attached to the pulleys above, the power of the -machine will be considerably increased. In the system -here represented the weight is twenty-six times the -power.</p> - -<p id="p281">(281.) In considering these several combinations of -pulleys, we have omitted to estimate the effects produced -by the weights of the sheaves and blocks. Without -entering into the details of this computation, it may be -observed generally, that in the systems represented in -<i><a href="#i_p214a">figs. 126.</a></i>, <i>127.</i> the weight of the wheel and blocks acts -against the power; but that in <i><a href="#i_p214a">figs. 128.</a></i> and <i>129.</i> they<span class="pagenum" id="Page_207">207</span> -assist the powers in supporting the weight. In the -systems represented in <i><a href="#i_p214a">fig. 123.</a></i> the weight of the pulleys, -to a certain extent, neutralise each other.</p> - -<p id="p282">(282.) It will in all cases be found, that that quantity -by which the weight exceeds the power is supported by -fixed points; and therefore, although it be commonly -stated that a small power supports a great weight, yet -in the pulley, as in all other machines, the power supports -no more of the weight than is exactly equal to its -own amount. It will not be necessary to establish this -in each of the examples which have been given: having -explained it in one instance, the student will find no -difficulty in applying the same reasoning to others. In -<i><a href="#i_p214a">fig. 126.</a></i>, the fixed pulley sustains a force equal to twice -the power, and by it the power giving tension to the first -rope sustains a part of the weight equal to itself. The -first hook sustains a portion of the weight equal to the -tension of the first string, or to the power. The second -hook sustains a force equal to twice the power; and the -third hook sustains a force equal to four times the -power. The three hooks therefore sustain a portion of -the weight equal to seven times the power; and the -weight itself being eight times the power, it is evident -that the part of the weight which remains to be supported -by the power is equal to the power itself.</p> - -<p id="p283">(283.) When a weight is raised by any of the systems -of pulleys which have been last described, the proportion -between the velocity of the weight and the velocity of -the power, so frequently noticed in other machines, will -always be observed. In the system of pulleys represented -in <i><a href="#i_p214a">fig. 126.</a></i> the weight being eight times the power, -the velocity of the power will be eight times that of the -weight. If the power be moved through eight feet, that -part of the rope between the fixed pulley and the first moveable -pulley will be shortened by eight feet. And since the -two parts which lie above the first moveable pulley must be -equally shortened, each will be diminished by four feet; -therefore the first pulley will rise through four feet while -the power moves through eight feet. In the same way<span class="pagenum" id="Page_208">208</span> -it may be shown, that while the first pulley moves -through four feet, the second moves through two; and -while the second moves through two, the third, to which -the weight is attached, is raised through one foot. While -the power, therefore, is carried through eight feet, the -weight is moved through one foot.</p> - -<p>By reasoning similar to this, it may be shown that -the space through which the power is moved in every -case is as many times greater than the height through -which the weight is raised, as the weight is greater than -the power.</p> - -<p id="p284">(284.) From its portable form, cheapness of construction, -and the facility with which it may be applied -in almost every situation, the pulley is one of the most -useful of the simple machines. The mechanical advantage, -however, which it appears in theory to possess is -considerably diminished in practice, owing to the stiffness -of the cordage, and the friction of the wheels and -blocks. By this means it is computed that in most cases -so great a proportion as two thirds of the power is lost. -The pulley is much used in building, where weights are -to be elevated to great heights. But its most extensive -application is found in the rigging of ships, where almost -every motion is accomplished by its means.</p> - -<p id="p285">(285.) In all the examples of pulleys, we have supposed -the parts of the rope sustaining the weight and -each of the moveable pulleys to be parallel to each other. -If they be subject to considerable obliquity, the relative -tensions of the different ropes must be estimated according -to the principle applied in (<a href="#p271">271</a>.)</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_209">209</span></p> - -<h2 class="nobreak" id="CHAP_XVI">CHAP. XVI.<br /> - -<span class="title">ON THE INCLINED PLANE, WEDGE, AND SCREW.</span></h2> -</div> - - -<p id="p286">(286.) <span class="smcap">The</span> inclined plane is the most simple of all -machines. It is a hard plane surface forming some -angle with a horizontal plane, that angle not being a -right angle. When a weight is placed on such a plane, -a two-fold effect is produced. A part of the effect of -the weight is resisted by the plane, and produces a pressure -upon it; and the remainder urges the weight down -the plane, and would produce a pressure against any -surface resisting its motion placed in a direction perpendicular -to the plane (<a href="#p131">131</a>.)</p> - -<p>Let A B, <i><a href="#i_p214a">fig. 130.</a></i>, be such a plane, B C its horizontal -base, A C its height, and A B C its angle of elevation. -Let W be a weight placed upon it. This weight acts -in the vertical direction W D, and is equivalent to two -forces, W F perpendicular to the plane, and W E directed -down the plane (<a href="#p74">74</a>.) If a plane be placed at right -angles to the inclined plane below W, it will resist the -descent of the weight, and sustain a pressure expressed -by W E. Thus, the weight W resting in the corner, -instead of producing one pressure in the direction -W D, will produce two pressures, one expressed by W F -upon the inclined plane, and the other expressed by -W E upon the resisting plane. These pressures respectively -have the same proportion to the entire weight -as W F and W E have to W D, or as D E and W E -have to W D, because D E is equal to W F. Now the -triangle W E D is in all respects similar to the triangle -A B C, the one differing from the other only in the scale -on which it is constructed. Therefore, the three lines -A C, C B, and B A, are in the same proportion to each -other as the lines W E, E D, and W D. Hence, A B -has to A C the same proportion as the whole weight -has to the pressure directed toward B, and A B has to<span class="pagenum" id="Page_210">210</span> -B C the same proportion as the whole weight has to the -pressure on the inclined plane.</p> - -<p>We have here supposed the weight to be sustained -upon the inclined plane by a hard plane fixed at right -angles to it. But the power necessary to sustain the -weight will be the same in whatever way it is applied, -provided it act in the direction of the plane. Thus, a -cord may be attached to the weight, and stretched towards -A, or the hands of men may be applied to the -weight below it, so as to resist its descent towards B. -But in whatever way it be applied, the amount of -the power will be determined in the same manner. Suppose -the weight to consist of as many pounds as there -are inches in A B, then the power requisite to sustain -it upon the plane will consist of as many pounds -as there are inches in A C, and the pressure on the plane -will amount to as many pounds as there are inches in B C.</p> - -<p>From what has been stated it may easily be inferred -that the less the elevation of the plane is, the less will -be the power requisite to sustain a given weight upon it, -and the greater will be the pressure upon it. Suppose -the inclined plane A B to turn upon a hinge at B, and to -be depressed so that its angle of elevation shall be diminished, -it is evident that as this angle decreases the -height of the plane decreases, and its base increases. -Thus, when it takes the position B <span class="ilb">A′</span>, the height <span class="ilb">A′</span> <span class="ilb">C′</span> -is less than the former height A C, while the base B <span class="ilb">C′</span> -is greater than the former base B C. The power requisite -to support the weight upon the plane in the position -B <span class="ilb">A′</span> is represented by <span class="ilb">A′</span> <span class="ilb">C′</span>, and is as much less than the -power requisite to sustain it upon the plane A B, as the -height <span class="ilb">A′</span> <span class="ilb">C′</span> is less than the height A C. On the other -hand, the pressure upon the plane in the position B A′ -is as much greater than the pressure upon the plane -B A, as the base B <span class="ilb">C′</span> is greater than the base B C.</p> - -<p id="p287">(287.) The power of an inclined plane, considered as -a machine, is therefore estimated by the proportion -which its length bears to its height. This power is -always increased by diminishing the elevation of the plane.</p> - -<p><span class="pagenum" id="Page_211">211</span></p> - -<p>Roads which are not level may be regarded as inclined -planes, and loads drawn upon them in carriages, considered -in reference to the powers which impel them, are -subject to all the conditions which have been established -for inclined planes. The inclination of the road is estimated -by the height corresponding to some proposed -length. Thus it is said to rise one foot in fifteen, one -foot in twenty, &c., meaning that if fifteen or twenty -feet of the road be taken as the length of an inclined -plane, such as A B, the corresponding height will be one -foot. Or the same may be expressed thus: that if -fifteen or twenty feet be measured upon the road, the -difference of the levels of the two extremities of the distance -measured is one foot. According to this method -of estimating the inclination of roads, the power requisite -to sustain a load upon them (setting aside the effect -of friction), is always proportional to that elevation. -Thus, if a road rise one foot in twenty, a power of one -ton will be sufficient to sustain twenty tons, and so on.</p> - -<p>On a horizontal plane the only resistance which the -power has to overcome is the friction of the load with -the plane, and the consideration of this being for the -present omitted, a weight once put in motion would continue -moving for ever, without any further action of the -power. But if the plane be inclined, the power will be -expended in raising the weight through the perpendicular -height of the plane. Thus, in a road which rises -one foot in ten, the power is expended in raising the -weight through one perpendicular foot for every ten feet -of the road over which it is moved. As the expenditure -of power depends upon the rate at which the weight is -raised perpendicularly, it is evident that the greater -the inclination of the road is, the slower the motion -must be with the same force. If the energy of -the power be such as to raise the weight at the rate of -one foot per minute, the weight may be moved in each -minute through that length of the road which corresponds -to a rise of one foot. Thus, if two roads rise -one at the rate of a foot in fifteen feet, and the other at<span class="pagenum" id="Page_212">212</span> -the rate of one foot in twenty feet, the same expenditure -of power will move the weight through fifteen feet of -the one, and twenty feet of the other at the same rate.</p> - -<p>From such considerations as these, it will readily -appear that it may often be more expedient to carry a -road through a circuitous route than to continue it in -the most direct course; for though the measured length -of road may be considerably greater than in the former -case, yet more may be gained in speed with the same -expenditure of power than is lost by the increase of -distance. By attending to these circumstances, modern -road-makers have greatly facilitated and expedited the -intercourse between distant places.</p> - -<p id="p288">(288.) If the power act obliquely to the plane, it will -have a twofold effect; a part being expended in supporting -or drawing the weight, and a part in diminishing -or increasing the pressure upon the plane. Let -W P, <i><a href="#i_p214a">fig. 130.</a></i>, be the power. This will be equivalent -to two forces, W <span class="ilb">F′</span>, perpendicular to the plane, and -W <span class="ilb">E′</span> in the direction of the plane. (<a href="#p74">74</a>.) In order -that the power should sustain the weight, it is necessary -that that part W <span class="ilb">E′</span> of the power which acts in the -direction of the plane should be equal to that part W E, -<i><a href="#i_p214a">fig. 130.</a></i>, of the weight which acts down the plane. The -other part W <span class="ilb">F′</span> of the power acting perpendicular to the -plane is immediately opposed to that part W F of the -weight which produces pressure. The pressure upon -the plane will therefore be diminished by the amount -of W <span class="ilb">F′</span>. The amount of the power which will equilibrate -with the weight may, in this case, be found as -follows. Take W <span class="ilb">E′</span> equal to W E, and draw <span class="ilb">E′</span> P -perpendicular to the plane, and meeting the direction of -the power. The proportion of the power to the -weight will be that of W P to W D. And the proportion -of the pressure to the weight will be that of the -difference between W F and W <span class="ilb">F′</span> to W D. If the -amount of the power have a less proportion to the weight -than W P has to W D, it will not support the body on -the plane, but will allow it to descend. And if it<span class="pagenum" id="Page_213">213</span> -have a greater proportion, it will draw the weight up -the plane towards A.</p> - -<p id="p289">(289.) It sometimes happens that a weight upon one -inclined plane is raised or supported by another weight -upon another inclined plane. Thus, if A B and A <span class="ilb">B′</span>, -<i><a href="#i_p214a">fig. 131.</a></i>, be two inclined planes forming an angle at A, -and W <span class="ilb">W′</span> be two weights placed upon these planes, -and connected by a cord passing over a pulley at A, the -one weight will either sustain the other, or one will -descend, drawing the other up. To determine the circumstances -under which these effects will ensue, draw -the lines W D and <span class="ilb">W′</span> <span class="ilb">D′</span> in the vertical direction, and -take upon them as many inches as there are ounces in -the weights respectively. W D and <span class="ilb">W′</span> <span class="ilb">D′</span> being the -lengths thus taken, and therefore representing the weights, -the lines W E and <span class="ilb">W′</span> <span class="ilb">E′</span> will represent the effects of -these weights respectively down the planes. If W E -and <span class="ilb">W′</span> <span class="ilb">E′</span> be equal, the weights will sustain each other -without motion. But if W E be greater than <span class="ilb">W′</span> <span class="ilb">E′</span>, -the weight W will descend, drawing the weight <span class="ilb">W′</span> up. -And if <span class="ilb">W′</span> <span class="ilb">E′</span> be greater than W E, the weight <span class="ilb">W′</span> will -descend, drawing the weight W up. In every case the -lines W F and <span class="ilb">W′</span> <span class="ilb">F′</span> will represent the pressures upon -the planes respectively.</p> - -<p>It is not necessary, for the effect just described, that -the inclined planes should, as represented in the figure, -form an angle with each other. They may be parallel, -or in any other position, the rope being carried over a -sufficient number of wheels placed so as to give it the -necessary deflection. This method of moving loads is -frequently applied in great public works where rail-roads -are used. Loaded waggons descend one inclined plane, -while other waggons, either empty or so loaded as to -permit the descent of those with which they are connected, -are drawn up the other.</p> - -<p id="p290">(290.) In the application of the inclined plane which -we have hitherto noticed, the machine itself is supposed -to be fixed in its position, while the weight or load is -moved upon it. But it frequently happens that resist<span class="pagenum" id="Page_214">214</span>ances -are to be overcome which do not admit of being -thus moved. In such cases, instead of moving the load -upon the planes, the plane is to be moved under or -against the load. Let D E, <i><a href="#i_p224a">fig. 132.</a></i>, be a heavy beam -secured in a vertical position between guides F G and -H I, so that it is free to move upwards and downwards, -but not laterally. Let A B C be an inclined plane, the -extremity of which is placed beneath the end of the -beam. A force applied to the back of this plane A C, in -the direction C B, will urge the plane under the beam so -as to raise the beam to the position represented in <i><a href="#i_p224a">fig. 133.</a></i> -Thus, while the inclined plane is moved through the -distance C B, the beam is raised through the height C A.</p> - -<p id="p291">(291.) When the inclined plane is applied in this -manner, it is called a <i>wedge</i>. And if the power applied -to the back were a continued pressure, its proportion to -the weight would be that of A C to C B. It follows, -therefore, that the more acute the angle B is, the more -powerful will be the wedge.</p> - -<p>In some cases, the wedge is formed of two inclined -planes, placed base to base, as represented in <i><a href="#i_p224a">fig. 134.</a></i> -The theoretical estimation of the power of this machine -is not applicable in practice with any degree of accuracy. -This is in part owing to the enormous proportion -which the friction in most cases bears to the theoretical -value of the power, but still more to the nature of the -power generally used. The force of a blow is of a -nature so wholly different from continued forces, such -as the pressure of weights, or the resistance offered by -the cohesion of bodies, that it admits of no numerical -comparison with them. Hence we cannot properly -state the proportion which the force of a blow bears to -the amount of a weight or resistance. The wedge is -almost invariably urged by percussion; while the resistances -which it has to overcome are as constantly -forces of the other kind. Although, however, no exact -numerical comparison can be made, yet it may be stated -in a general way that the wedge is more and more -powerful as its angle is more acute.</p> - -<div class="figcenter" id="i_p214a" style="max-width: 31.25em;"> - <img src="images/i_p214a.jpg" alt="" /> - <div class="caption"><p> -<span class="l-align"><i>C. Varley, del.</i></span> - -<span class="r-align"><i>H. Adlard, sc.</i></span></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<p><span class="pagenum" id="Page_215">215</span></p> - -<p>In the arts and manufactures, wedges are used where -enormous force is to be exerted through a very small -space. Thus it is resorted to for splitting masses of -timber or stone. Ships are raised in docks by wedges -driven under their keels. The wedge is the principal -agent in the oil-mill. The seeds from which the oil -is to be extracted are introduced into hair bags, and -placed between planes of hard wood. Wedges inserted -between the bags are driven by allowing heavy beams to -fall on them. The pressure thus excited is so intense, -that the seeds in the bags are formed into a mass nearly -as solid as wood. Instances have occurred in which -the wedge has been used to restore a tottering edifice to -its perpendicular position.</p> - -<p>All cutting and piercing instruments, such as knives, -razors, scissors, chisels, &c., nails, pins, needles, awls, -&c. are wedges. The angle of the wedge, in these -cases, is more or less acute, according to the purpose to -which it is to be applied. In determining this, two things -are to be considered—the mechanical power, which is -increased by diminishing the angle of the wedge; and -the strength of the tool, which is always diminished by -the same cause. There is, therefore, a practical limit -to the increase of the power, and that degree of sharpness -only is to be given to the tool which is consistent -with the strength requisite for the purpose to which it is -to be applied. In tools intended for cutting wood, the -angle is generally about 30°. For iron it is from 50° -to 60°; and for brass, from 80° to 90°. Tools which -act by pressure may be made more acute than those -which are driven by a blow; and in general the softer -and more yielding the substance to be divided is, and -the less the power required to act upon it, the more -acute the wedge may be constructed.</p> - -<p>In many cases the utility of the wedge depends on -that which is entirely omitted in its theory, viz. the -friction which arises between its surface and the substance -which it divides. This is the case when pins, -bolts, or nails are used for binding the parts of struc<span class="pagenum" id="Page_216">216</span>tures -together; in which case, were it not for the friction, -they would recoil from their places, and fail to -produce the desired effect. Even when the wedge is -used as a mechanical engine, the presence of friction is -absolutely indispensable to its practical utility. The -power, as has already been stated, generally acts by successive -blows, and is therefore subject to constant intermission, -and but for the friction the wedge would recoil -between the intervals of the blows with as much force -as it had been driven forward. Thus the object of -the labour would be continually frustrated. The friction -in this case is of the same use as a ratchet wheel, -but is much more necessary, as the power applied to the -wedge is more liable to intermission than in the cases -where ratchet wheels are generally used.</p> - -<p id="p292">(292.) When a road directly ascends the side of a -hill, it is to be considered as an inclined plane; but it -will not lose its mechanical character, if, instead of -directly ascending towards the top of the hill, it winds -successively round it, and gradually ascends so as after -several revolutions to reach the top. In the same manner -a path may be conceived to surround a pillar by -which the ascent may be facilitated upon the principle -of the inclined plane. Winding stairs constructed in the -interior of great columns partake of this character; for -although the ascent be produced by successive steps, yet -if a floor could be made sufficiently rough to prevent the -feet from slipping, the ascent would be accomplished -with equal facility. In such a case the winding path -would be equivalent to an inclined plane, bent into such -a form as to accommodate it to the peculiar circumstances -in which it would be required to be used. It will not be -difficult to trace the resemblance between such an adaptation -of the inclined plane and the appearances presented -by the thread of a <i>screw</i>: and it may hence be easily -understood that a screw is nothing more than an inclined -plane constructed upon the surface of a cylinder.</p> - -<p>This will, perhaps, be more apparent by the following -contrivance: Let A B, <i><a href="#i_p224a">fig. 135.</a></i>, be a common round<span class="pagenum" id="Page_217">217</span> -ruler, and let C D E be a piece of white paper cut in -the form of an inclined plane, whose height C D is equal -to the length of the ruler A B, and let the edge C E -of the paper be marked with a broad black line: let the -edge C D be applied to the ruler A B, and being attached -thereto, let the paper be rolled round the ruler; the -ruler will then present the appearance of a screw, <i><a href="#i_p224a">fig. 136.</a></i> -the thread of the screw being marked by the black line -C E, winding continually round the ruler. Let D F, -<i><a href="#i_p224a">fig. 135.</a></i>, be equal to the circumference of the ruler, and -draw F G parallel to D C, and G H parallel to D E, the -part C G F D of the paper will exactly surround the -ruler once: the part C G will form one convolution of the -thread, and may be considered as the length of one inclined -plane surrounding the cylinder, C H being the -corresponding height, and G H the base. The power of -the screw does not, as in the ordinary cases of the inclined -plane, act parallel to the plane or thread, but at right -angles to the length of the cylinder A B, or, what is to -the same effect, parallel to the base H G; therefore the -proportion of the power to the weight will be, according -to principles already explained, the same as that of C H -to the space through which the power moves parallel to -H G in one revolution of the screw. H C is evidently -the distance between the successive positions of the thread -as it winds round the cylinder; and it appears from what -has been just stated, that the less this distance is, or, in -other words, the finer the thread is, the more powerful -the machine will be.</p> - -<p id="p293">(293.) In the application of the screw the weight or -resistance is not, as in the inclined plane and wedge, -placed upon the surface of the plane or thread. The -power is usually transmitted by causing the screw to -move in a concave cylinder, on the interior surface of -which a spiral cavity is cut, corresponding exactly to -the thread of the screw, and in which the thread will -move by turning round the screw continually in the -same direction. This hollow cylinder is usually called -the <i>nut</i> or <i>concave screw</i>. The screw surrounded by its<span class="pagenum" id="Page_218">218</span> -spiral thread is represented in <i><a href="#i_p224a">fig. 137.</a></i>; and a section of -the same playing in the nut is represented in <i><a href="#i_p224a">fig. 138.</a></i></p> - -<p>There are several ways in which the effect of the -power may be conveyed to the resistance by this apparatus.</p> - -<p>First, let us suppose that the nut A B is fixed. If the -screw be continually turned on its axis, by a lever E F -inserted in one end of it, it will be moved in the direction -C D, advancing every revolution through a space -equal to the distance between two contiguous threads. -By turning the lever in an opposite direction, the screw -will be moved in the direction D C.</p> - -<p>If the screw be fixed, so as to be incapable either of -moving longitudinally or revolving on its axis, the nut -A B may be turned upon the screw by a lever, and will -move on the screw towards C or towards D, according to -the direction in which the lever is turned.</p> - -<p>In the former case we have supposed the nut to be -absolutely immoveable, and in the latter case the screw -to be absolutely immoveable. It may happen, however, -that the nut, though capable of revolving, is incapable -of moving longitudinally; and that the screw, though -incapable of revolving, is capable of moving longitudinally. -In that case, by turning the nut A B upon the -screw by the lever, the screw will be urged in the direction C D -or D C, according to the way in which the nut -is turned.</p> - -<p>The apparatus may, on the contrary, be so arranged, -that the nut, though incapable of revolving, is capable of -moving longitudinally; and the screw, though capable -of revolving, is incapable of moving longitudinally. In -this case, by turning the screw in the one direction or in -the other, the nut A B will be urged in the direction C D -or D C.</p> - -<p>All these various arrangements may be observed in -different applications to the machine.</p> - -<p id="p294">(294.) A screw may be cut upon a cylinder by -placing the cylinder in a turning lathe, and giving it -a rotatory motion upon its axis. The cutting point is<span class="pagenum" id="Page_219">219</span> -then presented to the cylinder, and moved in the direction -of its length, at such a rate as to be carried -through the distance between the intended thread, while -the cylinder revolves once. The relative motions of the -cutting point and the cylinder being preserved with -perfect uniformity, the thread will be cut from one end -to the other. The shape of the threads may be either -square, as in <i><a href="#i_p224a">fig. 137.</a></i>, or triangular, as in <i><a href="#i_p224a">fig. 139.</a></i></p> - -<p id="p295">(295.) The screw is generally used in cases where -severe pressure is to be excited through small spaces; it -is therefore the agent in most presses. In <i><a href="#i_p224a">fig. 140.</a></i>, the -nut is fixed, and by turning the lever, which passes -through the head of the screw, a pressure is excited -upon any substance placed upon the plate immediately -under the end of the screw. In <i><a href="#i_p224a">fig. 141.</a></i>, the screw is -incapable of revolving, but is capable of advancing in the -direction of its length. On the other hand, the nut is -capable of revolving, but does not advance in the direction -of the screw. When the nut is turned by means -of the screw inserted in it, the screw advances in the -direction of its length, and urges the board which is -attached to it upwards, so as to press any substance -placed between it and the fixed board above.</p> - -<p>In cases where liquids or juices are to be expressed -from solid bodies, the screw is the agent generally employed. -It is also used in coining, where the impression -of a die is to be made upon a piece of metal, and in the -same way in producing the impression of a seal upon -wax or other substance adapted to receive it. When -soft and light materials, such as cotton, are to be reduced -to a convenient bulk for transportation, the screw -is used to compress them, and they are thus reduced into -hard dense masses. In printing, the paper is urged by a -severe and sudden pressure upon the types, by means of -a screw.</p> - -<p id="p296">(296.) As the mechanical power of the screw depends -upon the relative magnitude of the circumference -through which the power revolves, and the distance between -the threads, it is evident, that, to increase the<span class="pagenum" id="Page_220">220</span> -efficacy of the machine, we must either increase the -length of the lever by which the power acts, or diminish -the magnitude of the thread. Although there is no -limit in theory to the increase of the mechanical efficacy -by these means, yet practical inconvenience arises which -effectually prevents that increase being carried beyond a -certain extent. If the lever by which the power acts be -increased, the same difficulty arises as was already explained -in the wheel and axle (<a href="#p254">254</a>.); the space -through which the power should act would be so unwieldy, -that its application would become impracticable. -If, on the other hand, the power of the machine be increased -by diminishing the size of the thread, the -strength of the thread will be so diminished, that a -slight resistance will tear it from the cylinder. The -cases in which it is necessary to increase the power of -the machine, being those in which the greatest resistances -are to be overcome, the object will evidently be defeated, -if the means chosen to increase that power deprive the -machine of the strength which is necessary to sustain -the force to which it is to be submitted.</p> - -<p id="p297">(297.) These inconveniences are removed by a contrivance -of Mr. Hunter, which, while it gives to the -machine all the requisite strength and compactness, -allows it to have an almost unlimited degree of mechanical -efficacy.</p> - -<p>This contrivance consists in the use of two screws, -the threads of which may have any strength and magnitude, -but which have a very small difference of -breadth. While the working point is urged forward by -that which has the greater thread, it is drawn back by -that which has the less; so that during each revolution -of the screw, instead of being advanced through a space -equal to the magnitude of either of the threads, it moves -through a space equal to their difference. The mechanical -power of such a machine will be the same as that of -a single screw having a thread, whose magnitude is -equal to the difference of the magnitudes of the two -threads just mentioned.</p> - -<p><span class="pagenum" id="Page_221">221</span></p> - -<p>Thus, without inconveniently increasing the sweep of -the power, on the one hand, or, on the other, diminishing -the thread until the necessary strength is lost, the -machine will acquire an efficacy limited by nothing -but the smallness of the difference between the two -threads.</p> - -<p>This principle was first applied in the manner represented -in <i><a href="#i_p224a">fig. 142.</a></i> A is the greater thread, playing in -the fixed nut; B is the lesser thread, cut upon a -smaller cylinder, and playing in a concave screw, cut -within the greater cylinder. During every revolution -of the screw, the cylinder A descends through a space -equal to the distance between its threads. At the same -time the smaller cylinder B ascends through a space -equal to the distance between the threads cut upon it: -the effect is, that the board D descends through a space -equal to the difference between the threads upon A and -the threads upon B, and the machine has a power proportionate -to the smallness of this difference.</p> - -<p>Thus, suppose the screw A has twenty threads in an -inch, while the screw B has twenty-one; during one -revolution, the screw A will descend through a space -equal to the 20th part of an inch. If, during this motion, -the screw B did not turn within A, the board D -would be advanced through the 20th of an inch; but -because the hollow screw within A turns upon B, the screw -B will, relatively to A, be raised in one revolution through -a space equal to the 21st part of an inch. Thus, -while the board D is depressed through the 20th of an -inch by the screw A, it is raised through the 21st of an -inch by the screw B. It is, therefore, on the whole, -depressed through a space equal to the excess of the -20th of an inch above the 21st of an inch, that is, -through the 420th of an inch.</p> - -<p>The power of this machine will, therefore, be expressed -by the number of times the 420th of an inch -is contained in the circumference through which the -power moves.</p> - -<p id="p298">(298.) In the practical application of this principle<span class="pagenum" id="Page_222">222</span> -at present the arrangement is somewhat different. The -two threads are usually cut on different parts of the same -cylinder. If nuts be supposed to be placed upon these, -which are capable of moving in the direction of the length, -but not of revolving, it is evident that by turning the -screw once round, each nut will be advanced through a -space equal to the breadth of the respective threads. By -this means the two nuts will either approach each other, or -mutually recede, according to the direction in which the -screw is turned, through a space equal to the difference -of the breadth of the threads, and they will exert a force -either in compressing or extending any substance placed -between them, proportionate to the smallness of that -difference.</p> - -<p id="p299">(299.) A toothed wheel is sometimes used instead of -a nut, so that the same quality by which the revolution -of the screw urges the nut forward is applied to make -the wheel revolve. The screw is in this case called an -endless screw, because its action upon the wheel may be -continued without limit. This application of the screw -is represented in <i><a href="#i_p242a">fig. 143.</a></i> P is the winch to which the -power is applied; and its effect at the circumference of -the wheel is estimated in the same manner as the effect -of the screw upon the nut. This effect is to be considered -as a power acting upon the circumference of the wheel; -and its proportion to the weight or resistance is to be -calculated in the same manner as the proportion of the -power to the weight in the wheel and axle.</p> - -<p id="p300">(300.) We have hitherto considered the screw as -an engine used to overcome great resistances. It is -also eminently useful in several departments of experimental -science, for the measurement of very minute motions -and spaces, the magnitude of which could scarcely -be ascertained by any other means. The very slow -motion which may be imparted to the end of a screw, -by a very considerable motion in the power, renders it -peculiarly well adapted for this purpose. To explain -the manner in which it is applied—suppose a screw to -be so cut as to have fifty threads in an inch, each revo<span class="pagenum" id="Page_223">223</span>lution -of the screw will advance its point through the -fiftieth part of an inch. Now, suppose the head of the -screw to be a circle, whose diameter is an inch, the circumference -of the head will be something more than three -inches: this may be easily divided into a hundred equal -parts distinctly visible. If a fixed index be presented -to this graduated circumference, the hundredth part of a -revolution of the screw may be observed, by noting the -passage of one division of the head under the index. -Since one entire revolution of the head moves the point -through the fiftieth of an inch, one division will correspond -to the five thousandth of an inch. In order to -observe the motion of the point of the screw in this case, -a fine wire is attached to it, which is carried across the -field of view of a powerful microscope, by which the -motion is so magnified as to be distinctly perceptible.</p> - -<p>A screw used for such purposes is called a <i>micrometer -screw</i>. Such an apparatus is usually attached to the -limbs of graduated instruments, for the purposes of -astronomical and other observation. Without the aid -of this apparatus, no observation could be taken with -greater accuracy than the amount of the smallest division -upon the limb. Thus, if an instrument for measuring -angles were divided into small arcs of one minute, and -an angle were observed which brought the index of the -instrument to some point between two divisions, we could -only conclude that the observed angle must consist of -a certain number of degrees and minutes, together with -an additional number of seconds, which would be unknown, -inasmuch as there would be no means of ascertaining -the fraction of a minute between the index and -the adjacent division of the instrument. But if a screw -be provided, the point of which moves through a space -equal to one division of the instrument, with sixty revolutions -of the head, and that the head itself be divided -into one hundred equal parts, each complete revolution -of the screw will correspond to the sixtieth part of a -minute, or to one second, and each division on the head -of the screw will correspond to the hundredth part of a<span class="pagenum" id="Page_224">224</span> -second. The index being attached to this screw, let the -head be turned until the index be moved from its observed -position to the adjacent division of the limb. The number -of complete revolutions of the screw necessary to accomplish -this will be the number of seconds; and the number -of parts of a revolution over the complete number of revolutions -will be the hundredth parts of a second necessary -to be added to the degrees and minutes primarily -observed.</p> - -<p>It is not, however, only to such instruments that the -micrometer screw is applicable; any spaces whatever -may be measured by it. An instance of its mechanical -application may be mentioned in a steel-yard, an instrument -for ascertaining the amount of weights by a given -weight, sliding on a long graduated arm of a lever. The -distance from the fulcrum, at which this weight counterpoises -the weight to be ascertained, serves as a measure -to the amount of that weight. When the sliding weight -happens to be placed between two divisions of the arm, -a micrometer screw is used to ascertain the fraction of -the division.</p> - -<p>Hunter’s screw, already described, seems to be well -adapted to micrometrical purposes; since the motion of -the point may be rendered indefinitely slow, without requiring -an exquisitely fine thread, such as in the single -screw would be necessary.</p> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<h2 class="nobreak" id="CHAP_XVII">CHAP. XVII.<br /> - -<span class="title">ON THE REGULATION AND ACCUMULATION OF FORCE.</span></h2> -</div> - - -<p id="p301">(301.) <span class="smcap">It</span> is frequently indispensable, and always desirable, -that the operation of a machine should be regular -and uniform. Sudden changes in its velocity, -and desultory variations in the effective energy of its -power, are often injurious or destructive to the apparatus -itself, and when applied to manufactures never fail<span class="pagenum" id="Page_225">225</span> -to produce unevenness in the work. To invent methods -for insuring the regular motion of machinery, by removing -those causes of inequality which may be avoided, -and by compensating others, has therefore been a problem -to which much attention and ingenuity have been -directed. This is chiefly accomplished by controlling, -and, as it were, measuring out the power according to -the exigencies of the machine, and causing its effective -energy to be always commensurate with the resistance -which it has to overcome.</p> - -<div class="figcenter" id="i_p224a" style="max-width: 31.25em;"> - <img src="images/i_p224a.jpg" alt="" /> - <div class="caption"><p> -<span class="l-align"><i>C. Varley, del.</i></span> - -<span class="r-align"><i>H. Adlard, sc.</i></span></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<p>Irregularity in the motion of machinery may proceed -from one or more of the following causes:—1. irregularity -in the prime mover; 2. occasional variation in the -amount of the load or resistance; and, 3. because, in the -various positions which the parts of the machine assume -during its motion, the power may not be transmitted -with equal effect to the working point.</p> - -<p>The energy of the prime mover is seldom if ever -regular. The force of water varies with the copiousness -of the stream. The power which impels the windmill -is proverbially capricious. The pressure of steam varies -with the intensity of the furnace. Animal power, the -result of will, temper, and health is difficult of control. -Human labour is most of all unmanageable; hence no -machine works so irregularly as one which is manipulated. -In some cases the moving force is subject, by the very -conditions of its existence, to constant variation, as in -the example of a spring, which gradually loses its energy -as it recoils. (<a href="#p255">255</a>.) In many instances the prime -mover is liable to regular intermission, and is actually -suspended for certain intervals of time. This is the case -in the single acting steam-engine, where the pressure of -the steam urges the descent of the piston, but is suspended -during its ascent.</p> - -<p>The load or resistance to which the machine is applied -is not less fluctuating. In mills there are a multiplicity -of parts which are severally liable to be -occasionally disengaged, and to have their operation -suspended. In large factories for spinning, weaving,<span class="pagenum" id="Page_226">226</span> -printing, &c. a great number of separate spinning machines, -looms, presses, or other engines, are usually -worked by one common mover, such as a water-wheel -or steam-engine. In these cases the number of machines -employed from time to time necessarily varies -with the fluctuating demand for the articles produced, -and from other causes. Under such circumstances the -velocity with which every part of the machinery is -moved would suffer corresponding changes, increasing -its rapidity with every augmentation of the moving -power or diminution of the resistance, or being retarded -in its speed by the contrary circumstances.</p> - -<p>But even when the prime mover and the resistance -are both regular, or rendered so by proper contrivances, -still it will rarely happen that the machine by which the -energy of the one is transmitted to the other conveys -this with unimpaired effect in all the phases of its operation. -To give a general notion of this cause of inequality -to those who have not been familiar with machinery -would not be easy, without having recourse to -an example. For the present we shall merely state, -that the several moving parts of every machine assume -in succession a variety of positions; that at regular periods -they return to their first position, and again undergo -the same succession of changes. In the different -positions through which they are carried in every period -of motion, the efficacy of the machine to transmit the -power to the resistance is different, and thus the effective -energy of the machine in acting upon the resistance -would be subject to continual fluctuation. This will be -more clearly understood when we come to explain the -methods of counteracting the defect or equalising the -action of the power upon the resistance.</p> - -<p>Such are the chief causes of the inequalities incidental -to the motion of machinery, and we now propose to describe -a few of the many ingenious contrivances which -the skill of engineers has produced to remove the consequent -inconveniences.</p> - -<p id="p302">(302.) Setting aside, for the present, the last cause<span class="pagenum" id="Page_227">227</span> -of inequality, and considering the machinery, whatever -it be, to transmit the power to the resistance without -irregular interruption, it is evident that every contrivance, -having for its object to render the velocity uniform, -can only accomplish this by causing the variations -of the power and resistance to be proportionate to each -other. This may be done either by increasing or diminishing -the power as the resistance increases or -diminishes; or by increasing or diminishing the resistance -as the power increases or diminishes.</p> - -<p>According to the facilities or convenience presented -by the peculiar circumstances of the case either of these -methods is adopted.</p> - -<p>The contrivances for effecting this are called <i>regulators</i>. -Most regulators act upon that part of the machine -which commands the supply of the power by means of -levers, or some other mechanical contrivance, so as to -check the quantity of the moving principle conveyed to -the machine when the velocity has a tendency to increase; -and, on the other hand, to increase that supply -upon any undue abatement of its speed. In a water-mill -this is done by acting upon the shuttle; in a wind-mill, -by an adjustment of the sail-cloth; and in a steam-engine, -by opening or closing, in a greater or less degree, -the valve by which the cylinder is supplied with steam.</p> - -<p id="p303">(303.) Of all the contrivances for regulating machinery, -that which is best known and most commonly -used is the <i>governor</i>. This regulator, which had been -long in use in mill-work and other machinery, has of -late years attracted more general notice by its beautiful -adaptation in the steam-engines of Watt. It consists -of heavy balls B B, <i><a href="#i_p242a">fig. 144.</a></i>, attached to the extremities -of rods B F. These rods play upon a joint at E, -passing through a mortise in the vertical stem D <span class="ilb">D′</span>. At -F they are united by joints to the short rods F H, which -are again connected by joints at H to a ring which slides -upon the vertical shaft D <span class="ilb">D′</span>. From this description it -will be apparent that when the balls B are drawn from -the axis, their upper arms E F are caused to increase<span class="pagenum" id="Page_228">228</span> -their divergence in the same manner as the blades of a -scissors are opened by separating the handles. These, -acting upon the ring by means of the short links F H, -draw it down the vertical axis from D towards E. A -contrary effect is produced when the balls B are brought -closer to the axis, and the divergence of the rods B E -diminished. A horizontal wheel W is attached to the -vertical axis D <span class="ilb">D′</span>, having a groove to receive a rope or -strap upon its rim. This strap passes round the wheel -or axis by which motion is transmitted to the machinery -to be regulated, so that the spindle or shaft D <span class="ilb">D′</span> will -always be made to revolve with a speed proportionate to -that of the machinery.</p> - -<p>As the shaft D <span class="ilb">D′</span> revolves, the balls B are carried -round it with a circular motion, and consequently acquire -a centrifugal force which causes them to recede -from the axle, and therefore to depress the ring H. -On the edge or rim of this ring is formed a groove, -which is embraced by the prongs of a fork I, at the extremity -of one arm of a lever whose fulcrum is at G. -The extremity K of the other arm is connected by some -means with the part of the machine which supplies the -power. In the present instance we shall suppose it a -steam-engine, in which case the rod K I communicates -with a flat circular valve V, placed in the principal -steam-pipe, and so arranged that, when K is elevated as -far as by their divergence the balls B have power over it, -the passage of the pipe will be closed by the valve V, -and the passage of steam entirely stopped; and, on the -other hand, when the balls subside to their lowest position, -the valve will be presented with its edge in the -direction of the tube, so as to intercept no part of the -steam.</p> - -<p>The property which renders this instrument so admirably -adapted to the purpose to which it is applied is, -that when the divergence of the balls is not very considerable, -they must always revolve with the same velocity, -whether they move at a greater or lesser distance -from the vertical axis. If any circumstance increases<span class="pagenum" id="Page_229">229</span> -that velocity, the balls instantly recede from the axis, -and closing the valve V, check the supply of steam, and -thereby diminishing the speed of the motion, restore the -machine to its former rate. If, on the contrary, that -fixed velocity be diminished, the centrifugal force being -no longer sufficient to support the balls, they descend -towards the axle, open the valve V, and, increasing the -supply of steam, restore the proper velocity of the -machine.</p> - -<p>When the governor is applied to a water-wheel it is -made to act upon the shuttle through which the water -flows, and controls its quantity as effectually, and upon -the same principle, as has just been explained in reference -to the steam-engine. When applied to a windmill -it regulates the sail-cloth so as to diminish the efficacy -of the power upon the arms as the force of the wind increases, -or <i>vice versâ</i>.</p> - -<p>In cases where the resistance admits of easy and convenient -change, the governor may act so as to accommodate -it to the varying energy of the power. This is -often done in corn-mills, where it acts upon the shuttle -which metes out the corn to the millstones. When the -power which drives the mill increases, a proportionally -increased feed of corn is given to the stones, so that the -resistance being varied in the ratio of the power, the same -velocity will be maintained.</p> - -<p id="p304">(304.) In some cases the centrifugal force of the -revolving balls is not sufficiently great to control the -power or the resistance, and regulators of a different -kind must be resorted to. The following contrivance is -called the <i>water-regulator</i>:—</p> - -<p>A common pump is worked by the machine, whose -motion is to be regulated, and water is thus raised and -discharged into a cistern. It is allowed to flow from -this cistern through a pipe of a given magnitude. When -the water is pumped up with the same velocity as it is -discharged by this pipe, it is evident that the level of the -water in the cistern will be stationary, since it receives -from the pump the exact quantity which it discharges<span class="pagenum" id="Page_230">230</span> -from the pipe. But if the pump throw in more water -in a given time than is discharged by the pipe, the cistern -will begin to be filled, and the level of the water -will rise. If, on the other hand, the supply from the -pump be less than the discharge from the pipe, the level -of the water in the cistern will subside. Since the rate at -which water is supplied from the pump will always be -proportional to the velocity of the machine, it follows that -every fluctuation in this velocity will be indicated by the -rising or subsiding of the level of the water in the cistern, -and that level never can remain stationary, except -at that exact velocity which supplies the quantity of -water discharged by the pipe. This pipe may be constructed -so as by an adjustment to discharge the water at -any required rate; and thus the cistern may be adapted -to indicate a constant velocity of any proposed amount.</p> - -<p>If the cistern were constantly watched by an attendant, -the velocity of the machine might be abated by -regulating the power when the level of the water is -observed to rise, or increased when it falls; but this -is much more effectually and regularly performed by -causing the surface of the water itself to perform the -duty. A float or large hollow metal ball is placed upon -the surface of the water in the cistern. This ball is -connected with a lever acting upon some part of the machinery, -which controls the power or regulates the amount -of resistance, as already explained in the case of the -governor. When the level of the water rises, the buoyancy -of the ball causes it to rise also with a force -equal to the difference between its own weight and the -weight of as much water as it displaces. By enlarging -the floating ball, a force may be obtained sufficiently -great to move those parts of the machinery -which act upon the power or resistance, and thus either -to diminish the supply of the moving principle or to -increase the amount of the resistance, and thereby retard -the motion and reduce the velocity to its proper limit. -When the level of the water in the cistern falls, the -floating ball being no longer supported on the liquid<span class="pagenum" id="Page_231">231</span> -surface, descends with the force of its own weight, and -producing an effect upon the power or resistance contrary -to the former, increases the effective energy of the one, or -diminishes that of the other, until the velocity proper to -the machine be restored.</p> - -<p>The sensibility of these regulators is increased by -making the surface of water in the cistern as small as -possible; for then a small change in the rate at which the -water is supplied by the pump will produce a considerable -change in the level of the water in the cistern.</p> - -<p>Instead of using a float, the cistern itself may be suspended -from the lever which controls the supply of the -power, and in this case a sliding weight may be placed -on the other arm, so that it will balance the cistern -when it contains that quantity of water which corresponds -to the fixed level already explained. If the -quantity of water in the cistern be increased by an undue -velocity of the machine, the weight of the cistern -will preponderate, draw down the arm of the lever, and -check the supply of the power. If, on the other hand, -the supply of water be too small, the cistern will no -longer balance the counterpoise, the arm by which it is -suspended will be raised, and the energy of the power -will be increased.</p> - -<p id="p305">(305.) In the steam-engine the self-regulating principle -is carried to an astonishing pitch of perfection. -The machine itself raises in due quantity the cold water -necessary to condense the steam. It pumps off the hot -water produced by the steam, which has been cooled, and -lodges it in a reservoir for the supply of the boiler. It -carries from this reservoir exactly that quantity of water -which is necessary to supply the wants of the boiler, and -lodges it therein according as it is required. It breathes -the boiler of redundant steam, and preserves that which -remains fit, both in quantity and quality, for the use of -the engine. It blows its own fire, maintaining its intensity, -and increasing or diminishing it according to the -quantity of steam which it is necessary to raise; so that -when much work is expected from the engine, the fire<span class="pagenum" id="Page_232">232</span> -is proportionally brisk and vivid. It breaks and prepares -its own fuel, and scatters it upon the bars at proper -times and in due quantity. It opens and closes its several -valves at the proper moments, works its own pumps, -turns its own wheels, and is only not alive. Among so -many beautiful examples of the self-regulating principle, -it is difficult to select. We shall, however, mention one -or two, and for others refer the reader to our treatise on -this subject<span class="nowrap">.<a id="FNanchor_3" href="#Footnote_3" class="fnanchor">3</a></span></p> - -<p>It is necessary in this machine that the water in the -boiler be maintained constantly at the same level, and, -therefore, that as much be supplied, from time to time, -as is consumed by evaporation. A pump which is -wrought by the engine itself supplies a cistern C, <i><a href="#i_p242a">fig. 145.</a></i>, -with hot water. At the bottom of this cistern is a -valve V opening into a tube which descends into the -boiler. This valve is connected by a wire with the arm -of a lever on the fulcrum D, the other arm E of which -is also connected by a wire with a stone float F, which -is partially immersed in the water of the boiler, and is -balanced by a sliding weight A. The weight A only -counterpoises the stone float F by the aid of its buoyance -in the water; for if the water be removed, the -stone F will preponderate, and raise the weight A. -When the water in the boiler is at its proper level, the -length of the wire connecting the valve V with the lever -is so adjusted that this valve shall be closed, the wire at -the same time being fully extended. When, by evaporation, -the water in the boiler begins to be diminished, -the level falls, and the stone weight F, being no longer -supported, overcomes the counterpoise A, raises the arm -of the lever, and, pulling the wire, opens the valve V. -The water in the cistern C then flows through the tube -into the boiler, and continues to flow until the level be -so raised that the stone weight F is again elevated, the -valve V closed, and the further supply of water from -the cistern C suspended.</p> - -<p>In order to render the operation of this apparatus<span class="pagenum" id="Page_233">233</span> -easily intelligible, we have here supposed an imperfection -which does not exist. According to what has just been -stated, the level of the water in the boiler descends from -its proper height, and subsequently returns to it. But, -in fact, this does not happen. The float F and valve V -adjust themselves, so that a constant supply of water -passes through the valve, which proceeds exactly at the -same rate as that at which the water in the boiler is -consumed.</p> - -<p id="p306">(306.) In the same machine there occurs a singularly -happy example of self-adjustment, in the method by which -the strength of the fire is regulated. The governor regulates -the supply of steam to the engine, and proportions -it to the work to be done. With this work, therefore, -the demands upon the boiler increase or diminish, and -with these demands the production of steam in the -boiler ought to vary. In fact, the rate at which steam -is generated in the boiler, ought to be equal to that at -which it is consumed in the engine, otherwise one of -two effects must ensue: either the boiler will fail to -supply the engine with steam, or steam will accumulate -in the boiler, being produced in undue quantity, and, -escaping at the safety valve, will thus be wasted. It is, -therefore, necessary to control the agent which generates -the steam, namely, the fire, and to vary its intensity -from time to time, proportioning it to the demands of -the engine. To accomplish this, the following contrivance -has been adopted:—Let T, <i><a href="#i_p242a">fig. 146.</a></i>, be a tube inserted -in the top of the boiler, and descending nearly to -the bottom. The pressure of the steam confined in the -boiler, acting upon the surface of the water, forces it to -a certain height in the tube T. A weight F, half immersed -in the water in the tube, is suspended by a chain, -which passes over the wheels P <span class="ilb">P′</span>, and is balanced by a -metal plate D, in the same manner as the stone float, -<i><a href="#i_p242a">fig. 145.</a></i>, is balanced by the weight A. The plate D passes -through the mouth of the flue E as it issues finally from the -boiler; so that when the plate D falls it stops the flue, -suspending thereby the draught of air through the furnace,<span class="pagenum" id="Page_234">234</span> -mitigating the intensity of the fire, and checking the production -of steam. If, on the contrary, the plate D be -drawn up, the draught is increased, the fire is rendered -more active, and the production of steam in the boiler -is stimulated. Now, suppose that the boiler produces -steam faster than the engine consumes it, either because -the load on the engine has been diminished, and, therefore, -its consumption of steam reduced, or because the -fire has become too intense; the consequence is, that the -steam, beginning to accumulate in the boiler, will press -upon the surface of the water with increased force, and -the water will be raised in the tube T. The weight F -will, therefore, be lifted, and the plate D will descend, -diminish, or stop the draught, mitigate the fire, and retard -the production of steam, and will continue to do so -until the rate at which steam is produced shall be commensurate -to the wants of the engine. If, on the -other hand, the production of steam be inadequate to -the exigency of the machine, either because of an increased -load, or of the insufficient force of the fire, the -steam in the boiler will lose its elasticity, and the surface -of the water not sustaining its wonted pressure, the -water in the tube T will fall; consequently the weight -F will descend, and the plate D will be raised. The -flue being thus opened, the draught will be increased, -and the fire rendered more intense. Thus the production -of steam becomes more rapid, and is rendered -sufficiently abundant for the purposes of the engine. -This apparatus is called the <i>self-acting damper</i>.</p> - -<p id="p307">(307.) When a perfectly uniform rate of motion -has not been attained, it is often necessary to indicate -small variations of velocity. The following contrivance, -called a <span class="nowrap"><i>tachometer</i><a id="FNanchor_4" href="#Footnote_4" class="fnanchor">4</a></span>, has been invented to accomplish this. -A cup, <i><a href="#i_p242a">fig. 147.</a></i>, is filled to the level C D with quicksilver, -and is attached to a spindle, which is whirled by the -machine in the same manner as the governor already -described. It is well known that the centrifugal force -produced by this whirling motion will cause the mer<span class="pagenum" id="Page_235">235</span>cury -to recede from the centre and rise upon the sides -of the cup, so that its surface will assume the concave -appearance represented in <i><a href="#i_p242a">fig. 148.</a></i> In this case the -centre of the surface will obviously have fallen below -its original level, <i><a href="#i_p242a">fig. 147.</a></i>, and the edges will have risen -above that level. As this effect is produced by the velocity -of the machine, so it is proportionate to that -velocity, and subject to corresponding variations. Any -method of rendering visible small changes in the central -level of the surface of the quicksilver will indicate minute -variations in the velocity of the machine.</p> - -<p>A glass tube A, open at both ends, and expanding at -one extremity into a bell B, is immersed with its wider -end in the mercury, the surface of which will stand at -the same level in the bell B, and in the cup C D. The -tube is so suspended as to be unconnected with the cup. -This tube is then filled to a certain height A, with spirits -tinged with some colouring matter, to render it easily -observable. When the cup is whirled by the machine -to which it is attached, the level of the quicksilver -in the bell falls, leaving more space for the spirits, -which, therefore, descends in the tube. As the motion -is continued, every change of velocity causes a corresponding -change in the level of the mercury, and, therefore, -also in the level A of the spirits. It will be -observed, that, in consequence of the capacity of the bell -B being much greater than that of the tube A, a very -small change in the level of the quicksilver in the bell -will produce a considerable change in the height of the -spirits in the tube. Thus this ingenious instrument -becomes a very delicate indicator of variations in the -motion of machinery.</p> - -<p id="p308">(308.) The governor, and other methods of regulating -the motion of machinery which have been just described, -are adapted principally to cases in which the -proportion of the resistance to the load is subject to certain -fluctuations or gradual changes, or at least to cases -in which the resistance is not at any time entirely withdrawn, -nor the energy of the power actually suspended.<span class="pagenum" id="Page_236">236</span> -Circumstances, however, frequently occur in which, while -the power remains in full activity, the resistance is at -intervals suddenly removed and as suddenly again returns. -On the other hand, cases also present themselves, -in which, while the resistance is continued, the impelling -power is subject to intermission at regular periods. -In the former case, the machine would be driven with a -ruinous rapidity during those periods at which it is -relieved from its load, and on the return of the load every -part would suffer a violent strain, from its endeavour to -retain the velocity which it had acquired, and the speedy -destruction of the engine could not fail to ensue. In the -latter case, the motion would be greatly retarded or -entirely suspended during those periods at which the -moving power is deprived of its activity, and, consequently, -the motion which it would communicate would -be so irregular as to be useless for the purposes of manufactures.</p> - -<p>It is also frequently desirable, by means of a weak -but continued power, to produce a severe but instantaneous -effect. Thus a blow may be required to be given -by the muscular action of a man’s arm with a force to -which, unaided by mechanical contrivance, its strength -would be entirely inadequate.</p> - -<p>In all these cases, it is evident that the object to be -attained is, an effectual method of accumulating the energy -of the power so as to make it available after the action -by which it has been produced has ceased. Thus, in the -case in which the load is at periodical intervals withdrawn -from the machine, if the force of the power could be -imparted to something by which it would be preserved, -so as to be brought against the load when it again -returned, the inconvenience would be removed. In like -manner, in the case where the power itself is subject to -intermission, if a part of the force which it exerts in its -intervals of action could be accumulated and preserved, -it might be brought to bear upon the machine during its -periods of suspension. By the same means of accumulating -force, the strength of an infant, by repeated efforts,<span class="pagenum" id="Page_237">237</span> -might produce effects which would be vainly attempted -by the single and momentary action of the strongest -man.</p> - -<p id="p309">(309.) The property of inertia, explained and illustrated -in the third and fourth chapters of this volume -furnishes an easy and effectual method of accomplishing -this. A mass of matter retains, by virtue of its inertia, -the whole of any force which may have been given -to it, except that part of which friction and the atmospheric -resistance deprives it. By contrivances which are -well known and present no difficulty, the part of the -moving force thus lost may be rendered comparatively -small, and the moving mass may be regarded as retaining -nearly the whole of the force impressed upon it. To -render this method of accumulating force fully intelligible, -let us first imagine a polished level plane on which a -heavy globe of metal, also polished, is placed. It is -evident that the globe will remain at rest on any part of -the plane without a tendency to move in any direction. -As the friction is nearly removed by the polish of the -surfaces, the globe will be easily moved by the least -force applied to it. Suppose a slight impulse given to -it, which will cause it to move at the rate of one foot in -a second. Setting aside the effects of friction, it will -continue to move at this rate for any length of time. -The same impulse repeated will increase its speed to two -feet per second. A third impulse to three feet, and so -on. Thus 10,000 repetitions of the impulse will cause -it to move at the rate of 10,000 feet per second. If the -body to which these impulses were communicated were -a cannon ball, it might, by a constant repetition of the -impelling force, be at length made to move with as much -force as if it were projected from the most powerful -piece of ordnance. The force with which the ball in -such a case would strike a building might be sufficient -to reduce it to ruins, and yet such force would be -nothing more than the accumulation of a number of -weak efforts not beyond the power of a child to exert, -which are stored up, and preserved, as it were, by the<span class="pagenum" id="Page_238">238</span> -moving mass, and thereby brought to bear, at the same -moment, upon the point to which the force is directed. -It is the sum of a number of actions exerted successively, -and, during a long interval, brought into operation at -one and the same moment.</p> - -<p>But the case which is here supposed cannot actually -occur; because we have not usually any practical -means of moving a body for any considerable time in -the same direction without much friction, and without -encountering numerous obstacles which would impede -its progress. It is not, however, essential to the effect -which is to be produced, that the motion should be in a -straight line. If a leaden weight be attached to the end -of a light rod or cord, and be whirled by the force of -the arm in a circle, it will gradually acquire increased -speed and force, and at length may receive an impetus -which would cause it to penetrate a piece of board as -effectually as if it were discharged from a musket.</p> - -<p>The force of a hammer or sledge depends partly on -its weight, but much more on the principle just explained. -Were it allowed merely to fall by the force of its weight -upon the head of a nail, or upon a bar of heated iron -which is to be flattened, an inconsiderable effect would -be produced. But when it is wielded by the arm of a -man, it receives at every moment of its motion increased -force, which is finally expended in a single instant on -the head of the nail, or on the bar of iron.</p> - -<p>The effects of flails in threshing, of clubs, whips, canes, -and instruments for striking, axes, hatchets, cleavers, -and all instruments which cut by a blow, depend on the -same principle, and are similarly explained.</p> - -<p>The bow-string which impels the arrow does not -produce its effect at once. It continues to act upon -the shaft until it resumes its straight position, and then -the arrow takes flight with the force accumulated during -the continuance of the action of the string, from the -moment it was disengaged from the finger of the bow-man.</p> - -<p>Fire-arms themselves act upon a similar principle,<span class="pagenum" id="Page_239">239</span> -as also the air-gun and steam-gun. In these instruments -the ball is placed in a tube, and suddenly exposed -to the pressure of a highly elastic fluid, either produced -by explosion as in fire-arms, by previous condensation -as in the air-gun, or by the evaporation of highly heated -liquids as in the steam-gun. But in every case this -pressure continues to act upon it until it leaves the mouth -of the tube, and then it departs with the whole force -communicated to it during its passage along the tube.</p> - -<p id="p310">(310.) From all these considerations it will easily be -perceived that a mass of inert matter may be regarded -as a magazine in which force may be deposited and accumulated, -to be used in any way which may be necessary. -For many reasons, which will be sufficiently -obvious, the form commonly given to the mass of matter -used for this purpose in machinery is that of a wheel, -in the rim of which it is principally collected. Conceive -a massive ring of metal, <i><a href="#i_p242a">fig. 149.</a></i>, connected with -a central box or nave by light spokes, and turning on -an axis with little friction. Such an apparatus is called -a fly-wheel. If any force be applied to it, with that -force (making some slight deduction for friction) it will -move, and will continue to move until some obstacle be -opposed to its motion, which will receive from it a part -of the force it has acquired. The uses of this apparatus -will be easily understood by examples of its application.</p> - -<p>Suppose that a heavy stamper or hammer is to be -raised to a certain height, and thence to be allowed to -fall, and that the power used for this purpose is a water-wheel. -While the stamper ascends, the power of the -wheel is nearly balanced by its weight, and the motion -of the machine is slow. But the moment the stamper -is disengaged and allowed to fall, the power of the wheel, -having no resistance, nor any object on which to expend -itself, suddenly accelerates the machine, which moves -with a speed proportioned to the amount of the power, -until it again engages the stamper, when its velocity is -as suddenly checked. Every part suffers a strain, and -the machine moves again slowly until it discharges its<span class="pagenum" id="Page_240">240</span> -load, when it is again accelerated, and so on. In this -case, besides the certainty of injury and wear, and the -probability of fracture from the sudden and frequent -changes of velocity, nearly the whole force exerted by -the power in the intervals between the commencement -of each descent of the stamper and the next ascent is -lost. These defects are removed by a fly-wheel. When -the stamper is discharged, the energy of the power is -expended in moving the wheel, which, by reason of its -great mass, will not receive an undue velocity. In the -interval between the descent and ascent of the stamper, -the force of the power is lodged in the heavy rim of the -fly-wheel. When the stamper is again taken up by the -machine, this force is brought to bear upon it, combined -with the immediate power of the water-wheel, and the -stamper is elevated with nearly the same velocity as that -with which the machine moved in the interval of its -descent.</p> - -<p id="p311">(311.) In many cases, when the moving power is not -subject to variation, the efficacy of the machine to transmit -it to the working point is subject to continual change. -The several parts of every machine have certain periods -of motion, in which they pass through a variety of positions, -to which they continually return after stated -intervals. In these different positions the effect of the -power transmitted to the working point is different; and -cases even occur in which this effect is altogether annihilated, -and the machine is brought into a predicament -in which the power loses all influence over the weight. -In such cases the aid of a fly-wheel is effectual and indispensable. -In those phases of the machine, which are -most favourable to the transmission of force, the fly-wheel -shares the effect of the power with the load, and -retaining the force thus received directs it upon the -load at the moments when the transmission of power by -the machine is either feeble or altogether suspended. -These general observations will, perhaps, be more clearly -apprehended by an example of an application of the fly-wheel, -in a case such as those now alluded to.</p> - -<p><span class="pagenum" id="Page_241">241</span></p> - -<p>Let A B C D E F, <i><a href="#i_p242a">fig. 150.</a></i>, be a <i>crank</i>, which is a -double winch (<a href="#p252">252</a>.) and <i><a href="#i_p182a">fig. 89.</a></i>), by which an axle, -A B E F, is to be turned. Attached to the middle of -C D by a joint is a rod, which is connected with a beam, -worked with an alternate motion on a centre, like the -brake of a pump, and driven by any constant power, -such as a steam-engine. The bar C D is to be carried -with a circular motion round the axis A E. Let the -machine, viewed in the direction A B E F of the axis, -be conceived to be represented in <i><a href="#i_p252a">fig. 151.</a></i>, where A represents -the centre round which the motion is to be produced, -and G the point where the connecting rod G H is -attached to the arm of the crank. The circle through -which G is to be urged by the rod is represented by the -dotted line. In the position represented in <i><a href="#i_p252a">fig. 151.</a></i>, the -rod acting in the direction H G has its full power to -turn the crank G A round the centre A. As the crank -comes into the position represented in <i><a href="#i_p252a">fig. 152.</a></i>, this -power is diminished, and when the point G comes immediately -below A, as in <i><a href="#i_p252a">fig. 153.</a></i>, the force in the direction -H G has no effect in turning the crank round A, but, on -the contrary, is entirely expended in pulling the crank -in the direction A G, and, therefore, only acts upon the -pivots or gudgeons which support the axle. At this -crisis of the motion, therefore, the whole effective energy -of the power is annihilated.</p> - -<p>After the crank has passed to the position represented -in <i><a href="#i_p252a">fig. 154.</a></i>, the direction of the force which acts -upon the connecting rod is changed, and now the crank -is drawn upward in the direction G H. In this position -the moving force has some efficacy to produce rotation -round A, which efficacy continually increases until the -crank attains the position shown in <i><a href="#i_p252a">fig. 155.</a></i>, when its -power is greatest. Passing from this position its efficacy -is continually diminished, until the point G comes immediately -above the axis A, <i><a href="#i_p252a">fig. 156.</a></i> Here again the -power loses all its efficacy to turn the axle. The force -in the direction G H or H G can obviously produce no -other effect than a strain upon the pivots or gudgeons.</p> - -<p><span class="pagenum" id="Page_242">242</span></p> - -<p>In the critical situations represented in <i><a href="#i_p252a">fig. 153.</a></i>, and -<i><a href="#i_p252a">fig. 156.</a></i>, the machine would be incapable of moving, -were the immediate force of the power the only impelling -principle. But having been previously in motion -by virtue of the inertia of its various parts, it has a -tendency to continue in motion; and if the resistance -of the load and the effects of friction be not too great, -this disposition to preserve its state of motion will extricate -the machine from the dilemma in which it is -involved in the cases just mentioned, by the peculiar -arrangement of its parts. In many cases, however, the -force thus acquired during the phases of the machine, in -which the power is active, is insufficient to carry it -through the dead points (<i><a href="#i_p252a">fig. 153.</a></i> and <i><a href="#i_p252a">fig. 156.</a></i>); and -in all cases the motion would be very unequal, being -continually retarded as it approached these points, and -continually accelerated after it passed them. A fly-wheel -attached to the axis A, or to some other part of the machinery, -will effectually remove this defect. When the -crank assumes the positions in <i><a href="#i_p252a">fig. 151.</a></i> and <i><a href="#i_p252a">fig. 155.</a></i>, the -power is in full play upon it, and a share of the effect is -imparted to the massive rim of the fly-wheel. When -the crank gets into the predicament exhibited in <i><a href="#i_p252a">fig. 153.</a></i> -and <i><a href="#i_p252a">fig. 156.</a></i>, the momentum which the fly-wheel received -when the crank acted with most advantage, immediately -extricates the machine, and, carrying the crank -beyond the dead point, brings the power again to bear -upon it.</p> - -<p>The astonishing effects of a fly-wheel, as an accumulator -of force, have led some into the error of supposing -that such an apparatus increases the actual power of a -machine. It is hoped, however, that after what has been -explained respecting the inertia of matter and the true -effects of machines, the reader will not be liable to a -similar mistake. On the contrary, as a fly cannot act -without friction, and as the amount of the friction, like -that of inertia, is in proportion to the weight, a portion of -the actual moving force must unavoidably be lost by the -use of a fly. In cases, however, where a fly is properly<span class="pagenum" id="Page_243">243</span> -applied this loss of power is inconsiderable, compared -with the advantageous distribution of what remains.</p> - -<div class="figcenter" id="i_p242a" style="max-width: 31.25em;"> - <img src="images/i_p242a.jpg" alt="" /> - <div class="caption"><p> -<span class="l-align"><i>C. Varley, del.</i></span> - -<span class="r-align"><i>H. Adlard, sc.</i></span></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<p>As an accumulator of force, a fly can never have more -force than has been applied to put it in motion. In this -respect it is analogous to an elastic spring, or the force -of condensed air, or any other power which derives its -existence from causes purely mechanical. In bending -a spring a gradual expenditure of power is necessary. -On the recoil this power is exerted in a much shorter -time than that consumed in its production, but its total -amount is not altered. Air is condensed by a succession -of manual efforts, one of which alone would be incapable -of projecting a leaden ball with any considerable force, -and all of which could not be immediately applied to the -ball at the same instant. But the reservoir of condensed -air is a magazine in which a great number of such efforts -are stored up, so as to be brought at once into action. If -a ball be exposed to their effect, it may be projected with -a destructive force.</p> - -<p>In mills for rolling metal the fly-wheel is used in this -way. The water-wheel or other moving power is allowed -for some time to act upon the fly-wheel alone, no -load being placed upon the machine. A force is thus -gained which is sufficient to roll a large piece of metal, -to which without such means the mill would be quite -inadequate. In the same manner a force may be gained -by the arm of a man acting on a fly for a few seconds, -sufficient to impress an image on a piece of metal by an -instantaneous stroke. The fly is, therefore, the principal -agent in coining presses.</p> - -<p id="p312">(312.) The power of a fly is often transmitted to the -working point by means of a screw. At the extremities -of the cross arm A B, <i><a href="#i_p252a">fig. 157.</a></i>, which works the screw, -two heavy balls of metal are placed. When the arm A B -is whirled round, those masses of metal acquire a momentum, -by which the screw, being driven downward, urges -the die with an immense force against the substance destined -to receive the impression.</p> - -<p>Some engines used in coining have flies with arms<span class="pagenum" id="Page_244">244</span> -four feet long, bearing one hundred weight at each of -their extremities. By turning such an arm at the rate -of one entire circumference in a second, the die will -be driven against the metal with the same force as that -with which 7500 pounds weight would fall from the -height of 16 feet; an enormous power, if the simplicity -and compactness of the machine be considered.</p> - -<p>The place to be assigned to a fly-wheel relatively to -the other parts of the machinery is determined by the -purpose for which it is used. If it be intended to equalise -the action, it should be near the working point. Thus, -in a steam-engine, it is placed on the crank which turns -the axle by which the power of the engine is transmitted -to the object it is finally designed to affect. On the -contrary, in handmills, such as those commonly used for -grinding coffee, &c., it is placed upon the axis of the -winch by which the machine is worked.</p> - -<p>The open work of fenders, fire-grates, and similar -ornamental articles constructed in metal, is produced by -the action of a fly, in the manner already described. -The cutting tool, shaped according to the pattern to be -executed, is attached to the end of the screw; and the -metal being held in a proper position beneath it, the fly -is made to urge the tool downwards with such force as -to stamp out pieces of the required figure. When the -pattern is complicated, and it is necessary to preserve -with exactness the relative situation of its different parts, -a number of punches are impelled together, so as to strike -the entire piece of metal at the same instant, and in this -manner the most elaborate open work is executed by a -single stroke.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_245">245</span></p> - -<h2 class="nobreak" id="CHAP_XVIII">CHAP. XVIII.<br /> - -<span class="title">MECHANICAL CONTRIVANCES FOR MODIFYING MOTION.</span></h2> -</div> - - -<p id="p313">(313.) <span class="smcap">The</span> classes of simple machines denominated -mechanic powers, have relation chiefly to the peculiar -principle which determines the action of the power on -the weight or resistance. In explaining this arrangement -various other reflections have been incidentally -mixed up with our investigations; yet still much -remains to be unfolded before the student can form a -just notion of those means by which the complex machinery -used in the arts and manufactures so effectually -attains the ends, to the accomplishment of which it is -directed.</p> - -<p>By a power of a given energy to oppose a resistance -of a different energy, or by a moving principle having a -given velocity to generate another velocity of a different -amount, is only one of the many objects to be effected -by a machine. In the arts and manufactures the <i>kind</i> -of motion produced is generally of greater importance -than its <i>rate</i>. The latter may affect the quantity of work -done in a given time, but the former is essential to the -performance of the work in any quantity whatever. In -the practical application of machines, the object to be -attained is generally to communicate to the working -point some peculiar sort of motion suitable to the uses -for which the machine is intended; but it rarely happens -that the moving power has this sort of motion. Hence, -the machine must be so contrived that, while that part -on which this power acts is capable of moving in obedience -to it, its connection with the other parts shall be -such that the working point may receive that motion -which is necessary for the purposes to which the machine -is applied.</p> - -<p>To give a perfect solution of this problem it would be -necessary to explain, first, all the varieties of moving<span class="pagenum" id="Page_246">246</span> -powers which are at our disposal; secondly, all the variety -of motions which it may be necessary to produce; -and, thirdly, to show all the methods by which each -variety of prime mover may be made to produce the -several species of motion in the working point. It is -obvious that such an enumeration would be impracticable, -and even an approximation to it would be unsuitable -to the present treatise. Nevertheless, so much -ingenuity has been displayed in many of the contrivances -for modifying motion, and an acquaintance -with some of them is so essential to a clear comprehension -of the nature and operation of complex machines, -that it would be improper to omit some account of those -at least which most frequently occur in machinery, or -which are most conspicuous for elegance and simplicity.</p> - -<p>The varieties of motion which most commonly present -themselves in the practical application of mechanics may -be divided into <i>rectilinear</i> and <i>rotatory</i>. In rectilinear -motion the several parts of the moving body proceed in -parallel straight lines with the same speed. In rotatory -motion the several points revolve round an axis, each -performing a complete circle, or similar parts of a circle, -in the same time.</p> - -<p>Each of these may again be resolved into continued -and reciprocating. In a continued motion, whether rectilinear -or rotatory, the parts move constantly in the same -direction, whether that be in parallel straight lines, or in -rotation on an axis. In reciprocating motion the several -parts move alternately in opposite directions, tracing the -same spaces from end to end continually. Thus, there -are four principal species of motion which more frequently -than any others act upon, or are required to be -transmitted by, machines:—</p> - -<p class="ml2em">1. <i>Continued rectilinear motion.</i><br /> - -2. <i>Reciprocating rectilinear motion.</i><br /> - -3. <i>Continued circular motion.</i><br /> - -4. <i>Reciprocating circular motion.</i></p> - -<p>These will be more clearly understood by examples of -each kind.</p> - -<p><span class="pagenum" id="Page_247">247</span></p> - -<p>Continued rectilinear motion is observed in the flowing -of a river, in a fall of water, in the blowing of the -wind, in the motion of an animal upon a straight road, -in the perpendicular fall of a heavy body, in the motion -of a body down an inclined plane.</p> - -<p>Reciprocating rectilinear motion is seen in the piston -of a common syringe, in the rod of a common pump, in -the hammer of a pavier, the piston of a steam-engine, -the stampers of a fulling mill.</p> - -<p>Continued circular motion is exhibited in all kinds of -wheel-work, and is so common, that to particularise it -is needless.</p> - -<p>Reciprocating circular motion is seen in the pendulum -of a clock, and in the balance-wheel of a watch.</p> - -<p>We shall now explain some of the contrivances by -which a power having one of these motions may be made -to communicate either the same species of motion -changed in its velocity or direction, or any of the other -three kinds of motion.</p> - -<p id="p314">(314.) By a continued rectilinear motion another continued -rectilinear motion in a different direction may be -produced, by one or more fixed pulleys. A cord passed -over these, one end of it being moved by the power, will -transmit the same motion unchanged to the other end. -If the directions of the two motions cross each other, one -fixed pulley will be sufficient; see <i><a href="#i_p204a">fig. 113.</a></i>, where the -hand takes the direction of the one motion, and the -weight that of the other. In this case the pulley must -be placed in the angle at which the directions of the two -motions cross each other. If this angle be distant from -the places at which the objects in motion are situate, an -inconvenient length of rope may be necessary. In this -case the same may be effected by the use of two pulleys, -as in <i><a href="#i_p252a">fig. 158.</a></i></p> - -<p>If the directions of the two motions be parallel, two -pulleys must be used as in <i><a href="#i_p252a">fig. 158.</a></i>, where <span class="ilb">P′</span> <span class="ilb">A′</span> is one -motion, and B W the other. In these cases the axles of -the two wheels are parallel.</p> - -<p>It may so happen that the directions of the two mo<span class="pagenum" id="Page_248">248</span>tions -neither cross each other nor are parallel. This -would happen, for example, if the direction of one were -upon the paper in the line P A, while the other were -perpendicular to the paper from the point O. In this -case two pulleys should be used, the axle of one <span class="ilb">O′</span> being -perpendicular to the paper, while the axle of the other O -should be on the paper. This will be evident by a little -reflection.</p> - -<p>In general, the axle of each pulley must be perpendicular -to the two directions in which the rope passes -from its groove; and by due attention to this condition -it will be perceived, that a continued rectilinear motion -may be transferred from any one direction to any other -direction, by means of a cord and two pulleys, without -changing its velocity.</p> - -<p>If it be necessary to change the velocity, any of the -systems of pulleys described in chap. <span class="lowercase smcap">XV</span>. may be used -in addition to the fixed pulleys.</p> - -<p>By the wheel and axle any one continued rectilinear -motion may be made to produce another in any other -direction, and with any other velocity. It has been -already explained (<a href="#p250">250</a>.) that the proportion of the velocity -of the power to that of the weight is as the diameter -of the wheel to the diameter of the axle. The thickness -of the axle being therefore regulated in relation to the -size of the wheel, so that their diameters shall have that -proportion which subsists between the proposed velocities, -one condition of the problem will be fulfilled. The -rope coiled upon the axle may be carried, by means of -one or more fixed pulleys, into the direction of one of the -proposed motions, while that which surrounds the wheel -is carried into the direction of the other by similar -means.</p> - -<p id="p315">(315.) By the wheel and axle a continued rectilinear -motion may be made to produce a continued rotatory -motion, or <i>vice versâ</i>. If the power be applied by a -rope coiled upon the wheel, the continued motion of the -power in a straight line will cause the machine to have -a rotatory motion. Again, if the weight be applied by<span class="pagenum" id="Page_249">249</span> -a rope coiled upon the axle, a power having a rotatory -motion applied to the wheel will cause the continued ascent -of the weight in a straight line.</p> - -<p>Continued rectilinear and rotatory motions may be -made to produce each other, by causing a toothed wheel -to work in a straight bar, called a <i>rack</i>, carrying teeth -upon its edge. Such an apparatus is represented in -<i><a href="#i_p252a">fig. 159.</a></i></p> - -<p>In some cases the teeth of the wheel work in the -links of a chain. The wheel is then called a <i>rag-wheel</i>, -<i><a href="#i_p252a">fig. 160.</a></i></p> - -<p>Straps, bands, or ropes, may communicate rotation -to a wheel, by their friction in a groove upon its edge.</p> - -<p>A continued rectilinear motion is produced by a continued -circular motion in the case of a screw. The -lever which turns the screw has a continued circular motion, -while the screw itself advances with a continued -rectilinear motion.</p> - -<p>The continued rectilinear motion of a stream of water -acting upon a wheel produces continued circular motion -in the wheel, <i><a href="#i_p182a">fig. 93</a></i>, <i><a href="#i_p188a">94</a></i>, <i>95</i>. In like manner the continued -rectilinear motion of the wind produces a continued -circular motion in the arms of a windmill.</p> - -<p>Cranes for raising and lowering heavy weights convert -a circular motion of the power into a continued rectilinear -motion of the weight.</p> - -<p id="p316">(316.) Continued circular motion may produce reciprocating -rectilinear motion, by a great variety of ingenious -contrivances.</p> - -<p>Reciprocating rectilinear motion is used when heavy -stampers are to be raised to a certain height, and allowed -to fall upon some object placed beneath them. This -may be accomplished by a wheel bearing on its edge -curved teeth, called <i>wipers</i>. The stamper is furnished -with a projecting arm or peg, beneath which the -wipers are successively brought by the revolution of -the wheel. As the wheel revolves the wiper raises the -stamper, until its extremity passes the extremity of the -projecting arm of the stamper, when the latter imme<span class="pagenum" id="Page_250">250</span>diately -falls by its own weight. It is then taken up by -the next wiper, and so the process is continued.</p> - -<p>A similar effect is produced if the wheel be partially -furnished with teeth, and the stamper carry a rack in -which these teeth work. Such an apparatus is represented -in <i><a href="#i_p252a">fig. 161.</a></i></p> - -<p>It is sometimes necessary that the reciprocating rectilinear -motion shall be performed at a certain varying -rate in both directions. This may be accomplished by -the machine represented in <i><a href="#i_p252a">fig. 162.</a></i> A wheel upon the -axle C turns uniformly in the direction A B D E. -A rod <i>mn</i> moves in guides, which only permit it to ascend -and descend perpendicularly. Its extremity <i>m</i> -rests upon a path or groove raised from the face of the -wheel, and shaped into such a curve that as the wheel -revolves the rod <i>mn</i> shall be moved alternately in opposite -directions through the guides, with the required -velocity. The manner in which the velocity varies -will depend on the form given to the groove or channel -raised upon the face of the wheel, and this may be -shaped so as to give any variation to the motion of the -rod <i>mn</i> which may be required for the purpose to which -it is to be applied.</p> - -<p>The <i>rose-engine</i> in the turning-lathe is constructed on -this principle. It is also used in spinning machinery.</p> - -<p>It is often necessary that the rod to which reciprocating -motion is communicated shall be urged by the -same force in both directions. A wheel partially furnished -with teeth, acting on two racks placed on different -sides of it, and both connected with the bar or -rod to which the reciprocating motion is to be communicated, -will accomplish this. Such an apparatus is -represented in <i><a href="#i_p252a">fig. 163.</a></i>, and needs no further explanation.</p> - -<p>Another contrivance for the same purpose is shown in -<i><a href="#i_p252a">fig. 164.</a></i>, where A is a wheel turned by a winch H, and -connected with a rod or beam moving in guides by the -joint <i>ab</i>. As the wheel A is turned by the winch H -the beam is moved between the guides alternately in<span class="pagenum" id="Page_251">251</span> -opposite directions, the extent of its range being governed -by the length of the diameter of the wheel. Such -an apparatus is used for grinding and polishing plane -surfaces, and also occurs in silk machinery.</p> - -<p>An apparatus applied by M. Zureda in a machine for -pricking holes in leather is represented in <i><a href="#i_p252a">fig. 165.</a></i> The -wheel A B has its circumference formed into teeth, the -shape of which may be varied according to the circumstances -under which it is to be applied. One extremity -of the rod <i>ab</i> rests upon the teeth of the wheel upon -which it is pressed by a spring at the other extremity. -When the wheel revolves, it communicates to this rod a -reciprocating rectilinear motion.</p> - -<p>Leupold has applied this mechanism to move the pistons -of pumps<span class="nowrap">.<a id="FNanchor_5" href="#Footnote_5" class="fnanchor">5</a></span> Upon the vertical axis of a horizontal -hydraulic wheel is fixed another horizontal wheel, -which is furnished with seven teeth in the manner of a -crown wheel (<a href="#p263">263</a>.). These teeth are shaped like inclined -planes, the intervals between them being equal to -the length of the planes. Projecting arms attached to -the piston rods rest upon the crown of this wheel; and, -as it revolves, the inclined surfaces of the teeth, being -forced under the arm, raise the rod upon the principle -of the wedge. To diminish the obstruction arising from -friction, the projecting arms of the piston rods are provided -with rollers, which run upon the teeth of the -wheel. In one revolution of the wheel each piston -makes as many ascents and descents as there are teeth.</p> - -<p id="p317">(317.) Wheel-work furnishes numerous examples of -continued circular motion round one axis, producing -continued circular motion round another. If the axles -be in parallel directions, and not too distant, rotation -may be transmitted from one to the other by two spur-wheels -(<a href="#p263">263</a>.); and the relative velocities may be determined -by giving a corresponding proportion to the -diameter of the wheels.</p> - -<p>If a rotary motion is to be communicated from one -axis to another parallel to it, and at any considerable<span class="pagenum" id="Page_252">252</span> -distance, it cannot in practice be accomplished by wheels -alone, for their diameters would be too large. In this -case a strap or chain is carried round the circumferences -of both wheels. If they are intended to turn in the -same direction, the strap is arranged as in <i><a href="#i_p188a">fig. 100.</a></i>; but -if in contrary directions it is crossed, as in <i><a href="#i_p188a">fig. 101.</a></i> In -this case, as with toothed wheels, the relative velocities -are determined by the proportion of the diameters of the -wheels.</p> - -<p>If the axles be distant and not parallel, the cord, by -which the motion is transmitted, must be passed over -grooved wheels, or fixed pulleys, properly placed between -the two axles.</p> - -<p>It may happen that the strain upon the wheel, to -which the motion is to be transmitted, is too great to -allow of a strap or cord being used. In this case, a -shaft extending from the one axis to another, and carrying -two bevelled wheels (<a href="#p263">263</a>.), will accomplish the object. -One of these bevelled wheels is placed upon the shaft -near to, and in connection with, the wheel from which -the motion is to be taken, and the other at a part of it -near to, and in connection with, that wheel to which -the motion is to be conveyed, <i><a href="#i_p260a">fig. 166.</a></i></p> - -<p>The methods of transmitting rotation from one axis -to another perpendicular to it, by crown and by bevelled -wheels, have been explained in (<a href="#p263">263</a>.).</p> - -<p>The endless screw (<a href="#p299">299</a>.) is a machine by which a -rotatory motion round one axis may communicate a -rotatory motion round another perpendicular to it. The -power revolves round an axis coinciding with the length -of the screw, and the axis of the wheel driven by the -screw is at right angles to this.</p> - -<p>The axis to which rotation is to be given, or from -which it is to be taken, is sometimes variable in its position. -In such cases, an ingenious contrivance, called -a <i>universal joint</i>, invented by the celebrated Dr. Hooke, -may be used. The two shafts or axles A B, <i><a href="#i_p260a">fig. 167.</a></i>, -between which the motion is to be communicated, terminate -in semicircles, the diameters of which, C D and<span class="pagenum" id="Page_253">253</span> -E F, are fixed in the form of a cross, their extremities -moving freely in bushes placed in the extremities of the -semicircles. Thus, while the central cross remains unmoved, -the shaft A and its semicircular end may revolve -round C D as an axis; and the shaft B and its semicircular -end may revolve round E F as an axis. If the -shaft A be made to revolve without changing its direction, -the points C D will move in a circle whose centre -is at the middle of the cross. The motion thus given -to the cross will cause the points E F to move in another -circle round the same centre, and hence the shaft B will -be made to revolve.</p> - -<div class="figcenter" id="i_p252a" style="max-width: 31.25em;"> - <img src="images/i_p252a.jpg" alt="" /> - <div class="caption"><p> -<span class="l-align"><i>C. Varley, del.</i></span> - -<span class="r-align"><i>H. Adlard, sc.</i></span></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<p>This instrument will not transmit the motion if the -angle under the directions of the shafts be less than 140°. -In this case a double joint, as represented in <i><a href="#i_p260a">fig. 168.</a></i>, -will answer the purpose. This consists of four semicircles -united by two crosses, and its principle and -operation is the same as in the last case.</p> - -<p>Universal joints are of great use in adjusting the position -of large telescopes, where, while the observer -continues to look through the tube, it is necessary to turn -endless screws or wheels whose axes are not in an accessible -position.</p> - -<p>The cross is not indispensably necessary in the universal -joint. A hoop, with four pins projecting from it -at four points equally distant from each other, or dividing -the circle of the hoop into four equal arcs, will -answer the purpose. These pins play in the bushes of -the semicircles in the same manner as those of the cross.</p> - -<p>The universal joint is much used in cotton-mills, -where shafts are carried to a considerable distance from -the prime mover, and great advantage is gained by dividing -them into convenient lengths, connected by a -joint of this kind.</p> - -<p id="p318">(318.) In the practical application of machinery, it is -often necessary to connect a part having a continued circular -motion with another which has a reciprocating or -alternate motion, so that either may move the other.<span class="pagenum" id="Page_254">254</span> -There are many contrivances by which this may be -effected.</p> - -<p>One of the most remarkable examples of it is presented -in the scapements of watches and clocks. In this -case, however, it can scarcely be said with strict propriety -that it is the rotation of the scapement-wheel -(<a href="#p266">266</a>.) which <i>communicates</i> the vibration to the balance-wheel -or pendulum. That vibration is produced in the -one case by the peculiar nature of the spiral spring fixed -upon the axis of the balance-wheel, and in the other -case by the gravity of the pendulum. The force of the -scapement-wheel only <i>maintains</i> the vibration, and prevents -its decay by friction and atmospheric resistance. -Nevertheless, between the two parts thus moving there -exists a mechanical connection, which is generally -brought within the class of contrivances now under consideration.</p> - -<p>A beam vibrating on an axis, and driven by the piston -of a steam-engine, or any other power, may communicate -rotary motion to an axis by a connector and a crank. -This apparatus has been already described in (<a href="#p311">311</a>.). -Every steam-engine which works by a beam affords an -example of this. The working beam is generally placed -over the engine, the piston rod being attached to one -end of it, while the connecting rod unites the other end -with the crank. In boat-engines, however, this position -would be inconvenient, requiring more room than could -easily be spared. The piston rod, in these cases, is, -therefore, connected with the end of the beam by long -rods, and the beam is placed beside and below the engine. -The use of a fly-wheel here would also be objectionable. -The effect of the dead points explained in (<a href="#p311">311</a>.) is -avoided without the aid of a fly, by placing two cranks -upon the revolving axle, and working them by two pistons. -The cranks are so placed that when either is at its -dead point, the other is in its most favourable position.</p> - -<p>A wheel A, <i><a href="#i_p260a">fig. 169.</a></i>, armed with wipers, acting -upon a sledge-hammer B, fixed upon a centre or axle C, -will, by a continued rotatory motion, give the hammer<span class="pagenum" id="Page_255">255</span> -the reciprocating motion necessary for the purposes to -which it is applied. The manner in which this acts -must be evident on inspecting the figure.</p> - -<p>The treddle of the lathe furnishes an obvious example -of a vibrating circular motion producing a continued -circular one. The treddle acts upon a crank, which -gives motion to the principal wheel, in the same manner -as already described in reference to the working beam -and crank in the steam-engine.</p> - -<p>By the following ingenious mechanism an alternate -or vibrating force may be made to communicate a circular -motion continually in the same direction. Let -A B, <i><a href="#i_p260a">fig. 170.</a></i>, be an axis receiving an alternate motion -from some force applied to it, such as a swinging weight. -Two ratchet wheels (<a href="#p253">253</a>.) <i>m</i> and <i>n</i> are fixed on this -axle, their teeth being inclined in opposite directions. -Two toothed wheels C and D are likewise placed upon -it, but so arranged that they turn upon the axle with a -little friction. These wheels carry two catches <i>p</i>, <i>q</i>, -which fall into the teeth of the ratchet wheels <i>m</i>, <i>n</i>, but -fall on opposite sides conformably to the inclination of -the teeth already mentioned. The effect of these catches -is, that if the axis be made to revolve in one direction, -one of the two toothed wheels is always compelled (by -the catch <i>against</i> which the motion is directed) to revolve -with it, while the other is permitted to remain -stationary in obedience to any force sufficiently great to -overcome its friction with the axle on which it is placed. -The wheels C and D are both engaged by bevelled teeth -(<a href="#p263">263</a>.) with the wheel E.</p> - -<p>According to this arrangement, in whichever direction -the axis A B is made to revolve, the wheel E will continually -turn in the same direction, and, therefore, if the -axle A B be made to turn alternately in the one direction -and the other, the wheel E will not change the direction -of its motion. Let us suppose that the axle A B is turned -against the catch <i>p</i>. The wheel C will then be made to turn -with the axle. This will drive the wheel E in the same -direction. The teeth on the opposite side of the wheel E<span class="pagenum" id="Page_256">256</span> -being engaged with those of the wheel D, the latter will -be turned upon the axle, the friction, which alone resists -its motion in that direction, being overcome. Let the -motion of the axle A B be now reversed. Since the -teeth of the ratchet wheel <i>n</i> are moved against the -catch <i>q</i>, the wheel D will be compelled to revolve with -the axle. The wheel E will be driven in the same direction -as before, and the wheel C will be moved on the -axle A B, and in a contrary direction to the motion of -the axle, the friction being overcome by the force of the -wheel E. Thus, while the axle A B is turned alternately -in the one direction and the other, the wheel E is constantly -moved in the same direction.</p> - -<p>It is evident that the direction in which the wheel E -moves may be reversed by changing the position of the -ratchet wheels and catches.</p> - -<p id="p319">(319.) It is often necessary to communicate an alternate -circular motion, like that of a pendulum, by means of -an alternate motion in a straight line. A remarkable instance -of this occurs in the steam engine. The moving -force in this machine is the pressure of steam, which impels -a piston from end to end alternately in a cylinder. -The force of this piston is communicated to the working -beam by a strong rod, which passes through a collar in -one end of the piston. Since it is necessary that the steam -included in the cylinder should not escape between the -piston rod and the collar through which it moves, and yet, -that it should move as freely and be subject to as little resistance -as possible, the rod is turned so as to be truly -cylindrical, and is well polished. It is evident that, -under these circumstances, it must not be subject to any -lateral or cross strain, which would bend it towards one -side or the other of the cylinder. But the end of the beam -to which it communicates motion, if connected immediately -with the rod by a joint, would draw it alternately -to the one side and the other, since it moves in the arc -of a circle, the centre of which is at the centre of the -beam. It is necessary, therefore, to contrive some method -of connecting the rod and the end of the beam, so<span class="pagenum" id="Page_257">257</span> -that while the one shall ascend and descend in a straight -line, the other may move in the circular arc.</p> - -<p>The method which first suggests itself to accomplish -this is, to construct an arch-head upon the end of the -beam, as in <i><a href="#i_p260a">fig. 171.</a></i> Let C be the centre on which the -beam works, and let B D be an arch attached to the end -of the beam, being a part of a circle having C for its -centre. To the highest point B of the arch a chain is -attached, which is carried upon the face of the arch B A, -and the other end of which is attached to the piston rod. -Under these circumstances it is evident, that when the -force of the steam impels the piston downwards, the -chain P A B will draw the end of the beam down, and -will, therefore, elevate the other end.</p> - -<p>When the steam-engine is used for certain purposes, -such as pumping, this arrangement is sufficient. The -piston in that case is not forced upwards by the pressure -of steam. During its ascent it is not subject to -the action of any force of steam, and the other end of -the beam falls by the weight of the pump-rods drawing -the piston, at the opposite end A, to the top of the cylinder. -Thus the machine is in fact passive during the -ascent of the piston, and exerts its power only during -the descent.</p> - -<p>If the machine, however, be applied to purposes -in which a constant action of the moving force is necessary, -as is always the case in manufactures, the force of -the piston must drive the beam in its ascent as well as -in its descent. The arrangement just described cannot -effect this; for although a chain is capable of transmitting -any force, by which its extremities are drawn in opposite -directions, yet it is, from its flexibility, incapable -of communicating a force which drives one extremity of -it towards the other. In the one case the piston first <i>pulls</i> -down the beam, and then the beam <i>pulls</i> up the piston. -The chain, because it is inextensible, is perfectly capable -of both these actions; and being flexible, it applies itself -to the arch-head of the beam, so as to maintain the direction -of its force upon the piston continually in the<span class="pagenum" id="Page_258">258</span> -same straight line. But when the piston acts upon the -beam in both ways, in pulling it down and pushing it -up, the chain becomes inefficient, being from its flexibility -incapable of the latter action.</p> - -<p>The problem might be solved by extending the length -of the piston rod, so that its extremity shall be above -the beam, and using two chains; one connecting the -highest point of the rod with the lowest point of the -arch-head, and the other connecting the highest point of -the arch-head with a point on the rod below the point -which meets the arch-head when the piston is at the top -of the cylinder, <i><a href="#i_p260a">fig. 172.</a></i></p> - -<p>The connection required may also be made by arming -the arch-head with teeth, <i><a href="#i_p260a">fig. 173.</a></i>, and causing the piston -rod to terminate in a rack. In cases where, as in -the steam-engine, smoothness of motion is essential, this -method is objectionable; and under any circumstances -such an apparatus is liable to rapid wear.</p> - -<p>The method contrived by Watt, for connecting the -motion of the piston with that of the beam, is one of the -most ingenious and elegant solutions ever proposed for a -mechanical problem. He conceived the motion of two -straight rods A B, C D, <i><a href="#i_p260a">fig. 174.</a></i>, moving on centres or -pivots A and C, so that the extremities B and D would -move in the arcs of circles having their centres at A -and C. The extremities B and D of these rods he -conceived to be connected with a third rod B D united -with them by pivots on which it could turn freely. -To the system of rods thus connected let an alternate -motion on the centres A and C be communicated: the -points B and D will move upwards and downwards in -the arcs expressed by the dotted lines, but the middle -point P of the connecting rod B D will move upwards and -downwards without any sensible deviation from a straight -line.</p> - -<p>To prove this demonstratively would require some -abstruse mathematical investigation. It may, however, -be rendered in some degree apparent by reasoning of a -looser and more popular nature. As the point B is raised<span class="pagenum" id="Page_259">259</span> -to E, it is also drawn aside towards the right. At the -same time the other extremity D of the rod B D is -raised to <span class="ilb">E′</span>, and is drawn aside towards the left. The -ends of the rod B D being thus at the same time drawn -equally towards opposite sides, its middle point P will -suffer no lateral derangement, and will move directly -upwards. On the other hand, if B be moved downwards -to F, it will be drawn laterally to the right, while -D being moved to <span class="ilb">F′</span> will be drawn to the left. Hence, -as before, the middle point P sustains no lateral derangement, -but merely descends. Thus, as the extremities B -and D move upwards and downwards in circles, the -middle point P moves upwards and downwards in a -straight line<span class="nowrap">.<a id="FNanchor_6" href="#Footnote_6" class="fnanchor">6</a></span></p> - -<p>The application of this geometrical principle in the -steam-engine evinces much ingenuity. The same arm -of the beam usually works two pistons, that of the cylinder -and that of the <i>air-pump</i>. The apparatus is represented -on the arm of the beam in <i><a href="#i_p260a">fig. 175.</a></i> The -beam moves alternately upwards and downwards on its -axis A. Every point of it, therefore, describes a part of -a circle of which A is the centre. Let B be the point -which divides the arm A G into two equal parts A B -and B G; and let C D be a straight rod, equal in length -to G B, and fixed on a centre or pivot C on which it is -at liberty to play. The end D of this rod is connected by -a straight bar with the point B, by pivots on which the -rod B D turns freely. If the beam be now supposed -to rise and fall alternately, the points B and D will move -upwards and downwards in circular arcs, and, as already -explained with respect to the points B D, <i><a href="#i_p260a">fig. 174.</a></i>, the -middle point P of the connecting rod B D will move -upwards and downwards without lateral deflection. To -this point one of the piston rods which are to be worked -is attached.</p> - -<p><span class="pagenum" id="Page_260">260</span></p> - -<p>To comprehend the method of working the other piston, -conceive a rod G <span class="ilb">P′</span>, equal in length to B D, to be -attached to the end G of the beam by a pivot on which -it moves freely; and let its extremity <span class="ilb">P′</span> be connected -with D by another rod <span class="ilb">P′</span> D, equal in length to G B, -and playing on points at <span class="ilb">P′</span> and D. The piston rod of -the cylinder is attached to the point <span class="ilb">P′</span>, and this point -has a motion precisely similar to that of P, without any -lateral derangement, but with a range in the perpendicular -direction twice as great. This will be apparent -by conceiving a straight line drawn from the centre A -of the beam to <span class="ilb">P′</span>, which will also pass through P. -Since G <span class="ilb">P′</span> is always parallel to B P, it is evident that -the triangle <span class="ilb">P′</span> G A is always similar to P B A, and -has its sides and angles similarly placed, but those sides -are each twice the magnitude of the corresponding sides -of the other triangle. Hence the point <span class="ilb">P′</span> must be subject -to the same changes of position as the point P, with -this difference only, that in the same time it moves over -a space of twice the magnitude. In fact, the line traced -by <span class="ilb">P′</span> is the same as that traced by P, but on a scale -twice as large. This contrivance is usually called the -<i>parallel motion</i>, but the same name is generally applied -to all contrivances by which a circular motion is made -to produce a rectilinear one.</p> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<h2 class="nobreak" id="CHAP_XIX">CHAP. XIX.<br /> - -<span class="title">OF FRICTION AND THE RIGIDITY OF CORDAGE.</span></h2> -</div> - - -<p id="p320">(320.) <span class="smcap">With</span> a view to the simplification of the elementary -theory of machines, the consideration of several -mechanical effects of great practical importance has been -postponed, and the attention of the student has been -directed exclusively to the way in which the moving -power is modified in being transmitted to the resistance -independently of such effects. A machine has been re<span class="pagenum" id="Page_261">261</span>garded -as an instrument by which a moving principle, -inapplicable in its existing state to the purpose for which -it is required, may be changed either in its velocity or -direction, or in some other character, so as to be adapted -to that purpose. But in accomplishing this, the several -parts of the machine have been considered as possessing -in a perfect degree qualities which they enjoy only in an -imperfect degree; and accordingly the conclusions to -which by such reasoning we are conducted are infected -with errors, the amount of which will depend on the -degree in which the machinery falls short of perfection -in those qualities which theoretically are imputed to it.</p> - -<div class="figcenter" id="i_p260a" style="max-width: 31.25em;"> - <img src="images/i_p260a.jpg" alt="" /> - <div class="caption"> -<p><span class="l-align"><i>C. Varley, del.</i></span> - -<span class="r-align"><i>H. Adlard, sc.</i></span></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<p>Of the several parts of a machine, some are designed -to move, while others are fixed; and of those which -move, some have motions differing in quantity and direction -from those of others. The several parts, whether -fixed or movable, are subject to various strains and pressures, -which they are intended to resist. These forces -not only vary according to the load which the machine -has to overcome, but also according to the peculiar form -and structure of the machine itself. During the operation -the surfaces of the movable parts move in immediate -contact with the surfaces either of fixed parts or of parts -having other motions. If these surfaces were endued -with perfect smoothness or polish, and the several parts -subject to strains possessed perfect inflexibility and infinite -strength, then the effects of machinery might be -practically investigated by the principles already explained. -But the materials of which every machine is -formed are endued with limited strength, and therefore -the load which is placed upon it must be restricted accordingly, -else it will be liable to be distorted by the -flexure, or even to be destroyed by the fracture of those -parts which are submitted to an undue strain. The surfaces -of the movable parts, and those surfaces with which -they move in contact, cannot in practice be rendered so -smooth but that such roughness and inequality will remain -as sensibly to impede the motion. To overcome -such an impediment requires no inconsiderable part of<span class="pagenum" id="Page_262">262</span> -the moving power. This part is, therefore, intercepted -before its arrival at the working point, and the resistance -to be finally overcome is deprived of it. The property -thus depending on the imperfect smoothness of surfaces, -and impeding the motion of bodies whose surfaces are in -immediate contact, is called <i>friction</i>. Before we can -form a just estimate of the effects of machinery, it is -necessary to determine the force lost by this impediment, -and the laws which under different circumstances -regulate that loss.</p> - -<p>When cordage is engaged in the formation of any part -of a machine, it has hitherto been considered as possessing -perfect flexibility. This is not the case in practice; -and the want of perfect flexibility, which is called <i>rigidity</i>, -renders a certain quantity of force necessary to bend a -cord or rope over the surface of an axle or the groove of -a wheel. During the motion of the rope a different part -of it must thus be continually bent, and the force which -is expended in producing the necessary flexure must -be derived from the moving power, and is thus intercepted -on its way to the working point. In calculating -the effects of cordage, due regard must be had to this -waste of power; and therefore it is necessary to enquire -into the laws which govern the flexure of imperfectly -flexible ropes, and the way in which these affect the machines -in which ropes are commonly used.</p> - -<p>To complete, therefore, the elementary theory of machinery, -we propose in the present and following chapter -to explain the principal laws which determine the effects -of friction, the rigidity of cordage, and the strength of -materials.</p> - -<p id="p321">(321.) If a horizontal plane surface were perfectly -smooth, and free from the smallest inequalities, and a -body having a flat surface also perfectly smooth were -placed upon it, any force applied to the latter would -put it in motion, and that motion would continue undiminished -as long as the body would remain upon the -smooth horizontal surface. But if this surface, instead -of being every where perfectly even, had in particular<span class="pagenum" id="Page_263">263</span> -places small projecting eminences, a certain quantity of -force would be necessary to carry the moving body over -these, and a proportional diminution in its rate of motion -would ensue. Thus, if such eminences were of -frequent occurrence, each would deprive the body of a -part of its speed, so that between that and the next it -would move with a less velocity than it had between the -same and the preceding one. This decrease being continued -by a sufficient number of such eminences encountering -the body in succession, the velocity would at -last be so much diminished that the body would not -have sufficient force to carry it over the next eminence, -and its motion would thus altogether cease.</p> - -<p>Now, instead of the eminences being at a considerable -distance asunder, suppose them to be contiguous, and to -be spread in every direction over the horizontal plane, -and also suppose corresponding eminences to be upon the -surface of the moving body; these projections incessantly -encountering one another will continually obstruct -the motion of the body, and will gradually diminish its -velocity, until it be reduced to a state of rest.</p> - -<p>Such is the cause of friction. The amount of this -resisting force increases with the magnitude of these -asperities, or with the roughness of the surfaces; but it -does not solely depend on this. The surfaces remaining -the same, a little reflection on the method of illustration -just adopted, will show that the amount of -friction ought also to depend upon the force with which -the surfaces moving one upon the other are pressed together. -It is evident, that as the weight of the body -supposed to move upon the horizontal plane is increased, -a proportionally greater force will be necessary to carry -it over the obstacles which it encounters, and therefore -it will the more speedily be deprived of its velocity -and reduced to a state of rest.</p> - -<p id="p322">(322.) Thus we might predict with probability, that -which accurate experimental enquiry proves to be true, -that the resistance from friction depends conjointly on -the roughness of the surfaces and the force of the<span class="pagenum" id="Page_264">264</span> -pressure. When the surfaces are the same, a double -pressure will produce a double amount of friction, a -treble pressure a treble amount of friction, and so on.</p> - -<p>Experiment also, however, gives a result which, at -least at first view, might not have been anticipated from -the mode of illustration we have adopted. It is found -that the resistance arising from friction does not at all -depend on the magnitude of the surface of contact; but -provided the nature of the surfaces and the amount of -pressure remain the same, this resistance will be equal, -whether the surfaces which move one upon the other be -great or small. Thus, if the moving body be a flat -block of wood, the face of which is equal to a square -foot in magnitude, and the edge of which does not exceed -a square inch, it will be subject to the same amount of -friction, whether it move upon its broad face or upon its -narrow edge. If we consider the effect of the pressure in -each case, we shall be able to perceive why this must be -the case. Let us suppose the weight of the block to be -144 ounces. When it rests upon its face, a pressure to -this amount acts upon a surface of 144 square inches, so -that a pressure of one ounce acts upon each square inch. -The total resistance arising from friction will, therefore, -be 144 times that resistance which would be produced -by a surface of one square inch under a pressure of one -ounce. Now, suppose the block placed upon its edge, -there is then a pressure of 144 ounces upon a surface -equal to one square inch. But it has been already -shown, that when the surface is the same, the friction -must increase in proportion to the pressure. Hence we -infer that the friction produced in the present case is -144 times the friction which would be produced by a -pressure of one ounce acting on one square inch of -surface, which is the same resistance as that which the -body was proved to be subject to when resting on its -face.</p> - -<p>These two laws, that friction is independent of the -magnitude of the surface, and is proportional to the -pressure when the quality of the surfaces is the same,<span class="pagenum" id="Page_265">265</span> -are useful in practice, and <i>generally</i> true. In very extreme -cases they are, however, in error. When the -pressure is very intense, in proportion to the surface, -the friction is somewhat <i>less</i> than it would be by these -laws; and when it is very small in proportion to the -surface, it is somewhat <i>greater</i>.</p> - -<p id="p323">(323.) There are two methods of establishing by -experiment the laws of friction, which have been just -explained.</p> - -<p>First. The surfaces between which the friction is to -be determined being rendered perfectly flat, let one be -fixed in the horizontal position on a table T <span class="ilb">T′</span>, <i><a href="#i_p278a">fig. 176.</a></i>; -and let the other be attached to the bottom of a box B C, -adapted to receive weights, so as to vary the pressure. -Let a silken cord S P, attached to the box, be carried -parallel to the table over a wheel at P, and let a dish D -be suspended from it. If no friction existed between -the surfaces, the smallest weight appended to the cord -would draw the box towards P with a continually increasing -speed. But the friction which always exists -interrupts this effect, and a small weight may act upon -the string without moving the box at all. Let weights -be put in the dish D, until a sufficient force is obtained -to overcome the friction without giving the box an accelerated -motion. Such a weight is equivalent to the -amount of the friction.</p> - -<p>The amount of the weight of the box being previously -ascertained, let this weight be now doubled by -placing additional weights in the box. The pressure -will thus be doubled, and it will be found that the -weight of the dish D and its load, which before was -able to overcome the friction, is now altogether inadequate -to it. Let additional weights be placed in the -dish until the friction be counteracted as before, and it -will be observed, that the whole weight necessary to -produce this effect is exactly twice the weight which -produced it in the former case. Thus it appears that -a double amount of pressure produces a double amount -of friction; and in a similar way it may be proved,<span class="pagenum" id="Page_266">266</span> -that any proposed increase or decrease of the pressure -will be attended with a proportionate variation in the -amount of the friction.</p> - -<p>Second. Let one of the surfaces be attached to a flat -plane A B, <i><a href="#i_p278a">fig. 177.</a></i>, which can be placed at any inclination -with an horizontal plane B C, the other surface -being, as before, attached to the box adapted to receive -weights. The box being placed upon the plane, let the -latter be slightly elevated. The tendency of the box to -descend upon A B, will bear the same proportion to its -entire weight as the perpendicular A E bears to the -length of the plane A B (<a href="#p286">286</a>.). Thus if the length -A B be 36 inches, and the height A E be three inches, -that is a twelfth part of the length, then the tendency -of the weight to move down the plane is equal to a -twelfth part of its whole amount. If the weight were -twelve ounces, and the surfaces perfectly smooth, a force -of one ounce acting up the plane would be necessary to -prevent the descent of the weight.</p> - -<p>In this case also the pressure on the plane will be represented -by the length of the base B E (<a href="#p286">286</a>.), that is, -it will bear the same proportion to the whole weight -as B E bears to B A. The relative amounts of the -weight, the tendency to descend, and the pressure, will -always be exhibited by the relative lengths of A B, A E, -and B E.</p> - -<p>This being premised, let the elevation of the plane -A B be gradually increased until the tendency of the -weight to descend just overcomes the friction, but not -so much as to allow the box to descend with accelerated -speed. The proportion of the whole weight, which -then acts down the plane, will be found by measuring -the height A E, and the pressure will be determined by -measuring the base B E. Now let the weight in the -box be increased, and it will be found that the same -elevation is necessary to overcome the friction; nor will -this elevation suffer any change, however the pressure -or the magnitude of the surfaces which move in contact -may be varied.</p> - -<p><span class="pagenum" id="Page_267">267</span></p> - -<p>Since, therefore, in all these cases, the height A E -and the base B E remain the same, it follows that the -proportion between the friction and pressure is undisturbed.</p> - -<p id="p324">(324.) The law that friction is proportional to the -pressure, has been questioned by the late professor Vince -of Cambridge, who deduced from a series of experiments, -that although the friction increases with the pressure, -yet that it increases in a somewhat less ratio; and from -this it would follow, that the variation of the surface of -contact must produce some effect upon the amount of -friction. The law, as we have explained it, however, is -sufficiently near the truth for most practical purposes.</p> - -<p id="p325">(325.) There are several circumstances regarding the -quality of the surfaces which produce important effects -on the quantity of friction, and which ought to be -noticed here.</p> - -<p>This resistance is different in the surfaces of different -substances. When the surfaces are those of wood newly -planed, it amounts to about half the pressure, but is different -in different kinds of wood. The friction of metallic -surfaces is about one fourth of the pressure.</p> - -<p>In general the friction between the surfaces of bodies -of different kinds is less than between those of the same -kind. Thus, between wood and metal the friction is -about one fifth of the pressure.</p> - -<p>It is evident that the smoother the surfaces are the -less will be the friction. On this account, the friction -of surfaces, when first brought into contact, is often -greater than after their attrition has been continued for -a certain time, because that process has a tendency to -remove and rub off those minute asperities and projections -on which the friction depends. But this has a -limit, and after a certain quantity of attrition the friction -ceases to decrease. Newly planed surfaces of wood -have at first a degree of friction which is equal to half -the entire pressure, but after they are worn by attrition -it is reduced to a third.</p> - -<p>If the surfaces in contact be placed with their grains<span class="pagenum" id="Page_268">268</span> -in the same direction, the friction will be greater than -if the grains cross each other.</p> - -<p>Smearing the surfaces with unctuous matter diminishes -the friction, probably by filling the cavities between the -minute projections which produce the friction.</p> - -<p>When the surfaces are first placed in contact, the -friction is less than when they are suffered to rest so for -some time; this is proved by observing the force which -in each case is necessary to move the one upon the -other, that force being less if applied at the first moment -of contact than when the contact has continued. -This, however, has a limit. There is a certain time, -different in different substances, within which this resistance -attains its greatest amount. In surfaces of -wood this takes place in about two minutes; in metals -the time is imperceptibly short; and when a surface of -wood is placed upon a surface of metal, it continues to -increase for several days. The limit is larger when the -surfaces are great, and belong to substances of different -kinds.</p> - -<p>The velocity with which the surfaces move upon one -another produces but little effect upon the friction.</p> - -<p id="p326">(326.) There are several ways in which bodies may -move one upon the other, in which friction will produce -different effects. The principal of these are, first, the -case where one body <i>slides</i> over another; the second, -where a body having a round form <i>rolls</i> upon another; -and, <i>thirdly</i>, where an axis revolves within a hollow -cylinder, or the hollow cylinder revolves upon the axis.</p> - -<p>With the same amount of pressure and a like quality -of surface, the quantity of friction is greatest in the first -case and least in the second. The friction in the second -case also depends on the diameter of the body which -rolls, and is small in proportion as that diameter is great. -Thus a carriage with large wheels is less impeded by -the friction of the road than one with small wheels.</p> - -<p>In the third case, the leverage of the wheel aids the -power in overcoming the friction. Let <i><a href="#i_p278a">fig. 178.</a></i> represent -a section of the wheel and axle; let C be the centre<span class="pagenum" id="Page_269">269</span> -of the axle, and let B E be the hollow cylinder in the -nave of the wheel in which the axle is inserted. If B -be the part on which the axle presses, and the wheel -turn in the direction N D M, the friction will act at B in -the direction B F, and with the leverage B C. The -power acts against this at D in the direction D A, and -with the leverage D C. It is therefore evident, that as -D C is greater than B C, in the same proportion does -the power act with mechanical advantage on the friction.</p> - -<p id="p327">(327.) Contrivances for diminishing the effects of -friction depend on the properties just explained, the -motion of rolling being as much as possible substituted -for that of sliding; and where the motion of rolling -cannot be applied, that of a wheel upon its axle is used. -In some cases both these motions are combined.</p> - -<p>If a heavy load be drawn upon a plane in the manner -of a sledge, the motion will be that of sliding, the -species which is attended with the greatest quantity of -friction; but if the load be placed upon cylindrical -rollers, the nature of the motion is changed, and becomes -that in which there is the least quantity of friction. -Thus large blocks of stone, or heavy beams of -timber, which would require an enormous power to move -them on a level road, are easily advanced when rollers -are put under them.</p> - -<p>When very heavy weights are to be moved through -small spaces, this method is used with advantage; but -when loads ore to be transported to considerable distances, -the process is inconvenient and slow, owing to -the necessity of continually replacing the rollers in front -of the load as they are left behind by its progressive -advancement.</p> - -<p>The wheels of carriages may be regarded as rollers -which are continually carried forward with the load. -In addition to the friction of the rolling motion on the -road, they have, it is true, the friction of the axle in -the nave; but, on the other hand, they are free from the -friction of the rollers with the under surface of the load, -or the carriage in which the load is transported. The<span class="pagenum" id="Page_270">270</span> -advantages of wheel carriages in diminishing the effects -of friction is sometimes attributed to the slowness with -which that axle moves within the box, compared with -the rate at which the wheel moves over the road; but -this is erroneous. The quantity of friction does not in -any case vary considerably with the velocity of the motion, -but least of all does it in that particular kind of -motion here considered.</p> - -<p>In certain cases, where it is of great importance to -remove the effects of friction, a contrivance called <i>friction-wheels</i>, -or friction-rollers, is used. The axle of a -friction-wheel, instead of revolving within a hollow -cylinder, which is fixed, rests upon the edges of wheels -which revolve with it; the species of motion thus becomes -that in which the friction is of least amount.</p> - -<p>Let A B and D C, <i><a href="#i_p278a">fig. 179.</a></i>, be two wheels revolving -on pivots P Q with as little friction as possible, and so -placed that the axle O of a third wheel E F may rest -between their edges. As the wheel E F revolves, the -axle O, instead of grinding its surface on the surface on -which it presses, carries that surface with it, causing the -wheels A B, C D, to revolve.</p> - -<p>In wheel carriages, the roughness of the road is more -easily overcome by large wheels than by small ones. -The cause of this arises partly from the large wheels -not being so liable to sink into holes as small ones, but -more because, in surmounting obstacles, the load is -elevated less abruptly. This will be easily understood -by observing the curves in <i><a href="#i_p278a">fig. 180.</a></i>, which represent -the elevation of the axle in each case.</p> - -<p id="p328">(328.) If a carriage were capable of moving on a -road without friction, the most advantageous direction -in which a force could be applied to draw it would be -parallel to the road. When the motion is impeded by -friction, it is better, however, that the line of draught -should be inclined to the road, so that the drawing force -may be expended partly in lessening the pressure on the -road, and partly in advancing the load.</p> - -<p>Let W, <i><a href="#i_p278a">fig. 181.</a></i>, be a load which is to be moved<span class="pagenum" id="Page_271">271</span> -upon the plane surface A B. If the drawing force be -applied in the direction C D, parallel to the plane A B, -it will have to overcome the friction produced by the -pressure of the whole weight of the load upon the plane; -but if it be inclined upwards in the direction C E, it -will be equivalent to two forces expressed (<a href="#p74">74</a>.) by C G -and C F. The part C G has the effect of lightening the -pressure of the carriage upon the road, and therefore -of diminishing the friction in the same proportion. The -part C F draws the load along the plane. Since C F is -less than C E or C D the whole moving force, it is evident -that a part of the force of draught is lost by this -obliquity; but, on the other hand, a part of the opposing -resistance is also removed. If the latter exceed the -former, an advantage will be gained by the obliquity; -but if the former exceed the latter, force will be lost.</p> - -<p>By mathematical reasoning, founded on these considerations, -it is proved that the best angle of draught is -exactly that obliquity which should be given to the road -in order to enable the carriage to move of itself. This -obliquity is sometimes called the <i>angle of repose</i>, and is -that angle which determines the proportion of the friction -to the pressure in the second method, explained in -(<a href="#p323">323</a>.). The more rough the road is, the greater will -this angle be; and therefore it follows, that on bad roads -the obliquity of the traces to the road should be greater -than on good ones. On a smooth Macadamised way, a -very slight declivity would cause a carriage to roll by -its own weight: hence, in this case, the traces should be -nearly parallel to the road.</p> - -<p>In rail roads, for like reasons, the line of draught -should be parallel to the road, or nearly so.</p> - -<p id="p329">(329.) When ropes or cords form a part of machinery, -the effects of their imperfect flexibility are in a -certain degree counteracted by bending them over the -grooves of wheels. But although this so far diminishes -these effects as to render ropes practically useful, yet -still, in calculating the powers of machinery, it is necessary -to take into account some consequences of the<span class="pagenum" id="Page_272">272</span> -rigidity of cordage which even by these means are not -removed.</p> - -<p>To explain the way in which the stiffness of a rope -modifies the operation of a machine, we shall suppose it -bent over a wheel and stretched by weights A B, <i><a href="#i_p278a">fig. 182.</a></i>, -at its extremities. The weights A and B being equal, -and acting at C and D in opposite ways, balance the -wheel. If the weight A receive an addition, it will -overcome the resistance of B, and turn the wheel in the -direction D E C. Now, for the present, let us suppose -that the rope is perfectly inflexible; the wheel and -weights will be turned into the position represented in -<i><a href="#i_p278a">fig. 183.</a></i> The leverage by which A acts will be diminished, -and will become O F, having been before O C; -and the leverage by which B acts will be increased to -O G, having been before O D.</p> - -<p>But the rope not being inflexible will yield partially -to the effects of the weights A and B, and the parts A C -and B D will be bent into the forms represented in -<i><a href="#i_p278a">fig. 184.</a></i> The form of the curvature which the rope on -each side of the wheel receives is still such that the -descending weight A works with a diminished leverage -F O, while the ascending weight resists it with an increased -leverage G O. Thus so much of the moving -power is lost, by the stiffness of the rope, as is necessary -to compensate this disadvantageous change in the power -of the machine.</p> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<h2 class="nobreak" id="CHAP_XX">CHAP. XX.<br /> - -<span class="title">ON THE STRENGTH OF MATERIALS.</span></h2> -</div> - - -<p id="p330">(330.) <span class="smcap">Experimental</span> enquiries into the laws which -regulate the strength of solid bodies, or their power to -resist forces variously applied to tear or break them, are -obstructed by practical difficulties, the nature and extent -of which are so discouraging that few have ventured -to encounter them at all, and still fewer have had the<span class="pagenum" id="Page_273">273</span> -steadiness to persevere until any result showing a general -law has been obtained. These difficulties arise, partly -from the great forces which must be applied, but more -from the peculiar nature of the objects of those experiments. -The end to which such an enquiry must be -directed is the development of a <i>general law</i>; that is, -such a rule as would be rigidly observed if the materials, -the strength of which is the object of enquiry, were perfectly -uniform in their texture, and subject to no casual -inequalities. In proportion as these inequalities are frequent, -experiments must be multiplied, that a long average -may embrace cases varying in both extremes, so as -to eliminate each other’s effects in the final result.</p> - -<p>The materials of which structures and works of art -are composed are liable to so many and so considerable -inequalities of texture, that any rule which can be deduced, -even by the most extensive series of experiments, -must be regarded as a mean result, from which individual -examples will be found to vary in so great a degree, -that more than usual caution must be observed in its -practical application. The details of this subject belong -to engineering, more properly than to the elements of -mechanics. Nevertheless, a general view of the most -important principles which have been established respecting -the strength of materials will not be misplaced -in this treatise.</p> - -<p>A piece of solid matter may be submitted to the action -of a force tending to separate its parts in several ways; -the principal of which are,—</p> - -<p>1. To a <i>direct pull</i>,—as when a rope or wire is -stretched by a weight. When a tie-beam resists the -separation of the sides of a structure, &c.</p> - -<p>2. To a direct pressure or thrust,—as when a weight -rests upon a pillar.</p> - -<p>3. To a transverse strain,—as when weights on the -ends of a lever press it on the fulcrum.</p> - -<p id="p331">(331.) If a solid be submitted to a force which draws -it in the direction of its length, having a tendency to pull -its ends in opposite directions, its strength or power to<span class="pagenum" id="Page_274">274</span> -resist such a force is proportional to the magnitude of its -transverse section. Thus, suppose a square rod of metal -A B, <i><a href="#i_p278a">fig. 185.</a></i>, of the breadth and thickness of one inch, -be pulled by a force in the direction A B, and that a -certain force is found sufficient to tear it; a rod of the -same metal of twice the breadth and the same thickness -will require double the force to break it; one of treble -the breadth and the same thickness will require treble -the force to break it, and so on.</p> - -<p>The reason of this is evident. A rod of double or -treble the thickness, in this case, is equivalent to two or -three equal and similar rods which equally and separately -resist the drawing force, and therefore possess a degree -of strength proportionate to their number.</p> - -<p>It will easily be perceived, that whatever be the section, -the same reasoning will be applicable, and the power -of resistance will, in general, be proportional to its magnitude -or area.</p> - -<p>If the material were perfectly uniform throughout its -dimensions, the resistance to a direct pull would not be -affected by the length of the rod. In practice, however, -the increase of length is found to lessen the strength. -This is to be attributed to the increased chance of inequality.</p> - -<p id="p332">(332.) No satisfactory results have been obtained -either by theory or experiment respecting the laws by -which solids resist compression. The power of a perpendicular -pillar to support a weight placed upon it -evidently depends on its thickness, or the magnitude of its -base, and on its height. It is certain that when the height -is the same, the strength increases with every increase of -the base, but it seems doubtful whether the strength be -exactly proportional to the base. That is, if two columns -of the same material have equal heights, and the base of -one be double the base of the other, the strength of one -will be greater, but it is not certain whether it will exactly -double that of the other. According to the theory -of Euler, which is in a certain degree verified by the -experiments of Musschenbrock, the strength will be in<span class="pagenum" id="Page_275">275</span>creased -in a greater proportion than the base, so that, if -the base be doubled, the strength will be more than -doubled.</p> - -<p>When the base is the same, the strength is diminished -by increasing the height, and this decrease of strength is -proportionally greater than the increase of height. According -to Euler’s theory, the decrease of strength is proportional -to the square of the height; that is, when the -height is increased in a two-fold proportion, the strength -is diminished in a four-fold proportion.</p> - -<p id="p333">(333.) The strain to which solids forming the parts -of structures of every kind are most commonly exposed -is the lateral or transverse strain, or that which acts at -right angles to their lengths. If any strain act obliquely -to the direction of their length it may be resolved into -two forces (<a href="#p76">76</a>.), one in the direction of the length, and -the other at right angles to the length. That part which -acts in the direction of the length will produce either -compression or a direct pull, and its effect must be investigated -accordingly.</p> - -<p>Although the results of theory, as well as those of -experimental investigations, present great discordances -respecting the transverse strength of solids, yet there are -some particulars, in which they, for the most part, agree; -to this it is our object here to confine our observations, -declining all details relating to disputed points.</p> - -<p>Let A B C D, <i><a href="#i_p278a">fig. 186.</a></i>, be a beam, supported at its -ends A and B. Its strength to support a weight at E -pressing downwards at right angles to its length is evidently -proportional to its breadth, the other things being -the same. For a beam of double or treble breadth, and -of the same thickness, is equivalent to two or three -equal and similar beams placed side by side. Since -each of these would possess the same strength, the whole -taken together would possess double or treble the strength -of any one of them.</p> - -<p>When the breadth and length are the same the -strength obviously increases with the depth, but not in -the same proportion. The increase of strength is found<span class="pagenum" id="Page_276">276</span> -to be much greater in proportion than the increase of -depth. By the theory of Galileo, a double or treble -thickness ought to increase the strength in a four-fold -or nine-fold proportion, and experiments in most cases -do not materially vary from this rule.</p> - -<p>If while the breadth and depth remain the same, the -length of the beam, or rather, the distance between the -points of support, vary, the strength will vary accordingly, -decreasing in the same proportion as the -length increases.</p> - -<p>From these observations it appears, that the transverse -strength of a beam depends more on its thickness -than its breadth. Hence we find that a broad -thin board is much stronger when its edge is presented -upwards. On this principle the joists or rafters of -floors and roofs are constructed.</p> - -<p>If two beams be in all respects similar, their strengths -will be in the proportion of the squares of their lengths. -Let the length, breadth, and depth of the one be respectively -double the length, breadth, and depth of the -other. By the double breadth the beam doubles its -strength, but by doubling the length half this strength -is lost. Thus the increase of length and breadth counteract -each other’s effects, and as far as they are concerned -the strength of the beam is not changed. But by -doubling the thickness the strength is increased in a -four-fold proportion, that is, as the square of the length. -In the same manner it may be shown, that when all the -dimensions are trebled, the strength is increased in a -nine-fold proportion, and so on.</p> - -<p id="p334">(334.) In all structures the materials have to support -their own weight, and therefore their available -strength is to be estimated by the excess of their absolute -strength above that degree of strength which is just -sufficient to support their own weight. This consideration -leads to some conclusions, of which numerous and -striking illustrations are presented in the works of nature -and art.</p> - -<p>We have seen that the absolute strength with which<span class="pagenum" id="Page_277">277</span> -a lateral strain is resisted is in the proportion of the -square of the linear dimensions of similar parts of a -structure, and therefore the amount of this strength increases -rapidly with every increase of the dimensions of -a body. But at the same time the weight of the body -increases in a still more rapid proportion. Thus, if the -several dimensions be doubled, the strength will be increased -in a four-fold but the weight in an eight-fold -proportion. If the dimensions be trebled, the strength -will be multiplied nine times, but the weight twenty-seven -times. Again, if the dimensions be multiplied -four times, the strength will be multiplied sixteen times, -and the weight sixty-four times, and so on.</p> - -<p>Hence it is obvious, that although the strength of a -body of small dimensions may greatly exceed its weight, -and, therefore, it may be able to support a load many -times its own weight; yet by a great increase in the dimensions -the weight increasing in a much greater degree -the available strength may be much diminished, and -such a magnitude may be assigned, that the weight of -the body must exceed its strength, and it not only -would be unable to support any load, but would actually -fall to pieces by its own weight.</p> - -<p>The strength of a structure of any kind is not, therefore, -to be determined by that of its model, which will -always be much stronger in proportion to its size. All -works natural and artificial have limits of magnitude -which, while their materials remain the same, they cannot -surpass.</p> - -<p>In conformity with what has just been explained, it -has been observed, that small animals are stronger in -proportion than large ones; that the young plant has more -available strength in proportion than the large forest -tree; that children are less liable to injury from accident -than men, &c. But although to a certain extent these -observations are just, yet it ought not to be forgotten, -that the mechanical conclusions which they are brought -to illustrate are founded on the supposition, that the -smaller and greater bodies which are compared are com<span class="pagenum" id="Page_278">278</span>posed -of precisely similar materials. This is not the -case in any of the examples here adduced.</p> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<h2 class="nobreak" id="CHAP_XXI">CHAP. XXI.<br /> - -<span class="title">ON BALANCES AND PENDULUMS.</span></h2> -</div> - - -<p id="p335">(335.) <span class="smcap">The</span> preceding chapters have been confined almost -wholly to the consideration of the laws of mechanics, -without entering into a particular description of the machinery -and instruments dependant upon those laws. -Such descriptions would have interfered too much with -the regular progress of the subject, and it therefore appeared -preferable to devote a chapter exclusively to this -portion of the work.</p> - -<p>Perhaps there are no ideas which man receives through -the medium of sense which may not be referred ultimately -to matter and motion. In proportion, therefore, -as he becomes acquainted with the properties of the one -and the laws of the other, his knowledge is extended, -his comforts are multiplied; he is enabled to bend the -powers of nature to his will, and to construct machinery -which effects with ease that which the united labour of -thousands would in vain be exerted to accomplish.</p> - -<p>Of the properties of matter, one of the most important -is its weight, and the element which mingles inseparably -with the laws of motion is time.</p> - -<p>In the present chapter it is our intention to describe -such instruments as are usually employed for determining -the weight of bodies. To attempt a description of -the various machines which are used for the measurement -of time, would lead us into too wide a field for the -present occasion, and we shall, therefore, confine ourselves -to an account of the methods which have been practised -to perfect, to perfect that instrument which affords the most -correct means of measuring time, the pendulum.</p> - -<p>The instrument by which we are enabled to determine, -with greater accuracy than by any other means,<span class="pagenum" id="Page_279">279</span> -the relative weight of a body, compared with the weight -of another body assumed as a standard, is the balance.</p> - -<div class="figcenter" id="i_p278a" style="max-width: 31.25em;"> - <img src="images/i_p278a.jpg" alt="" /> - <div class="caption"> -<p class="tar"><i>H. Adlard, sc.</i></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - - -<p class="tac"><i>Of the Balance.</i></p> - -<p>The balance may be described as consisting of an inflexible -rod or lever, called the beam, furnished with -three axes; one, the fulcrum or centre of motion situated -in the middle, upon which the beam turns, and the other -two near the extremities, and at equal distances from the -middle. These last are called the points of support, and -serve to sustain the pans or scales.</p> - -<p>The points of support and the fulcrum are in the -same right line, and the centre of gravity of the whole -should be a little below the fulcrum when the position -of the beam is horizontal.</p> - -<p>The arms of the lever being equal, it follows that if -equal weights be put into the scales no effect will be -produced on the position of the balance, and the beam -will remain horizontal.</p> - -<p>If a small addition be made to the weight in one of -the scales, the horizontality of the beam will be disturbed; -and after oscillating for some time, it will, on -attaining a state of rest, form an angle with the horizon, -the extent of which is a measure of the delicacy or sensibility -of the balance.</p> - -<p>As the sensibility of a balance is of the utmost importance -in nice scientific enquiries, we shall enter somewhat -at large into a consideration of the circumstances -by which this property is influenced.</p> - -<p>In <i><a href="#i_p302a">fig. 187.</a></i> let A B represent the beam drawn from -the horizontal position by a very small weight placed in -the scale suspended from the point of support B; then -the force tending to draw the beam from the horizontal -position may be expressed by P B, multiplied by such -very small weight acting upon the point B.</p> - -<p>Let the centre of gravity of the whole be at G; then -the force acting against the former will be G P multiplied -into the weight of the beam and scales, and when -these forces are equal, the beam will rest in an inclined<span class="pagenum" id="Page_280">280</span> -position. Hence we may perceive that as the centre of -gravity is nearer to or further from the fulcrum S, (every -thing else remaining the same) the sensibility of the -balance will be increased or diminished.</p> - -<p>For, suppose the centre of gravity were removed to <i>g</i>, -then to produce an opposing force equal to that acting -upon the extremity of the beam, the distance <i>g p</i> from -the perpendicular line must be increased until it becomes -nearly equal to G P; but for this purpose the -end of the beam B must descend, which will increase the -angle H S B.</p> - -<p>As all weights placed in the scales are referred to the -line joining the points of support, and as this line is -above the centre of gravity of the beam when not -loaded, such weights will raise the centre of gravity; but -it will be seen that the sensibility of the balance, as far -as it depends upon this cause, will remain unaltered.</p> - -<p>For, calling the distance S G unity, the distance of -the centre of gravity from the point S (to which the -weight which has been added is referred) will be expressed -by the reciprocal of the weight of the beam so -increased; that is, if the weight of the beam be doubled -by weights placed in the scales, S <i>g</i> will be one half of -S G; and if the weight of the beam be in like manner -trebled, S <i>g</i> will be one third of S G, and so on. And -as G P varies as S G, <i>g p</i> will be inversely proportionate -to the increased weight of the beam, and consequently, -the product obtained by multiplying <i>g p</i> by the -weight of the beam and its load will be a constant -quantity, and the sensibility of the balance, as before -stated, will suffer no alteration.</p> - -<p>We will now suppose that the fulcrum S, <i><a href="#i_p302a">fig. 188.</a></i>, -is situated below the line joining the points of support, -and that the centre of gravity of the beam when not -loaded is at G. Also that when a very small weight is -placed in the scale suspended from the point B, the -beam is drawn from its horizontal position, the deviation -being a measure of the sensibility of the balance. Then, -as before stated, G P multiplied by the weight of the<span class="pagenum" id="Page_281">281</span> -beam will be equal to <span class="ilb">P′</span> B multiplied by the very small -additional weight acting on the point B.</p> - -<p>Now if we place equal weights in both scales, such -additional weights will be referred to the point W, and -the resulting distance of the centre of gravity from the -point W, calling W G unity, will be expressed as before -by the reciprocal of the increased weight of the loaded -beam. But G P will decrease in a greater proportion -than W G: thus, supposing the weight of the beam to -be doubled, W <i>g</i> would be one half of W G; but <i>g p</i>, as -will be evident on an inspection of the figure, will be -less than half of G P; and the same small weight which -was before applied to the point B, if now added, would -depress the point B, until the distance <i>g p</i> became such -as that, when multiplied by the weight of the whole, the -product would be as before equal to <span class="ilb">P′</span> B, multiplied by -the before mentioned very small added weight. The -sensibility of the balance, therefore, in this case would be -increased.</p> - -<p>If the beam be sufficiently loaded, the centre of gravity -will at length be raised to the fulcrum S, and the -beam will rest indifferently in any position. If more -weight be then added, the centre of gravity will be -raised above the fulcrum, and the beam will turn over.</p> - -<p>Lastly, if the fulcrum S, <i><a href="#i_p302a">fig. 189.</a></i>, is above the -line joining the two points of support, as any additional -weights placed in the scales will be referred to the -point W, in the line joining A and B, if the weight -of the beam be doubled by such added weights, and the -centre of gravity be consequently raised to <i>g</i>, W <i>g</i> -will become equal to half of W G. But <i>g p</i>, being -greater than one half of G P, the end of the beam B -will rise until <i>g p</i> becomes such as to be equal, when -multiplied by the whole increased weight of the beam, -to P B, multiplied by the small weight, which we suppose -to have been placed as in the preceding examples, -in the scale.</p> - -<p>From what has been said it will be seen that there -are three positions of the fulcrum which influence the<span class="pagenum" id="Page_282">282</span> -sensibility of the balance: first, when the fulcrum and -the points of support are in a right line, when the sensibility -of the balance will remain the same, though the -weight with which the beam is loaded should be varied: -secondly, when the fulcrum is below the line joining -the two points of support, in which case the sensibility -of the balance will be increased by additional weights, -until at length the centre of gravity is raised above the -fulcrum, when the beam will turn over; and, thirdly, -when the fulcrum is above the line joining the two -points of support, in which case the sensibility of the -balance will be diminished as the weight with which -the beam is loaded is increased.</p> - -<p>The sensibility of a balance, as here defined, is the -angular deviation of the beam occasioned by placing an -additional constant small weight in one of the scales; -but it is frequently expressed by the proportion which -such small additional weight bears to the weight of the -beam and its load, and sometimes to the weight the -value of which is to be determined.</p> - -<p>This proportion, however, will evidently vary with -different weights, except in the case where the centre of -gravity of the beam is in the line joining the points -supporting the scales, the fulcrum being above this line, -and it is therefore necessary, in every other case, when -speaking of the sensibility of the balance, to designate -the weight with which it is loaded: thus, if a balance -has a troy pound in each scale, and the horizontality of -the beam varies a certain small quantity, just perceptible -on the addition of one hundredth of a grain, we -say that the balance is sensible to <span class="nowrap"> <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">1152000</span></span></span> part of its -load with a pound in each scale, or that it will determine -the weight of a troy pound within <span class="nowrap"> <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">576000</span></span></span> part of -the whole.</p> - -<p>The nearer the centre of gravity of a balance is to its -fulcrum the slower will be the oscillations of the beam. -The number of oscillations, therefore, made by the beam -in a given time (a minute for example), affords the -most accurate method of judging of the sensibility of the<span class="pagenum" id="Page_283">283</span> -balance, which will be the greater as the oscillations are -fewer.</p> - -<p>Balances of the most perfect kind, and of such only -it is our present object to treat, are usually furnished -with adjustments, by means of which the length of the -arms, or the distances of the fulcrum from the points of -support, may be equalised, and the fulcrum and the two -points of support be placed in a right line; but these -adjustments, as will hereafter be seen, are not absolutely -necessary.</p> - -<p>The beam is variously constructed, according to the -purposes to which the balance is to be applied. Sometimes -it is made of a rod of solid steel; sometimes of -two hollow cones joined at their bases; and, in some -balances, the beam is a frame in the form of a rhombus: -the principal object in all, however, is to combine -strength and inflexibility with lightness.</p> - -<p>A balance of the best kind, made by Troughton, is so -contrived as to be contained, when not in use, in a -drawer below the case; and when in use, it is protected -from any disturbance from currents of air, by being enclosed -in the case above the drawer, the back and front -of which are of plate glass. There are doors in the -sides, through which the scale-pans are loaded, and there -is a door at the top through which the beam may be -taken out.</p> - -<p>A strong brass pillar, in the centre of the box, supports -a square piece, on the front and back of which -rise two arches, nearly semicircular, on which are fixed -two horizontal planes of agate, intended to support -the fulcrum. Within the pillar is a cylindrical tube, -which slides up and down by means of a handle on the -outside of the case. To the top of this interior tube is -fixed an arch, the terminations of which pass beneath -and outside of the two arches before described. These -terminations are formed into Y <i>s</i>, destined to receive -the ends of the fulcrum, which are made cylindrical for -this purpose, when the interior tube is elevated in order -to relieve the axis when the balance is not in use. On<span class="pagenum" id="Page_284">284</span> -depressing the interior tube, the Y <i>s</i> quit the axis, and -leave it in its proper position on the agate planes. The -beam is about eighteen inches long, and is formed of -two hollow cones of brass, joined at their bases. The -thickness of the brass does not exceed 0·02 of an inch, -but by means of circular rings driven into the cones at -intervals they are rendered almost inflexible. Across -the middle of the beam passes a cylinder of steel, the -lower side of which is formed into an edge, having an -angle of about thirty degrees, which, being hardened and -well polished, constitutes the fulcrum, and rests upon the -agate planes for the length of about 0·05 of an inch.</p> - -<p>Each point of suspension is formed of an axis having -two sharp concave edges, upon which rest at right angles -two other sharp concave edges formed in the spur-shaped -piece to which the strings carrying the scale-pan are -attached. The two points are adjustable, the one horizontally, -for the purpose of equalising the arms of the -beam, and the other vertically, for bringing the points -of suspension and the fulcrum into a right line.</p> - -<p>Such is the form of Troughton’s balance: we shall -now give the description of a balance as constructed by -Mr. Robinson of Devonshire Street, Portland Place:—</p> - -<p>The beam of this balance is only ten inches long. It -is a frame of bell-metal in the form of a rhombus. The -fulcrum is an equilateral triangular prism of steel one -inch in length; but the edge on which the beam vibrates -is formed to an angle of 120°, in order to prevent any -injury from the weight with which it may be loaded. The -chief peculiarity in this balance consists in the knife-edge -which forms the fulcrum bearing upon an agate plane -throughout its whole length, whereas we have seen in -the balance before described that the whole weight is -supported by portions only of the knife-edge, amounting -together to one tenth of an inch. The supports for the -scales are knife-edges each six tenths of an inch long. -These are each furnished with two pressing screws, by -means of which they may be made parallel to the central -knife-edge.</p> - -<p><span class="pagenum" id="Page_285">285</span></p> - -<p>Each end of the beam is sprung obliquely upwards -and towards the middle, so as to form a spring through -which a pushing screw passes, which serves to vary the -distance of the point of support from the fulcrum, and, -at the same time, by its oblique action to raise or depress -it, so as to furnish a means of bringing the points of -support and the fulcrum into a right line.</p> - -<p>A piece of wire, four inches long, on which a screw -is cut, proceeds from the middle of the beam downwards. -This is pointed to serve as an index, and a -small brass ball moves on the screw, by changing the -situation of which the place of the centre of gravity may -be varied at pleasure.</p> - -<p>The fulcrum, as before remarked, rests upon an agate -plane throughout its whole length, and the scale-pans are -attached to planes of agate which rest upon the knife-edges -forming the points of support. This method of -supporting the scale-pans, we have reason to believe, is -due to Mr. Cavendish. Upon the lower half of the pillar -to which the agate plane is fixed, a tube slides up and -down by means of a lever which passes to the outside of -the case. From the top of this tube arms proceed -obliquely towards the ends of the balance, serving to -support a horizontal piece, carrying at each extremity -two sets of Y <i>s</i>, one a little above the other. The upper -Y <i>s</i> are destined to receive the agate planes to which the -scale-pans are attached, and thus to relieve the knife-edges -from their pressure; the lower to receive the -knife-edges which, form the points of support, consequently -these latter Y <i>s</i>, when in action, sustain the -whole beam.</p> - -<p>When the lever is freed from a notch in which it is -lodged, a spring is allowed to act upon the tube we have -mentioned, and to elevate it. The upper Y <i>s</i> first meet -the agate planes carrying the scale-pans and free them -from the knife-edges. The lower Y <i>s</i> then come into -action and raise the whole beam, elevating the central -knife-edge above the agate plane. This is the usual state -of the balance when not in use: when it is to be brought<span class="pagenum" id="Page_286">286</span> -into action, the reverse of what we have described takes -place. On pressing down the lever, the central knife-edge -first meets the agate plane, and afterwards the two -agate planes carrying the scale-pans are deposited upon -their supporting knife-edges.</p> - -<p>A balance of this construction was employed by the -writer of this article in adjusting the national standard -pound. With a pound troy in each scale, the addition -of one hundredth of a grain caused the index to vary -one division, equal to one tenth of an inch, and Mr. Robinson -adjusts these balances so that with one thousand -grains in each scale, the index varies perceptibly on the -addition of one thousandth of a grain, or of one-millionth -part of the weight to be determined.</p> - -<p>It may not be uninteresting to subjoin, from the Philosophical -Transactions for 1826, the description of a -balance perhaps the most sensible that has yet been -made, constructed for verifying the national standard -bushel. The author says,—</p> - -<p>“The weight of the bushel measure, together with -the 80 lbs. of water it should contain, was about 250 lbs.; -and as I could find no balance capable of determining so -large a weight with sufficient accuracy, I was under the -necessity of constructing one for this express purpose.</p> - -<p>“I first tried cast iron; but though the beam was -made as light as was consistent with the requisite degree -of strength, the inertia of such a mass appeared to be so -considerable, that much time must have been lost before -the balance would have answered to the small differences -I wished to ascertain. Lightness was a property essentially -necessary, and bulk was very desirable, in order -to preclude such errors as might arise from the beam -being partially affected by sudden alterations of temperature. -I therefore determined to employ wood, a material -in which the requisites I sought were combined. -The beam was made of a plank of mahogany, about 7O -inches long, 22 inches wide, and <span class="nowrap">2<span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">4</span></span></span> thick, tapering from -the middle to the extremities. An opening was cut in -the centre, and strong blocks screwed to each side of the<span class="pagenum" id="Page_287">287</span> -plank, to form a bearing for the back of a knife-edge -which passed through the centre. Blocks were also -screwed to each side at the extremities of the beam on -which rested the backs of the knife-edges for supporting -the pans. The opening in the centre was made sufficiently -large to admit the support hereafter to be described, -upon which the knife-edge rested.</p> - -<p>“In all beams which I have seen, with the exception -of those made by Mr. Robinson, the whole weight is -sustained by short portions at the extremities of the -knife-edge; and the weight being thus thrown upon a -few points, the knife-edge becomes more liable to change -its figure and to suffer injury.</p> - -<p>“To remedy this defect, the central knife-edge of the -beam I am describing was made 6 inches, and the two -others 5 inches long. They were triangular prisms -with equal sides of three fourths of an inch, very carefully -finished, and the edges ultimately formed to an -angle of 120°.</p> - -<p>“Each knife-edge was screwed to a thick plate of -brass, the surfaces in contact having been previously -ground together; and these plates were screwed to the -beam, the knife-edges being placed in the same plane, -and as nearly equidistant and parallel to each other as -could be done by construction.</p> - -<p>“The support upon which the central knife-edge -rested throughout its whole length was formed of a plate -of polished hard steel, screwed to a block of cast iron. -This block was passed through the opening before mentioned -in the centre of the beam, and properly attached -to a frame of cast iron.</p> - -<p>“The stirrups to which the scales were hooked rested -upon plates of polished steel to which they were attached, -and the under surfaces of which were formed by -careful grinding into cylindrical segments. These were -in contact with the knife-edges their whole length, and -were known to be in their proper position by the correspondence -of their extremities with those of the knife-edges. -A well imagined contrivance was applied by<span class="pagenum" id="Page_288">288</span> -Mr. Bate for raising the beam when loaded, in order to -prevent unnecessary wear of the knife-edge, and for the -purpose of adjusting the place of the centre of gravity, -when the beam was loaded with the weight required to -be determined, a screw carrying a movable ball projected -vertically from the middle of die beam.</p> - -<p>“The performance of this balance fully equalled my -expectations. With two hundred and fifty pounds in -each scale, the addition of a single grain occasioned an -immediate variation in the index of one twentieth of an -inch, the radius being fifty inches.”</p> - -<p>From the preceding account it appears that this balance -is sensible to <span class="nowrap"> <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">1750000</span></span></span> part of the weight which -was to be determined.</p> - -<p>We shall now describe the method to be pursued in -adjusting a balance.</p> - -<p>1. To bring the points of suspension and the fulcrum -into a right line.</p> - -<p>Make the vibrations of the balance very slow by moving -the weight which influences the centre of gravity, -and bring the beam into a horizontal position, by means -of small bits of paper thrown into the scales. Then -load the scales with nearly the greatest weight the -beam is fitted to carry. If the vibrations are performed -in the same time as before, no further adjustment -is necessary; but if the beam vibrates quicker, or if it -oversets, cause it to vibrate in the same time as at first, -by moving the adjusting weight, and note the distance -through which the weight has passed. Move the weight -then in the contrary direction through double this distance, -and then produce the former slow motion by -means of the screw acting vertically on the point of support. -Repeat this operation until the adjustment is -perfect.</p> - -<p>2. To make the arms of the beam of an equal -length.</p> - -<p>Put weights in the scales as before; bring the beam -as nearly as possible to a horizontal position, and note -the division at which the index stands; unhook the<span class="pagenum" id="Page_289">289</span> -scales, and transfer them with their weights to the other -ends of the beam, when, if the index points to the same -division, the arms are of an equal length; but if not, -bring the index to the division which had been noted, -by placing small weights in one or the other scale. Take -away half these weights, and bring the index again to -the observed division by the adjusting screw, which acts -horizontally on the point of support. If the scale-pans -are known to be of the same weight, it will not be necessary -to change the scales, but merely to transfer the -weights from one scale-pan to the other.</p> - - -<p class="tac"><i>Of the Use of the Balance.</i></p> - -<p>Though we have described the method of adjusting -the balance, these adjustments, as we have before -remarked, may be dispensed with. Indeed, in all delicate -scientific operations, it is advisable never to rely -upon adjustments, which, after every care has been employed -in effecting them, can only be considered as -approximations to the truth. We shall, therefore, now -describe the best method of ascertaining the weight of a -body, and which does not depend on the accuracy of -these adjustments.</p> - -<p>Having levelled the case which contains the balance, -and thrown the beam out of action, place a weight in -each scale-pan nearly equal to the weight which is to be -determined. Lower the beam very gently till it is in -action, and by means of the adjustment for raising or -lowering the centre of gravity, cause the beam to vibrate -very slowly. Remove these weights, and place the substance, -the weight of which is to be determined in one -of the scale-pans; carefully counterpoise it by means of -any convenient substances put into the other scale-pan, -and observe the division at which the index stands; -remove the body, the weight of which is to be ascertained, -and substitute standard weights for it so as to -bring the index to the same division as before. These -weights will be equal to the weight of the body.</p> - -<p>If it be required to compare two weights together<span class="pagenum" id="Page_290">290</span> -which are intended to be equal, and to ascertain their -difference, if any, the method of proceeding will be -nearly the same. The standard weight is to be carefully -counterpoised, and the division at which the index -stands, noted. And now it will be convenient to add in -either of the scales some small weight, such as one or -two hundredths of a grain, and mark the number of divisions -passed over in consequence by the index, by which -the value of one division of the scale will be known. -This should be repeated a few times, and the mean taken -for greater certainty.</p> - -<p>Having noted the division at which the index rests, -the standard weight is to be removed, and the weight -which is to be compared with it substituted for it. The -index is then again to be noted, and the difference between -this and the former indication will give the difference -between the weights in parts of a grain.</p> - -<p>If the balance is adjusted so as to be very sensible, it -will be long before it comes to a state of rest. It may, -therefore, sometimes be advisable to take the mean of -the extent of the vibrations of the index as the point -where it would rest, and this may be repeated several -times for greater accuracy. It must, however, be remembered, -that it is not safe to do this when the extent -of the vibrations is beyond one or two divisions of the -scale; but with this limitation it is, perhaps, as good -a method as can be pursued.</p> - -<p>Many precautions are necessary to ensure a satisfactory -result. The weights should never be touched by -the hand; for not only would this oxydate the weight, -but by raising its temperature it would appear lighter, -when placed in the scale-pan, than it should do, in consequence -of the ascent of the heated air. For the larger -weights a wooden fork or tongs, according to the form -of the weight, should be employed; and for the smaller, -a pair of forceps made of copper will be found the most -convenient. This metal possessing sufficient elasticity to -open the forceps on their being released from pressure, and -yet not opposing a resistance sufficient to interfere with<span class="pagenum" id="Page_291">291</span> -that delicacy of touch which is desirable in such operations.</p> - - -<p class="tac"><i>Of Weights.</i></p> - -<p>It must be obvious, that the excellence of the balance -would be of little use, unless the weights employed were -equally to be depended upon. The weights may either be -accurately adjusted, or the difference between each weight -and the standard may be determined, and, consequently, -its true value ascertained. It has been already shown -how the latter may be effected, in the instructions which -have been given for comparing two weights together; -and we shall now show the readiest mode of adjusting -weights to an exact equality with a given standard.</p> - -<p>The material of the weight may be either brass or -platina, and its form may be cylindrical: the diameter -being nearly twice the height. A small spherical knob -is screwed into the centre, a space being left under the -screw to receive the portions of fine wire used in the -adjustment. It will be convenient to form a cavity in -the bottom of each weight to receive the knob of the -weight upon which it may be placed.</p> - -<p>Each weight is now to be compared with the standard, -and should it be too heavy, it is to be reduced till it -becomes in a very small degree too light, when the -amount of the deficiency is to be carefully determined.</p> - -<p>Some very fine silver wire is now to be taken, and -the weight of three or four feet of it ascertained. From -this it will be known what length of the wire is equal -to the error of the weight to be adjusted; and this -length being cut off is to be enclosed under the screw. -To guard against any possible error, it will be advisable -before the screw is firmly fixed in its place, again to -compare the weight with the standard.</p> - -<p>The most approved method of making weights expressing -the decimal parts of a grain, is to determine, as -before, with great care, the weight of a certain length of -fine wire, and then to cut off such portions as are equal -to the weights required.</p> - -<p><span class="pagenum" id="Page_292">292</span></p> - -<p>Before we conclude this article we shall give a description, -from the Annals of Philosophy for 1825, of -“a very sensible balance,” used by the late Dr. Black:—</p> - -<p>“A thin piece of fir wood, not thicker than a shilling, and -a foot long, three tenths of an inch broad in the middle, and -one tenth and a half at each end, is divided by transverse -lines into twenty parts; that is, ten parts on each side of -the middle. These are the principal divisions, and each of -them is subdivided into halves and quarters. Across the -middle is fixed one of the smallest needles I could procure, -to serve as an axis, and it is fixed in its place by -means of a little sealing wax. The numeration of the -divisions is from the middle to each end of the beam. -The fulcrum is a bit of plate brass, the middle of which -lies flat on my table when I use the balance, and the -two ends are bent up to a right angle so as to stand -upright. These two ends are ground at the same time -on a flat hone, that the extreme surfaces of them may -be in the same plane; and their distance is such that -the needle, when laid across them, rests on them at a -small distance from the sides of the beam. They rise -above the surface of the table only one tenth and a half -or two tenths of an inch, so that the beam is very limited -in its play. See <i><a href="#i_p302a">fig. 190.</a></i></p> - -<p>“The weights I use are one globule of gold, which -weighs one grain, and two or three others which weigh -one tenth of a grain each; and also a number of small -rings of fine brass wire, made in the manner first mentioned -by Mr. Lewis, by appending a weight to the -wire, and coiling it with the tension of that weight -round a thicker brass wire in a close spiral, after which, -the extremity of the spiral being tied hard with waxed -thread, I put the covered wire into a vice, and applying -a sharp knife, which is struck with a hammer, I cut -through a great number of the coils at one stroke, and -find them as exactly equal to one another as can be -desired. Those I use happen to be the <span class="nowrap"> <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">30</span></span></span> part of a -grain each, or 300 of them weigh ten grains; but -I have others much lighter.</p> - -<p><span class="pagenum" id="Page_293">293</span></p> - -<p>“You will perceive that by means of these weights -placed on different parts of the beam, I can learn the -weight of any little mass from one grain, or a little -more, to the <span class="nowrap"> <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">1200</span></span></span> of a grain. For if the thing to be -weighed weighs one grain, it will, when placed on one -extremity of the beam, counterpoise the large gold -weight at the other extremity. If it weighs half a -grain it will counterpoise the heavy gold weight placed -at 5. If it weigh <span class="nowrap"> <span class="fraction"><span class="fnum">6</span><span class="bar">/</span><span class="fden">10</span></span></span> of a grain, you must place the -heavy gold weight at 5, and one of the lighter ones at -the extremity to counterpoise it, and if it weighs only -one or two, or three or four hundredths of a grain, -it will be counterpoised by one of the small gold weights -placed at the first or second, or third or fourth division. -If, on the contrary, it weighs one grain and a fraction, it -will be counterpoised by the heavy gold weight at the -extremity, and one or more of the lighter ones placed -in some other part of the beam.</p> - -<p>“This beam has served me hitherto for every purpose; -but had I occasion for a more delicate one, I -could make it easily by taking a much thinner and -lighter slip of wood, and grinding the needle to give it an -edge. It would also be easy to make it carry small -scales of paper for particular purposes.”</p> - -<p>The writer of this article has used a balance of this -kind, and finds that it is sensible to <span class="nowrap"> <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">1000</span></span></span> of a grain -when loaded with ten grains. It is necessary, however, -where accuracy is required, to employ a scale-pan. -This may be made of thin card paper, shaped as in -<i><a href="#i_p302a">fig. 191.</a></i></p> - -<p>A thread is to be passed through the two ends, by -tightening which they may be brought near each other.</p> - -<p>The most convenient weights for this beam appear to -be two of one grain each, and one of one tenth of a -grain. They should be made of straight wire; and if -the beam be notched at the divisions, they may be -lodged in these notches very conveniently. Ten divisions -on each side of the middle will be sufficient. The -weight of the scale-pan must first be carefully ascertained,<span class="pagenum" id="Page_294">294</span> -in order that it may be deducted from the weight, afterwards -determined, of the scale-pan and the substance it -may contain.</p> - -<p>If the scale-pan be placed at the tenth division of the -beam, it is evident that by means of the two grain -weights, a greater weight cannot be determined than -one grain and nine tenths; but if the scale-pan be placed -at any other division of the beam, the resulting apparent -weight must be increased by multiplying it by ten, and -dividing by the number of the division at which the -scale-pan is placed; and in this manner it is evident that -if the scale-pan be placed at the division numbered 1, -a weight amounting to nineteen grains may be determined.</p> - -<p>We have been tempted to describe this little apparatus, -because it is extremely simple in its construction, -may be easily made, and may be very usefully employed -on many occasions where extreme accuracy is not necessary.</p> - - -<p class="tac"><i>Description of the Steelyard.</i></p> - -<p>The steelyard is a lever, having unequal arms; and -in its most simple form it is so arranged, that one weight -alone serves to determine a great variety of others, by -sliding it along the longer arm of the lever, and thus -varying its distance from the fulcrum.</p> - -<p>It has been demonstrated, chapter <a href="#CHAP_XIII">xiii</a>., that in the -lever the proportion of the power to the weight will be -always the same as that of their distances from the fulcrum, -taken in a reverse order; consequently, when a -constant weight is used, and an equilibrium established -by sliding this weight on the longer arm of the lever, -the relative weight of the substance weighed, to the -constant weight, will be in the same proportion as the -distance of the constant weight from the fulcrum is to -the length of the shorter arm.</p> - -<p>Thus, suppose the length of the shorter arm, or the -distance of the fulcrum from the point from which the -weight to be determined is suspended, to be one inch;<span class="pagenum" id="Page_295">295</span> -let the longer arm of the lever be divided into parts of -one inch each, beginning at the fulcrum. Now let the -constant weight be equal to one pound, and let the -steelyard be so constructed that the shorter arm shall be -sufficiently heavy to counterpoise the longer when the -bar is unloaded. Then suppose a substance, the weight -of which is five pounds, to be suspended from the -shorter arm. It will be found that when the constant -weight is placed at the distance of five inches from the -fulcrum, the weights will be in equilibrium, and the -bar consequently horizontal. In this steelyard, therefore, -the distance of each inch from the fulcrum indicates a -weight of one pound. An instrument of this form was -used by the Romans, and it is usually described as the -Roman statera or steelyard. A representation of it is -given at <i><a href="#i_p302a">fig. 192.</a></i></p> - -<p>The steelyard is in very general use for the coarser -purposes of commerce, but constructed differently from -that which we have described. The beam with the -scales or hooks is seldom in equilibrium upon the point -F, when the weight P is removed; but the longer arm -usually preponderates, and the commencement of the -graduations, therefore, is not at F, but at some point -between B and F. The common steelyard, which we -have represented at <i><a href="#i_p302a">fig. 193.</a></i>, is usually furnished with -two points, from either of which the substance, the -weight of which is to be determined, may be suspended. -The value of the divisions is in this case -increased in proportion as the length of the shorter -arm is decreased. Thus, in the steelyard which we -have described, if there be a second point of suspension -at the distance of half an inch from the fulcrum, each -division of the longer arm will indicate two pounds -instead of one, and these divisions are usually marked -upon the opposite edge of the steelyard, which is made -to turn over.</p> - -<p>This instrument is very convenient, because it requires -but one weight; and the pressure on the fulcrum is less -than in the balance, when the substance to be weighed<span class="pagenum" id="Page_296">296</span> -is heavier than the constant weight. But, on the contrary, -when the constant weight exceeds the substance -to be weighed, the pressure on the fulcrum is greater in -the steelyard than in the balance, and the balance is, -therefore, preferable in determining small weights. -There is also an advantage in the balance, because the -subdivision of weights can be effected with a greater -degree of precision than the subdivision of the arm of -the steelyard.</p> - - -<p class="tac"><i>C. Paul’s Steelyard.</i></p> - -<p>A steelyard has been constructed by Mr. C. Paul, -inspector of weights and measures at Geneva, which -is much to be preferred to that in common use. Mr. C. -Paul states, that steelyards have two advantages over -balances: 1. That their axis of suspension is not loaded -with any other weight than that of the merchandise, -the constant weight of the apparatus itself excepted; -while the axis of the balance, besides the weight of the instrument, -sustains a weight double to that of the merchandise. -2. The use of the balance requires a considerable -assortment of weights, which causes a proportional -increase in the price of the apparatus, independently of -the chances of error which it multiplies, and of the time -employed in producing an equilibrium.</p> - -<p>1. In C. Paul’s steelyard the centres of the movement -of suspension, or the two constant centres, are placed on -the exact line of the divisions of the beam; an elevation -almost imperceptible in the axis of the beam, destined -to compensate for the very slight flexion of the bar, -alone excepted.</p> - -<p>2. The apparatus, by the construction of the beam, -is balanced below its centre of motion, so that when no -weight is suspended the beam naturally remains horizontal, -and resumes that position when removed from it, as -also when the steelyard is loaded, and the weight is at -the division which ought to show how much the merchandise -weighs. The horizontal situation in this steelyard, -as well as in the others, is known by means of the<span class="pagenum" id="Page_297">297</span> -tongue which rises vertically above the axis of suspension.</p> - -<p>3. It may be discovered, that the steelyard is deranged -if, when not loaded, the beam does not remain -horizontal.</p> - -<p>4. The advantage of a great and a small side (which in -the other augments the extent of their power of weighing) -is supplied by a very simple process, which accomplishes -the same end with some additional advantages. -This process is to employ on the same division different -weights. The numbers of the divisions on the bar, point -out the degree of heaviness expressed by the corresponding -weights. For example, when the large weight of the large -steelyard weighs 16 lbs., each division it passes over on -the bar is equivalent to a pound; the small weight, -weighing sixteen times less than the large one, will represent -on each of these divisions the sixteenth part of -a pound, or one ounce; and the opposite face of the bar -is marked by pounds at each sixteenth division. In -this construction, therefore, we have the advantage of being -able, by employing both weights at once, to ascertain, -for example, almost within an ounce, the weight of -500 pounds of merchandise. It will be sufficient to -add what is indicated by the small weight in ounces, to -that of the large one in pounds, after an equilibrium has -been obtained by the position of the two weights, viz. -the large one placed at the next pound below its real -weight, and the small one at the division which determines -the number of ounces to be added.</p> - -<p>5. As the beam is graduated only on one edge, it -may have the form of a thin bar, which renders it much -less susceptible of being bent by the action of the weight, -and affords room for making the figures more visible on -both the faces.</p> - -<p>6. In these steelyards the disposition of the axes is not -only such that the beam represents a mathematical lever -without weight, but in the principle of its division, the -interval between every two divisions is a determined and -aliquot part of the distance between the two fixed points<span class="pagenum" id="Page_298">298</span> -of suspension; and each of the two weights employed -has for its absolute weight the unity of the weight it represents, -multiplied by the number of the divisions contained -in the interval between the two centres of -motion.</p> - -<p>Thus, supposing the arms of the steelyard divided in -such a manner that ten divisions are exactly contained -in the distance between the two constant centres of motion, -a weight to express the pounds on each division of -the beam must really weigh ten pounds; that to point -out the ounces on the same divisions must weigh ten -ounces, &c. So that the same steelyard may be adapted -to any system of measures whatever, and in particular -to the decimal system, by varying the absolute heaviness -of the weights, and their relation with each other.</p> - -<p>But to trace out, in a few words, the advantages of -the steelyards constructed by C. Paul for commercial -purposes, we shall only observe,—</p> - -<p>1. That the buyer and seller are certain of the correctness -of the instrument, if the beam remains horizontal -when it is unloaded and in its usual position. 2. That -these steelyards have one suspension less than the old -ones, and are so much more simple. 3. That by these -means we obtain, with the greatest facility, by employing -two weights, the exact weight of merchandise, with -all the approximation that can be desired, and even with -a greater precision than that given by common balances. -There are few of these which, when loaded with 500 -pounds at each end, give decided indication of an ounce -variation; and the steelyards of C. Paul possess that -advantage, and cost one half less than balances of equal -dominion. 4. In the last place, we may verify at pleasure -the justness of the weights, by the transposition -which their ratio to each other will permit; for example, -by observing whether, when the weight of one -pound is brought back one division, and the weight of -one ounce carried forward sixteen divisions, the equilibrium -still remains.</p> - -<p>It is on this simple and advantageous principle that<span class="pagenum" id="Page_299">299</span> -C. Paul has constructed his universal steelyard. It -serves for weighing in the usual manner, and according -to any system of weights, all ponderable bodies to the -precision of half a grain in the weight of a hundred -ounces; that is to say, of a ten-thousandth part. It is -employed, besides, for ascertaining the specific gravity -of solids, of liquids, and of air, by processes extremely -simple, and which do not require many subdivisions in -the weights.</p> - -<p>We think the description above given will be sufficiently -intelligible without a representation of this instrument. -An account of its application to the determination -of specific gravities will be found in vol. iii. -of the Philosophical Magazine.</p> - - -<p class="tac"><i>The Chinese Steelyard.</i></p> - -<p>This instrument is used in China and the East Indies -for weighing gems, precious metals, &c. The -beam is a small rod of ivory, about a foot in length. -Upon this are three lines of divisions, marked by fine -silver studs, all beginning from the end of the beam, -whence the first is extended 8 inches, the second <span class="nowrap">6<span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>, and -the third <span class="nowrap">8<span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>. The first is European weight, and the -other two Chinese. At the other end of the beam -hangs a round scale, and at three several distances from -this end are holes, through which pass so many fine -strings, serving as different points of suspension. The -first distance makes <span class="nowrap">1<span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">5</span></span></span> inches, the second <span class="nowrap">3<span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">5</span></span></span>, or double -the former, and the third <span class="nowrap">4<span class="fraction"><span class="fnum">4</span><span class="bar">/</span><span class="fden">5</span></span></span>, or triple the same. The -instrument, when used, is held by one of the strings, -and a sealed weight of about <span class="nowrap">1<span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">4</span></span></span> oz. troy, is slid upon -the beam until an equilibrium is produced; the weight -of the body is then indicated by the graduated scale -above mentioned.</p> - - -<p class="tac"><i>The Danish Balance.</i></p> - -<p>The Danish balance is a straight bar or lever, having -a heavy weight fixed to one end, and a hook or scale-pan -to receive the substance, the weight of which is to<span class="pagenum" id="Page_300">300</span> -be determined, suspended from the other end. The fulcrum -is moveable, and is made to slide upon the bar, -till the beam rests in a horizontal position, when the -place of the fulcrum indicates the weight required. In -order to construct a balance of this kind, let the distance -of the centre of gravity from that point to which the -substance to be weighed is suspended be found by experiment, -when the beam is unloaded. Multiply this -distance by the weight of the whole apparatus, and divide -the product by the weight of the apparatus increased -by the weight of the body. This will give the distance -from the point of suspension, at which the fulcrum -being placed, the whole will be in equilibrio: for example, -supposing the distance of the centre of gravity -from the point of suspension to be 10 inches, and the -weight of the whole apparatus to be ten pounds; suppose, -also, it were required to mark the divisions which -should indicate weights of one, two, or three pounds, &c. -First, for the place of the division indicating one pound -we have <span class="nowrap"> <span class="fraction2"><span class="fnum">10 × 10</span><span class="bar">/</span><span class="fden2">10 + 1</span></span></span> = <span class="nowrap"> <span class="fraction2"><span class="fnum">100</span><span class="bar">/</span><span class="fden2">10 + 1</span></span></span> = <span class="nowrap">9<span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">11</span></span></span> inches, the place of -the division marking one pound. For two pounds we -have <span class="nowrap"> <span class="fraction2"><span class="fnum">100</span><span class="bar">/</span><span class="fden2">10 + 2</span></span></span> = <span class="nowrap">8<span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">3</span></span></span> inches, the place of the division indicating -two pounds; and for three pounds <span class="nowrap"> <span class="fraction2"><span class="fnum">100</span><span class="bar">/</span><span class="fden2">10 + 3</span></span></span> = <span class="nowrap">7<span class="fraction"><span class="fnum">9</span><span class="bar">/</span><span class="fden">13</span></span></span> -inches for the place of the division indicating three -pounds, and so on.</p> - -<p>This balance is subject to the inconvenience of the -divisions becoming much shorter as the weight increases. -The distance between the divisions indicating one and -two pounds being, in the example we have given, about -seven tenths of an inch, whilst that between 20 and 21 -pounds is only one tenth of an inch; consequently a -very small error in the place of the divisions indicating -the larger weights would occasion very inaccurate results. -The Danish balance is represented at <i><a href="#i_p302a">fig. 194.</a></i></p> - -<p><span class="pagenum" id="Page_301">301</span></p> - - -<p class="tac"><i>The Bent Lever Balance.</i></p> - -<p>This instrument is represented at <i><a href="#i_p302a">fig. 195.</a></i> The -weight at C, is fixed at the end of the bent lever -A B C, which is supported by its axis B on the pillar -I H. A scale-pan E, is suspended from the other end of -the lever at A. Through the centre of motion B draw -the horizontal line K B G, upon which, from A and C -let fall the perpendiculars A K and C D. Then if B K -and B D are reciprocally proportional to the weights at -A and C, they will be in equilibrio, but if not, the weight -C will move upwards or downwards along the arc F G -till that ratio is obtained. If the lever be so bent that -when A coincides with the line G K, C coincides with -the vertical B H, then as C moves from F to G, its -momentum will increase while that of the weight in the -scale-pan E will decrease. Hence the weight in E, corresponding -to different positions of the balance, may be -expressed on the graduated arc F G.</p> - - -<p class="tac"><i>Brady’s Balance, or Weighing Apparatus.</i></p> - -<p>This partakes of the properties both of the bent -lever balance and of the steelyard. It is represented, -at <i><a href="#i_p302a">fig. 196.</a></i> A B C is a frame of cast iron having a -great part of its weight towards A. F is a fulcrum, and -E a moveable suspender, having a scale and hook at its -lower extremity. E K G are three distinct places, to -which the suspender E may be applied, and to which -belong respectively the three graduated scales of division -expressing weights, <i>f</i> C, <i>c d</i>, and <i>a b</i>. When the scale -and suspender are applied at G, the apparatus is in equilibrio, -with the edge A B horizontal, and the suspender -cuts the zero on the scale <i>a b</i>. Now, any substance, the -weight of which is to be ascertained, being put into the -scale, the whole apparatus turns about F, and the part -towards B descends till the equilibrium is again established, -when the weight of the body is read off from the -scale <i>a b</i>, which registers to ounces and extends to two -pounds. If the weight of the body exceed two pounds,<span class="pagenum" id="Page_302">302</span> -and be less than eleven pounds, the suspender is placed -at K; and when the scale is empty, the number 2 is -found to the right of the index of the suspender. If now -weights exceeding two pounds be placed in the scale, the -whole again turns about F, and the weight of the body -is shown on the graduated arc <i>c d</i>, which extends to -eleven pounds, and registers to every two ounces.</p> - -<p>If the weight of the body exceed eleven pounds, the -suspender is hung on at E, and the weights are ascertained -in the same manner on the scale <i>f</i> C to thirty -pounds, the subdivisions being on this scale quarters of -pounds. The same principles would obviously apply to -weights greater or less than the above. To prevent -mistake, the three points of support G, K, E, are numbered -1, 2, 3; and the corresponding arcs are respectively -numbered in the same manner. When the hook -is used instead of the scale, the latter is turned upwards, -there being a joint at <i>m</i> for that purpose.</p> - - -<p class="tac"><i>The Weighing Machine for Turnpike Roads.</i></p> - -<p>This machine is for the purpose of ascertaining the -weight of heavy bodies, such as wheel carriages. It consists -of a wooden platform placed over a pit made in the -line of the road, and which contains the machinery. The -pit is walled withinside, and the platform is fitted to the -walls of the pit, but without touching them, and it is -therefore at liberty to move freely up and down. The -platform is supported by levers placed beneath it, and is -exactly level with the surface of the road, so that a carriage -is easily drawn on it, the wheels being upon the -platform whilst the horses are upon the solid ground -beyond it. The construction of this machine will be -readily understood by reference to <i><a href="#i_p308a">fig. 197.</a></i>, in which -the platform is supposed to be transparent so as to allow -of the levers being seen below it.</p> - -<p>A, B, C, D, represent four levers tending towards the -centre of the platform, and each moveable on its fulcrum -at A, B, C, D; the fulcrum of each rests upon a piece -securely fixed in the corner of the pit. The platform is<span class="pagenum" id="Page_303">303</span> -supported upon the cross pins <i>a</i>, <i>b</i>, <i>c</i>, <i>d</i>, by means of -pieces of iron which project from it near its corners, and -which are represented in the plate by the short dark -lines crossing the pins <i>a</i>, <i>b</i>, <i>c</i>, <i>d</i>. The four levers are -connected under the centre of the platform, but not so -as to prevent their free motion, and are supported by a -long lever at the point F, the fulcrum of which rests -upon a piece of masonry at E: the end of this last lever -passes below the surface of the road into the turnpike -house, and is there attached to one arm of a balance, or, -as in Salmon’s patent weighing machine, to a strap -passing round a cylinder which winds up a small weight -round a spiral, and indicates, by means of an index, the -weight placed upon the platform.</p> - -<div class="figcenter" id="i_p302a" style="max-width: 31.25em;"> - <img src="images/i_p302a.jpg" alt="" /> - <div class="caption"> -<p><span class="l-align"><i>Captn. Kater, del.</i></span> - -<span class="r-align"><i>H. Adlard, sc.</i></span></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<p>Suppose the distance from A to F to be ten times as -great as that from A to <i>a</i>, then a force of one pound -applied beneath F would balance ten pounds applied at -<i>a</i>, or upon the platform. Again: let the distance from -E to G be also ten times greater than the distance from -the fulcrum E to F; then a force of one pound applied -to raise up the end of the lever G would counterpoise a -weight of ten pounds placed upon F. Now, as we gain -ten times the power by the first levers, and ten times -more by the lever E G, it follows, that a force of one -pound tending to elevate G, would balance 100 lbs. -placed on the platform; so that if the end of the lever -G be attached to one arm of a balance, a weight of 10 lbs. -placed in a scale suspended from the other arm, will -express the value of 1000 lbs. placed upon the platform. -The levers are counterpoised, when the platform is not -loaded, by a weight H applied to the end of the last -lever, continued beyond the fulcrum for that purpose.</p> - - -<p class="tac"><i>Of Instruments for weighing by means of a Spring.</i></p> - -<p>The spring is well adapted to the construction of a -weighing machine, from the property it possesses of -yielding in proportion to the force impressed, and consequently -giving a scale of equal parts for equal additions -of weight. It is liable, however, to suffer injury, unless<span class="pagenum" id="Page_304">304</span> -the steel of which it is composed be very well tempered, -from a want of perfect elasticity, and, consequently, from -not returning to its original place after it has been forcibly -compressed. This, however, must be considered to -arise, in a great measure, from imperfection of workmanship, -or of the material employed, or from its having -been subjected to too great a force.</p> - - -<p class="tac"><i>The Spring Steelyard.</i></p> - -<p>The little instrument known by this name is in very -general use, and is particularly convenient where great -accuracy is not necessary, as a spring which will ascertain -weights from one pound to fifty, is contained in a -cylinder only 4 inches long and <span class="nowrap"> <span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">4</span></span></span> inch diameter.</p> - -<p>This instrument is represented at <i><a href="#i_p308a">fig. 198.</a></i> It consists -of a tube of iron, of the dimensions just stated, -closed at the bottom, to which is attached an iron hook -for supporting the substance to be weighed; a rod of -iron <i>a b</i>, four tenths of an inch wide and one tenth -thick, is firmly fixed in the circular plate <i>c d</i>, which -slides smoothly in the iron tube.</p> - -<p>A strong steel spring is also fastened to this plate, and -passed round the rod <i>a b</i> without touching it, and -without coming in contact with the interior of the cylindrical -tube. The tube is closed at the top by a circular -piece of iron through which the piece <i>a b</i> passes.</p> - -<p>Upon the face of <i>a b</i> the weight is expressed by -divisions, each of which indicates one pound, and five of -such divisions in the instrument now before us occupy -two tenths of an inch. The divisions, notwithstanding, -are of sufficient size to enable them to be subdivided by -the eye.</p> - -<p>To use this instrument, the substance to be weighed -is suspended by the hook, the instrument being held by -a ring passing through the rod at the other end. The -spring then suffers a compression proportionate to the -weight, and the number of pounds is indicated by the -division on the rod which is cut by the top of the cylindrical -tube.</p> - -<p><span class="pagenum" id="Page_305">305</span></p> - - -<p class="tac"><i>Salter’s improved Spring Balance.</i></p> - -<p>A very neat form of the instrument last described has -been recently brought before the public by Mr. Salter, -under the name of the Improved Spring Balance. It -is represented at <i><a href="#i_p308a">fig. 199.</a></i> The spring is contained in -the upper half of a cylinder behind the brass plate -forming the face of the instrument; and the rod is fixed -to the lower extremity of the spring, which is consequently -extended, instead of being compressed, by the -application of the weight. The divisions, each indicating -half a pound, are engraved upon the face of the brass plate, -and are pointed out by an index attached to the rod.</p> - - -<p class="tac"><i>Marriott’s Patent Dial Weighing Machine.</i></p> - -<p>The exterior of this instrument is represented at -<i><a href="#i_p308a">fig. 200.</a></i>, and the interior at <i><a href="#i_p308a">fig. 201.</a></i> A B C is a shallow -brass box, having a solid piece as represented at A, to -which the spring D E F is firmly fixed by a nut at -D. The other end of the spring at F is pinned to -the brass piece G H, to the part of which at G is also -fixed the iron racked plate I. A screw L serves as a -stop to keep this rack in its place. The teeth of the -rack fit into those of the pinion M, the axis of which -passes through the centre of the dial-plate, and carries -an index which points out the weight. The brass piece -G H is merely a plate where it passes over the spring, -and the tail piece H, to which the weight is suspended, -passes through an opening in the side of the box.</p> - - -<p class="tac"><i>Of the Dynamometer.</i></p> - -<p>This is an important instrument in mechanics, calculated -to measure the muscular strength exerted by -men and animals. It consists essentially of a spring -steelyard, such as that we first described. This is sometimes -employed alone, and sometimes in combination -with various levers, which allow of the spring being -made more delicate, and consequently increase the extent -of the divisions indicating the weight.</p> - -<p><span class="pagenum" id="Page_306">306</span></p> - -<p>The first instrument of this kind appears to have been -invented by Mr. Graham, but it was too bulky and inconvenient -for use. M. le Roy made one of a more -simple construction. It consisted of a metal tube, about -a foot long, placed vertically upon a stand, and containing -in the inside a spiral spring, having above it a graduated -rod terminating in a globe. This rod entered -the tube more or less in proportion to the force applied -to the globe, and the divisions indicated the quantity of -this force. Therefore, when a man pressed upon the globe -with all his strength, the divisions upon the rod showed -the number of pounds weight to which it was equal.</p> - -<p>An instrument of this kind for determining the force -of a blow struck by a man with his fist was lately exhibited -at the National Repository. It was fixed to a -wall, from which it projected horizontally. In place of -the globe there was a cushion to receive the blow, and -as the suddenness with which the spring returned rendered -it impossible to read the division upon the rod, -another rod similarly divided was forced in by the plate -forming the basis of the cushion, and remained stationary -when the spring returned. The common spring -steelyard, however, which we first described, is in principle -the same as M. le Roy’s dynamometer, and is -much more conveniently constructed for the purpose we -are considering. The ring at one end may be fixed to -an immovable object, and the hook at the other attached -to a man, or to an animal, and the extent to which the -graduated rod is drawn out of the cylinder shows at -once the force which is applied. Though this is perhaps -the best, and certainly the most simple dynamometer, -others have been contrived, which are, however, but -modifications of the spring steelyard. One of these is -represented at <i><a href="#i_p308a">fig. 202.</a></i> The spiral spring acts in the -manner before described, but its divisions are increased -in size, and therefore rendered more perceptible by -means of a rack fixed to the plate, acting against the -spiral spring, the teeth of which move a pinion upon -which the arm I is fixed, pointing to the graduated arc K.</p> - -<p><span class="pagenum" id="Page_307">307</span></p> - -<p>Another dynamometer has been invented by Mr. Salmon; -it is represented at <i><a href="#i_p308a">fig. 203.</a></i> and is a combination -of levers with the spring. By means of these -levers a much more delicate spring, and which is therefore -more sensible, may be employed than in the dynamometer -last described.</p> - -<p>The manner in which these levers and spring act will -be readily understood by an inspection of the figure. -Like the weighing machine for carriages, the fulcrum of -each lever is at one end, and the force is diminished in -passing to the spring, in the ratio of the length of its -arms. The spring moves a pinion by means of a rack, -upon which pinion a hand is placed, indicating by divisions -upon a circular dial-plate, the amount of the -force employed.</p> - -<p>The spring used in this machine is calculated to weigh -only about 50 lbs. instead of about 5 cwt., as in the -last described; but by means of the levers which intervene -between it and the force applied, it will serve to -estimate a force equal to 6 cwt., and might obviously be -made to go to a much greater extent, by varying the -ratio of the length of the arms of the levers.</p> - - -<p class="tac">ON COMPENSATION PENDULUMS.</p> - -<p id="p336">(336.) It is said of Galileo that, when very young, he -observed a lamp suspended from the roof of a church at -Pisa, swinging backwards and forwards with a pendulous -motion. This, if it had been remarked at all by an -uneducated mind, would, most probably, have been passed -by as a common occurrence, unworthy of the slightest -notice; but to the mind imbued with science no incident -is insignificant; and a circumstance apparently the -most trivial, when subjected to the giant force of expanded -intellect, may become of immense importance to -the improvement and to the well-being of man. The -fall of an apple, it is said, suggested to Newton the -theory of gravitation, and his powerful mind speedily -extended to all creation that great law which brings an<span class="pagenum" id="Page_308">308</span> -apple to the ground. The swinging of a lamp in a -church at Pisa, viewed by the piercing intellect of Galileo, -gave rise to an instrument which affords the most perfect -measure of time, which serves to determine the figure -of the earth, and which is inseparably connected with all -the refinements of modern astronomy.</p> - -<p>The properties of the pendulum, and the manner in -which it serves to measure time, have been fully explained -in chapter <a href="#CHAP_XI">xi</a>.; and if a substance could be -found not susceptible of any change in its dimensions -from a change of temperature, nothing more would be -necessary, as the centre of oscillation would always remain -at the same distance from the point of suspension. As -every known substance, however, expands with heat, -and contracts with cold, the length of the pendulum will -vary with every alteration of temperature, and thus the -time of its vibration will suffer a corresponding change. -The effect of a difference of temperature of 25°, or -that which usually occurs between winter and summer, -would occasion a clock furnished with a pendulum having -an iron rod to gain or lose six seconds in twenty-four -hours.</p> - -<p>It became, then, highly important to discover some -means of counteracting this variation to which the length -of the pendulum was liable, or, in other words, to devise -a method by which the centre of oscillation should, under -every change of temperature, remain at the same distance -from the point of suspension: happily, the difference in -the rate of expansion of different metals presented a ready -means of effecting this.</p> - -<p>Graham, in the year 1715, made several experiments -to ascertain the relative expansions of various metals, -with a view of availing himself of the difference of the -expansions of two or more of them when opposed to -each other, to construct a compensating pendulum. But -the difference he found was so small, that he gave up all -hope of being able to accomplish his object in that way. -Knowing, however, that mercury was much more affected -by a given change of temperature than any other sub<span class="pagenum" id="Page_309">309</span>stance, -he saw that if the mercury could be made to -ascend while the rod of the pendulum became longer, -and <i>vice versâ</i>, the centre of oscillation might always be -kept at the same distance from the point of suspension. -This idea happily gave birth to the mercurial pendulum, -which is now in very general use.</p> - -<div class="figcenter" id="i_p308a" style="max-width: 31.25em;"> - <img src="images/i_p308a.jpg" alt="" /> - <div class="caption"> -<p><span class="l-align"><i>Captn. Kater, del.</i></span> - -<span class="r-align"><i>H. Adlard, sc.</i></span></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<p>In the mean time, Graham’s suggestion excited the -ingenuity of Harrison, originally a carpenter at Barton -in Lincolnshire, who, in 1726, produced a pendulum -formed of parallel brass and steel rods, known by the -name of the gridiron pendulum.</p> - -<p>In the mercurial pendulum, the bob or weight is the -material affording the compensation; but in the gridiron -pendulum the object is attained by the greater expansion -of the brass rods, which raise the bob upwards towards -the point of suspension as much as the steel rods elongate -downwards.</p> - -<p>In the present article, we shall describe such compensation -pendulums as appear to us likely to answer best -in practice; and we trust we shall be able to simplify -the subject so as to render a knowledge of mathematics -in the construction of this important instrument unnecessary.</p> - -<p>The following table contains the linear expansion of -various substances in parts of their length, occasioned by -a change of temperature amounting to one degree. We -have taken the liberty of extracting it from a very valuable -paper by F. Bailey, Esq., on the mercurial compensation -pendulum, published in the Memoirs of the -Astronomical Society of London for 1824.</p> - -<p><span class="pagenum" id="Page_310">310</span></p> - - -<p class="tac"><a id="TABLE_I"></a>TABLE I.</p> - -<p class="tac"><i>Linear Expansion of various Substances for One Degree -of Fahrenheit’s Thermometer.</i></p> - -<div class="center"> -<table width="450" class="" cellpadding="2" summary=""> -<tr> -<td class="tac ball" colspan="2"><div>Substances.</div></td> -<td class="tac ball"><div>Expansions.</div></td> -<td class="tac ball" colspan="2"><div>Authors.</div></td> -</tr> -<tr> -<td class="tal bl" rowspan="2">White Deal,</td> -<td class="tar vab" rowspan="2"><img src="images/31x6bl.png" width="6" height="31" alt="" /></td> -<td class="tal brl">·0000022685</td> -<td class="tal"></td> -<td class="tal br">Captain Kater.</td> -</tr> -<tr> - -<td class="tal brl">·0000028444</td> -<td class="tal"></td> -<td class="tal br">Dr. Struve.</td> -</tr> -<tr> -<td class="tal bl">English Flint Glass,</td> -<td class="tar"></td> -<td class="tal brl">·0000047887</td> -<td class="tal"></td> -<td class="tal br">Dulong and Petit.</td> -</tr> -<tr> -<td class="tal bl" rowspan="2">Iron (cast),</td> -<td class="tar vab" rowspan="2"><img src="images/31x6bl.png" width="6" height="31" alt="" /></td> -<td class="tal brl">·0000061700</td> -<td class="tal"></td> -<td class="tal br">General Roy.</td> -</tr> -<tr> -<td class="tal brl">·0000065668</td> -<td class="tal"></td> -<td class="tal br">Dulong and Petit.</td> -</tr> -<tr> -<td class="tal bl">Iron (wire),</td> -<td class="tar"></td> -<td class="tal brl">·0000068613</td> -<td class="tal"></td> -<td class="tal br">Lavoisier and L.</td> -</tr> -<tr> -<td class="tal bl">Iron (bar),</td> -<td class="tar"></td> -<td class="tal brl">·0000069844</td> -<td class="tal"></td> -<td class="tal br">Hasslar.</td> -</tr> -<tr> -<td class="tal bl">Steel (rod),</td> -<td class="tar"></td> -<td class="tal brl">·0000063596</td> -<td class="tal"></td> -<td class="tal br">General Roy.</td> -</tr> -<tr> -<td class="tal bl">Brass,</td> -<td class="tar"></td> -<td class="tal brl">·0000104400</td> -<td class="tal vab"><img src="images/65x6bl.png" width="6" height="65" alt="" /></td> -<td class="tal br">Commissioners of<br />Weights and Measures<br />—mean of several<br />experiments.</td> -</tr> -<tr> -<td class="tal bl">Lead,</td> -<td class="tar"></td> -<td class="tal brl">·0000159259</td> -<td class="tal"></td> -<td class="tal br">Smeaton.</td> -</tr> -<tr> -<td class="tal bl">Zinc,</td> -<td class="tar"></td> -<td class="tal brl">·0000163426</td> -<td class="tal"></td> -<td class="tal br">Ditto.</td> -</tr> -<tr> -<td class="tal bl">Zinc (hammered),</td> -<td class="tar"></td> -<td class="tal brl">·0000172685</td> -<td class="tal"></td> -<td class="tal br">Ditto.</td> -</tr> -<tr> -<td class="tal bbl" colspan="2">Mercury <i>in bulk</i>,</td> -<td class="tal bbrl">·00010010</td> -<td class="tal bb"></td> -<td class="tal bbr">Dulong and Petit.</td> -</tr> -</table> -</div> - -<p>From this table it is easy to determine the length of -a rod of any substance the expansion of which shall be -equal to that of a rod of given length of any other substance.</p> - -<p>The lengths of such rods will be inversely proportionate -to their expansions. If, therefore, we divide the lesser -expansion by the greater (supposing the rod the length -of which is given to be made of the lesser expansible -material), and multiply the given length by this quotient, -we shall have the required length of a rod, the expansion -of which will be equal to that of the rod given. For -example:—The expansion of a rod of steel being, -from the above table, ·0000063596, and that of brass,<span class="pagenum" id="Page_311">311</span> -·0000104400; if it were required to determine the -length of a rod of brass which should expand as much as -a rod of steel of 39 inches in length, we have <span class="nowrap"><span class="fraction"><span class="fnum">·0000063596</span><span class="bar">/</span><span class="fden">·0000104400</span></span></span> -= ·6091, which, multiplied by 39, gives 23·75 inches -for the length of brass required.</p> - -<p>We shall here, in order to facilitate calculation, give -the ratio of the lengths of such substances as may be employed -in the construction of compensation pendulums.</p> - - -<p class="tac"><a id="TABLE_II"></a>TABLE II.</p> - -<div class="center"> -<table width="400" cellpadding="2" summary=""> -<tr> -<td class="tal btl">Steel rod and brass compensation, as 1:</td> -<td class="tal btr">·6091</td> -</tr> -<tr> -<td class="tal bl">Iron wire rod and lead compensation,</td> -<td class="tal br">·4308</td> -</tr> -<tr> -<td class="tal bl">Steel rod and lead compensation,</td> -<td class="tal br">·3993</td> -</tr> -<tr> -<td class="tal bl">Iron wire rod and zinc compensation,</td> -<td class="tal br">·3973</td> -</tr> -<tr> -<td class="tal bl">Steel rod and zinc compensation,</td> -<td class="tal br">·3682</td> -</tr> -<tr> -<td class="tal bl">Glass rod and lead compensation,</td> -<td class="tal br">·3007</td> -</tr> -<tr> -<td class="tal bl">Glass rod and zinc compensation,</td> -<td class="tal br">·2773</td> -</tr> -<tr> -<td class="tal bl">Deal rod and lead compensation,</td> -<td class="tal br">·1427</td> -</tr> -<tr> -<td class="tal bl">Deal rod and zinc compensation,</td> -<td class="tal br">·1313</td> -</tr> -<tr> -<td class="tal bl">Steel rod and mercury in a steel cylinder,</td> -<td class="tal br">·0728</td> -</tr> -<tr> -<td class="tal bl">Steel rod and mercury in a glass cylinder,</td> -<td class="tal br">·0703</td> -</tr> -<tr> -<td class="tal bbl">Glass rod and mercury in a glass cylinder,</td> -<td class="tal bbr">·0529</td> -</tr> -</table> -</div> - -<p>It is evident that in this table the decimals express -the length of a rod of the compensating material, the -expansion of which is equal to that of a pendulum rod -whose length is unity.</p> - -<p>As we are not aware of the existence of any work -which contains instructions that might enable an artist -or an amateur to make a compensation pendulum, we -shall endeavour to give such detailed information as may -free the subject from every difficulty.</p> - -<p>The pendulum of a clock is generally suspended by a -spring, fixed to its upper extremity, and passing through -a slit made in a piece which is called the cock of the -pendulum. The point of suspension is, therefore, that -part of the spring which meets the lower surface of the -cock. Now the distance of the centre of oscillation of -the pendulum from this point may be varied in two -ways; the one by drawing up the spring through this<span class="pagenum" id="Page_312">312</span> -slit, and the other by raising the bob of the pendulum. -Either of these methods may be practised in the compensation -pendulum, but the former is subject to objections -from which the latter is exempt.</p> - -<p>Suppose it were required to compensate a pendulum -of 39 inches in length, of steel, by means of the expansion -of a brass rod. Here, referring to <i><a href="#i_p314a">fig. 204.</a></i>, -we have S C 39 inches (which is to remain constant) -of steel; the pendulum spring, passing through the cock -at S, is attached to another rod of steel, which is fixed -to the cross piece R A at A. The other end of the cross -piece at R is fastened to a brass rod, the lower extremity -of which is fixed to the cock of the pendulum at B. -Now the brass rod B R must expand upwards, as much -as the steel rod A C expands downwards; and the length -of the brass must be such as to effect this, leaving 39 -inches of the steel rod below the cock of the pendulum.</p> - -<p>Let us first try 80 inches of steel. Multiplying -this by ·6091, we have 48·73 inches for the length -of brass, which compensates 80 inches of steel. But as -48·73 inches of the steel, equal in length to the brass, -would in this case be above the cock of the pendulum, -it would leave only 31·27 inches below it, instead of -39 inches.</p> - -<p>Let us now try 100 inches of steel. This, multiplied -as before by ·6091, gives 60·91 inches, according to the -expansions which we have used, for the length of the brass -rod, and leaves 39·09 inches below the cock of the -pendulum, which is sufficiently near for our present -purpose.</p> - -<p>From what has been said we may perceive that the -total length of the material of which the pendulum rod -is composed must be always equal to the length of the -pendulum added to the length of the compensation.</p> - -<p>In this instance we have effected our object, by drawing -the pendulum-spring through the slit; but we will -now show how the same thing may be done by moving -the bob of the pendulum. At <i><a href="#i_p314a">fig. 205.</a></i>, let S C, as before, -be equal to 39 inches. Let the steel rod S D turn off<span class="pagenum" id="Page_313">313</span> -at right angles at D, and let a rod of brass B R, of 61 -inches in length, ascend perpendicularly from this cross -piece to R. To the upper part of the brass rod fix another -cross piece R A, and from the extremity A let a -steel rod descend to E, bending it as in the figure till it -reaches C. Now the total length of the pieces of steel -expanding downwards is equal to S D, D F, and F C -(amounting together to 39 inches), to which must be -added a length of steel equal to that of the brass rod B R, -(61 inches), making together 100 inches of steel as before, -the expansion of which downwards is compensated -by that of the brass rod, of 61 inches in length, -expanding upwards.</p> - -<p>This form, however, is evidently inconvenient, from -the great length of brass and steel which is carried above -the cock of the pendulum; but it is the same thing whether -the brass and steel be each in one piece, or divided -into several, provided the pieces of steel be all so arranged -as to expand downwards, and those of brass upwards. -Thus, at <i><a href="#i_p314a">fig. 206.</a></i>, the portions of steel expanding -downwards are together equal, as before, to 100 inches, -and the two brass pieces expanding upwards are together -equal to 61 inches. So that, in fact, the two last forms -of compensation which we have described differ in no -respect from each other in principle, but only in the -arrangement of the materials. The last is the half of -the gridiron pendulum, the remaining bars being merely -duplicates of those we have described, and serving no -other purpose but to form a secure frame-work.</p> - - -<p class="tac"><i>Harrison’s Gridiron Pendulum.</i></p> - -<p>After what has been said, little more is necessary than -to give a representation of this pendulum. This is done -at <i><a href="#i_p314a">fig. 207.</a></i>, in which the darker lines represent the steel -rods, and the lighter those of brass. The central rod is -fixed at its lower extremity to the middle of the third -cross piece from the bottom, and passes freely through -holes in the cross pieces which are above, whilst the -other rods are secured near their extremities to the cross<span class="pagenum" id="Page_314">314</span> -pieces by pins passing through them. In order to render -the whole more secure, the bars pass freely through -holes made in two other cross pieces, the extremities of -which are fixed to the exterior steel wires. As different -kinds of the same metal vary in their rate of -expansion, the pendulum when finished may be found -upon trial to be not duly compensated. In this case one -or more of the cross pieces is shifted higher or lower -upon the bars, and secured by pins passed through fresh -holes.</p> - - -<p class="tac"><i>Troughton’s Tubular Pendulum.</i></p> - -<p>This is an admirable modification of Harrison’s gridiron -pendulum. It is represented at <i><a href="#i_p318a">fig. 208.</a></i>, where it -may be seen that it has the appearance of a simple pendulum, -as the whole compensation is concealed within -a tube six tenths of an inch in diameter.</p> - -<p>A steel wire, about one tenth of an inch in diameter, -is fixed in the usual manner to the spring by which the -pendulum is suspended. This wire passes to the bottom -of an interior brass tube, in the centre of which it is -firmly screwed. The top of this tube is closed, the steel -rod passing freely through a hole in the centre. Into -the top of this interior tube two steel wires, of one tenth -of an inch in diameter, are screwed into holes made in -that diameter, which is at right angles to the motion of -the pendulum. These wires pass down the tube without -touching either it or the central rod, through holes made -in the piece which closes the bottom of the interior tube. -The lower extremities of these wires, which project a -little beyond the inner tube, are securely fixed in a piece -which closes the bottom of an exterior brass tube, which -is of such a diameter as just to allow the interior tube -to pass freely through it, and of a sufficient length to -extend a little above it. The top of the exterior tube is -closed like that of the interior, having also a hole in its -centre, to allow the first steel rod to pass freely through -it. Into the top of the exterior tube, in that diameter -which coincides with the motion of the pendulum, a<span class="pagenum" id="Page_315">315</span> -second pair of steel wires of the same diameter as the -former are screwed, their distance from the central rod -being equal to the distance of each from the first pair. -They consequently pass down within the interior tube, -and through holes made in the pieces closing the lower -ends of both the interior and exterior tubes. The lower -ends of these wires are fastened to a short cylindrical -piece of brass of the same diameter as the exterior tube, -to which the bob is suspended by its centre.</p> - -<div class="figcenter" id="i_p314a" style="max-width: 31.25em;"> - <img src="images/i_p314a.jpg" alt="" /> - <div class="caption"> -<p><span class="l-align"><i>Captn. Kater, del.</i></span> - -<span class="r-align"><i>H. Adlard, sc.</i></span></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<p><i>Fig. 209.</i> is a full sized section of the rod; the three -concentric circles represent the two tubes, and the rectangular -position of the two pair of wires round the -middle one is shown by the five small circles.</p> - -<p><i>Fig. 210.</i> is the part which closes the upper end of the -interior tube. The two small circles are the two wires -which proceed from it, and the three large circles show -the holes through which the middle wire and the other -pair of wires pass.</p> - -<p><i>Fig. 211.</i> is the bottom of the interior tube. The small -circle in the centre is where the central rod is fastened -to it, the others the holes for the other four wires to pass -through.</p> - -<p><i>Fig. 212.</i> is the part which closes the top of the external -tube. In the large circle in the centre a small brass tube -is fixed, which serves as a covering for the upper part of -the middle wire, and the two small circles are to receive -the wires of the last expansion.</p> - -<p><i>Fig. 213.</i> represents the bottom of the exterior tube, in -which the small circles show the places where the wires -of the second expansion are fastened, and the larger ones -the holes for the other pair of wires to pass through.</p> - -<p><i>Fig. 214.</i> is a cylindrical piece of brass, showing the -manner in which the lower ends of the wires of the last -expansion are fastened to it, and the hole in the middle -is that by which it is pinned to the centre of the bob. -The upper ends of the two pair of wires are, as we have -observed, fastened by screwing them into the pieces -which stop up the ends of the tubes, but at the lower ends -they are all fixed as represented in <i><a href="#i_p318a">fig. 214.</a></i> The pieces<span class="pagenum" id="Page_316">316</span> -represented by <i>figs. 213.</i> and <i>214.</i> have each a jointed -motion, by means of which the fellow wires of each pair -would be equally stretched, although they were not exactly -of the same length.</p> - -<p>The action of this pendulum is evidently the same as -that of the gridiron pendulum, as we have three lengths -of steel expanding downwards, and two of brass expanding -upwards. The weight of the pendulum has a tendency -to straighten the steel rods, and the tubular form -of the brass compensation effectually precludes the fear -of its bending; an advantage not possessed by the gridiron -pendulum, in which brass rods are employed.</p> - -<p>Mr. Troughton, to the account he has given of this -pendulum in Nicholson’s Journal, for December, 1804, -has added the lengths of the different parts of which it -was composed, and the expansions of brass and steel -from which these lengths were computed. The length -of the interior tube was 31·9 inches, and that of the exterior -one 32·8 inches, to which must be added 0·4, the -quantity by which in this pendulum the centre of oscillation -is higher than the centre of the bob. These are -all of brass. The parts which are of steel are,—the middle -wire, which, including 0·6, the length of the suspension -spring, is 39·3 inches. The first pair of wires 32·5 -inches; and the second pair, 33·2 inches. The expansions -used were, for brass ·00001666, and for steel -·00000661, in parts of their length for one degree of -temperature.</p> - - -<p class="tac"><i>Benzenberg’s Pendulum.</i></p> - -<p>This pendulum is mentioned in Nicholson’s Journal -for April, 1804, and is taken from Voigt’s Magazin für -den Neuesten Zustande der Naturkunde, vol. iv. p. 787. -The compensation appears to have been effected by a -single rod of lead in the centre, of about half an inch -thick; the descending rods were made of the best thick -iron wire.</p> - -<p>As this pendulum deserves attention from the ease -with which it may be made, and as others which have<span class="pagenum" id="Page_317">317</span> -since been produced resemble it in principle, we have -given a representation of it at <i><a href="#i_p318a">fig. 215.</a></i>, where A B C D -are two rods of iron wire riveted into the cross -pieces A C B D. E F is a rod of lead pinned to the -middle of the piece B D, and also at its upper extremity -to the cross piece G H, into which the second pair of -iron wires are fixed, which pass downwards freely -through holes made in the cross piece B D. The lower -extremities of these last iron wires are fastened into the -piece K L, which carries the bob of the pendulum.</p> - -<p>To determine the length of lead necessary for the -compensation, we must recollect, as before, that the -distance from the point of suspension to the centre of -the bob (speaking always of a pendulum intended to -vibrate seconds) must be 39 inches. Let us suppose -the total length of the iron wire to be 60 inches; -then, from the table which we have given, we have -·4308 for the length of a rod of lead, the expansion of -which is equivalent to that of an iron rod whose length -is unity. Multiplying 60 inches by ·4308, we have -25·84 inches of lead, which would compensate 60 -inches of iron; but this, taken from 60 inches, leaves -only 34·16 instead of 39 inches. Trying again, in -like manner, 68·5 inches of iron, we find 29·5 inches -of lead for the length, affording an equivalent compensation, -and which, taken from 68·5 inches, leaves 39 -inches.</p> - -<p>The length of the rod of lead then required as a -compensation in this pendulum is about <span class="nowrap">29<span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> inches.</p> - -<p>The writer of this article would suggest another form -for this pendulum, which has the advantage of greater -simplicity of construction.</p> - -<p>S A, <i><a href="#i_p318a">fig. 216.</a></i>, is a rod of iron wire, to which the -pendulum spring is attached. Upon this passes a cylindrical -tube of lead, <span class="nowrap">29<span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> inches long, which is either -pinned at its lower extremity to the end of the iron rod -S A, or rests upon a nut firmly screwed upon the extremity -of this rod.</p> - -<p>A tube of sheet iron passes over the tube of lead, and<span class="pagenum" id="Page_318">318</span> -is furnished at top with a flanche, by which it is supported -upon the leaden tube; or it may be fastened to -the top of this tube in any manner that may be thought -convenient.</p> - -<p>The bob of the pendulum may be either passed upon -the iron tube (continued to a sufficient length) and -secured by a pin passing through the centre of the bob, -or the iron tube may be terminated by an iron wire -serving the same purpose.</p> - -<p>Here we have evidently the same expansions upwards -and downwards as in the gridiron form, given to this -pendulum by Mr. Benzenberg, joined to the compactness -of Troughton’s tubular pendulum.</p> - - -<p class="tac"><i>Ward’s Compensation Pendulum.</i></p> - -<p>In the year 1806, Mr. Henry Ward, of Blandford in -Dorsetshire, received the silver medal of the Society of -Arts for the compensation pendulum which we are about -to describe.</p> - -<p><i>Fig. 217.</i> is a side view of the pendulum rod when -together. H H and I I are two flat rods of iron about -an eighth of an inch thick. K K is a bar of zinc -placed between them, and is nearly a quarter of an inch -thick. The corners of the iron bars are bevelled off, -which gives them a much lighter appearance. These bars -are kept together by means of three screws, O O O, which -pass through oblong holes in the bars H H and K K, and -screw into the rod I I. The bar H H is fastened to the -bar of zinc K K, by the screw <i>m</i>, which is called the -adjusting screw. This screw is tapped into H H, and -passes just through K K; but that part of the screw -which passes K K has its threads turned off. The iron -bar I I has a shoulder at its upper end, and rests on the -top of the zinc bar K K and is wholly supported by -it. There are several holes for the screw <i>m</i>, in order to -adjust the compensation.</p> - -<p>The action of this pendulum is similar to that last -described, the zinc expanding upwards as much as -the iron rods expand downwards, and consequently the<span class="pagenum" id="Page_319">319</span> -instance from the point of suspension to the centre of -oscillation remains the same.</p> - -<div class="figcenter" id="i_p318a" style="max-width: 31.25em;"> - <img src="images/i_p318a.jpg" alt="" /> - <div class="caption"> -<p><span class="l-align"><i>Captn. Kater, del.</i></span> - -<span class="r-align"><i>H. Adlard, sc.</i></span></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<p>Mr. Ward states that the expansion of the zinc he -used (hammered zinc) was greater than that given in -the tables. He found that the true length of the zinc -bar should be about 23 inches; our computation would -make it nearly 26.</p> - - -<p class="tac"><i>The Compensation Tube of Julien le Roy.</i></p> - -<p>We mention this merely to state that it is similar in -principal to the apparatus represented at <i><a href="#i_p314a">fig. 204.</a></i>, -with merely this difference, that, instead of the steel -rod being fixed to a cross piece proceeding from the -brass bar B R, it is attached to a cap fixed upon a -brass tube (through which it passes) of the same length -as that of the brass rod B R. Cassini spoke well of -this pendulum, and it was used in the observatory of -Cluny about the year 1748.</p> - - -<p class="tac"><i>Deparcieux’s Compensation.</i></p> - -<p>This was contrived in the same year as that invented -by Julien le Roy. It is represented at <i><a href="#i_p322a">fig. 218.</a></i>, where -A B D F is a steel bar, the ends of which are to be -fixed to the lower sides of pieces forming a part of -the cock of the pendulum. G E I H is of brass, and stands -with its extremities resting on the horizontal part B D -of the steel frame. The upper part E I of the brass -frame passes above the cock of the pendulum, and -admits the tapped wire K, to which the pendulum spring -is fixed through a squared hole in the middle. A nut -upon this tapped wire gives the adjustment for time. -The spring passes through the slit in the cock in the -usual manner.</p> - -<p>It may be easily perceived that this pendulum is in -principle the same as that of Le Roy; the expansion -of the total length of steel A B S C downwards being -compensated by the equivalent expansion of the brass bar -G E upwards. It is, however, preferable to Le Roy’s, because -the compensation is contained in the clock case.</p> - -<p><span class="pagenum" id="Page_320">320</span></p> - -<p>Deparcieux had previously published, in the year 1739, -an improvement of an imperfectly compensating pendulum, -proposed in the year 1733 by Regnauld, a -clockmaker of Chalons. In this pendulum Deparcieux -employed a lever with unequal arms to increase the -effect of the expansion of the brass rod, which was too -short.</p> - -<p>We may here remark, that all fixed compensations -are liable to the same objection, namely, that of not -moving with the pendulum, and therefore not taking -precisely the same temperature.</p> - - -<p class="tac"><i>Captain Kater’s Compensation Pendulum.</i></p> - -<p>In Nicholson’s Journal, for July, 1808, is the description -of a compensation pendulum by the writer of this -article. In this pendulum the rod is of white deal, -three quarters of an inch wide, and a quarter of an inch -thick. It was placed in an oven, and suffered to remain -there for a long time until it became a little charred. -The ends were then soaked in melted sealing-wax; and -the rod, being cleaned, was coated several times with copal -varnish. To the lower extremity of the rod a cap of -brass was firmly fixed, from which a strong steel screw -proceeded for the purpose of regulating the pendulum -for time in the usual manner.</p> - -<p>A square tube of zinc was cast, seven inches long and -three quarters of an inch square; the internal dimensions -being four tenths of an inch. The lower part of -the pendulum rod was cut away on the two sides, so as -to slide with perfect freedom within the tube of zinc. -To the bottom of this zinc tube a piece of brass a quarter -of an inch thick was soldered, in which a circular hole -was made nearly four tenths of an inch in diameter, -having a screw on the inside. A cylinder of zinc, furnished -with a corresponding screw on its surface, fitted -into this aperture, and a thin plate of brass screwed upon -the cylinder, served as a clamp to prevent any shake -after the length of zinc necessary for compensation -should have been determined. A hole was made through<span class="pagenum" id="Page_321">321</span> -the axis of the cylinder, through which passed the steel -screw terminating the pendulum rod.</p> - -<p>An opening was made through the bob of the pendulum, -extending to its centre, to admit the square tube -of zinc which was fixed at its upper extremity to the -centre of the bob. The pendulum rod passed through -the bob in the usual manner, and the whole was supported -by a nut on the steel screw at the extremity.</p> - -<p>In this form the compensation acts immediately upon -the centre of the bob, elevating it along the rod as -much as the rod elongates downwards: the method of -calculating the length of the required compensation is -precisely the same as that we have before given.</p> - -<p>Assuming the length of the deal rod to be 43 inches, -and multiplying this by ·1313 from <a href="#TABLE_II">Table II</a>., we have -5·64 inches for the length of the zinc necessary to counteract -the expansion of the deal. The length of the -steel screw between the termination of the pendulum -rod and the nut was two inches, and that of the suspension -spring one inch. Now, 3 inches of steel multiplied -by ·3682 would give 1·10 inches for the length -of zinc which would compensate the steel, and, adding -this to 5·64 inches, we have 6·74 inches for the whole -length of zinc required.</p> - -<p>In this pendulum, the length of the compensating -part may be varied by means of the zinc cylinder furnished -with a screw for that purpose. The bob of this -pendulum and its compensation are represented at -<i><a href="#i_p322a">fig. 219.</a></i></p> - -<p>It has been objected to the use of wooden pendulum -rods, that it is difficult, if not impossible, to secure them -from the action of moisture, which would at once be -fatal to their correct performance. The pendulum now -before us has, however, been going with but little intermission -since it was first constructed: it is attached -to a sidereal clock, not of a superior description, and -exposed to very considerable variations of moisture and -dryness; yet the change in its rate has been so very -trifling as to authorize the belief that moisture has little<span class="pagenum" id="Page_322">322</span> -or no effect upon a wooden rod prepared in the manner -we have described. Its rate, under different temperatures, -shows that it is over-compensated; the length of the -zinc remaining, as stated in Nicholson’s Journal 7·42 -inches, instead of which it appears, by our present compensation, -that it should be 6·78 inches.</p> - - -<p class="tac"><i>Reid’s Compensation Pendulum.</i></p> - -<p>Mr. Adam Reid of Woolwich presented to the Society -of Arts, in 1809, a compensation pendulum, for which -he was rewarded with fifteen guineas. This pendulum -is the same in principle with that last described; the -rod, however, is of steel instead of wood, and the compensation -possesses no means of adjustment. This pendulum -is represented at <i><a href="#i_p322a">fig. 220.</a></i>, where S B is the steel -rod, a little thicker where it enters the bob C, and of -a lozenge shape to prevent the bob turning, but above -and below it is cylindrical.</p> - -<p>A tube of zinc D passes to the centre of the bob from -below, and the bob is supported upon it by a piece which -crosses its centre, and which meets the upper end of the -tube.</p> - -<p>The rod being passed through the bob and zinc tube, -a nut is applied upon a screw at the lower extremity of -the rod in the usual manner. If the compensation -should be too much, the zinc tube is to be shortened -until it is correct.</p> - -<p>The length of the zinc tube will be the same in this -pendulum as in that of Mr. Ward—about 23 inches, -if his experiments are to be relied upon.</p> - -<p>The objection to this pendulum appears to be its -great length, which amounts to 62 inches. We conceive -it would be preferable to place the zinc above the bob, -as in the modification which we have suggested of Benzenberg’s -pendulum.</p> - - -<p class="tac"><i>Ellicott’s Pendulum.</i></p> - -<p>It appears that the idea of combining the expansions -of different metals with a lever, so as to form a com<span class="pagenum" id="Page_323">323</span>pensation -pendulum, originated with Mr. Graham; for -Mr. Short, in the Philosophical Transactions for 1752, -states that he was informed by Mr. Shelton, that Mr. -Graham, in the year 1737, made a pendulum, consisting -of three bars, one of steel between two of brass; and -that the steel bar acted upon a lever so as to raise the -pendulum when lengthened by heat, and to let it down -when shortened by cold.</p> - -<div class="figcenter" id="i_p322a" style="max-width: 31.25em;"> - <img src="images/i_p322a.jpg" alt="" /> - <div class="caption"> -<p><span class="l-align"><i>Captn. Kater, del.</i></span> - -<span class="r-align"><i>H. Adlard, sc.</i></span></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<p>This pendulum, however, was found upon trial to -move by jerks, and was therefore laid aside by the inventor -to make way for the mercurial pendulum.</p> - -<p>Mr. Short also says that Mr. Fotheringham, a quaker -of Lincolnshire, caused a pendulum to be made, in the -year 1738 or 1739, consisting of two bars, one of brass -and the other of steel, fastened together by screws with -levers to raise or let down the bob, and that these levers -were placed above the bob.</p> - -<p>Mr. John Ellicott of London had made very, accurate -experiments on the relative expansions of seven -different metals, which, however, will be found to differ -more or less from the results of the experiments of -others. It is not, however, from this to be concluded -that Ellicott’s determinations were erroneous; for the -expansion of a metal will suffer considerable change even -by the processes to which it is necessarily subjected in -the construction of a pendulum. It is therefore desirable, -whenever a compensation pendulum is to be -made, that the expansions of the materials employed -should be determined after the processes of drilling, -filing, and hammering have been gone through.</p> - -<p>It had been objected to Harrison’s gridiron pendulum, -that the adjustments of the rods was inconvenient, and -that the expansion of the bob supported at its lower -edge would, unless taken into the account, vitiate the -compensation. These considerations, it is supposed, -gave rise to Ellicott’s pendulum, which is nearly similar -to those we have just mentioned.</p> - -<p>Ellicott’s pendulum is thus constructed:—A bar of -brass and a bar of iron are firmly fixed together at their<span class="pagenum" id="Page_324">324</span> -upper ends, the bar of brass lying upon the bar of iron, -which is the rod of the pendulum. These bars are held -near each other by screws passing through oblong holes -in the brass, and tapped into the iron, and thus the -brass is allowed to expand or contract freely upon the -iron with any change of temperature. The brass bar -passes to the centre of the bob of the pendulum, a little -above and below which the iron is left broader for the -purpose of attaching the levers to it, and the iron is -made of a sufficient length to pass quite through the -bob of the pendulum.</p> - -<p>The pivots of two strong steel levers turn in two holes -drilled in the broad part of the iron bar. The short -arms of these levers are in contact with the lower extremity -of the brass bar, and their longer arms support -the bob of the pendulum by meeting the heads of two -screws which pass horizontally from each side of the -bob towards its centre. By advancing these screws towards -the centre of the bob, the longer arms of the -lever are shortened, and thus the compensation may be -readily adjusted. At the lower end of the iron rod, -under the bob, a strong double spring is fixed, to support -the greater part of the weight of the bob by its -pressure upwards against two points at equal distances -from the pendulum rod. Mr. Ellicott gave a description -of this pendulum to the Royal Society in 1752, but he -says the thought was executed in 1738. As this pendulum -is very seldom met with, we think it unnecessary -to give a representation of it.</p> - - -<p class="tac"><i>Compensation by means of a Compound Bar of Steel -and Brass.</i></p> - -<p>Several compensations for pendulums have been proposed, -by means of a compound bar formed of steel and -brass soldered together. In a bar of this description, the -brass expanding more than the steel, the bar becomes -curved by a change of temperature, the brass side becoming -convex and the steel concave with heat. Now, -if a bar of this description have its ends resting on<span class="pagenum" id="Page_325">325</span> -supports on each side the cock of the pendulum, the -bar passing above the cock with the brass uppermost, if -the pendulum spring be attached to the middle of the -bar, and it pass in the usual manner through the slit of -the cock, it is evident that, by an increase of temperature, -the bar will become curved upwards, and the pendulum -spring be drawn upwards through the slit, and thus the -elongation of the pendulum downwards will be compensated. -The compensation may be adjusted by varying -the distance of the points of support from the -middle of the bar.</p> - -<p>Such was one of the modes of compensation proposed -by Nicholson. Others of the same description (that is, -with compound bars) have been brought before the -public by Mr. Thomas Doughty and Mr. David Ritchie; -but as they are supposed to be liable to many practical -objections, we do not think it requisite to describe them -more particularly.</p> - -<p>There is, however, a mode of compensation by means -of a compound bar, described by M. Biot in the first -volume of his Traité de Physique, which appears to -possess considerable merit, of which he mentions having -first witnessed the successful employment by the -inventor, a clockmaker named Martin. At <i><a href="#i_p334a">fig. 221.</a></i>, -S C, is the rod of the pendulum, made, in the usual -manner, of iron or steel; this rod passes through the -middle of a compound bar of brass and steel (the brass -being undermost), which should be furnished with a -short tube and screws, by means of which, or by passing -a pin through the tube and rod, it may be securely fixed -at any part of the pendulum rod.</p> - -<p>Two small equal weights W W slide along the compound -bar, and, when their proper position has been -determined, may be securely clamped.</p> - -<p>The manner in which this compensation acts is thus:—Suppose -the temperature to increase, the brass expanding -more than the steel, the bar becomes curved, -and its extremities carrying the weights W and W are -elevated, and thus the place of the centre of oscillation<span class="pagenum" id="Page_326">326</span> -is made to approach the point of suspension as much, -when the compensation is properly adjusted, as it had -receded from it by the elongation of the pendulum rod.</p> - -<p>There are three methods of adjusting this compensation: -the first, by increasing or diminishing the weights -W and W; the second, by varying the distance of the -weights W and W from the middle of the bar; and the -third, by varying the distance of the bar from the bob -of the pendulum, taking care not to pass the middle of -the rod. The effect of the compensation is greater as -the weights W and W are greater or more distant from -the centre of the bar, and also as the bar is nearer to -the bob of the pendulum.</p> - -<p>M. Biot says that he and M. Matthieu employed a -pendulum of this kind for a long time in making astronomical -observations in which they were desirous of -attaining an extreme degree of precision, and that they -found its rate to be always perfectly regular.</p> - -<p>In all the pendulums which we have described, the -bob is supposed to be fixed to the rod by a pin passing -through its centre, and the adjustment for time is to be -made by means of a small weight sliding upon the rod.</p> - - -<p class="tac"><i>Of the Mercurial Pendulum.</i></p> - -<p>We have been guided, in our arrangement of the -pendulums which we have described, by the similarity -in the mode of compensation employed; and we have -now to treat of that method of compensation which is -effected by the expansion of the material of which the -bob itself of the pendulum is composed.</p> - -<p>On this subject, as we have before observed, an -admirable paper, from the pen of Mr. Francis Baily, -may be found in the Memoirs of the Astronomical -Society of London, which leaves nothing to be desired -by the mathematical reader. But as our object is to simplify, -and to render our subjects as popular as may be, -we must endeavour to substitute for the perfect accuracy -which Mr. Baily’s paper presents, such rules as may be -found not only readily intelligible, but practically appli<span class="pagenum" id="Page_327">327</span>cable, -within the limits of those inevitable errors which -arise from a want of knowledge of the exact expansion -of the materials employed.</p> - -<p>At <i><a href="#i_p334a">fig. 222.</a></i>, let S B represent the rod of a pendulum, -and F C B a metallic tube or cylinder, supported -by a nut at the extremity of the pendulum rod, in -the usual manner, and having a greater expansibility -than that of the rod. Now C, the centre of gravity, -supposing the rod to be without weight, will be in the -middle of the cylinder; and if C B, or half the cylinder, -be of such a length as to expand upwards as much as -the pendulum rod S B expands downwards, it is evident -that the centre of gravity C will remain, under any -change of temperature, at the same distance from the -point of suspension S. M. Biot imagined that, in -effecting this, a compensation sufficiently accurate would -be obtained; but Mr. Baily has shown that this is by -no means the fact.</p> - -<p>Let us suppose the place of the centre of oscillation -to be at O, about three or four tenths of an inch, in a -pendulum of the usual construction, below the centre of -gravity. Now, the object of the compensation is to -preserve the distance from S to O invariable, and not -the distance from S to C.</p> - -<p>The distance of the centre of oscillation varies with -the length of the cylinder F B, and hence suffers an -alteration in its distance from the point of suspension -by the elongation of the cylinder, although the distance -of the centre of gravity C from the point of suspension -remains unaltered.</p> - -<p>We shall endeavour to render this perfectly familiar. -Suppose a metallic cylinder, 6 inches long, to be suspended -by a thread 36 inches long, thus forming a pendulum -in which the distance of the centre of gravity -from the point of suspension is 39 inches: the centre of -oscillation in such a pendulum will be nearly one tenth -of an inch below the centre of gravity. Now let us -imagine cylindrical portions of equal lengths to be added -to each end of the cylinder, until it reaches the point of<span class="pagenum" id="Page_328">328</span> -suspension; we shall then have a cylinder of 78 inches -in length, the centre of gravity of which will still be at -the distance of 39 inches from the point of suspension. -But it is well known that the centre of oscillation of -such a cylinder is at the distance of about two thirds of -its length from the point of suspension. The centre of -oscillation, therefore, has been removed, by the elongation -of the cylinder, about 13 inches below the centre -of gravity, whilst the centre of gravity has remained -stationary.</p> - -<p>Now the same thing as that which we have just -described takes place, though in a very minor degree, -with our former cylinder, employed as a compensating -bob to a pendulum. The rod expands downwards, the -centre of gravity remains at the same distance from the -point of suspension, and the cylinder elongates both -above and below this point; the consequence of which -is, that though the centre of gravity has remained stationary, -the distance of the centre of oscillation from the -point of suspension has increased. It is, therefore, evident -that the length of the compensation must be such -as to carry the centre of gravity a little nearer to the -point of suspension than it was before the expansion -took place; by which means the centre of oscillation -will be restored to its former distance from the point of -suspension.</p> - -<p>Let us suppose the expansions to have taken place, -and that the centre of gravity, remaining at the same -distance from the point of suspension, the centre of -oscillation is removed to a greater distance, as we have -before explained. It is well known that the product -obtained by multiplying the distance from the point of -suspension to the centre of gravity, by the distance from -the centre of gravity to the centre of oscillation, is a -constant quantity; if, therefore, the distance from the -centre of gravity to the point of suspension be lessened, -the distance from the centre of gravity to the centre of -oscillation will be proportionally, though not equally, increased, -and the centre of oscillation will, therefore, be<span class="pagenum" id="Page_329">329</span> -elevated. We see, then, if we elevate the centre of -gravity precisely the requisite quantity, by employing a -sufficient length of the compensating material, that -although the distance from the centre of gravity to the -point of suspension is lessened, yet the distance from -the point of suspension to the centre of oscillation will -suffer no change.</p> - -<p>The following rule for finding the length of the compensating -material in a pendulum of the kind we have -been considering will be found sufficiently accurate for -all practical purposes:—</p> - -<p><i>Find in the manner before directed the length of the -compensating material, the expansion of which will be -equal to that of the rod of the pendulum. Double this -length, and increase the product by its one-tenth part, -which will give the total length required.</i> We shall give -examples of this as we proceed.</p> - - -<p class="tac"><i>Graham’s Mercurial Pendulum.</i></p> - -<p>It was in the year 1721 that Graham first put up a -pendulum of this description, and subjected it to the test -of experiment; but it appears to have been afterwards -set aside to make way for Harrison’s gridiron pendulum, -or for others of a similar description. For some years -past, however, its merits have been more generally -known, and it is not surprising that it should be considered -as preferable to others, both from the simplicity -of its construction, and the perfect ease with which the -compensation may be adjusted.</p> - -<p>We have already alluded to Mr. Baily’s very able -paper on this pendulum, and we shall take the liberty of -extracting from it the following description:—</p> - -<p>At <i><a href="#i_p334a">fig. 223.</a></i> is a drawing of the mercurial pendulum, -as constructed in the manner proposed by Mr. Baily.</p> - -<p>“The rod S F is made of steel, and perfectly straight; -its form may be either cylindrical, of about a quarter of an -inch in diameter, or a flat bar, three eighths of an inch -wide, and one eighth of an inch thick: its length from S to -F, that is, from the bottom of the spring to the bottom of<span class="pagenum" id="Page_330">330</span> -the rod at F, should be 34 inches. The lower part of this -rod, which passes through the top of the stirrup, and -about half an inch above and below the same, must be -formed into a <i>coarse</i> and <i>deep</i> screw, about two tenths of -an inch in diameter, and having about thirty turns in an -inch. A steel nut with a milled head must be placed at -the end of the rod, in order to support the stirrup; and a -similar nut must also be placed on the rod <i>above</i> the head -of the stirrup, in order to screw firmly down on the same, -and thus secure it in its position, after it has been adjusted -<i>nearly</i> to the required rate. These nuts are represented -at B and C. A small slit is cut in the rod, where it passes -through the head of the stirrup, through which a steel -pin E is screwed, in order to keep the stirrup from turning -round on the rod. The stirrup itself is also made of -steel, and the side pieces should be of the same form as -the rod, in order that they may readily acquire the same -temperature. The top of the stirrup consists of a flat -piece of steel, shaped as in the drawing, somewhat more -than three eighths of an inch thick. Through the middle -of the top (which at this part is about one inch deep) -a hole must be drilled sufficiently large to enable the -screw of the rod to pass <i>freely</i>, but without <i>shaking</i>. -The inside height of the stirrup from A to D may be -<span class="nowrap">8<span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> inches, and the inside width between the bars about -three inches. The bottom piece should be about three -eighths of an inch thick, and hollowed out nearly a quarter -of an inch deep, so as to admit the glass cylinder -freely. This glass cylinder should have a brass or iron -cover G, which should fit the mouth of it freely, with a -shoulder projecting on each side, by means of which it -should be screwed to the side bars of the stirrup, and thus -be secured always in the same position. This cap should -not <i>press</i> on the glass cylinder, so as to prevent its expansion. -The measures above given may require a slight -modification, according to the weight of the mercury -employed, and the magnitude of the cylinder: the final -adjustment, however, may be safely left to the artist. -Some persons have recommended that a circular piece of<span class="pagenum" id="Page_331">331</span> -thick plate glass should float on the mercury, in order to -preserve its surface uniformly level<span class="nowrap">.<a id="FNanchor_7" href="#Footnote_7" class="fnanchor">7</a></span> The part at the -bottom marked H is a piece of brass fastened with -screws to the front of the bottom of the stirrup, through -a small hole, in which a steel wire or common needle is -passed, in order to indicate (on a scale affixed to the -case of the clock) the arc of vibration. This wire should -merely rest in the hole, whereby it may be easily removed -when it is required to detach the pendulum from -the clock, in order that the stirrup might then stand -securely on its base. One of the screw holes should be -rather larger than the body of the screw, in order to admit -of a small adjustment, in case the steel wire should -not stand exactly perpendicular to the axis of motion. -The scale should be divided into <i>degrees</i>, and not <i>inches</i>, -observing that with a radius of 44 inches (the estimated -distance from the bend of the spring to the end of the -steel wire) the length of each degree on the scale must -be 0·768 inch.”</p> - -<p>In order to determine the length of the mercurial -column necessary to form the compensation for this pendulum, -we must proceed in the following manner:—</p> - -<p>Let us suppose the length of the steel rod and stirrup -together to be 42 inches. The absolute expansion of -the mercury is ·00010010; but it is not the absolute -expansion, but the vertical expansion in a glass cylinder, -which is required, and this will evidently be influenced -by the expansion of the base of this cylinder. It is -easily demonstrable that, if we multiply the linear expansion -of any substance (always supposed to be a very -small part of its length) by 3, we may in all cases take -the result for the cubical or absolute expansion of such<span class="pagenum" id="Page_332">332</span> -substance. In like manner, if we multiply the linear expansion -by 2, we shall have the superficial expansion.</p> - -<p>If we want the apparent expansion of mercury, the -absolute or cubical expansion of the glass vessel must -be deducted from the absolute expansion of the mercury, -which will leave its excess or apparent expansion. -In like manner, deducting the superficial expansion of -glass from the absolute expansion of mercury, we shall -have its relative vertical expansion. Now, taking the -rate of expansion of glass to be ·00000479, and multiplying -it by 2, the relative vertical expansion of the -mercury in the glass cylinder will be ·00010010 - -·00000958 = ·00009052.</p> - -<p>The expansion of a steel rod, according to our table, -is ·0000063596; which, divided by ·00009052, gives -·0703 for the length of a column of mercury, the expansion -of which is equal to that of a steel rod whose -length is unity.</p> - -<p>We have now to multiply 42 inches by ·0703, which -gives 2·95 inches; and this, deducted from 42, leaves -39·1 inches; so that the length of rod we have chosen -is sufficiently near the truth. Now, double 2·95 inches, -and add one tenth of its product, and we shall have 6·49 -inches for the length of the mercurial column forming -the requisite compensation. Mr. Baily’s more accurate -calculation gives 6·31 inches.</p> - -<p>A mercurial compensation pendulum may be formed, -having a cylinder of steel or iron, with its top constructed -in the same manner as the top of the stirrup, -so as to receive the screw of the rod. To find the -length of the mercurial column necessary in a pendulum -of this description (that is, with a cylinder made of -steel), we must double the linear expansion of steel, and -take it from the absolute expansion of mercury to obtain -the relative vertical expansion of the mercury. This -will be ·00010010 - ·00001272 = ·00008738; and, -proceeding as before, we have <span class="nowrap"><span class="fraction"><span class="fnum">·0000063596</span><span class="bar">/</span><span class="fden">·00008738</span></span></span> = ·07279.</p> - -<p>Let the length of the steel rod be, as before, 42 inches.<span class="pagenum" id="Page_333">333</span> -Multiplying this by ·07279, we have 3·057, which -being doubled, and one tenth of the product added, we -obtain 6·72 inches for the length of the compensating -mercurial column; which Mr. Baily states to be 6·59.</p> - -<p>A mercurial compensation pendulum having a rod of -glass has been employed by the writer of this article, -who has had reason to think well of its performance. -Its cheapness and simplicity much recommend it. It is -merely a cylinder of glass of about 7 inches in depth, -and <span class="nowrap">2<span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> inches diameter, terminated by a long neck, -which forms the rod of the pendulum, the whole blown -in one piece. A cap of brass is clamped by means of -screws to the top of the rod, and to this the pendulum -spring is pinned.</p> - -<p>We have unquestionable authority for saying, that -the mercurial pendulum of the usual construction, that -is, with a steel rod and glass cylinder, is not affected by -a change of temperature simultaneously in all its parts. -Now, the pendulum of which we are treating being -formed throughout of the same material in a single -piece, and in every part of the same thickness, it is presumed -it cannot expand in a linear direction, until the -temperature has penetrated to the whole interior surface -of the glass, when it is rapidly diffused through the -mass of mercury. M. Biot mentions that a pendulum -of this kind was formerly used in France, and expresses -his surprise that it was no longer employed, as he had -heard it very highly spoken of. The writer of this -article has also used a pendulum with a glass rod, which -differs from that we have just mentioned, in having the -lower end of the rod firmly fixed in a socket attached to -the centre of a circular iron plate, on the circumference -of which a screw is cut, which fits into a collar of iron, -supporting the cylinder (to which it is cemented) by -means of a circular lip.</p> - -<p>This arrangement, though perhaps less perfect than -that we have just described, the pendulum not being in -one piece, has the advantage of allowing a circular plate of -glass to be placed upon the surface of the mercury, as<span class="pagenum" id="Page_334">334</span> -practised by Mr. Browne. To determine the length of -a column of mercury for a glass pendulum, let us suppose -the glass, including the cylinder, to be 41 inches -in length. Multiplying this by ·0529, the number -taken from Table II. for a glass rod and mercury in a -glass cylinder, we have 2·17 inches for the uncorrected -length of mercury, which compensates 41 inches of -glass. Suppose the steel spring to be one inch and a -half long: multiplying this by ·0703, the appropriate -decimal taken from <a href="#TABLE_II">Table II</a>., we have 0·1, the length of -mercury due to the steel, making with the former 2·27 -inches, which, being doubled, and the product increased -by its one-tenth part, we obtain five inches for the -length of the required column of mercury.</p> - - -<p class="tac"><i>Compensation Pendulum of Wood and Lead, on the -Principle of the Mercurial Pendulum.</i></p> - -<p>If by any contrivance wood could be rendered impervious -to moisture, it would afford one of the most convenient -substances known for a compensation pendulum. -It does not appear that sufficient experiments have been -made upon this subject to decide the question. Mr. -Browne of Portland Place, who has devoted much of -his time and attention to the most delicate enquiries of -this kind, has, we believe, found that if a teak rod is -well gilded, it will not afterwards be affected by -moisture. At all events, it makes a far superior pendulum, -when thus prepared, to what it does when such -preparation is omitted.</p> - -<p>Mr. Baily, in the paper we have before alluded to, -proposes an economical pendulum to be constructed by -means of a leaden cylinder and a deal rod. He prefers -lead to zinc, on account of its inferior price, and the ease -with which it may be formed into the required shape; -and as there is no considerable difference in their rates -of expansion, it is equally applicable to the purpose.</p> - -<p>Let the length of the deal rod be taken at 46 inches. -Then, to find the length of the cylinder of lead to compensate -this, we have, in Table II., ·1427 for such a<span class="pagenum" id="Page_335">335</span> -pendulum; which, being multiplied by 46, the product -doubled, and one tenth of the result added to it, gives -14·44 inches for the length of the leaden cylinder. -Mr. Baily’s compensation gives 14·3 inches.</p> - -<div class="figcenter" id="i_p334a" style="max-width: 31.25em;"> - <img src="images/i_p334a.jpg" alt="" /> - <div class="caption"> -<p><span class="l-align"><i>Captn. Kater, del.</i></span> - -<span class="r-align"><i>H. Adlard, sc.</i></span></p> - -<p class="tac"><i>London, Pubd. by Longman & Co.</i></p></div> -</div> - -<p>The rod is recommended to be made of about three -eighths of an inch in diameter: the leaden cylinder is -to be cast with a hole through its centre, which will admit -with perfect freedom the cylindrical end of the rod. -The cylinder is supported upon a nut, which screws on -the end of the rod in the usual manner. This pendulum -is represented at <i><a href="#i_p334a">fig. 224.</a></i></p> - -<p>Mr. Baily proposes that the pendulum should be -adjusted nearly to the given rate by means of the screw -at the bottom, and that the final adjustment be made -by means of a slider moving along the rod. Indeed, -this is a means of adjustment which we would recommend -to be employed in every pendulum.</p> - - -<p class="tac"><i>Smeaton’s Pendulum.</i></p> - -<p>We shall conclude our account of compensation pendulums -with a description of that invented by Mr. -Smeaton. The compensation for temperature in this -pendulum is effected by combining the two modes, which -have been so fully described in the preceding part of -this article.</p> - -<p>The pendulum rod is of solid glass, and is furnished -with a steel screw and nut at the bottom in the usual -manner. Upon the glass rod a hollow cylinder of zinc, -about the eighth of an inch thick, and about 12 inches -long, passes freely, and rests upon the nut at the bottom -of the pendulum rod.</p> - -<p>Over the zinc cylinder passes a tube made of sheet-iron. -The edge of this tube at the top is turned inwards, -and is notched so as to allow of this being -effected. A flanche is thus formed, by which the iron -tube is supported, upon the zinc cylinder. The lower -edge of the iron tube is turned outwards, so as to form -a base destined to support a leaden cylinder, which we -are about to describe.</p> - -<p><span class="pagenum" id="Page_336">336</span></p> - -<p>A cylinder of lead, rather more than 12 inches long, -is cast with a hole through its axis, of such a diameter -as to allow of its sliding freely, but without shake, upon -the iron tube over which it passes, and by the lower -extremity of which it is supported.</p> - -<p>Now the zinc, resting upon the nut and expanding -upwards, will raise the whole of the remaining part of -the compensation. This expansion upwards will be -slightly counteracted by the lesser expansion downwards -of the iron tube, which carries with it the leaden -cylinder. The cylinder of lead now acts upon the -principle of the mercurial pendulum, and, expanding -upwards, contributes that which was wanting to restore -the centre of oscillation to its proper distance from the -point of suspension.</p> - -<p>This pendulum, we have been informed, does well in -practice, and we are not aware that any description of -it has been before published.</p> - -<p>The method of calculating the length of the tubes -required to form the compensation is very simple; -nothing more is necessary than to find the length of -zinc, the expansion of which is equal to that of the -pendulum rod.</p> - -<p>Let the pendulum rod be composed of 43 inches of -glass, the spring being an inch and a half long, and the -screw between the end of the glass rod and the nut half -an inch, making in the whole two inches of steel and -43 inches of glass.</p> - -<p>Now to find the length of zinc that will compensate -the glass, we have, from <a href="#TABLE_II">Table II</a>., for glass and zinc -·2773, which, multiplied by 43, gives 11·92 inches. -In like manner we obtain as a compensation for two -inches of steel 0·74 of zinc, which, added to 11·92, gives -12·66 inches for the total length of the zinc cylinder.</p> - -<p>Now if the iron tube and the lead cylinder be each -made of the same length as the zinc, and arranged as -we have described, the compensation will be perfect.</p> - -<p>To prove this, find, by means of the expansions given -in <a href="#TABLE_I">Table I</a>., the actual expansion of each of the sub<span class="pagenum" id="Page_337">337</span>stances -employed in the pendulum, and we shall have -the following results:—</p> - -<div class="center"> -<table width="550" summary="pendulum temperature compensation"> -<tr> - <td class="tal pl1hi1 pr1">The expansion of 12·66 inches of zinc - expanding upwards is</td> - <td class="vab">·0002186</td> -</tr> -<tr> - <td class="tal pl1hi1 pr1">Deduct that of 12·66 inches of iron - expanding downwards</td> - <td class="vab">·0000869</td> -</tr> -<tr> - <td></td> - <td>──────</td> -</tr> -<tr> - <td class="tal pl1hi1 pr1">Remaining effect of expansion upwards, - referred to the lower extremity of the iron tube</td> - <td class="vab">·0001317</td> -</tr> -<tr> - <td class="tal pl1hi1 pr1">Now, for the lead.—On the principle - of the mercurial compensation, subtract one - tenth part of the length of the cylinder, - and take half the remainder, and we shall - have six inches of lead, the expansion of - which upwards is</td> - <td class="vab">·0000955</td> -</tr> -<tr> - <td></td> - <td>──────</td> -</tr> -<tr> - <td class="tal pl1hi1 pr1">Total expansion of the compensation upwards</td> - <td class="vab">·0002272</td></tr> -<tr> - <td></td> - <td>──────</td> -</tr> -<tr> - <td class="tal pl1hi1 pr1">To find the expansion of the rod, we have - the expansion of 43 inches of glass</td> - <td class="vab">·0002059</td> -</tr> -<tr> - <td class="tal pl1hi1 pr1">Of two inches of steel</td><td>·0000127</td> -</tr> -<tr> - <td></td><td>──────</td> -</tr> -<tr> - <td class="tal pl1hi1 pr1">Total expansion of the pendulum rod</td> - <td class="vab">·0002186</td> -</tr> -</table> -</div> - - -<p>Agreeing near enough with that of the compensation -before found.</p> - -<p>As we conceive we have been sufficiently explicit in -our description of this pendulum, in the construction of -which no difficulty presents itself, we think an engraved -representation of it would be superfluous.</p> - -<p>We have hitherto treated only of compensations for -temperature; but there is another kind of error, which -has been sometimes insisted upon, arising from a variation -in the density of the atmosphere. If the density -of the atmosphere be increased, the pendulum will experience -a greater resistance, the arc of vibration will in -consequence be diminished, and the pendulum will<span class="pagenum" id="Page_338">338</span> -vibrate faster. This, however, is in some measure -counteracted by the increased buoyancy of the atmosphere, -which, acting in opposition to gravity, occasions -the pendulum to vibrate slower. If the one effect -exactly equalled the other, it is evident no error would -arise; and in a paper by Mr. Davies Gilbert, President -of the Royal Society of London, published in the Quarterly -Journal for 1826, he has proved that, by a -happy chance, the arc in which pendulums of clocks are -usually made to vibrate is the arc at which this compensation -of error takes place. This arc, for a pendulum -having a brass bob, is 1° 56′ 30″ on each side of the -perpendicular; and for a mercurial pendulum, 1° 31′ 44″, -or about one degree and a half.</p> - -<p>It is well known that, if a pendulum vibrates in a -circular arc, the times of vibration will vary nearly as -the squares of the arcs; but if the pendulum could be -made to vibrate in a cycloid, the time of its vibration in -arcs of different extent would then remain the same. -Huygens and others, therefore, endeavoured to effect -this by placing the spring of the pendulum between -cheeks of a cycloidal form.</p> - -<p>When escapements are employed which do not insure -an unvarying impulse to the pendulum, the force may -be unequally transmitted through the train of the clock -in consequence of unavoidable imperfections of workmanship, -and the arc of vibration may suffer some increase -or diminution from this cause. To discover a -remedy for this is certainly desirable.</p> - -<p>The writer of this article some years ago imagined a -mode, which he believes has also been suggested by -others, by which he conceived a pendulum might be -made to describe an arc approaching in form to that of -a cycloid. The pendulum spring was of a triangular -form, and the point or vertex was pinned into the top -of the pendulum rod, the base of the triangle forming -the axis of suspension. Now it is evident that when -the pendulum is in motion, the spring will resist bending<span class="pagenum" id="Page_339">339</span> -at the axis of suspension, with a force in some sort -proportionate to the base of the triangle.</p> - -<p>Suppose the pendulum to have arrived at the extent -of its vibrations; the spring will present a curved appearance; -and if the distance from the point of suspension -to the centre of oscillation be then measured, it -will evidently, in consequence of the curvature of the -spring, be shorter than the distance from the point of -suspension to the centre of oscillation, measured when -the pendulum is in a perpendicular position, and consequently -when the spring is perfectly straight.</p> - -<p>The base of the triangle may be diminished, or the -spring be made thinner; either of which will lessen its -effect. We cannot say how this plan might answer -upon further trial, as sufficient experiments were not -made at the time to authorize a decisive conclusion.</p> - -<p>We have thus completed our account of compensation -pendulums; but before we conclude, it may not be unacceptable -if we offer a few remarks on some points which -may be found of practical utility.</p> - -<p>The cock of the pendulum should be firmly fixed -either to the wall or to the case of the clock, and not to -the clock itself, as is sometimes done, and which has -occasioned much irregularity in its rate, from the motion -communicated to the point of suspension. We prefer a -bracket or shelf of cast iron or brass, upon which the -clock may be fixed, and the cock carrying the pendulum -attached to its perpendicular back. This bracket may -either be screwed to the back of the clock-case, or, which -is the better mode, securely fixed to the wall; and if the -latter be adopted, the whole may be defended from the -atmosphere, or from dust, by the clock-case, which thus -has no connection either with the clock or with the pendulum.</p> - -<p>The point of suspension should be distinctly defined -and immovable. This may be readily effected, after the -pendulum shall have taken the direction of gravity, by -means of a strong screw entering the cock (which should<span class="pagenum" id="Page_340">340</span> -be very stout) on one side, and pressing a flat piece of -brass into firm contact with the spring.</p> - -<p>The impulse should be given in that plane of the rod -which coincides with the plane of vibration passing -through the axis of the rod. If the impulse be given at -any point either before or behind this plane, the probable -result will be a tremulous unsteady motion of the pendulum.</p> - -<p>A few rough trials, and moving the weight, will bring -the pendulum near its intended time of vibration, which -should be left a little too slow; when the bob should be -firmly fixed to the rod, if the form of the pendulum -will admit of it, by a pin or screw passing through its -centre.</p> - -<p>The more delicate adjustment may be completed by -shifting the place of the slider with which the pendulum -is supposed to be furnished on the rod.</p> - -<p>Mr. Browne (of whom we have before spoken) practises -the following very delicate mode of adjustment for -rate, which will be found extremely convenient, as it is -not necessary to stop the pendulum in order to make the -required alteration. Having ascertained, by experiment, -the effect produced on the rate of the clock, by placing -a weight upon the bob equal to a given number of grains, -he prepares certain smaller weights of sheet-lead, which -are turned up at the corners, that they may be conveniently -laid hold of by a pair of forceps, and the effect -of these small weights on the rate of the clock will be, of -course, known by proportion. The rate being supposed -to be in defect, the weights necessary to correct this may -be deposited, without difficulty, upon the bob of the -pendulum, or upon some convenient plane surface, placed -in order to receive them: and should it be necessary to -remove any one of the weights, this may readily be done -by employing a delicate pair of forceps, without producing -the slightest disturbance in the motion of the -pendulum.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_341">341</span></p> - -<h2 class="nobreak" id="INDEX">INDEX.</h2> -</div> - -<p> -A.<br /> -Action and reaction, <a href="#Page_34">34</a>.<br /> -Aeriform fluids, <a href="#Page_26">26</a>.<br /> -Animalcules, <a href="#Page_12">12</a>.<br /> -Atmosphere, impenetrability of, <a href="#Page_22">22</a>.<br /> - Compressibility and elasticity of, <a href="#Page_23">23</a>.<br /> -Atoms, <a href="#Page_6">6</a>.<br /> - Coherence of, <a href="#Page_7">7</a>.<br /> -Attraction, magnetic, of gravitation, <a href="#Page_8">8</a>, <a href="#Page_50">50</a>, <a href="#Page_64">64</a>.<br /> - Molecular or atomic, <a href="#Page_69">69</a>.<br /> - Cohesion, <a href="#Page_70">70</a>.<br /> -Attwood, machine of, <a href="#Page_92">92</a>.<br /> -Axes, principal, <a href="#Page_138">138</a>.<br /> -Axis, mechanical properties of, <a href="#Page_128">128</a>.<br /> -<br /> -B.<br /> -Balance, <a href="#Page_279">279</a>.<br /> - Of Bates, <a href="#Page_288">288</a>.<br /> - Use of, <a href="#Page_289">289</a>.<br /> - Danish, <a href="#Page_299">299</a>.<br /> - Bent-lever of Brady, <a href="#Page_301">301</a>.<br /> -Bodies, <a href="#Page_2">2</a>.<br /> - Lines, surfaces, edges, area, length of, <a href="#Page_4">4</a>.<br /> - Figure, volume, shape of, <a href="#Page_5">5</a>.<br /> - Porosity of, <a href="#Page_17">17</a>.<br /> - Compressibility of, <a href="#Page_18">18</a>.<br /> - Elasticity, dilatibility of, <a href="#Page_19">19</a>.<br /> - Inertia of, <a href="#Page_27">27</a>.<br /> - Rule for determining velocity of; motion of two bodies after impact, <a href="#Page_38">38</a>.<br /> -<br /> -C.<br /> -Capillary attraction, <a href="#Page_73">73</a>.<br /> -Capstan, <a href="#Page_179">179</a>.<br /> -Cause and effect, <a href="#Page_7">7</a>.<br /> -Circle of curvature, <a href="#Page_99">99</a>.<br /> -Cog, hunting, <a href="#Page_191">191</a>.<br /> -Components, <a href="#Page_51">51</a>.<br /> -Cord, <a href="#Page_163">163</a>.<br /> -Cordage, friction and rigidity of, <a href="#Page_260">260</a>.<br /> -Crank, <a href="#Page_241">241</a>.<br /> -Crystallisation, <a href="#Page_14">14</a>.<br /> -Cycloid, <a href="#Page_158">158</a>.<br /> -<br /> -D.<br /> -Damper, self-acting, <a href="#Page_234">234</a>.<br /> -Deparcieux’s compensation pendulum, <a href="#Page_319">319</a>.<br /> -Diagonal, <a href="#Page_51">51</a>.<br /> -Dynamics, <a href="#Page_160">160</a>.<br /> -Dynamometer, <a href="#Page_305">305</a>.<br /> -<br /> -E.<br /> -Electricity, <a href="#Page_76">76</a>.<br /> -Electro-magnetism, <a href="#Page_76">76</a>.<br /> -Equilibrium, neutral, instable, and stable, <a href="#Page_118">118</a>.<br /> -<br /> -F.<br /> -Figure, <a href="#Page_5">5</a>.<br /> -Fly-wheel, <a href="#Page_239">239</a>.<br /> -Force, <a href="#Page_6">6</a>.<br /> - Composition and resolution of, <a href="#Page_49">49</a>.<br /> - Centrifugal, <a href="#Page_98">98</a>.<br /> - Moment of; leverage of, <a href="#Page_135">135</a>.<br /> - Regulation and accumulation of, <a href="#Page_224">224</a>.<br /> -Friction, effects of, <a href="#Page_96">96</a>.<br /> - Laws of, <a href="#Page_264">264</a>.<br /> -<br /> -G.<br /> -Governor, <a href="#Page_227">227</a>.<br /> -Gravitation, attraction of, <a href="#Page_77">77</a>.<br /> - Terrestrial, <a href="#Page_84">84</a>.<br /> -Gravity, centre of, <a href="#Page_107">107</a>.<br /> -Gyration, radius of, centre of, <a href="#Page_137">137</a>.<br /> -<br /> -H.<br /> -Hooke’s universal joint, <a href="#Page_252">252</a>.<br /> -Hydrophane, porosity of, <a href="#Page_18">18</a>.<br /> -<br /> -I.<br /> -Impact, <a href="#Page_39">39</a>.<br /> -Impulse, <a href="#Page_65">65</a>.<br /> -Inclined plane, <a href="#Page_163">163–209</a>.<br /> -Inclined roads, <a href="#Page_211">211</a>.<br /> -Inertia, <a href="#Page_27">27</a>.<br /> - Laws of, <a href="#Page_32">32</a>.<br /> - Moment of, <a href="#Page_137">137</a>.<br /> -<br /> -J.<br /> -Julien le Roy, compensation tube of, <a href="#Page_319">319</a>.<br /> -<br /> -L.<br /> -Lever, <a href="#Page_163">163</a>.<br /> - Fulcrum of; three kinds of, <a href="#Page_167">167</a>.<br /> - Equivalent, <a href="#Page_176">176</a>.<br /> -Line of direction, <a href="#Page_110">110</a>.<br /> -Liquids, compressibility of, <a href="#Page_24">24</a>.<br /> -Loadstone, <a href="#Page_68">68</a>.<br /> -<br /> -M.<br /> -Machines, simple, <a href="#Page_160">160</a>.<br /> - Power of, <a href="#Page_175">175</a>.<br /> - Regulation of, <a href="#Page_225">225</a>.<br /> -Magnet, <a href="#Page_68">68</a>.<br /> -Magnetic attraction, <a href="#Page_8">8</a>.<br /> -Magnetism, <a href="#Page_76">76</a>.<br /> -Magnitude, <a href="#Page_4">4</a>.<br /> -Marriott’s patent weighing machine, <a href="#Page_305">305</a>.<br /> -Materials, strength of, <a href="#Page_272">272</a>.<br /> -Matter, properties of, <a href="#Page_2">2</a>.<br /> - Impenetrability of, <a href="#Page_4">4</a>.<br /> - Atoms of; molecules of, <a href="#Page_6">6</a>.<br /> - Divisibility of, <a href="#Page_9">9</a>.<br /> - Examples of the subtilty of, <a href="#Page_12">12</a>.<span class="pagenum" id="Page_342">342</span><br /> - Limit to the divisibility of, <a href="#Page_13">13</a>.<br /> - Porosity of; density of, <a href="#Page_17">17</a>.<br /> - Compressibility of, <a href="#Page_18">18</a>.<br /> - Elasticity and dilatability of, <a href="#Page_19">19</a>.<br /> - Impenetrability of, <a href="#Page_22">22</a>.<br /> - Inertia of, <a href="#Page_27">27</a>.<br /> -Mechanical science, foundation of, <a href="#Page_16">16</a>.<br /> -Metronomes, principles of, <a href="#Page_153">153</a>.<br /> -Molecules, <a href="#Page_6">6</a>.<br /> -Motion, laws of, <a href="#Page_46">46</a>.<br /> - Uniformly accelerated, <a href="#Page_87">87</a>.<br /> - Table illustrative of, <a href="#Page_90">90</a>.<br /> - Retarded; of bodies on inclined planes and curves, <a href="#Page_94">94</a>.<br /> - Rotary and progressive, <a href="#Page_127">127</a>.<br /> - Mechanical contrivances for the modification of, <a href="#Page_245">245</a>.<br /> - Continued rectilinear; reciprocatory rectilinear; continued circular; reciprocating circular, <a href="#Page_246">246</a>.<br /> -<br /> -N.<br /> -Newton, method of, for determining the thickness of transparent substances, <a href="#Page_10">10</a>.<br /> - Laws of motion of, <a href="#Page_46">46</a>.<br /> -<br /> -O.<br /> -Oscillation, <a href="#Page_129">129</a>.<br /> - Of the pendulum, <a href="#Page_145">145</a>.<br /> - Centre of, <a href="#Page_152">152</a>.<br /> -<br /> -P.<br /> -Parallelogram, <a href="#Page_51">51</a>.<br /> -Particle, <a href="#Page_6">6</a>.<br /> -Pendulum, oscillation or vibration of, <a href="#Page_145">145</a>.<br /> - Isochronism of, <a href="#Page_147">147</a>.<br /> - Centre of oscillation of, <a href="#Page_152">152</a>.<br /> - Of Troughton, <a href="#Page_284">284</a>.<br /> - Compensation, <a href="#Page_307">307</a>.<br /> - Of Harrison, <a href="#Page_313">313</a>.<br /> - Tubular, of Troughton, <a href="#Page_314">314</a>.<br /> - Of Benzenberg, <a href="#Page_316">316</a>.<br /> - Ward’s compensation, <a href="#Page_318">318</a>.<br /> - Captain Kater’s compensation, <a href="#Page_320">320</a>.<br /> - Reid’s; Ellicott’s compensation, <a href="#Page_322">322</a>.<br /> - Steel and brass compensation, <a href="#Page_324">324</a>.<br /> - Mercurial, <a href="#Page_326">326</a>.<br /> - Graham’s mercurial, <a href="#Page_329">329</a>.<br /> - Wood and lead, <a href="#Page_334">334</a>.<br /> - Smeaton’s, <a href="#Page_335">335</a>.<br /> -Percussion, <a href="#Page_130">130</a>.<br /> - Centre of, <a href="#Page_144">144</a>.<br /> -Planes of cleavage, <a href="#Page_15">15</a>.<br /> -Porosity, <a href="#Page_17">17</a>.<br /> -Power, <a href="#Page_161">161</a>.<br /> -Properties, <a href="#Page_2">2</a>.<br /> -Projectiles, curvilinear path of, <a href="#Page_82">82</a>.<br /> -Pulley, <a href="#Page_164">164</a>.<br /> - Tackle; fixed, <a href="#Page_198">198</a>.<br /> - Single moveable, <a href="#Page_200">200</a>.<br /> - Called a runner; Spanish bartons, <a href="#Page_205">205</a>.<br /> -<br /> -R.<br /> -Rail-roads, <a href="#Page_213">213</a>.<br /> -Regulating damper, <a href="#Page_233">233</a>.<br /> -Regulators, <a href="#Page_227">227</a>.<br /> -Repulsion, <a href="#Page_8">8</a>.<br /> - Molecular, <a href="#Page_74">74</a>.<br /> -Resultant, <a href="#Page_51">51</a>.<br /> -Rose-engine, <a href="#Page_250">250</a>.<br /> -<br /> -S.<br /> -Salters, spring balance of, <a href="#Page_305">305</a>.<br /> -Screw, <a href="#Page_209">209</a>.<br /> - Concave, <a href="#Page_217">217</a>.<br /> - Micrometer, <a href="#Page_223">223</a>.<br /> -Shape, <a href="#Page_5">5</a>.<br /> -Siphon, capillary, <a href="#Page_73">73</a>.<br /> -Spring, <a href="#Page_304">304</a>.<br /> -Statics, <a href="#Page_160">160</a>.<br /> -Steelyard, <a href="#Page_294">294</a>.<br /> - C. Paul’s, <a href="#Page_296">296</a>.<br /> - Chinese, <a href="#Page_299">299</a>.<br /> -<br /> -T.<br /> -Table, whirling, <a href="#Page_99">99</a>.<br /> -Tachometer, <a href="#Page_234">234</a>.<br /> -Tread-mill, <a href="#Page_179">179</a>.<br /> -<br /> -V.<br /> -Velocity, angular, <a href="#Page_99">99</a>.<br /> -Vibration, <a href="#Page_129">129</a>.<br /> - Of the pendulum, <a href="#Page_145">145</a>.<br /> - Centre of, <a href="#Page_152">152</a>.<br /> -Volume, <a href="#Page_5">5–17</a>.<br /> -<br /> -W.<br /> -Watch, mainspring of; balance wheel of, <a href="#Page_195">195</a>.<br /> -Water regulator, <a href="#Page_229">229</a>.<br /> -Wedge, <a href="#Page_209">209</a>.<br /> - Use of, <a href="#Page_215">215</a>.<br /> -Weight, <a href="#Page_161">161–291</a>.<br /> -Weighing machines, <a href="#Page_278">278</a>.<br /> - For turnpike roads, <a href="#Page_302">302</a>.<br /> - By means of a spring, <a href="#Page_303">303</a>.<br /> -Wheels, spur, crown, bevelled, <a href="#Page_189">189</a>.<br /> - Escapement, <a href="#Page_194">194</a>.<br /> -Wheel and axle, <a href="#Page_177">177</a>.<br /> -Wheel-work, <a href="#Page_176">176</a>.<br /> -Winch, <a href="#Page_179">179</a>.<br /> -Windlass, <a href="#Page_178">178</a>.<br /> -Wollaston’s wire, <a href="#Page_10">10</a>.<br /> -<br /> -Z.<br /> -Zureda, apparatus of; Leupold’s application of, <a href="#Page_251">251</a>.</p> - - -<p class="tac fs80 mt3em">END OF MECHANICS.</p> - -<p class="tac fs80 mt3em"> -<span class="smcap">London</span>:<br /> -<span class="smcap">Spottiswoodes</span> and <span class="smcap">Shaw</span><br /> -New-street-Square.</p> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<h2 class="nobreak" id="FOOTNOTES">FOOTNOTES:</h2> -</div> - -<div class="footnote"> - -<p><a id="Footnote_1" href="#FNanchor_1" class="label">1</a> -More exactly through <span class="nowrap">16<span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">12</span></span></span> feet, or 193 inches.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_2" href="#FNanchor_2" class="label">2</a> -This ratio is that of 31,416 to 10,000 very nearly.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_3" href="#FNanchor_3" class="label">3</a> -Lardner on the Steam-Engine, Steam-Navigation, Roads, and Railways. -8th edition. 1851.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_4" href="#FNanchor_4" class="label">4</a> -From the Greek words <i>tachos</i> speed, and <i>metron</i> measure.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_5" href="#FNanchor_5" class="label">5</a> -Theatrum Machinarum, tom. ii. pl. 36. fig. 3.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_6" href="#FNanchor_6" class="label">6</a> -In a strictly mathematical sense, the path of the point P is a curve, -and not a straight line; but in the play given to it in its application to the -steam-engine, it moves through a part only of its entire locus, and this part -extending equally on each side of a point of inflection, the radius of curvature -is infinite, so that in practice the deviation from a straight line, when -proper proportions are observed in the rods, is imperceptible.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_7" href="#FNanchor_7" class="label">7</a> -The variation produced in the height of the column of mercury (supposed -to be <span class="nowrap">6<span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> inches high) by an alteration of ± 16° in the temperature -will be only ± <span class="nowrap"> <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">100</span></span></span> of an inch, or in other words, <span class="nowrap"> <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">100</span></span></span> of an inch will be the -total variation from its <i>mean</i> state, by an alteration of 32° in the temperature. -It is therefore probable that, in most cases of moderate alteration -in the temperature, the <i>centre</i> only of the column of mercury is subject to -elevation and depression, whilst the exterior parts remain attached to the -sides of the glass vessel. It was with a view to obviate this inconvenience -that Henry Browne, Esq. of Portland Place (I believe) first suggested the -piece of floating glass.</p> - -</div> - - - - -<div style='display:block; margin-top:4em'>*** END OF THE PROJECT GUTENBERG EBOOK A TREATISE ON MECHANICS ***</div> -<div style='text-align:left'> - -<div style='display:block; margin:1em 0'> -Updated editions will replace the previous one—the old editions will -be renamed. -</div> - -<div style='display:block; margin:1em 0'> -Creating the works from print editions not protected by U.S. copyright -law means that no one owns a United States copyright in these works, -so the Foundation (and you!) can copy and distribute it in the United -States without permission and without paying copyright -royalties. Special rules, set forth in the General Terms of Use part -of this license, apply to copying and distributing Project -Gutenberg™ electronic works to protect the PROJECT GUTENBERG™ -concept and trademark. 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