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diff --git a/old/42473.txt b/old/42473.txt deleted file mode 100644 index de16531..0000000 --- a/old/42473.txt +++ /dev/null @@ -1,21815 +0,0 @@ -The Project Gutenberg EBook of Encyclopaedia Britannica, 11th Edition, -Volume 17, Slice 8, by Various - -This eBook is for the use of anyone anywhere 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 - - -Title: Encyclopaedia Britannica, 11th Edition, Volume 17, Slice 8 - "Matter" to "Mecklenburg" - -Author: Various - -Release Date: April 7, 2013 [EBook #42473] - -Language: English - -Character set encoding: ASCII - -*** START OF THIS PROJECT GUTENBERG EBOOK ENCYCLOPAEDIA BRITANNICA *** - - - - -Produced by Marius Masi, Don Kretz and the Online -Distributed Proofreading Team at http://www.pgdp.net - - - - - - - -Transcriber's notes: - -(1) Numbers following letters (without space) like C2 were originally - printed in subscript. Letter subscripts are preceded by an - underscore, like C_n. - -(2) Characters following a carat (^) were printed in superscript. - -(3) Side-notes were relocated to function as titles of their respective - paragraphs. - -(4) Macrons and breves above letters and dots below letters were not - inserted. - -(5) [root] stands for the root symbol; [alpha], [beta], etc. for greek - letters, [oo] for infinity and [dP] for partial differential - symbol. - -(6) The following typographical errors have been corrected: - - ARTICLE MAURITIUS: "... in 1893 a great part of Port Louis was - destroyed by fire." 'a' added. - - ARTICLE MAXIMA AND MINIMA: "If d^2u/dx^2 vanishes, then there is no - maximum or minimum unless d^2u/dx^2 vanishes ..." 'minimum' amended - from 'minimun.' - - ARTICLE MAYOR: "Any female servant or slave in the household of a - barbarian, whose business it was to overlook other female servants - or slaves, would be quite naturally called a majorissa." - 'household' amended from 'houselold'. - - ARTICLE MAZANDARAN: "They speak a marked Persian dialect, but a - Turki idiom closely akin to the Turkoman is still current amongst - the tribes, although they have mostly already passed from the nomad - to the settled state." 'idiom' amended from 'idion'. - - ARTICLE MAZARIN, JULES: "But he began to wish for a wider sphere - than papal negotiations, and, seeing that he had no chance of - becoming a cardinal except by the aid of some great power ..." - 'sphere' amended from 'shpere'. - - ARTICLE MAZZINI, GIUSEPPE: "he did not actually hinder more than he - helped the course of events by which the realization of so much of - the great dream of his life was at last brought about." 'hinder' - amended from 'binder'. - - ARTICLE MEAUX: "The building, which is 275 ft. long and 105 ft. - high, consists of a short nave, with aisles, a fine transept, a - choir and a sanctuary." 'sanctuary' amended from 'sanctury'. - - ARTICLE MECHANICS: "The simplest case is that of a frame of three - bars, when the three joints A, B, C fall into a straight [** - amended from straght] line ..." 'straight' amended from 'straght'. - - ARTICLE MECHANICS: "... a determinate series of quantities having - to one another the above-mentioned ratios, whilst the constants C - ..." 'quantities' amended from 'quantites'. - - ARTICLE MECHANICS: "Then assuming that the acceleration of one - point of a particular link of the mechanism is known together with - the corresponding configuration of the mechanism ..." 'particular' - amended from 'particuar'. - - ARTICLE MECKLENBURG: "... were succeeded in the 6th century by some - Slavonic tribes, one of these being the Obotrites, whose chief - fortress was Michilenburg ..." 'Slavonic' amended from 'Salvonic'. - - - - - ENCYCLOPAEDIA BRITANNICA - - A DICTIONARY OF ARTS, SCIENCES, LITERATURE - AND GENERAL INFORMATION - - ELEVENTH EDITION - - - VOLUME XVII, SLICE VIII - - Matter to Mecklenburg - - - - -ARTICLES IN THIS SLICE: - - - MATTER MAX MULLER, FRIEDRICH - MATTERHORN MAXWELL - MATTEUCCI, CARLO MAXWELL, JAMES CLERK - MATTHEW, ST MAXWELLTOWN - MATTHEW, TOBIAS MAY, PHIL - MATTHEW, GOSPEL OF ST MAY, THOMAS - MATTHEW CANTACUZENUS MAY, WILLIAM - MATTHEW OF PARIS MAY (month) - MATTHEW OF WESTMINSTER MAY, ISLE OF - MATTHEWS, STANLEY MAYA - MATTHIAE, AUGUST HEINRICH MAYAGUEZ - MATTHIAS (disciple) MAYAVARAM - MATTHIAS (Roman emperor) MAYBOLE - MATTHIAS I., HUNYADI MAYEN - MATTHISSON, FRIEDRICH VON MAYENNE, CHARLES OF LORRAINE - MATTING MAYENNE (department of France) - MATTOCK MAYENNE (town of France) - MATTO GROSSO MAYER, JOHANN TOBIAS - MATTOON MAYER, JULIUS ROBERT - MATTRESS MAYFLOWER - MATURIN, CHARLES ROBERT MAY-FLY - MATVYEEV, ARTAMON SERGYEEVICH MAYHEM - MAUBEUGE MAYHEW, HENRY - MAUCH CHUNK MAYHEW, JONATHAN - MAUCHLINE MAYHEW, THOMAS - MAUDE, CYRIL MAYMYO - MAULE MAYNARD, FRANCOIS DE - MAULEON, SAVARI DE MAYNE, JASPER - MAULSTICK MAYNOOTH - MAUNDY THURSDAY MAYO, RICHARD SOUTHWELL BOURKE - MAUPASSANT, HENRI GUY DE MAYO - MAUPEOU, RENE NICOLAS AUGUSTIN MAYOR, JOHN EYTON BICKERSTETH - MAUPERTUIS, PIERRE MOREAU DE MAYOR - MAU RANIPUR MAYOR OF THE PALACE - MAUREL, ABDIAS MAYORUNA - MAUREL, VICTOR MAYO-SMITH, RICHMOND - MAURENBRECHER, KARL WILHELM MAYOTTE - MAUREPAS, JEAN PHELYPEAUX MAYOW, JOHN - MAURER, GEORG LUDWIG VON MAYSVILLE - MAURETANIA MAZAGAN - MAURIAC MAZAMET - MAURICE, ST MAZANDARAN - MAURICE (Roman emperor) MAZARIN, JULES - MAURICE (elector of Saxony) MAZAR-I-SHARIF - MAURICE, JOHN FREDERICK DENISON MAZARRON - MAURICE OF NASSAU MAZATLAN - MAURISTS MAZE - MAURITIUS MAZEPA-KOLEDINSKY, IVAN STEPANOVICH - MAURY, JEAN SIFFREIN MAZER - MAURY, LOUIS FERDINAND ALFRED MAZURKA - MAURY, MATTHEW FONTAINE MAZZARA DEL VALLO - MAUSOLEUM MAZZINI, GIUSEPPE - MAUSOLUS MAZZONI, GIACOMO - MAUVE, ANTON MAZZONI, GUIDO - MAVROCORDATO MEAD, LARKIN GOLDSMITH - MAWKMAI MEAD, RICHARD - MAXENTIUS, MARCUS VALERIUS MEAD - MAXIM, SIR HIRAM STEVENS MEADE, GEORGE GORDON - MAXIMA AND MINIMA MEADE, WILLIAM - MAXIMIANUS MEADVILLE - MAXIMIANUS, MARCUS VALERIUS MEAGHER, THOMAS FRANCIS - MAXIMILIAN I. (elector of Bavaria) MEAL - MAXIMILIAN I. (king of Bavaria) MEALIE - MAXIMILIAN II. (king of Bavaria) MEAN - MAXIMILIAN I. (Roman emperor) MEASLES - MAXIMILIAN II. (Roman emperor) MEAT - MAXIMILIAN (emperor of Mexico) MEATH - MAXIMINUS, GAIUS JULIUS VERUS MEAUX - MAXIMINUS, GALERIUS VALERIUS MECCA - MAXIMS, LEGAL MECHANICS - MAXIMUS MECHANICVILLE - MAXIMUS, ST MECHITHARISTS - MAXIMUS OF SMYRNA MECKLENBURG - MAXIMUS OF TYRE - - - - -MATTER. Our conceptions of the nature and structure of matter have been -profoundly influenced in recent years by investigations on the -Conduction of Electricity through Gases (see CONDUCTION, ELECTRIC) and -on Radio-activity (q.v.). These researches and the ideas which they have -suggested have already thrown much light on some of the most fundamental -questions connected with matter; they have, too, furnished us with far -more powerful methods for investigating many problems connected with the -structure of matter than those hitherto available. There is thus every -reason to believe that our knowledge of the structure of matter will -soon become far more precise and complete than it is at present, for now -we have the means of settling by testing directly many points which are -still doubtful, but which formerly seemed far beyond the reach of -experiment. - -The Molecular Theory of Matter--the only theory ever seriously -advocated--supposes that all visible forms of matter are collocations of -simpler and smaller portions. There has been a continuous tendency as -science has advanced to reduce further and further the number of the -different kinds of things of which all matter is supposed to be built -up. First came the molecular theory teaching us to regard matter as made -up of an enormous number of small particles, each kind of matter having -its characteristic particle, thus the particles of water were supposed -to be different from those of air and indeed from those of any other -substance. Then came Dalton's Atomic Theory which taught that these -molecules, in spite of their almost infinite variety, were all built up -of still smaller bodies, the atoms of the chemical elements, and that -the number of different types of these smaller bodies was limited to the -sixty or seventy types which represent the atoms of the substance -regarded by chemists as elements. - -In 1815 Prout suggested that the atoms of the heavier chemical elements -were themselves composite and that they were all built up of atoms of -the lightest element, hydrogen, so that all the different forms of -matter are edifices built of the same material--the atom of hydrogen. If -the atoms of hydrogen do not alter in weight when they combine to form -atoms of other elements the atomic weights of all elements would be -multiples of that of hydrogen; though the number of elements whose -atomic weights are multiples or very nearly so of hydrogen is very -striking, there are several which are universally admitted to have -atomic weights differing largely from whole numbers. We do not know -enough about gravity to say whether this is due to the change of weight -of the hydrogen atoms when they combine to form other atoms, or whether -the primordial form from which all matter is built up is something other -than the hydrogen atom. Whatever may be the nature of this primordial -form, the tendency of all recent discoveries has been to emphasize the -truth of the conception of a common basis of matter of all kinds. That -the atoms of the different elements have a common basis, that they -behave as if they consisted of different numbers of small particles of -the same kind, is proved to most minds by the Periodic Law of Mendeleeff -and Newlands (see ELEMENT). This law shows that the physical and -chemical properties of the different elements are determined by their -atomic weights, or to use the language of mathematics, the properties of -an element are functions of its atomic weight. Now if we constructed -models of the atoms out of different materials, the atomic weight would -be but one factor out of many which would influence the physical and -chemical properties of the model, we should require to know more than -the atomic weight to fix its behaviour. If we were to plot a curve -representing the variation of some property of the substance with the -atomic weight we should not expect the curve to be a smooth one, for -instance two atoms might have the same atomic weight and yet if they -were made of different materials have no other property in common. The -influence of the atomic weight on the properties of the elements is -nowhere more strikingly shown than in the recent developments of physics -connected with the discharge of electricity through gases and with -radio-activity. The transparency of bodies to Rontgen rays, to cathode -rays, to the rays emitted by radio-active substances, the quality of the -secondary radiation emitted by the different elements are all determined -by the atomic weight of the element. So much is this the case that the -behaviour of the element with respect to these rays has been used to -determine its atomic weight, when as in the case of Indium, uncertainty -as to the valency of the element makes the result of ordinary chemical -methods ambiguous. - -The radio-active elements indeed furnish us with direct evidence of this -unity of composition of matter, for not only does one element uranium, -produce another, radium, but all the radio-active substances give rise -to helium, so that the substance of the atoms of this gas must be -contained in the atoms of the radio-active elements. - -It is not radio-active atoms alone that contain a common constituent, -for it has been found that all bodies can by suitable treatment, such as -raising them to incandescence or exposing them to ultra-violet light, be -made to emit negatively electrified particles, and that these particles -are the same from whatever source they may be derived. These particles -all carry the same charge of negative electricity and all have the same -mass, this mass is exceedingly small even when compared with the mass of -an atom of hydrogen, which until the discovery of these particles was -the smallest mass known to science. These particles are called -corpuscles or electrons; their mass according to the most recent -determinations is only about 1/1700 of that of an atom of hydrogen, and -their radius is only about one hundred-thousandth part of the radius of -the hydrogen atom. As corpuscles of this kind can be obtained from all -substances, we infer that they form a constituent of the atoms of all -bodies. The atoms of the different elements do not all contain the same -number of corpuscles--there are more corpuscles in the atoms of the -heavier elements than in the atoms of the lighter ones; in fact, many -different considerations point to the conclusion that the number of -corpuscles in the atom of any element is proportional to the atomic -weight of the element. Different methods of estimating the exact number -of corpuscles in the atom have all led to the conclusion that this -number is of the same order as the atomic weight; that, for instance, -the number of corpuscles in the atom of oxygen is not a large multiple -of 16. Some methods indicate that the number of corpuscles in the atom -is equal to the atomic weight, while the maximum value obtained by any -method is only about four times the atomic weight. This is one of the -points on which further experiments will enable us to speak with greater -precision. Thus one of the constituents of all atoms is the negatively -charged corpuscle; since the atoms are electrically neutral, this -negative charge must be accompanied by an equal positive one, so that on -this view the atoms must contain a charge of positive electricity -proportional to the atomic weight; the way in which this positive -electricity is arranged is a matter of great importance in the -consideration of the constitution of matter. The question naturally -arises, is the positive electricity done up into definite units like the -negative, or does it merely indicate a property acquired by an atom when -one or more corpuscles leave it? It is very remarkable that we have up -to the present (1910), in spite of many investigations on this point, no -direct evidence of the existence of positively charged particles with a -mass comparable with that of a corpuscle; the smallest positive particle -of which we have any direct indication has a mass equal to the mass of -an atom of hydrogen, and it is a most remarkable fact that we get -positively charged particles having this mass when we send the electric -discharge through gases at low pressures, whatever be the kind of gas. -It is no doubt exceedingly difficult to get rid of traces of hydrogen in -vessels containing gases at low pressures through which an electric -discharge is passing, but the circumstances under which the positively -electrified particles just alluded to appear, and the way in which they -remain unaltered in spite of all efforts to clear out any traces of -hydrogen, all seem to indicate that these positively electrified -particles, whose mass is equal to that of an atom of hydrogen, do not -come from minute traces of hydrogen present as an impurity but from the -oxygen, nitrogen, or helium, or whatever may be the gas through which -the discharge passes. If this is so, then the most natural conclusion we -can come to is that these positively electrified particles with the mass -of the atom of hydrogen are the natural units of positive electricity, -just as the corpuscles are those of negative, and that these positive -particles form a part of all atoms. - -Thus in this way we are led to an electrical view of the constitution of -the atom. We regard the atom as built up of units of negative -electricity and of an equal number of units of positive electricity; -these two units are of very different mass, the mass of the negative -unit being only 1/1700 of that of the positive. The number of units of -either kind is proportional to the atomic weight of the element and of -the same order as this quantity. Whether this is anything besides the -positive and negative electricity in the atom we do not know. In the -present state of our knowledge of the properties of matter it is -unnecessary to postulate the existence of anything besides these -positive and negative units. - -The atom of a chemical element on this view of the constitution of -matter is a system formed by n corpuscles and n units of positive -electricity which is in equilibrium or in a state of steady motion under -the electrical forces which the charged 2n constituents exert upon each -other. Sir J. J. Thomson (_Phil. Mag._, March 1904, "Corpuscular Theory -of Matter") has investigated the systems in steady motion which can be -formed by various numbers of negatively electrified particles immersed -in a sphere of uniform positive electrification, a case, which in -consequence of the enormous volume of the units of positive electricity -in comparison with that of the negative has much in common with the -problem under consideration, and has shown that some of the properties -of n systems of corpuscles vary in a periodic way suggestive of the -Periodic Law in Chemistry as n is continually increased. - -_Mass on the Electrical Theory of Matter._--One of the most -characteristic things about matter is the possession of mass. When we -take the electrical theory of matter the idea of mass takes new and -interesting forms. This point may be illustrated by the case of a single -electrified particle; when this moves it produces in the region around -it a magnetic field, the magnetic force being proportional to the -velocity of the electrified particle.[1] In a magnetic field, however, -there is energy, and the amount of energy per unit volume at any place -is proportional to the square of the magnetic force at that place. Thus -there will be energy distributed through the space around the moving -particle, and when the velocity of the particle is small compared with -that of light we can easily show that the energy in the region around -the charged particle is ([mu]e^2)/(3a), when v is the velocity of the -particle, e its charge, a its radius, and [mu] the magnetic permeability -of the region round the particle. If m is the ordinary mass of the -particle, the part of the kinetic energy due to the motion of this mass -is (1/2)mv^2, thus the total kinetic energy is (1/2)[m + -(2/3)[mu]e^2/a]. Thus the electric charge on the particle makes it -behave as if its mass were increased by (2/3)[mu]e^2/a. Since this -increase in mass is due to the energy in the region outside the charged -particle, it is natural to look to that region for this additional mass. -This region is traversed by the tubes of force which start from the -electrified body and move with it, and a very simple calculation shows -that we should get the increase in the mass which is due to the -electrification if we suppose that these tubes of force as they move -carry with them a certain amount of the ether, and that this ether had -mass. The mass of ether thus carried along must be such that the amount -of it in unit volume at any part of the field is such that if this were -to move with the velocity of light its kinetic energy would be equal to -the potential energy of the electric field in the unit volume under -consideration. When a tube moves this mass of ether only participates in -the motion at right angles to the tube, it is not set in motion by a -movement of the tube along its length. We may compare the mass which a -charged body acquires in virtue of its charge with the additional mass -which a ball apparently acquires when it is placed in water; a ball -placed in water behaves as if its mass were greater than its mass when -moving in vacuo; we can easily understand why this should be the case, -because when the ball in the water moves the water around it must move -as well; so that when a force acting on the ball sets it in motion it -has to move some of the water as well as the ball, and thus the ball -behaves as if its mass were increased. Similarly in the case of the -electrified particle, which when it moves carries with it its lines of -force, which grip the ether and carry some of it along with them. When -the electrified particle is moved a mass of ether has to be moved too, -and thus the apparent mass of the particle is increased. The mass of the -electrified particle is thus resident in every part of space reached by -its lines of force; in this sense an electrified body may be said to -extend to an infinite distance; the amount of the mass of the ether -attached to the particle diminishes so rapidly as we recede from it that -the contributions of regions remote from the particle are quite -insignificant, and in the case of a particle as small as a corpuscle not -one millionth part of its mass will be farther away from it than the -radius of an atom. - -The increase in the mass of a particle due to given charges varies as we -have seen inversely as the radius of the particle; thus the smaller the -particle the greater the increase in the mass. For bodies of appreciable -size or even for those as small as ordinary atoms the effect of any -realizable electric charge is quite insignificant, on the other hand for -the smallest bodies known, the corpuscle, there is evidence that the -whole of the mass is due to the electric charge. This result has been -deduced by the help of an extremely interesting property of the mass due -to a charge of electricity, which is that this mass is not constant but -varies with the velocity. This comes about in the following way. When -the charged particle, which for simplicity we shall suppose to be -spherical, is at rest or moving very slowly the lines of electric force -are distributed uniformly around it in all directions; when the sphere -moves, however, magnetic forces are produced in the region around it, -while these, in consequence of electro-magnetic induction in a moving -magnetic field, give rise to electric forces which displace the tubes of -electric force in such a way as to make them set themselves so as to be -more at right angles to the direction in which they are moving than they -were before. Thus if the charged sphere were moving along the line AB, -the tubes of force would, when the sphere was in motion, tend to leave -the region near AB and crowd towards a plane through the centre of the -sphere and at right angles to AB, where they would be moving more nearly -at right angles to themselves. This crowding of the lines of force -increases, however, the potential energy of the electric field, and -since the mass of the ether carried along by the lines of force is -proportional to the potential energy, the mass of the charged particle -will also be increased. The amount of variation of the mass with the -velocity depends to some extent on the assumptions we make as to the -shape of the corpuscle and the way in which it is electrified. The -simplest expression connecting the mass with the velocity is that when -the velocity is v the mass is equal to (2/3)[mu]e^2/a [1/(1 - -v^2/c^2)^(1/2)] where c is the velocity of light. We see from this that -the variation of mass with velocity is very small unless the velocity of -the body approaches that of light, but when, as in the case of the -[beta] particles emitted by radium, the velocity is only a few per cent -less than that of light, the effect of velocity on the mass becomes very -considerable; the formula indicates that if the particles were moving -with a velocity equal to that of light they would behave as if their -mass were infinite. By observing the variation in the mass of a -corpuscle as its velocity changes we can determine how much of the mass -depends upon the electric charge and how much is independent of it. For -since the latter part of the mass is independent of the velocity, if it -predominates the variation with velocity of the mass of a corpuscle will -be small; if on the other hand it is negligible the variation in mass -with velocity will be that indicated by theory given above. The -experiment of Kaufmann (_Gottingen Nach._, Nov. 8, 1901), Bucherer -(_Ann. der Physik._, xxviii. 513, 1909) on the masses of the [beta] -particles shot out by radium, as well as those by Hupka (_Berichte der -deutsch. physik. Gesell._, 1909, p. 249) on the masses of the corpuscle -in cathode rays are in agreement with the view that the _whole_ of the -mass of these particles is due to their electric charge. - -The alteration in the mass of a moving charge with its velocity is -primarily due to the increase in the potential energy which accompanies -the increase in velocity. The connexion between potential energy and -mass is general and holds for any arrangement of electrified particles; -thus if we assume the electrical constitution of matter, there will be a -part of the mass of any system dependent upon the potential energy and -in fact proportional to it. Thus every change in potential energy, such -for example as occurs when two elements combine with evolution or -absorption of heat, must be attended by a change in mass. The amount of -this change can be calculated by the rule that if a mass equal to the -change in mass were to move with the velocity of light its kinetic -energy would equal the change in the potential energy. If we apply this -result to the case of the combination of hydrogen and oxygen, where the -evolution of heat, about 1.6 X 10^11 ergs per gramme of water, is -greater than in any other known case of chemical combination, we see -that the change in mass would only amount to one part in 3000 million, -which is far beyond the reach of experiment. The evolution of energy by -radio-active substances is enormously larger than in ordinary chemical -transformations; thus one gramme of radium emits per day about as much -energy as is evolved in the formation of one gramme of water, and goes -on doing this for thousands of years. We see, however, that even in this -case it would require hundreds of years before the changes in mass -became appreciable. - -The evolution of energy from the gaseous emanation given off by radium -is more rapid than that from radium itself, since according to the -experiments of Rutherford (Rutherford, _Radio-activity_, p. 432) a -gramme of the emanation would evolve about 2.1 X 10^16 ergs in four -days; this by the rule given above would diminish the mass by about one -part in 20,000; but since only very small quantities of the emanation -could be used the detection of the change of mass does not seem feasible -even in this case. - -On the view we have been discussing the existence of potential energy -due to an electric field is always associated with mass; wherever there -is potential energy there is mass. On the electro-magnetic theory of -light, however, a wave of light is accompanied by electric forces, and -therefore by potential energy; thus waves of light must behave as if -they possessed mass. It may be shown that it follows from the same -principles that they must also possess momentum, the direction of the -momentum being the direction along which the light is travelling; when -the light is absorbed by an opaque substance the momentum in the light -is communicated to the substance, which therefore behaves as if the -light pressed upon it. The pressure exerted by light was shown by -Maxwell (_Electricity and Magnetism_, 3rd ed., p. 440) to be a -consequence of his electro-magnetic theory, its existence has been -established by the experiment of Lebedew, of Nichols and Hull, and of -Poynting. - - - Weight. - -We have hitherto been considering mass from the point of view that the -constitution of matter is electrical; we shall proceed to consider the -question of weight from the same point of view. The relation between -mass and weight is, while the simplest in expression, perhaps the most -fundamental and mysterious property possessed by matter. The weight of a -body is proportional to its mass, that is if the weights of a number of -substances are equal the masses will be equal, whatever the substances -may be. This result was verified to a considerable degree of -approximation by Newton by means of experiments with pendulums; later, -in 1830 Bessel by a very extensive and accurate series of experiments, -also made on pendulums, showed that the ratio of mass to weight was -certainly to one part in 60,000 the same for all the substances examined -by him, these included brass, silver, iron, lead, copper, ivory, water. - -The constancy of this ratio acquires new interest when looked at from -the point of view of the electrical constitution of matter. We have seen -that the atoms of all bodies contain corpuscles, that the mass of a -corpuscle is only 1/1700 of the mass of an atom of hydrogen, that it -carries a constant charge of negative electricity, and that its mass is -entirely due to this charge, and can be regarded as arising from ether -gripped by the lines of force starting from the electrical charge. The -question at once suggests itself, Is this kind of mass ponderable? does -it add to the weight of the body? and, if so, is the proportion between -mass and weight the same as for ordinary bodies? Let us suppose for a -moment that this mass is not ponderable, so that the corpuscles increase -the mass but not the weight of an atom. Then, since the mass of a -corpuscle is 1/1700 that of an atom of hydrogen, the addition or removal -of one corpuscle would in the case of an atom of atomic weight x alter -the mass by one part in 1700 x, without altering the weight, this would -produce an effect of the same magnitude on the ratio of mass to weight -and would in the case of the atoms of the lighter elements be easily -measurable in experiments of the same order of accuracy as those made by -Bessel. If the number of corpuscles in the atom were proportional to the -atomic weight, then the ratio of mass to weight would be constant -whether the corpuscles were ponderable or not. If the number were not -proportional there would be greater discrepancies in the ratio of mass -to weight than is consistent with Bessel's experiments if the corpuscles -had no weight. We have seen there are other grounds for concluding that -the number of corpuscles in an atom is proportional to the atom weight, -so that the constancy of the ratio of mass to weight for a large number -of substances does not enable us to determine whether or not mass due to -charges of electricity is ponderable or not. - -There seems some hope that the determination of this ratio for -radio-active substances may throw some light on this point. The enormous -amount of heat evolved by these bodies may indicate that they possess -much greater stores of potential energy than other substances. If we -suppose that the heat developed by one gramme of a radio-active -substance in the transformations which it undergoes before it reaches -the non-radio-active stage is a measure of the excess of the potential -energy in a gramme of this substance above that in a gramme of -non-radio-active substance, it would follow that a larger part of the -mass was due to electric charges in radio-active than in -non-radio-active substances; in the case of uranium this difference -would amount to at least one part in 20,000 of the total mass. If this -extra mass had no weight the ratio of mass to weight for uranium would -differ from the normal amount by more than one part in 20,000, a -quantity quite within the range of pendulum experiments. It thus appears -very desirable to make experiments on the ratio of mass to weight for -radio-active substances. Sir J. J. Thomson, by swinging a small pendulum -whose bob was made of radium bromide, has shown that this ratio for -radium does not differ from the normal by one part in 2000. The small -quantity of radium available prevented the attainment of greater -accuracy. Experiments just completed (1910) by Southerns at the -Cavendish Laboratory on this ratio for uranium show that it is normal to -an accuracy of one part in 200,000; indicating that in non-radio-active, -as in radio-active, substances the electrical mass is proportional to -the atomic weight. - -Though but few experiments have been made in recent years on the value -of the ratio of mass to weight, many important investigations have been -made on the effect of alterations in the chemical and physical -conditions on the weight of bodies. These have all led to the conclusion -that no change which can be detected by our present means of -investigation occurs in the weight of a body in consequence of any -physical or chemical changes yet investigated. Thus Landolt, who devoted -a great number of years to the question whether any change in weight -occurs during chemical combination, came finally to the conclusion that -in no case out of the many he investigated did any measurable change of -weight occur during chemical combination. Poynting and Phillips (_Proc. -Roy. Soc._, 76, p. 445), as well as Southerns (78, p. 392), have shown -that change in temperature produces no change in the weight of a body; -and Poynting has also shown that neither the weight of a crystal nor the -attraction between two crystals depends at all upon the direction in -which the axis of the crystal points. The result of these laborious and -very carefully made experiments has been to strengthen the conviction -that the weight of a given portion of matter is absolutely independent -of its physical condition or state of chemical combinations. It should, -however, be noticed that we have as yet no accurate investigation as to -whether or not any changes of weight occur during radio-active -transformations, such for example as the emanation from radium undergoes -when the atoms themselves of the substance are disrupted. - -It is a matter of some interest in connexion with a discussion of any -views of the constitution of matter to consider the theories of -gravitation which have been put forward to explain that apparently -invariable property of matter--its weight. It would be impossible to -consider in detail the numerous theories which have been put forward to -account for gravitation; a concise summary of many of these has been -given by Drude (Wied. _Ann._ 62, p. 1);[2] there is no dearth of -theories as to the cause of gravitation, what is lacking is the means of -putting any of them to a decisive test. - -There are, however, two theories of gravitation, both old, which seem to -be especially closely connected with the idea of the electrical -constitution of matter. The first of these is the theory, associated -with the two fluid theory of electricity, that gravity is a kind of -residual electrical effect, due to the attraction between the units of -positive and negative electricity being a little greater than the -repulsion between the units of electricity of the same kind. Thus on -this view two charges of equal magnitude, but of opposite sign, would -exert an attraction varying inversely as the square of the distance on a -charge of electricity of either sign, and therefore an attraction on a -system consisting of two charges equal in magnitude but opposite in sign -forming an electrically neutral system. Thus if we had two neutral -systems, A and B, A consisting of m positive units of electricity and an -equal number of negative, while B has n units of each kind, then the -gravitational attraction between A and B would be inversely proportional -to the square of the distance and proportional to n m. The connexion -between this view of gravity and that of the electrical constitution of -matter is evidently very close, for if gravity arose in this way the -weight of a body would only depend upon the number of units of -electricity in the body. On the view that the constitution of matter is -electrical, the fundamental units which build up matter are the units of -electric charge, and as the magnitude of these charges does not change, -whatever chemical or physical vicissitudes matter, the weight of matter -ought not to be affected by such changes. There is one result of this -theory which might possibly afford a means of testing it: since the -charge on a corpuscle is equal to that on a positive unit, the weights -of the two are equal; but the mass of the corpuscle is only 1/1700 of -that of the positive unit, so that the acceleration of the corpuscle -under gravity will be 1700 times that of the positive unit, which we -should expect to be the same as that for ponderable matter or 981. - -The acceleration of the corpuscle under gravity on this view would be -1.6 X 10^6. It does not seem altogether impossible that with methods -slightly more powerful than those we now possess we might measure the -effect of gravity on a corpuscle if the acceleration were as large as -this. - -The other theory of gravitation to which we call attention is that due -to Le Sage of Geneva and published in 1818. Le Sage supposed that the -universe was thronged with exceedingly small particles moving with very -great velocities. These particles he called ultra-mundane corpuscles, -because they came to us from regions far beyond the solar system. He -assumed that these were so penetrating that they could pass through -masses as large as the sun or the earth without being absorbed to more -than a very small extent. There is, however, some absorption, and if -bodies are made up of the same kind of atoms, whose dimensions are small -compared with the distances between them, the absorption will be -proportional to the mass of the body. So that as the ultra-mundane -corpuscles stream through the body a small fraction, proportional to the -mass of the body, of their momentum is communicated to it. If the -direction of the ultra-mundane corpuscles passing through the body were -uniformly distributed, the momentum communicated by them to the body -would not tend to move it in one direction rather than in another, so -that a body, A, alone in the universe and exposed to bombardment by the -ultra-mundane corpuscles would remain at rest. If, however, there were a -second body, B, in the neighbourhood of A, B will shield A from some of -the corpuscles moving in the direction BA; thus A will not receive as -much momentum in this direction as when it was alone; but in this case -it only received just enough to keep it in equilibrium, so that when B -is present the momentum in the opposite direction will get the upper -hand and A will move in the direction AB, and will thus be attracted by -B. Similarly, we see that B will be attracted by A. Le Sage proved that -the rate at which momentum was being communicated to A or B by the -passage through them of his corpuscles was proportional to the product -of the masses of A and B, and if the distance between A and B was large -compared with their dimensions, inversely proportional to the square of -the distance between them; in fact, that the forces acting on them would -obey the same laws as the gravitational attraction between them. Clerk -Maxwell (article "ATOM," _Ency. Brit._, 9th ed.) pointed out that this -transference of momentum from the ultra-mundane corpuscles to the body -through which they passed involved the loss of kinetic energy by the -corpuscles, and if the loss of momentum were large enough to account for -the gravitational attraction, the loss of kinetic energy would be so -large that if converted into heat it would be sufficient to keep the -body white hot. We need not, however, suppose that this energy is -converted into heat; it might, as in the case where Rontgen rays are -produced by the passage of electrified corpuscles through matter, be -transformed into the energy of a still more penetrating form of -radiation, which might escape from the gravitating body without heating -it. It is a very interesting result of recent discoveries that the -machinery which Le Sage introduced for the purpose of his theory has a -very close analogy with things for which we have now direct experimental -evidence. We know that small particles moving with very high speeds do -exist, that they possess considerable powers of penetrating solids, -though not, as far as we know at present, to an extent comparable with -that postulated by Le Sage; and we know that the energy lost by them as -they pass through a solid is to a large extent converted into a still -more penetrating form of radiation, Rontgen rays. In Le Sage's theory -the only function of the corpuscles is to act as carriers of momentum, -any systems which possessed momentum, moved with a high velocity and had -the power of penetrating solids, might be substituted for them; now -waves of electric and magnetic force, such as light waves or Rontgen -rays, possess momentum, move with a high velocity, and the latter at any -rate possess considerable powers of penetration; so that we might -formulate a theory in which penetrating Rontgen rays replaced Le Sage's -corpuscles. Rontgen rays, however, when absorbed do not, as far as we -know, give rise to more penetrating Rontgen rays as they should to -explain attraction, but either to less penetrating rays or to rays of -the same kind. - -We have confined our attention in this article to the view that the -constitution of matter is electrical; we have done so because this view -is more closely in touch with experiment than any other yet advanced. -The units of which matter is built up on this theory have been isolated -and detected in the laboratory, and we may hope to discover more and -more of their properties. By seeing whether the properties of matter are -or are not such as would arise from a collection of units having these -properties, we can apply to this theory tests of a much more definite -and rigorous character than we can apply to any other theory of matter. - (J. J. T.) - - -FOOTNOTES: - - [1] We may measure this velocity with reference to any axes, provided - we refer the motion of all the bodies which come into consideration - to the same axes. - - [2] A theory published after Drude's paper in that of Professor - Osborne Reynolds, given in his Rede lecture "On an Inversion of Ideas - as to the Structure of the Universe." - - - - -MATTERHORN, one of the best known mountains (14,782 ft.) in the Alps. It -rises S.W. of the village of Zermatt, and on the frontier between -Switzerland (canton of the Valais) and Italy. Though on the Swiss side -it appears to be an isolated obelisk, it is really but the butt end of a -ridge, while the Swiss slope is not nearly as steep or difficult as the -grand terraced walls of the Italian slope. It was first conquered, after -a number of attempts chiefly on the Italian side, on the 14th of July -1865, by Mr E. Whymper's party, three members of which (Lord Francis -Douglas, the Rev. C. Hudson and Mr Hadow) with the guide, Michel Croz, -perished by a slip on the descent. Three days later it was scaled from -the Italian side by a party of men from Val Tournanche. Nowadays it is -frequently ascended in summer, especially from Zermatt. - - - - -MATTEUCCI, CARLO (1811-1868), Italian physicist, was born at Forli on -the 20th of June 1811. After attending the Ecole Polytechnique at -Paris, he became professor of physics successively at Bologna (1832), -Ravenna (1837) and Pisa (1840). From 1847 he took an active part in -politics, and in 1860 was chosen an Italian senator, at the same time -becoming inspector-general of the Italian telegraph lines. Two years -later he was minister of education. He died near Leghorn on the 25th of -June 1868. - - He was the author of four scientific treatises: _Lezioni di fisica_ (2 - vols., Pisa, 1841), _Lezioni sui fenomeni fisicochimici dei corpi - viventi_ (Pisa, 1844), _Manuale di telegrafia elettrica_ (Pisa, 1850) - and _Cours special sur l'induction, le magnetisme de rotation_, &c. - (Paris, 1854). His numerous papers were published in the _Annales de - chimie et de physique_ (1829-1858); and most of them also appeared at - the time in the Italian scientific journals. They relate almost - entirely to electrical phenomena, such as the magnetic rotation of - light, the action of gas batteries, the effects of torsion on - magnetism, the polarization of electrodes, &c., sufficiently complete - accounts of which are given in Wiedemann's _Galvanismus_. Nine - memoirs, entitled "Electro-Physiological Researches," were published - in the _Philosophical Transactions_, 1845-1860. See Bianchi's _Carlo - Matteucci e l'Italia del suo tempo_ (Rome, 1874). - - - - -MATTHEW, ST ([Greek: Maththaios] or [Greek: Matthaios], probably a -shortened form of the Hebrew equivalent to Theodorus), one of the twelve -apostles, and the traditional author of the First Gospel, where he is -described as having been a tax-gatherer or customs-officer ([Greek: -telones], x. 3), in the service of the tetrarch Herod. The circumstances -of his call to become a follower of Jesus, received as he sat in the -"customs house" in one of the towns by the Sea of Galilee--apparently -Capernaum (Mark ii. 1, 13), are briefly related in ix. 9. We should -gather from the parallel narrative in Mark ii. 14, Luke v. 27, that he -was at the time known as "Levi the son of Alphaeus" (compare Simon -Cephas, Joseph Barnabas): if so, "James the son of Alphaeus" may have -been his brother. Possibly "Matthew" (Yahweh's gift) was his Christian -surname, since two native names, neither being a patronymic, is contrary -to Jewish usage. It must be noted, however, that Matthew and Levi were -sometimes distinguished in early times, as by Heracleon (c. 170 A.D.), -and more dubiously by Origen (c. _Celsum_, i. 62), also apparently in -the Syriac _Didascalia_ (sec. iii.), V. xiv. 14. It has generally been -supposed, on the strength of Luke's account (v. 29), that Matthew gave a -feast in Jesus' honour (like Zacchaeus, Luke xix. 6 seq.). But Mark (ii. -15), followed by Matthew (ix. 10), may mean that the meal in question -was one in Jesus' own home at Capernaum (cf. v. 1). In the lists of the -Apostles given in the Synoptic Gospels and in Acts, Matthew ranks third -or fourth in the second group of four--a fair index of his relative -importance in the apostolic age. The only other facts related of Matthew -on good authority concern him as Evangelist. Eusebius (_H.E._ iii. 24) -says that he, like John, wrote only at the spur of necessity. "For -Matthew, after preaching to Hebrews, when about to go also to others, -committed to writing in his native tongue the Gospel that bears his -name; and so by his writing supplied, for those whom he was leaving, the -loss of his presence." The value of this tradition, which may be based -on Papias, who certainly reported that "Matthew compiled the Oracles (of -the Lord) in Hebrew," can be estimated only in connexion with the study -of the Gospel itself (see below). No historical use can be made of the -artificial story, in _Sanhedrin_ 43a, that Matthew was condemned to -death by a Jewish court (see Laihle, _Christ in the Talmud_, 71 seq.). -According to the Gnostic Heracleon, quoted by Clement of Alexandria -(_Strom._ iv. 9), Matthew died a natural death. The tradition as to his -ascetic diet (in Clem. Alex. _Paedag._ ii. 16) maybe due to confusion -with Matthias (cf. _Mart. Matthaei_, i.). The earliest legend as to his -later labours, one of Syrian origin, places them in the Parthian -kingdom, where it represents him as dying a natural death at Hierapolis -(= Mabog on the Euphrates). This agrees with his legend as known to -Ambrose and Paulinus of Nola, and is the most probable in itself. The -legends which make him work with Andrew among the Anthropophagi near the -Black Sea, or again in Ethiopia (Rufinus, and Socrates, _H.E._ i. 19), -are due to confusion with Matthias, who from the first was associated in -his Acts with Andrew (see M. Bonnet, _Acta Apost. apocr._, 1808, II. i. -65). Another legend, his _Martyrium_, makes him labour and suffer in -Mysore. He is commemorated as a martyr by the Greek Church on the 16th -of November, and by the Roman on the 21st of September, the scene of his -martyrdom being placed in Ethiopia. The Latin Breviary also affirms that -his body was afterwards translated to Salerno, where it is said to lie -in the church built by Robert Guiscard. In Christian art (following -Jerome) the Evangelist Matthew is generally symbolized by the "man" in -the imagery of Ezek. i. 10, Rev. iv. 7. - - For the historical Matthew, see _Ency. Bibl._ and Zahn, _Introd. to - New Test._, ii. 506 seq., 522 seq. For his legends, as under MARK. - (J. V. B.) - - - - -MATTHEW, TOBIAS, or TOBIE (1546-1628), archbishop of York, was the son -of Sir John Matthew of Ross in Herefordshire, and of his wife Eleanor -Crofton of Ludlow. He was born at Bristol in 1546. He was educated at -Wells, and then in succession at University College and Christ Church, -Oxford. He proceeded B.A. in 1564, and M.A. in 1566. He attracted the -favourable notice of Queen Elizabeth, and his rise was steady though not -very rapid. He was public orator in 1569, president of St John's -College, Oxford, in 1572, dean of Christ Church in 1576, vice-chancellor -of the university in 1579, dean of Durham in 1583, bishop of Durham in -1595, and archbishop of York in 1606. In 1581 he had a controversy with -the Jesuit Edmund Campion, and published at Oxford his arguments in 1638 -under the title, _Piissimi et eminentissimi viri Tobiae Matthew, -archiepiscopi olim Eboracencis concio apologetica adversus Campianam_. -While in the north he was active in forcing the recusants to conform to -the Church of England, preaching hundreds of sermons and carrying out -thorough visitations. During his later years he was to some extent in -opposition to the administration of James I. He was exempted from -attendance in the parliament of 1625 on the ground of age and -infirmities, and died on the 29th of March 1628. His wife, Frances, was -the daughter of William Barlow, bishop of Chichester. - -His son, SIR TOBIAS, or TOBIE, MATTHEW (1577-1655), is remembered as the -correspondent and friend of Francis Bacon. He was educated at Christ -Church, and was early attached to the court, serving in the embassy at -Paris. His debts and dissipations were a great source of sorrow to his -father, from whom he is known to have received at different times -L14,000, the modern equivalent of which is much larger. He was chosen -member for Newport in Cornwall in the parliament of 1601, and member for -St Albans in 1604. Before this time he had become the intimate friend of -Bacon, whom he replaced as member for St Albans. When peace was made -with Spain, on the accession of James I., he wished to travel abroad. -His family, who feared his conversion to Roman Catholicism, opposed his -wish, but he promised not to go beyond France. When once safe out of -England he broke his word and went to Italy. The persuasion of some of -his countrymen in Florence, one of whom is said to have been the Jesuit -Robert Parsons, and a story he heard of the miraculous liquefaction of -the blood of San Januarius at Naples, led to his conversion in 1606. -When he returned to England he was imprisoned, and many efforts were -made to obtain his reconversion without success. He would not take the -oath of allegiance to the king. In 1608 he was exiled, and remained out -of England for ten years, mostly in Flanders and Spain. He returned in -1617, but went abroad again in 1619. His friends obtained his leave to -return in 1621. At home he was known as the intimate friend of Gondomar, -the Spanish ambassador. In 1623 he was sent to join Prince Charles, -afterwards Charles I., at Madrid, and was knighted on the 23rd of -October of that year. He remained in England till 1640, when he was -finally driven abroad by the parliament, which looked upon him as an -agent of the pope. He died in the English college in Ghent on the 13th -of October 1655. In 1618 he published an Italian translation of Bacon's -essays. The "Essay on Friendship" was written for him. He was also the -author of a translation of _The Confessions of the Incomparable Doctor -St Augustine_, which led him into controversy. His correspondence was -published in London in 1660. - - For the father, see John Le Neve's _Fasti ecclesiae anglicanae_ - (London, 1716), and Anthony Wood's _Athenae oxonienses_. For the son, - the notice in _Athenae oxonienses_, an abridgment of his - autobiographical _Historical Relation_ of his own life, published by - Alban Butler in 1795, and A. H. Matthew and A. Calthrop, _Life of Sir - Tobie Matthew_ (London, 1907). - - - - -MATTHEW, GOSPEL OF ST, the first of the four canonical Gospels of the -Christian Church. The indications of the use of this Gospel in the two -or three generations following the Apostolic Age (see GOSPEL) are more -plentiful than of any of the others. Throughout the history of the -Church, also, it has held a place second to none of the Gospels alike in -public instruction and in the private reading of Christians. The reasons -for its having impressed itself in this way and become thus familiar are -in large part to be found in the characteristics noticed below. But in -addition there has been from an early time the belief that it was the -work of one of those publicans whose heart Jesus touched and of whose -call to follow Him the three Synoptics contain an interesting account, -but who is identified as Matthew (q.v.) only in this one (Matt. ix. 9-13 -= Mark ii. 13-17 = Luke v. 27-32). - -1. _The Connexion of our Greek Gospel of Matthew with the Apostle whose -name it bears._--The earliest reference to a writing by Matthew occurs -in a fragment taken by Eusebius from the same work of Papias from which -he has given an account of the composition of a record by Mark (Euseb. -_Hist. Eccl._ iii. 39; see MARK, GOSPEL OF ST). The statement about -Matthew is much briefer and is harder to interpret. In spite of much -controversy, the same measure of agreement as to its meaning cannot be -said to have been attained. This is the fragment: "Matthew, however, put -together and wrote down the Oracles ([Greek: ta logia synegrapsen]) in -the Hebrew language, and each man interpreted them as he was able." -Whether "the elder" referred to in the passage on Mark, or some other -like authority, was the source of this statement also does not appear; -but it is probable that this was the case from the context in which -Eusebius gives it. Conservative writers on the Gospels have frequently -maintained that the writing here referred to was virtually the Hebrew -original of our Greek Gospel which bears his name. And it is indeed -likely that Papias himself closely associated the latter with the Hebrew -(or Aramaic) work by Matthew, of which he had been told, since the -traditional connexion of this Greek Gospel with Matthew can hardly have -begun later than this time. It is reasonable also to suppose that there -was some ground for it. The description, however, of what Matthew did -suits better the making of a collection of Christ's discourses and -sayings than the composition of a work corresponding in form and -character to our Gospel of Matthew. - -The next reference in Christian literature to a Gospel-record by Matthew -is that of Irenaeus in his famous passage on the four Gospels (_Adv. -haer._ iii. i. r). He says that it was written in Hebrew; but in all -probability he regarded the Greek Gospel, which stood first in his, as -it does in our, enumeration, as in the strict sense a translation of the -Apostle's work; and this was the view of it universally taken till the -16th century, when some of the scholars of the Reformation maintained -that the Greek Gospel itself was by Matthew. - -The actual phenomena, however, of this Gospel, and of its relation to -sources that have been used in it, cannot be explained consistently with -either of the two views just mentioned. It is a composite work in which -two chief sources, known in Greek to the author of our present Gospel, -have, together with some other matter, been combined. It is -inconceivable that one of the Twelve should have proceeded in this way -in giving an account of Christ's ministry. One of the chief documents, -however, here referred to seems to correspond in character with the -description given in Papias' fragment of a record of the compilation of -"the divine utterances" made by Matthew; and the use made of it in our -first Gospel may explain the connexion of this Apostle's name with it. -In the Gospel of Luke also, it is true, this same source has been used -for the teaching of Jesus. But the original Aramaic Logian document may -have been more largely reproduced in our Greek Matthew. Indeed, in the -case of one important passage (v. 17-48) this is suggested by a -comparison with Luke itself, and there are one or two others where from -the character of the matter it seems not improbable, especially vi. 1-18 -and xxiii. 1-5, 7b-10, 15-22. On the whole, as will be seen below, what -appears to be a Palestinian form of the Gospel-tradition is most fully -represented in this Gospel; but in many instances at least this may well -be due to some other cause than the use of the original Logian document. - -2. _The Plan on which the Contents is arranged._--In two respects the -arrangement of the book itself is significant. - - (a) As to the general outline in the first half of the account of the - Galilean ministry (iv. 23-xi. 30). Immediately after relating the call - of the first four disciples (iv. 18-22) the evangelist gives in iv. 23 - a comprehensive summary of Christ's work in Galilee under its two - chief aspects, teaching and healing. In the sequel both these are - illustrated. First, he gives in the Sermon on the Mount (v.-vii.) a - considerable body of teaching, of the kind required by the disciples - of Jesus generally, and a large portion of which probably also stood - not far from the beginning of the Logian document. After this he turns - to the other aspect. Up to this point he has mentioned no miracle. He - now describes a number in succession, introducing all but the first of - those told between Mark i. 23 and ii. 12, and also four specially - remarkable ones, which occurred a good deal later according to Mark's - order (Matt. viii. 23-34 = Mark iv. 35-v. 20; Matt. ix. 18-26 = Mark - v. 21-43); and he also adds some derived from another source, or other - sources (viii. 5-13; ix. 27-34). Then, after another general - description at ix. 35, similar to that at iv. 23, he brings strikingly - before us the needs of the masses of the people and Christ's - compassion for them, and so introduces the mission of the Twelve - (which again occurs later according to Mark's order, viz. at vi. 7 - seq.), whereby the ministry both of teaching and of healing was - further extended (ix. 36-x. 42). Finally, the message of John the - Baptist, and the reply of Jesus, and the reflections that follow - (xi.), bring out the significance of the preceding narrative. It - should be observed that examples have been given of every kind of - mighty work referred to in the reply of Jesus to the messengers of the - Baptist; and that in the discourse which follows their departure the - perversity and unbelief of the people generally are condemned, and the - faith of the humble-minded is contrasted therewith. The greater part - of the matter from ix. 37 to end of xi. is taken from the Logian - document. After this point, i.e. from xii. 1 onwards, the first - evangelist follows Mark almost step by step down to the point (Mark - xvi. 8), after which Mark's Gospel breaks off, and another ending has - been supplied; and gives in substance almost the whole of Mark's - contents, with the exception that he passes over the few narratives - that he has (as we have seen) placed earlier. At the same time he - brings in additional matter in connexion with most of the Marcan - sections. - - (b) With the accounts of the words of Jesus spoken on certain - occasions, which our first evangelist found given in one or another of - his sources, he has combined other pieces, taken from other parts of - the same source or from different sources, which seemed to him - connected in subject, e.g. into the discourse spoken on a mountain, - when crowds from all parts were present, given in the Logian document, - he has introduced some pieces which, as we infer from Luke, stood - separately in that document (cf. Matt. vi. 19-21 with Luke xii. 33, - 34; Matt. vi. 22, 23 with Luke xi. 34-36; Matt. vi. 24 with Luke xvi. - 13; Matt. vi. 25-34 with Luke xii. 22-32; Matt. vii. 7-11 with Luke - xi. 9-13). Again, the address to the Twelve in Mark vi. 7-11, which in - Matthew is combined with an address to disciples, from the Logian - document, is connected by Luke with the sending out of seventy - disciples (Luke x. 1-16). Our first evangelist has also added here - various other sayings (Matt. x. 17-39, 42). Again, with the Marcan - account of the charge of collusion with Satan and Christ's reply (Mark - iii. 22-30), the first evangelist (xii. 24-45) combines the parallel - account in the Logian document and adds Christ's reply to another - attack (Luke xi. 14-16, 17-26, 29-32). These are some examples. He has - in all in this manner constructed eight discourses or collections of - sayings, into which the greater part of Christ's teaching is gathered: - (1) On the character of the heirs of the kingdom (v.-vii.); (2) The - Mission address (x.); (3) Teaching suggested by the message of John - the Baptist (xi.); (4) The reply to an accusation and a challenge - (xii. 22-45); (5) The teaching by parables (xiii.); (6) On offences - (xviii.); (7) Concerning the Scribes and Pharisees (xxiii.); (8) On - the Last Things (xxiv., xxv.). In this arrangement of his material the - writer has in many instances disregarded chronological considerations. - But his documents also gave only very imperfect indications of the - occasions of many of the utterances; and the result of his method of - procedure has been to give us an exceedingly effective representation - of the teaching of Jesus. - - In the concluding verses of the Gospel, where the original Marcan - parallel is wanting, the evangelist may still have followed in part - that document while making additions as before. The account of the - silencing of the Roman guard by the chief priests is the sequel to the - setting of this guard and their presence at the Resurrection, which at - an earlier point arc peculiar to Matthew (xxvii. 62-66, xxviii. 4). - And, further, this matter seems to belong to the same cycle of - tradition as the story of Pilate's wife and his throwing the guilt of - the Crucifixion of Jesus upon the Jews, and the testimony borne by - the Roman guard (as well as the centurion) who kept watch by the cross - (xxvii. 15-26, 54), all which also are peculiar to this Gospel. It - cannot but seem probable that these are legendary additions which had - arisen through the desire to commend the Gospel to the Romans. - - On the other hand, the meeting of Jesus with the disciples in Galilee - (Matt. xxviii. 16 seq.) is the natural sequel to the message to them - related in Mark xvi. 7, as well as in Matt, xxviii. 7. Again, the - commission to them to preach throughout the world is supported by Luke - xxiv. 47, and by the present ending of Mark (xvi. 15), though neither - of these mention Galilee as the place where it was given. The - baptismal formula in Matt. xxviii. 19, is, however, peculiar, and in - view of its non-occurrence in the Acts and Epistles of the New - Testament must be regarded as probably an addition in accordance with - Church usage at the time the Gospel was written. - -3. _The Palestinian Element._--Teaching is preserved in this Gospel -which would have peculiar interest and be specially required in the home -of Judaism. The best examples of this are the passages already referred -to near end of S 1, as probably derived from the Logian document. There -are, besides, a good many turns of expression and sayings peculiar to -this Gospel which have a Semitic cast, or which suggest a point of view -that would be natural to Palestinian Christians, e.g. "kingdom of -heaven" frequently for "kingdom of God"; xiii. 52 ("every scribe"); -xxiv. 20 ("neither on a Sabbath"). See also v. 35 and xix. 9; x. 5, 23. -Again, several of the quotations which are peculiar to this Gospel are -not taken from the LXX., as those in the other Gospels and in the -corresponding contexts in this Gospel commonly are, but are wholly or -partly independent renderings from the Hebrew (ii. 6, 15, 18; viii. 17, -xii. 17-21, &c.). Once more, there is somewhat more parallelism between -the fragments of the Gospel according to the Hebrews and this Gospel -than is the case with Luke, not to say Mark. - -4. _Doctrinal Character._--In this Gospel, more decidedly than in either -of the other two Synoptics, there is a doctrinal point of view from -which the whole history is regarded. Certain aspects which are of -profound significance are dwelt upon, and this without there being any -great difference between this Gospel and the two other Synoptics in -respect to the facts recorded or the beliefs implied. The effect is -produced partly by the comments of the evangelist, which especially take -the form of citations from the Old Testament; partly by the frequency -with which certain expressions are used, and the prominence that is -given in this and other ways to particular traits and topics. - -He sets forth the restriction of the mission of Jesus during His life on -earth to the people of Israel in a way which suggests at first sight a -spirit of Jewish exclusiveness. But there are various indications that -this is not the true explanation. In particular the evangelist brings -out more strongly than either Mark or Luke the national rejection of -Jesus, while the Gospel ends with the commission of Jesus to His -disciples after His resurrection to "make disciples of all the peoples." -One may divine in all this an intention to "justify the ways of God" to -the Jew, by proving that God in His faithfulness to His ancient people -had given them the first opportunity of salvation through Christ, but -that now their national privilege had been rightly forfeited. He was -also specially concerned to show that prophecy is fulfilled in the life -and work of Jesus, but the conception of this fulfilment which is -presented to us is a large one; it is to be seen not merely in -particular events or features of Christ's ministry, but in the whole new -dispensation, new relations between God and men, and new rules of -conduct which Christ has introduced. The divine meaning of the work of -Jesus is thus made apparent, while of the majesty and glory of His -person a peculiarly strong impression is conveyed. - -Some illustrations in detail of these points are subjoined. Where there -are parallels in the other Gospels they should be compared and the words -in Matthew noted which in many instances serve to emphasize the points -in question. - - (a) _The Ministry of Jesus among the Jewish People as their promised - Messiah, their rejection of Him, and the extension of the Gospel to - the Gentiles._ The mission to Israel: Matt. i. 21; iv. 23 (note in - these passages the use of [Greek: ho laos], which here, as generally - in Matthew, denotes the chosen nation), ix. 33, 35, xv. 31. For the - rule limiting the work of Jesus while on earth see xv. 24 (and note - [Greek: ixelthousa] in verse 22, which implies that Jesus had not - himself entered the heathen borders), and for a similar rule - prescribed to the disciples, x. 5, 6 and 23. - - The rejection of Jesus by the people in Galilee, xi. 21; xiii. 13-15, - and by the heads of "the nation," xxvi. 3, 47 and by "the whole - nation," xxvii. 25; their condemnation xxiii. 38. - - Mercy to the Gentiles and the punishment of "the sons of the kingdom" - is foretold viii. 11, 12. The commission to go and convert Gentile - peoples ([Greek: ethne]) is given after Christ's resurrection (xxviii. - 19). - - (b) _The Fulfilment of Prophecy._--In the birth and childhood of - Jesus, i. 23; ii. 6, 15, 18, 23. By these citations attention is drawn - to the lowliness of the beginnings of the Saviour's life, the - unexpected and secret manner of His appearing, the dangers to which - from the first He was exposed and from which He escaped. - - The ministry of Christ's forerunner, iii. 3. (The same prophecy, Isa. - xl. 3, is also quoted in the other Gospels.) - - The ministry of Jesus. The quotations serve to bring out the - significance of important events, especially such as were - turning-points, and also to mark the broad features of Christ's life - and work, iv. 15, 16; viii. 17; xii. 18 seq.; xiii. 35; xxi. 5; xxvii. - 9. - - (c) _The Teaching on the Kingdom of God._--Note the collection of - parables "of the Kingdom" in xiii.; also the use of [Greek: he - basileia] ("the Kingdom") without further definition as a term the - reference of which could not be misunderstood, especially in the - following phrases peculiar to this Gospel: [Greek: to euangelion tes - basileias] ("the Gospel of the Kingdom") iv. 23, ix. 35, xxiv. 14; and - [Greek: ho logos tes basileias] ("the word of the kingdom") xiii. 19. - The following descriptions of the kingdom, peculiar to this Gospel, - are also interesting [Greek: he basileia tou patros auton] ("the - kingdom of their father") xiii. 43 and [Greek: tou patros mou]("of my - father") xxvi. 29. - - (d) _The Relation of the New Law to the Old._--Verses 17-48, cf. also, - addition at xxii. 40 and xix. 19b. Further, his use of [Greek: - dikaiosyne] ("righteousness") and [Greek: dikaios]("righteous") - (specially frequent in this Gospel) is such as to connect the New with - the Old; the standard in mind is the law which "fulfilled" that - previously given. - - (e) _The Christian Ecclesia._--Chap. xvi. 18, xviii. 17. - - (f) _The Messianic Dignity and Glory of Jesus._--The narrative in i. - and ii. show the royalty of the new-born child. The title "Son of - David" occurs with special frequency in this Gospel. The following - instances are without parallels in the other Gospels: ix. 27; xii. 23; - xv. 22; xxi. 9; xxi. 15. The title "Son of God" is also used with - somewhat greater frequency than in Mark and Luke: ii. 15; xiv. 33; - xvi. 16; xxii. 2 seq. (where it is implied); xxvii. 40, 43. - - The thought of the future coming of Christ, and in particular of the - judgment to be executed by Him then, is much more prominent in this - Gospel than in the others. Some of the following predictions are - peculiar to it, while in several others there are additional touches: - vii. 22, 23; x. 23, 32, 33; xiii. 39-43; xvi. 27, 28; xix. 28; xxiv. - 3, 27, 30, 31, 37, 39; xxv. 31-46; xxvi. 64. - - The majesty of Christ is also impressed upon us by the signs at His - crucifixion, some of which are related only in this Gospel, xxvii. - 51-53, and by the sublime vision of the Risen Christ at the close, - xxviii. 16-20. - -(5) _Time of Composition and Readers addressed._--The signs of dogmatic -reflection in this Gospel point to its having been composed somewhat -late in the 1st century, probably after Luke's Gospel, and this is in -accord with the conclusion that some insertions had been made in the -Marcan document used by this evangelist which were not in that used by -Luke (see LUKE, GOSPEL OF ST). We may assign A.D. 80-100 as a probable -time for the composition. - -The author was in all probability a Jew by race, and he would seem to -have addressed himself especially to Jewish readers; but they were Jews -of the Dispersion. For although he was in specially close touch with -Palestine, either personally or through the sources at his command, or -both, his book was composed in Greek by the aid of Greek documents. - - See commentaries by Th. Zahn (1903) and W. C. Allen (in the series of - International Critical Commentaries, 1907); also books on the Four - Gospels or the Synoptic Gospels cited at the end of GOSPEL. - (V. H. S.) - - - - -MATTHEW CANTACUZENUS, Byzantine emperor, was the son of John VI. -Cantacuzenus (q.v.). In return for the support he gave to his father -during his struggle with John V. he was allowed to annex part of Thrace -under his own dominion and in 1353 was proclaimed joint emperor. From -his Thracian principality he levied several wars against the Servians. -An attack which he prepared in 1350 was frustrated by the defection of -his Turkish auxiliaries. In 1357 he was captured by his enemies, who -delivered him to the rival emperor, John V. Compelled to abdicate, he -withdrew to a monastery, where he busied himself with writing -commentaries on the Scriptures. - - - - -MATTHEW OF PARIS (d. 1259), English monk and chronicler known to us only -through his voluminous writings. In spite of his surname, and of his -knowledge of the French language, his attitude towards foreigners -attests that he was of English birth. He may have studied at Paris in -his youth, but the earliest fact which he records of himself is his -admission as a monk at St Albans in the year 1217. His life was mainly -spent in this religious house. In 1248, however, he was sent to Norway -as the bearer of a message from Louis IX. of France to Haakon VI.; he -made himself so agreeable to the Norwegian sovereign that he was -invited, a little later, to superintend the reformation of the -Benedictine monastery of St Benet Holme at Trondhjem. Apart from these -missions, his activities were devoted to the composition of history, a -pursuit for which the monks of St Albans had long been famous. Matthew -edited anew the works of Abbot John de Cella and Roger of Wendover, -which in their altered form constitute the first part of his most -important work, the _Chronica majora_. From 1235, the point at which -Wendover dropped his pen, Matthew continued the history on the plan -which his predecessors had followed. He derived much of his information -from the letters of important personages, which he sometimes inserts, -but much more from conversation with the eye-witnesses of events. Among -his informants were Earl Richard of Cornwall and Henry III. With the -latter he appears to have been on terms of intimacy. The king knew that -Matthew was writing a history, and showed some anxiety that it should be -as exact as possible. In 1257, in the course of a week's visit to St -Albans, Henry kept the chronicler beside him night and day, "and guided -my pen," says Paris, "with much good will and diligence." It is -therefore curious that the _Chronica majora_ should give so unfavourable -an account of the king's policy. Luard supposes that Matthew never -intended his work to see the light in its present form, and many -passages of the autograph have against them the note _offendiculum_, -which shows that the writer understood the danger which he ran. On the -other hand, unexpurgated copies were made in Matthew's lifetime; though -the offending passages are duly omitted or softened in his abridgment of -his longer work, the _Historia Anglorum_ (written about 1253), the real -sentiments of the author must have been an open secret. In any case -there is no ground for the old theory that he was an official -historiographer. - - Matthew Paris was unfortunate in living at a time when English - politics were peculiarly involved and tedious. His talent is for - narrative and description. Though he took a keen interest in the - personal side of politics he has no claim to be considered a judge of - character. His appreciations of his contemporaries throw more light on - his own prejudices than on their aims and ideas. His work is always - vigorous, but he imputes motives in the spirit of a partisan who never - pauses to weigh the evidence or to take a comprehensive view of the - situation. His redeeming feature is his generous admiration for - strength of character, even when it goes along with a policy of which - he disapproves. Thus he praises Grosseteste, while he denounces - Grosseteste's scheme of monastic reform. Matthew is a vehement - supporter of the monastic orders against their rivals, the secular - clergy and the mendicant friars. He is violently opposed to the court - and the foreign favourites. He despises the king as a statesman, - though for the man he has some kindly feeling. The frankness with - which he attacks the court of Rome for its exactions is remarkable; - so, too, is the intense nationalism which he displays in dealing with - this topic. His faults of presentment are more often due to - carelessness and narrow views than to deliberate purpose. But he is - sometimes guilty of inserting rhetorical speeches which are not only - fictitious, but also misleading as an account of the speaker's - sentiments. In other cases he tampers with the documents which he - inserts (as, for instance, with the text of Magna Carta). His - chronology is, for a contemporary, inexact; and he occasionally - inserts duplicate versions of the same incident in different places. - Hence he must always be rigorously checked where other authorities - exist and used with caution where he is our sole informant. None the - less, he gives a more vivid impression of his age than any other - English chronicler; and it is a matter for regret that his great - history breaks off in 1259, on the eve of the crowning struggle - between Henry III and the baronage. - - AUTHORITIES.--The relation of Matthew Paris's work to those of John de - Cella and Roger of Wendover may best be studied in H. R. Luard's - edition of the _Chronica majora_ (7 vols., Rolls series, 1872-1883), - which contains valuable prefaces. The _Historia_ _Anglorum sive - historia minor_ (1067-1253) has been edited by F. Madden (3 vols., - Rolls series, 1866-1869). Matthew Paris is often confused with - "Matthew of Westminster," the reputed author of the _Flores - historiarum_ edited by H. R. Luard (3 vols., Rolls series, 1890). This - work, compiled by various hands, is an edition of Matthew Paris, with - continuations extending to 1326. Matthew Paris also wrote a life of - Edmund Rich (q.v.), which is probably the work printed in W. Wallace's - _St Edmund of Canterbury_ (London, 1893) pp. 543-588, though this is - attributed by the editor to the monk Eustace; _Vitae abbatum S Albani_ - (up to 1225) which have been edited by W. Watts (1640, &c.); and - (possibly) the _Abbreviatio chronicorum_ (1000-1255), edited by F. - Madden, in the third volume of the _Historia Anglorum_. On the value - of Matthew as an historian see F. Liebermann in G. H. Pertz's - _Scriptores_ xxviii. pp. 74-106; A. Jessopp's _Studies by a Recluse_ - (London, 1893); H. Plehn's _Politische Character Matheus Parisiensis_ - (Leipzig, 1897). (H. W. C. D.) - - - - -MATTHEW OF WESTMINSTER, the name of an imaginary person who was long -regarded as the author of the _Flores Historiarum_. The error was first -discovered in 1826 by Sir F. Palgrave, who said that Matthew was "a -phantom who never existed," and later the truth of this statement was -completely proved by H. R. Luard. The name appears to have been taken -from that of Matthew of Paris, from whose _Chronica majora_ the earlier -part of the work was mainly copied, and from Westminster, the abbey in -which the work was partially written. - - The _Flores historiarum_ is a Latin chronicle dealing with English - history from the creation to 1326, although some of the earlier - manuscripts end at 1306; it was compiled by various persons, and - written partly at St Albans and partly at Westminster. The part from - 1306 to 1326 was written by Robert of Reading (d. 1325) and another - Westminster monk. Except for parts dealing with the reign of Edward I. - its value is not great. It was first printed by Matthew Parker, - archbishop of Canterbury, in 1567, and the best edition is the one - edited with introduction by H. R. Luard for the Rolls series (London, - 1890). It has been translated into English by C. D. Yonge (London, - 1853). See Luard's introduction, and C. Bemont in the _Revue critique - d'histoire_ (Paris, 1891). - - - - -MATTHEWS, STANLEY (1824-1889), American jurist, was born in Cincinnati, -Ohio, on the 21st of July 1824. He graduated from Kenyon College in -1840, studied law, and in 1842 was admitted to the bar of Maury county, -Tennessee. In 1844 he became assistant prosecuting attorney of Hamilton -county, Ohio; and in 1846-1849 edited a short-lived anti-slavery paper, -the _Cincinnati Herald_. He was clerk of the Ohio House of -Representatives in 1848-1849, a judge of common pleas of Hamilton county -in 1850-1853, state senator in 1856-1858, and U.S. district-attorney for -the southern district of Ohio in 1858-1861. First a Whig and then a -Free-Soiler, he joined the Republican party in 1861. After the outbreak -of the Civil War he was commissioned a lieutenant of the 23rd Ohio, of -which Rutherford B. Hayes was major; but saw service only with the 57th -Ohio, of which he was colonel, and with a brigade which he commanded in -the Army of the Cumberland. He resigned from the army in 1863, and was -judge of the Cincinnati superior court in 1863-1864. He was a Republican -presidential elector in 1864 and 1868. In 1872 he joined the Liberal -Republican movement, and was temporary chairman of the Cincinnati -convention which nominated Horace Greeley for the presidency, but in the -campaign he supported Grant. In 1877, as counsel before the Electoral -Commission, he opened the argument for the Republican electors of -Florida and made the principal argument for the Republican electors of -Oregon. In March of the same year he succeeded John Sherman as senator -from Ohio, and served until March 1879. In 1881 President Hayes -nominated him as associate justice of the Supreme Court, to succeed Noah -H. Swayne; there was much opposition, especially in the press, to this -appointment, because Matthews had been a prominent railway and -corporation lawyer and had been one of the Republican "visiting -statesmen" who witnessed the canvass of the vote of Louisiana[1] in -1876; and the nomination had not been approved when the session of -Congress expired. Matthews was renominated by President Garfield on the -15th of March, and the nomination was confirmed by the Senate (22 for, -21 against) on the 12th of May. He was an honest, impartial and -conscientious judge. He died in Washington, on the 22nd of March 1889. - - -FOOTNOTE: - - [1] It seems certain that Matthews and Charles Foster of Ohio gave - their written promise that Hayes, if elected, would recognize the - Democratic governors in Louisiana and South Carolina. - - - - -MATTHIAE, AUGUST HEINRICH (1769-1835), German classical scholar, was -born at Gottingen, on the 25th of December 1769, and educated at the -university. He then spent some years as a tutor in Amsterdam. In 1798 he -returned to Germany, and in 1802 was appointed director of the -Friedrichsgymnasium at Altenburg, which post he held till his death, on -the 6th of January 1835. Of his numerous important works the best-known -are his _Greek Grammar_ (3rd ed., 1835), translated into English by E. -V. Blomfield (5th ed., by J. Kenrick, 1832), his edition of _Euripides_ -(9 vols., 1813-1829), _Grundriss der Geschichte der griechischen und -romischen Litteratur_ (3rd ed., 1834, Eng. trans., Oxford, 1841) -_Lehrbuch fur den ersten Unterricht in der Philosophie_ (3rd ed., 1833), -_Encyklopadie und Methodologie der Philologie_ (1835). His _Life_ was -written by his son Constantin (1845). - -His brother, FRIEDRICH CHRISTIAN MATTHIAE (1763-1822), rector of the -Frankfort gymnasium, published valuable editions of Seneca's _Letters_, -Aratus, and Dionysius Periegetes. - - - - -MATTHIAS, the disciple elected by the primitive Christian community to -fill the place in the Twelve vacated by Judas Iscariot (Acts i. 21-26). -Nothing further is recorded of him in the New Testament. Eusebius -(_Hist. Eccl._, I. xii.) says he was, like his competitor, Barsabas -Justus, one of the seventy, and the Syriac version of Eusebius calls him -throughout not Matthias but Tolmai, i.e. Bartholomew, without confusing -him with the Bartholomew who was originally one of the Twelve, and is -often identified with the Nathanael mentioned in the Fourth Gospel -(_Expository Times_, ix. 566). Clement of Alexandria says some -identified him with Zacchaeus, the Clementine _Recognitions_ identify -him with Barnabas, Hilgenfeld thinks he is the same as Nathanael. - - Various works--a Gospel, Traditions and Apocryphal Words--were - ascribed to him; and there is also extant _The Acts of Andrew and - Matthias_, which places his activity in "the city of the cannibals" in - Ethiopia. Clement of Alexandria quotes two sayings from the - Traditions: (1) Wonder at the things before you (suggesting, like - Plato, that wonder is the first step to new knowledge); (2) If an - elect man's neighbour sin, the elect man has sinned. - - - - -MATTHIAS (1557-1619), Roman emperor, son of the emperor Maximilian II. -and Maria, daughter of the emperor Charles V., was born in Vienna, on -the 24th of February 1557. Educated by the diplomatist O. G. de Busbecq, -he began his public life in 1577, soon after his father's death, when he -was invited to assume the governorship of the Netherlands, then in the -midst of the long struggle with Spain. He eagerly accepted this -invitation, although it involved a definite breach with his Spanish -kinsman, Philip II., and entering Brussels in January 1578 was named -governor-general; but he was merely a cipher, and only held the position -for about three years, returning to Germany in October 1581. Matthias -was appointed governor of Austria in 1593 by his brother, the emperor -Rudolph II.; and two years later, when another brother, the archduke -Ernest, died, he became a person of more importance as the eldest -surviving brother of the unmarried emperor. As governor of Austria -Matthias continued the policy of crushing the Protestants, although -personally he appears to have been inclined to religious tolerance; and -he dealt with the rising of the peasants in 1595, in addition to -representing Rudolph at the imperial diets, and gaining some fame as a -soldier during the Turkish War. A few years later the discontent felt by -the members of the Habsburg family at the incompetence of the emperor -became very acute, and the lead was taken by Matthias. Obtaining in May -1605 a reluctant consent from his brother, he took over the conduct of -affairs in Hungary, where a revolt had broken out, and was formally -recognized by the Habsburgs as their head in April 1606, and was -promised the succession to the Empire. In June 1606 he concluded the -peace of Vienna with the rebellious Hungarians, and was thus in a better -position to treat with the sultan, with whom peace was made in November. -This pacific policy was displeasing to Rudolph, who prepared to renew -the Turkish War; but having secured the support of the national party in -Hungary and gathered an army, Matthias forced his brother to cede to him -this kingdom, together with Austria and Moravia, both of which had -thrown in their lot with Hungary (1608). The king of Hungary, as -Matthias now became, was reluctantly compelled to grant religious -liberty to the inhabitants of Austria. The strained relations which had -arisen between Rudolph and Matthias as a result of these proceedings -were temporarily improved, and a formal reconciliation took place in -1610; but affairs in Bohemia soon destroyed this fraternal peace. In -spite of the letter of majesty (_Majestatsbrief_) which the Bohemians -had extorted from Rudolph, they were very dissatisfied with their ruler, -whose troops were ravaging their land; and in 1611 they invited Matthias -to come to their aid. Accepting this invitation, he inflicted another -humiliation upon his brother, and was crowned king of Bohemia in May -1611. Rudolph, however, was successful in preventing the election of -Matthias as German king, or king of the Romans, and when he died, in -January 1612, no provision had been made for a successor. Already king -of Hungary and Bohemia, however, Matthias obtained the remaining -hereditary dominions of the Habsburgs, and in June 1612 was crowned -emperor, although the ecclesiastical electors favoured his younger -brother, the archduke Albert (1559-1621). - -The short reign of the new emperor was troubled by the religious -dissensions of Germany. His health became impaired and his indolence -increased, and he fell completely under the influence of Melchior Klesl -(q.v.), who practically conducted the imperial business. By Klesl's -advice he took up an attitude of moderation and sought to reconcile the -contending religious parties; but the proceedings at the diet of -Regensburg in 1613 proved the hopelessness of these attempts, while -their author was regarded with general distrust. Meanwhile the younger -Habsburgs, led by the emperor's brother, the archduke Maximilian, and -his cousin, Ferdinand, archduke of Styria, afterwards the emperor -Ferdinand II., disliking the peaceful policy of Klesl, had allied -themselves with the unyielding Roman Catholics, while the question of -the imperial succession was forcing its way to the front. In 1611 -Matthias had married his cousin Anna (d. 1618), daughter of the archduke -Ferdinand (d. 1595), but he was old and childless and the Habsburgs were -anxious to retain his extensive possessions in the family. Klesl, on the -one hand, wished the settlement of the religious difficulties to precede -any arrangement about the imperial succession; the Habsburgs, on the -other, regarded the question of the succession as urgent and vital. -Meanwhile the disputed succession to the duchies of Cleves and Julich -again threatened a European war; the imperial commands were flouted in -Cologne and Aix-la-Chapelle, and the Bohemians were again becoming -troublesome. Having decided that Ferdinand should succeed Matthias as -emperor, the Habsburgs had secured his election as king of Bohemia in -June 1617, but were unable to stem the rising tide of disorder in that -country. Matthias and Klesl were in favour of concessions, but Ferdinand -and Maximilian met this move by seizing and imprisoning Klesl. Ferdinand -had just secured his coronation as king of Hungary when there broke out -in Bohemia those struggles which heralded the Thirty Years' War; and on -the 20th of March 1619 the emperor died at Vienna. - - For the life and reign of Matthias the following works may be - consulted: J. Heling, _Die Wahl des romischen Konigs Matthias_ - (Belgrade, 1892); A. Gindely, _Rudolf II. und seine Zeit_ (Prague, - 1862-1868); F. Stieve, _Die Verhandlungen uber die Nachfolge Kaisers - Rudolf II._ (Munich, 1880); P. von Chlumecky, _Karl von Zierotin und - seine Zeit_ (Brunn, 1862-1879); A. Kerschbaumer, _Kardinal Klesel_ - (Vienna, 1865); M. Ritter, _Quellenbeitrage zur Geschichte des Kaisers - Rudolf II._ (Munich, 1872); _Deutsche Geschichte im Zeitalter der - Gegenreformation und des dreissigjahrigen Krieges_ (Stuttgart, 1887, - seq.); and the article on Matthias in the _Allgemeine deutsche - Biographie_, Bd. XX. (Leipzig, 1884); L. von Ranke, _Zur deutschen - Geschichte vom Religionsfrieden bis zum 30-jahrigen Kriege_ (Leipzig, - 1888); and J. Janssen, _Geschichte des deutschen Volks seit dem - Ausgang des Mittelalters_ (Freiburg, 1878 seq.), Eng. trans. by M. A. - Mitchell and A. M. Christie (London, 1896, seq.). - - - - -MATTHIAS I., HUNYADI (1440-1490), king of Hungary, also known as -Matthias Corvinus, a surname which he received from the raven (_corvus_) -on his escutcheon, second son of Janos Hunyadi and Elizabeth Szilagyi, -was born at Kolozsvar, probably on - -the 23rd of February 1440. His tutors were the learned Janos Vitez, -bishop of Nagyvarad, whom he subsequently raised to the primacy, and the -Polish humanist Gregory Sanocki. The precocious lad quickly mastered the -German, Latin and principal Slavonic languages, frequently acting as his -father's interpreter at the reception of ambassadors. His military -training proceeded under the eye of his father, whom he began to follow -on his campaigns when only twelve years of age. In 1453 he was created -count of Bistercze, and was knighted at the siege of Belgrade in 1454. -The same care for his welfare led his father to choose him a bride in -the powerful Cilli family, but the young Elizabeth died before the -marriage was consummated, leaving Matthias a widower at the age of -fifteen. On the death of his father he was inveigled to Buda by the -enemies of his house, and, on the pretext of being concerned in a purely -imaginary conspiracy against Ladislaus V., was condemned to -decapitation, but was spared on account of his youth, and on the king's -death fell into the hands of George Podebrad, governor of Bohemia, the -friend of the Hunyadis, in whose interests it was that a national king -should sit on the Magyar throne. Podebrad treated Matthias hospitably -and affianced him with his daughter Catherine, but still detained him, -for safety's sake, in Prague, even after a Magyar deputation had -hastened thither to offer the youth the crown. Matthias was the elect of -the Hungarian people, gratefully mindful of his father's services to the -state and inimical to all foreign candidates; and though an influential -section of the magnates, headed by the palatine Laszlo Garai and the -voivode of Transylvania, Miklos Ujlaki, who had been concerned in the -judicial murder of Matthias's brother Laszlo, and hated the Hunyadis as -semi-foreign upstarts, were fiercely opposed to Matthias's election, -they were not strong enough to resist the manifest wish of the nation, -supported as it was by Matthias's uncle Mihaly Szilagyi at the head of -15,000 veterans. On the 24th of January 1458, 40,000 Hungarian noblemen, -assembled on the ice of the frozen Danube, unanimously elected Matthias -Hunyadi king of Hungary, and on the 14th of February the new king made -his state entry into Buda. - -The realm at this time was environed by perils. The Turks and the -Venetians threatened it from the south, the emperor Frederick III. from -the west, and Casimir IV. of Poland from the north, both Frederick and -Casimir claiming the throne. The Czech mercenaries under Giszkra held -the northern counties and from thence plundered those in the centre. -Meanwhile Matthias's friends had only pacified the hostile dignitaries -by engaging to marry the daughter of the palatine Garai to their -nominee, whereas Matthias not unnaturally refused to marry into the -family of one of his brother's murderers, and on the 9th of February -confirmed his previous nuptial contract with the daughter of George -Podebrad, who shortly afterwards was elected king of Bohemia (March 2, -1458). Throughout 1458 the struggle between the young king and the -magnates, reinforced by Matthias's own uncle and guardian Szilagyi, was -acute. But Matthias, who began by deposing Garai and dismissing -Szilagyi, and then proceeded to levy a tax, without the consent of the -Diet, in order to hire mercenaries, easily prevailed. Nor did these -complications prevent him from recovering the fortress of Galamboc from -the Turks, successfully invading Servia, and reasserting the suzerainty -of the Hungarian crown over Bosnia. In the following year there was a -fresh rebellion, when the emperor Frederick was actually crowned king by -the malcontents at Vienna-Neustadt (March 4, 1459); but Matthias drove -him out, and Pope Pius II. intervened so as to leave Matthias free to -engage in a projected crusade against the Turks, which subsequent -political complications, however, rendered impossible. From 1461 to 1465 -the career of Matthias was a perpetual struggle punctuated by truces. -Having come to an understanding with his father-in-law Podebrad, he was -able to turn his arms against the emperor Frederick, and in April 1462 -Frederick restored the holy crown for 60,000 ducats and was allowed to -retain certain Hungarian counties with the title of king; in return for -which concessions, extorted from Matthias by the necessity of coping -with a simultaneous rebellion of the Magyar noble in league with -Podebrad's son Victorinus, the emperor recognized Matthias as the actual -sovereign of Hungary. Only now was Matthias able to turn against the -Turks, who were again threatening the southern provinces. He began by -defeating Ali Pasha, and then penetrated into Bosnia, and captured the -newly built fortress of Jajce after a long and obstinate defence (Dec. -1463). On returning home he was crowned with the holy crown on the 29th -of March 1464, and, after driving the Czechs out of his northern -counties, turned southwards again, this time recovering all the parts of -Bosnia which still remained in Turkish hands. - -A political event of the first importance now riveted his attention upon -the north. Podebrad, who had gained the throne of Bohemia with the aid -of the Hussites and Utraquists, had long been in ill odour at Rome, and -in 1465 Pope Paul II. determined to depose the semi-Catholic monarch. -All the neighbouring princes, the emperor, Casimir IV. of Poland and -Matthias, were commanded in turn to execute the papal decree of -deposition, and Matthias gladly placed his army at the disposal of the -Holy See. The war began on the 31st of May 1468, but, as early as the -27th of February 1469, Matthias anticipated an alliance between George -and Frederick by himself concluding an armistice with the former. On the -3rd of May the Czech Catholics elected Matthias king of Bohemia, but -this was contrary to the wishes of both pope and emperor, who preferred -to partition Bohemia. But now George discomfited all his enemies by -suddenly excluding his own son from the throne in favour of Ladislaus, -the eldest son of Casimir IV., thus skilfully enlisting Poland on his -side. The sudden death of Podebrad on the 22nd of March 1471 led to -fresh complications. At the very moment when Matthias was about to -profit by the disappearance of his most capable rival, another dangerous -rebellion, headed by the primate and the chief dignitaries of the state, -with the object of placing Casimir, son of Casimir IV., on the throne, -paralysed Matthias's foreign policy during the critical years 1470-1471. -He suppressed this domestic rebellion indeed, but in the meantime the -Poles had invaded the Bohemian domains with 60,000 men, and when in 1474 -Matthias was at last able to take the field against them in order to -raise the siege of Breslau, he was obliged to fortify himself in an -entrenched camp, whence he so skilfully harried the enemy that the -Poles, impatient to return to their own country, made peace at Breslau -(Feb. 1475) on an _uti possidetis_ basis, a peace subsequently confirmed -by the congress of Olmutz (July 1479). During the interval between these -peaces, Matthias, in self-defence, again made war on the emperor, -reducing Frederick to such extremities that he was glad to accept peace -on any terms. By the final arrangement made between the contending -princes, Matthias recognized Ladislaus as king of Bohemia proper in -return for the surrender of Moravia, Silesia and Upper and Lower -Lusatia, hitherto component parts of the Czech monarchy, till he should -have redeemed them for 400,000 florins. The emperor promised to pay -Matthias 100,000 florins as a war indemnity, and recognized him as the -legitimate king of Hungary on the understanding that he should succeed -him if he died without male issue, a contingency at this time somewhat -improbable, as Matthias, only three years previously (Dec. 15, 1476), -had married his third wife, Beatrice of Naples, daughter of Ferdinand of -Aragon. - -The endless tergiversations and depredations of the emperor speedily -induced Matthias to declare war against him for the third time (1481), -the Magyar king conquering all the fortresses in Frederick's hereditary -domains. Finally, on the 1st of June 1485, at the head of 8000 veterans, -he made his triumphal entry into Vienna, which he henceforth made his -capital. Styria, Carinthia and Carniola were next subdued, and Trieste -was only saved by the intervention of the Venetians. Matthias -consolidated his position by alliances with the dukes of Saxony and -Bavaria, with the Swiss Confederation, and the archbishop of Salzburg, -and was henceforth the greatest potentate in central Europe. His -far-reaching hand even extended to Italy. Thus, in 1480, when a Turkish -fleet seized Otranto, Matthias, at the earnest solicitation of the pope, -sent Balasz Magyar to recover the fortress, which surrendered to him on -the 10th of May 1481. Again in 1488, Matthias took Ancona under his -protection for a time and occupied it with a Hungarian garrison. - -Though Matthias's policy was so predominantly occidental that he soon -abandoned his youthful idea of driving the Turks out of Europe, he at -least succeeded in making them respect Hungarian territory. Thus in 1479 -a huge Turkish army, on its return home from ravaging Transylvania, was -annihilated at Szaszvaros (Oct. 13), and in 1480 Matthias recaptured -Jajce, drove the Turks from Servia and erected two new military banates, -Jajce and Srebernik, out of reconquered Bosnian territory. On the death -of Mahommed II. in 1481, a unique opportunity for the intervention of -Europe in Turkish affairs presented itself. A civil war ensued in Turkey -between his sons Bayezid and Jem, and the latter, being worsted, fled to -the knights of Rhodes, by whom he was kept in custody in France (see -BAYEZID II.). Matthias, as the next-door neighbour of the Turks, claimed -the custody of so valuable a hostage, and would have used him as a means -of extorting concessions from Bayezid. But neither the pope nor the -Venetians would hear of such a transfer, and the negotiations on this -subject greatly embittered Matthias against the Curia. The last days of -Matthias were occupied in endeavouring to secure the succession to the -throne for his illegitimate son Janos (see CORVINUS, JANOS); but Queen -Beatrice, though childless, fiercely and openly opposed the idea and the -matter was still pending when Matthias, who had long been crippled by -gout, expired very suddenly on Palm Sunday, the 4th of April 1490. - -Matthias Hunyadi was indisputably the greatest man of his day, and one -of the greatest monarchs who ever reigned. The precocity and -universality of his genius impress one the most. Like Napoleon, with -whom he has often been compared, he was equally illustrious as a -soldier, a statesman, an orator, a legislator and an administrator. But -in all moral qualities the brilliant adventurer of the 15th was -infinitely superior to the brilliant adventurer of the 19th century. -Though naturally passionate, Matthias's self-control was almost -superhuman, and throughout his stormy life, with his innumerable -experiences of ingratitude and treachery, he never was guilty of a -single cruel or vindictive action. His capacity for work was -inexhaustible. Frequently half his nights were spent in reading, after -the labour of his most strenuous days. There was no branch of knowledge -in which he did not take an absorbing interest, no polite art which he -did not cultivate and encourage. His camp was a school of chivalry, his -court a nursery of poets and artists. Matthias was a middle-sized, -broad-shouldered man of martial bearing, with a large fleshy nose, hair -reaching to his heels, and the clean-shaven, heavy chinned face of an -early Roman emperor. - - See Vilmos Fraknoi, _King Matthias Hunyadi_ (Hung., Budapest, 1890, - German ed., Freiburg, 1891); Ignacz Acsady, _History of the Hungarian - Realm_ (Hung. vol. i., Budapest, 1904); Jozsef Teleki, _The Age of the - Hunyadis in Hungary_ (Hung., vols. 3-5, Budapest, 1852-1890); V. - Fraknoi, _Life of Janos Vitez_ (Hung. Budapest 1879); Karl Schober, - _Die Eroberung Niederosterreichs durch Matthias Corvinus_ (Vienna, - 1879); Janos Huszar, _Matthias's Black Army_ (Hung. Budapest, 1890); - Antonio Bonfini, _Rerum hungaricarum decades_ (7th ed., Leipzig, - 1771); Aeneas Sylvius, _Opera_ (Frankfort, 1707); _The Correspondence - of King Matthias_ (Hung. and Lat., Budapest, 1893); V. Fraknoi, _The - Embassies of Cardinal Carvajal to Hungary_ (Hung., Budapest, 1889); - Marzio Galeotti, _De egregie sapienter et jocose, dictis ac factis - Matthiae regis (Script. reg. hung. I.)_ (Vienna, 1746). Of the above - the first is the best general sketch and is rich in notes; the second - somewhat chauvinistic but excellently written; the third the best work - for scholars; the seventh, eighth and eleventh are valuable as being - by contemporaries. (R. N. B.) - - - - -MATTHISSON, FRIEDRICH VON (1761-1831), German poet, was born at -Hohendodeleben near Magdeburg, the son of the village pastor, on the -23rd of January 1761. After studying theology and philology at the -university of Halle, he was appointed in 1781 master at the classical -school Philanthropin in Dessau. This once famous seminary was, however, -then rapidly decaying in public favour, and in 1784 Matthisson was glad -to accept a travelling tutorship. He lived for two years with the Swiss -author Bonstetten at Nyon on the lake of Geneva. In 1794 he was -appointed reader and travelling companion to the princess Louisa of -Anhalt-Dessau. In 1812 he entered the service of the king of -Wurttemberg, was ennobled, created counsellor of legation, appointed -intendant of the court theatre and chief librarian of the royal library -at Stuttgart. In 1828 he retired and settled at Worlitz near Dessau, -where he died on the 12th of March 1831. Matthisson enjoyed for a time a -great popularity on account of his poems, _Gedichte_ (1787; 15th ed., -1851; new ed., 1876), which Schiller extravagantly praised for their -melancholy sweetness and their fine descriptions of scenery. The verse -is melodious and the language musical, but the thought and sentiments -they express are too often artificial and insincere. His _Adelaide_ has -been rendered famous owing to Beethoven's setting of the song. Of his -elegies, _Die Elegie in den Ruinen eines alten Bergschlosses_ is still a -favourite. His reminiscences, _Erinnerungen_ (5 vols., 1810-1816), -contain interesting accounts of his travels. - - Matthisson's _Schriften_ appeared in eight volumes (1825-1829), of - which the first contains his poems, the remainder his _Erinnerungen_; - a ninth volume was added in 1833 containing his biography by H. - Doring. His _Literarischer Nachlass_, with a selection from his - correspondence, was published in four volumes by F. R. Schoch in 1832. - - - - -MATTING, a general term embracing many coarse woven or plaited fibrous -materials used for covering floors or furniture, for hanging as screens, -for wrapping up heavy merchandise and for other miscellaneous purposes. -In the United Kingdom, under the name of "coir" matting, a large amount -of a coarse kind of carpet is made from coco-nut fibre; and the same -material, as well as strips of cane, Manila hemp, various grasses and -rushes, is largely employed in various forms for making door mats. Large -quantities of the coco-nut fibre are woven in heavy looms, then cut up -into various sizes, and finally bound round the edges by a kind of rope -made from the same material. The mats may be of one colour only, or they -may be made of different colours and in different designs. Sometimes the -names of institutions are introduced into the mats. Another type of mat -is made exclusively from the above-mentioned rope by arranging alternate -layers in sinuous and straight paths, and then stitching the parts -together. It is also largely used for the outer covering of ships' -fenders. Perforated and otherwise prepared rubber, as well as wire-woven -material, are also largely utilized for door and floor mats. Matting of -various kinds is very extensively employed throughout India for floor -coverings, the bottoms of bedsteads, fans and fly-flaps, &c.; and a -considerable export trade in such manufactures is carried on. The -materials used are numerous; but the principal substances are straw, the -bulrushes _Typha elephantina_ and _T. angustifolia_, leaves of the date -palm (_Phoenix sylvestris_), of the dwarf palm (_Chamaerops Ritchiana_), -of the Palmyra palm (_Borassus flabelliformis_), of the coco-nut palm -(_Cocos nucifera_) and of the screw pine (_Pandanus odoratissimus_), the -munja or munj grass (_Saccharum Munja_) and allied grasses, and the mat -grasses _Cyperus textilis_ and _C. Pangorei_, from the last of which the -well-known Palghat mats of the Madras Presidency are made. Many of these -Indian grass-mats are admirable examples of elegant design, and the -colours in which they are woven are rich, harmonious and effective in -the highest degree. Several useful household articles are made from the -different kinds of grasses. The grasses are dyed in all shades and -plaited to form attractive designs suitable for the purposes to which -they are to be applied. This class of work obtains in India, Japan and -other Eastern countries. Vast quantities of coarse matting used for -packing furniture, heavy and coarse goods, flax and other plants, &c., -are made in Russia from the bast or inner bark of the lime tree. This -industry centres in the great forest governments of Viatka, -Nizhniy-Novgorod, Kostroma, Kazan, Perm and Simbirsk. - - - - -MATTOCK (O.E. _mattuc_, of uncertain origin), a tool having a double -iron head, of which one end is shaped like an adze, and the other like a -pickaxe. The head has a socket in the centre in which the handle is -inserted transversely to the blades. It is used chiefly for grubbing and -rooting among tree stumps in plantations and copses, where the roots are -too close for the use of a spade, or for loosening hard soil. - - - - -MATTO GROSSO, an inland state of Brazil, bounded N. by Amazonas and -Para, E. by Goyaz, Minas Geraes, Sao Paulo and Parana, S. by Paraguay -and S.W. and W. by Bolivia. It ranks next to Amazonas in size, its area, -which is largely unsettled and unexplored, being 532,370 sq. m., and its -population only 92,827 in 1890 and 118,025 in 1900. No satisfactory -estimate of its Indian population can be made. The greater part of the -state belongs to the western extension of the Brazilian plateau, across -which, between the 14th and 16th parallels, runs the watershed which -separates the drainage basins of the Amazon and La Plata. This elevated -region is known as the plateau of Matto Grosso, and its elevations so -far as known rarely exceed 3000 ft. The northern slope of this great -plateau is drained by the Araguaya-Tocantins, Xingu, Tapajos and -Guapore-Mamore-Madeira, which flow northward, and, except the first, -empty into the Amazon; the southern slope drains southward through a -multitude of streams flowing into the Parana and Paraguay. The general -elevation in the south part of the state is much lower, and large areas -bordering the Paraguay are swampy, partially submerged plains which the -sluggish rivers are unable to drain. The lowland elevations in this part -of the state range from 300 to 400 ft. above sea-level, the climate is -hot, humid and unhealthy, and the conditions for permanent settlement -are apparently unfavourable. On the highlands, however, which contain -extensive open _campos_, the climate, though dry and hot, is considered -healthy. The basins of the Parana and Paraguay are separated by low -mountain ranges extending north from the _sierras_ of Paraguay. In the -north, however, the ranges which separate the river valleys are -apparently the remains of the table-land through which deep valleys have -been eroded. The resources of Matto Grosso are practically undeveloped, -owing to the isolated situation of the state, the costs of -transportation and the small population. - -The first industry was that of mining, gold having been discovered in -the river valleys on the southern slopes of the plateau, and diamonds on -the head-waters of the Paraguay, about Diamantino and in two or three -other districts. Gold is found chiefly in placers, and in colonial times -the output was large, but the deposits were long ago exhausted and the -industry is now comparatively unimportant. As to other minerals little -is definitely known. Agriculture exists only for the supply of local -needs, though tobacco of a superior quality is grown. Cattle-raising, -however, has received some attention and is the principal industry of -the landowners. The forest products of the state include fine woods, -rubber, ipecacuanha, sarsaparilla, jaborandi, vanilla and copaiba. There -is little export, however, the only means of communication being down -the Paraguay and Parana rivers by means of subsidized steamers. The -capital of the state is Cuyaba, and the chief commercial town is Corumba -at the head of navigation for the larger river boats, and 1986 m. from -the mouth of the La Plata. Communication between these two towns is -maintained by a line of smaller boats, the distance being 517 m. - -The first permanent settlements in Matto Grosso seem to have been made -in 1718 and 1719, in the first year at Forquilha and in the second at or -near the site of Cuyaba, where rich placer mines had been found. At this -time all this inland region was considered a part of Sao Paulo, but in -1748 it was made a separate _capitania_ and was named Matto Grosso -("great woods"). In 1752 its capital was situated on the right bank of -the Guapore river and was named Villa Bella da Santissima Trindade de -Matto Grosso, but in 1820 the seat of government was removed to Cuyaba -and Villa Bella has fallen into decay. In 1822 Matto Grosso became a -province of the empire and in 1889 a republican state. It was invaded by -the Paraguayans in the war of 1860-65. - - - - -MATTOON, a city of Coles county, Illinois, U.S.A., in the east central -part of the state, about 12 m. south-east of Peoria. Pop. (1890), 6833; -(1900), 9622, of whom 430 were foreign-born; (1910 census) 11,456. It is -served by the Illinois Central and Cleveland, Cincinnati, Chicago & St -Louis railways, which have repair shops here, and by inter-urban -electric lines. The city has a public library, a Methodist Episcopal -Hospital, and an Old Folks' Home, the last supported by the Independent -Order of Odd Fellows. Mattoon is an important shipping point for Indian -corn and broom corn, extensively grown in the vicinity, and for fruit -and livestock. Among its manufactures are foundry and machine shop -products, stoves and bricks; in 1905 the factory product was valued at -$1,308,781, an increase of 71.2% over that in 1900. The municipality -owns the waterworks and an electric lighting plant. Mattoon was first -settled about 1855, was named in honour of William Mattoon, an early -landowner, was first chartered as a city in 1857, and was reorganized -under a general state law in 1879. - - - - -MATTRESS (O.Fr. _materas_, mod. _matelas_; the origin is the Arab. -_al-materah_, cushion, whence Span. and Port. _almadraque_, Ital. -_materasso_), the padded foundation of a bed, formed of canvas or other -stout material stuffed with wool, hair, flock or straw; in the last case -it is properly known as a "palliasse" (Fr. _paille_, straw; Lat. -_palea_); but this term is often applied to an under-mattress stuffed -with substances other than straw. The padded mattress on which lay the -feather-bed has been replaced by the "wire-mattress," a network of wire -stretched on a light wooden or iron frame, which is either a separate -structure or a component part of the bedstead itself. The -"wire-mattress" has taken the place of the "spring mattress," in which -spiral springs support the stuffing. The term "mattress" is used in -engineering for a mat of brushwood, faggots, &c., corded together and -used as a foundation or as surface in the construction of dams, jetties, -dikes, &c. - - - - -MATURIN, CHARLES ROBERT (1782-1824), Irish novelist and dramatist, was -born in Dublin in 1782. His grandfather, Gabriel Jasper Maturin, had -been Swift's successor in the deanery of St Patrick. Charles Maturin was -educated at Trinity College, Dublin, and became curate of Loughrea and -then of St Peter's, Dublin. His first novels, _The Fatal Revenge; or, -the Family of Montorio_ (1807), _The Wild Irish Boy_ (1808), _The -Milesian Chief_ (1812), were issued under the pseudonym of "Dennis -Jasper Murphy." All these were mercilessly ridiculed, but the irregular -power displayed in them attracted the notice of Sir Walter Scott, who -recommended the author to Byron. Through their influence Maturin's -tragedy of _Bertram_ was produced at Drury Lane in 1816, with Kean and -Miss Kelly in the leading parts. A French version by Charles Nodier and -Baron Taylor was produced in Paris at the Theatre Favart. Two more -tragedies, _Manuel_ (1817) and _Fredolfo_ (1819), were failures, and his -poem _The Universe_ (1821) fell flat. He wrote three more novels, -_Women_ (1818), _Melmoth, the Wanderer_ (1820), and _The Albigenses_ -(1824). _Melmoth_, which forms its author's title to remembrance, is the -best of them, and has for hero a kind of "Wandering Jew." Honore de -Balzac wrote a sequel to it under the title of _Melmoth reconcilie a -l'eglise_ (1835). Maturin died in Dublin on the 30th of October 1824. - - - - -MATVYEEV, ARTAMON SERGYEEVICH ( -1682), Russian statesman and reformer, -was one of the greatest of the precursors of Peter the Great. His -parentage and the date of his birth are uncertain. Apparently his birth -was humble, but when the obscure figure of the young Artamon emerges -into the light of history we find him equipped at all points with the -newest ideas, absolutely free from the worst prejudices of his age, a -ripe scholar, and even an author of some distinction. In 1671 the tsar -Alexius and Artamon were already on intimate terms, and on the -retirement of Orduin-Nashchokin Matvyeev became the tsar's chief -counsellor. It was at his house, full of all the wondrous, -half-forbidden novelties of the west, that Alexius, after the death of -his first consort, Martha, met Matvyeev's favourite pupil, the beautiful -Natalia Naruishkina, whom he married on the 21st of January 1672. At the -end of the year Matvyeev was raised to the rank of _okolnichy_, and on -the 1st of September 1674 attained the still higher dignity of _boyar_. -Matvyeev remained paramount to the end of the reign and introduced -play-acting and all sorts of refining western novelties into Muscovy. -The deplorable physical condition of Alexius's immediate successor, -Theodore III. suggested to Matvyeev the desirability of elevating to -the throne the sturdy little tsarevich Peter, then in his fourth year. -He purchased the allegiance of the _stryeltsi_, or musketeers, and then, -summoning the boyars of the council, earnestly represented to them that -Theodore, scarce able to live, was surely unable to reign, and urged the -substitution of little Peter. But the reactionary boyars, among whom -were the near kinsmen of Theodore, proclaimed him tsar and Matvyeev was -banished to Pustozersk, in northern Russia, where he remained till -Theodore's death (April 27, 1682). Immediately afterwards Peter was -proclaimed tsar by the patriarch, and the first _ukaz_ issued in Peter's -name summoned Matvyeev to return to the capital and act as chief adviser -to the tsaritsa Natalia. He reached Moscow on the 15th of May, prepared -"to lay down his life for the tsar," and at once proceeded to the head -of the Red Staircase to meet and argue with the assembled stryeltsi, who -had been instigated to rebel by the anti-Petrine faction. He had already -succeeded in partially pacifying them, when one of their colonels began -to abuse the still hesitating and suspicious musketeers. Infuriated, -they seized and flung Matvyeev into the square below, where he was -hacked to pieces by their comrades. - - See R. Nisbet Bain, _The First Romanovs_ (London, 1905); M. P. - Pogodin, _The First Seventeen Years of the Life of Peter the Great_ - (Rus.), (Moscow, 1875); S. M. Solovev, _History of Russia_ (Rus.), - (vols. 12, 13, (St Petersburg, 1895, &c.); L. Shehepotev, _A. S. - Matvyeev as an Educational and Political Reformer_ (Rus.), (St - Petersburg, 1906). (R. N. B.) - - - - -MAUBEUGE, a town of northern France, in the department of Nord, situated -on both banks of the Sambre, here canalized, 23(1/2) m. by rail E. by S. -of Valenciennes, and about 2 m. from the Belgian frontier. Pop. (1906), -town 13,569, commune 21,520. As a fortress Maubeuge has an old enceinte -of bastion trace which serves as the centre of an important entrenched -camp of 18 m. perimeter, constructed for the most part after the war of -1870, but since modernized and augmented. The town has a board of trade -arbitration, a communal college, a commercial and industrial school; and -there are important foundries, forges and blast-furnaces, together with -manufactures of machine-tools, porcelain, &c. It is united by electric -tramway with Hautmont (pop. 12,473), also an important metallurgical -centre. - -Maubeuge (_Malbodium_) owes its origin to a double monastery, for men -and women, founded in the 7th century by St Aldegonde relics of whom are -preserved in the church. It subsequently belonged to the territory of -Hainault. It was burnt by Louis XI., by Francis I., and by Henry II., -and was finally assigned to France by the Treaty of Nijmwegen. It was -fortified at Vauban by the command of Louis XIV., who under Turenne -first saw military service there. Besieged in 1793 by Prince Josias of -Coburg, it was relieved by the victory of Wattignies, which is -commemorated by a monument in the town. It was unsuccessfully besieged -in 1814, but was compelled to capitulate, after a vigorous resistance, -in the Hundred Days. - - - - -MAUCH CHUNK, a borough and the county-seat of Carbon county, -Pennsylvania, U.S.A., on the W. bank of the Lehigh river and on the -Lehigh Coal and Navigation Company's Canal, 46 m. by rail W.N.W. of -Easton. Pop. (1800), 4101; (1900), 4029 (571 foreign-born); (1910), -3952. Mauch Chunk is served by the Central of New Jersey railway and, at -East Mauch Chunk, across the river, connected by electric railway, by -the Lehigh Valley railway. The borough lies in the valley of the Lehigh -river, along which runs one of its few streets and in another deeply cut -valley at right angles to the river; through this second valley east and -west runs the main street, on which is an electric railway; parallel to -it on the south is High Street, formerly an Irish settlement; half way -up the steep hill, and on the north at the top of the opposite hill is -the ward of Upper Mauch Chunk, reached by the electric railway. An -incline railway, originally used to transport coal from the mines to the -river and named the "Switch-Back," now carries tourists up the steep -slopes of Mount Pisgah and Mount Jefferson, to Summit Hill, a rich -anthracite coal region, with a famous "burning mine," which has been on -fire since 1832, and then back. An electric railway to the top of -Flagstaff Mountain, built in 1900, was completed in 1901 to Lehighton, 4 -m. south-east of Mauch Chunk, where coal is mined and silk and stoves -are manufactured, and which had a population in 1900 of 4629, and in -1910 of 5316. Immediately above Mauch Chunk the river forms a horseshoe; -on the opposite side, connected by a bridge, is the borough of East -Mauch Chunk (pop. 1900, 3458; 1910, 3548); and 2 m. up the river is Glen -Onoko, with fine falls and cascades. The principal buildings in Mauch -Chunk are the county court house, a county gaol, a Young Men's Christian -Association building, and the Dimmick Memorial Library (1890). The -borough was long a famous shipping point for coal. It now has ironworks -and foundries, and in East Mauch Chunk there are silk mills. The name is -Indian and means "Bear Mountain," this English name being used for a -mountain on the east side of the river. The borough was founded by the -Lehigh Coal and Navigation Company in 1818. This company began in 1827 -the operation of the "Switch-Back," probably the first railway in the -country to be used for transporting coal. In 1831 the town was opened to -individual enterprise, and in 1850 it was incorporated as a borough. -Mauch Chunk was for many years the home of Asa Packer, the projector and -builder of the Lehigh Valley railroad from Mauch Chunk to Easton. - - - - -MAUCHLINE, a town in the division of Kyle, Ayrshire, Scotland. Pop. -(1901), 1767. It lies 8 m. E.S.E. of Kilmarnock and 11 m. E. by N. of -Ayr by the Glasgow and South-Western railway. It is situated on a gentle -slope about 1 m. from the river Ayr, which flows through the south of -the parish of Mauchline. It is noted for its manufacture of snuff-boxes -and knick-knacks in wood, and of curling-stones. There is also some -cabinet-making, besides spinning and weaving, and its horse fairs and -cattle markets have more than local celebrity. The parish church, dating -from 1829, stands in the middle of the village, and on the green a -monument, erected in 1830, marks the spot where five Covenanters were -killed in 1685. Robert Burns lived with his brother Gilbert on the farm -of Mossgiel, about a mile to the north, from 1784 to 1788. Mauchline -kirkyard was the scene of the "Holy Fair"; at "Poosie Nansie's" (Agnes -Gibson's)--still, though much altered, a popular inn--the "Jolly -Beggars" held their high jinks; near the church (in the poet's day an -old, barn-like structure) was the Whiteford Arms inn, where on a pane of -glass Burns wrote the epitaph on John Dove, the landlord; "auld Nanse -Tinnock's" house, with the date of 1744 above the door, nearly faces the -entrance to the churchyard; the Rev. William Auld was minister of -Mauchline, and "Holy Willie," whom the poet scourged in the celebrated -"Prayer," was one of "Daddy Auld's" elders; behind the kirkyard stands -the house of Gavin Hamilton, the lawyer and firm friend of Burns, in -which the poet was married. The braes of Ballochmyle, where he met the -heroine of his song, "The Lass o' Ballochmyle," lie about a mile to the -south-east. Adjoining them is the considerable manufacturing town of -CATRINE (pop. 2340), with cotton factories, bleach fields and brewery, -where Dr Matthew Stewart (1717-1785), the father of Dugald Stewart--had -a mansion, and where there is a big water-wheel said to be inferior in -size only to that of Laxey in the Isle of Man. Barskimming House, 2 m. -south by west of Mauchline, the seat of Lord-President Miller -(1717-1789), was burned down in 1882. Near the confluence of the Fail -and the Ayr was the scene of Burns's parting with Highland Mary. - - - - -MAUDE, CYRIL (1862- ), English actor, was born in London and educated -at Charterhouse. He began his career as an actor in 1883 in America, and -from 1896 to 1905 was co-manager with F. Harrison of the Haymarket -Theatre, London. There he became distinguished for his quietly humorous -acting in many parts. In 1906 he went into management on his own -account, and in 1907 opened his new theatre The Playhouse. In 1888 he -married the actress Winifred Emery (b. 1862), who had made her London -debut as a child in 1875, and acted with Irving at the Lyceum between -1881 and 1887. She was a daughter of Samuel Anderson Emery (1817-1881) -and granddaughter of John Emery (1777-1822), both well-known actors in -their day. - - - - -MAULE, a coast province of central Chile, bounded N. by Talea, E. by -Linares and Nuble, and S. by Concepcion, and lying between the rivers -Maule and Itata, which form its northern and southern boundaries. Pop. -(1895), 119,791; area, 2475 sq. m. Maule is traversed from north to -south by the coast range and its surfaces are much broken. The -Buchupureo river flows westward across the province. The climate is mild -and healthy. Agriculture and stock-raising are the principal -occupations, and hides, cattle, wheat and timber are exported. Transport -facilities are afforded by the Maule and the Itata, which are navigable, -and by a branch of the government railway from Cauquenes to Parral, an -important town of southern Linares. The provincial capital, Cauquenes -(pop., in 1895, 8574; 1902 estimate, 9895), is centrally situated on the -Buchupureo river, on the eastern slopes of the coast cordilleras. The -town and port of Constitucion (pop., in 1900, about 7000) on the south -bank of the Maule, one mile above its mouth, was formerly the capital of -the province. The port suffers from a dangerous bar at the mouth of the -river, but is connected with Talca by rail and has a considerable trade. - -The Maule river, from which the province takes its name, is of historic -interest because it is said to have marked the southern limits of the -Inca Empire. It rises in the Laguna del Maule, an Andean lake near the -Argentine frontier, 7218 ft. above sea-level, and flows westward about -140 m. to the Pacific, into which it discharges in 35 deg. 18' S. The -upper part of its drainage basin, to which the _Anuario Hydrografico_ -gives an area of 8000 sq. m., contains the volcanoes of San Pedro -(11,800 ft.), the Descabezado (12,795 ft.), and others of the same group -of lower elevations. The upper course and tributaries of the Maule, -principally in the province of Linares, are largely used for irrigation. - - - - -MAULEON, SAVARI DE (d. 1236), French soldier, was the son of Raoul de -Mauleon, vicomte de Thouars and lord of Mauleon (now Chatillon-sur-Sevre). -Having espoused the cause of Arthur of Brittany, he was captured at -Mirebeau (1202), and imprisoned in the chateau of Corfe. But John set him -at liberty in 1204, gained him to his side and named him seneschal of -Poitou (1205). In 1211 Savari de Mauleon assisted Raymond VI. count of -Toulouse, and with him besieged Simon de Montfort in Castelnaudary. Philip -Augustus bought his services in 1212 and gave him command of a fleet which -was destroyed in the Flemish port of Damme. Then Mauleon returned to John, -whom he aided in his struggle with the barons in 1215. He was one of those -whom John designated on his deathbed for a council of regency (1216). Then -he went to Egypt (1219), and was present at the taking of Damietta. -Returning to Poitou he was a second time seneschal for the king of -England. He defended Saintonge against Louis VIII. in 1224, but was -accused of having given La Rochelle up to the king of France, and the -suspicions of the English again threw him back upon the French. Louis -VIII. then turned over to him the defence of La Rochelle and the coast of -Saintonge. In 1227 he took part in the rising of the barons of Poitiers -and Anjou against the young Louis IX. He enjoyed a certain reputation for -his poems in the _langue d'oc_. - - See Chilhaud-Dumaine, "Savari de Mauleon," in _Positions des Theses - des eleves de l'Ecole des Chartes_ (1877); _Histoire litteraire de la - France_, xviii. 671-682. - - - - -MAULSTICK, or MAHLSTICK, a stick with a soft leather or padded head, -used by painters to support the hand that holds the brush. The word is -an adaptation of the Dutch _maalstok_, i.e. the painter's stick, from -_malen_, to paint. - - - - -MAUNDY THURSDAY (through O.Fr. _mande_ from Lat. _mandatum_, -commandment, in allusion to Christ's words: "A new commandment give I -unto you," after he had washed the disciples' feet at the Last Supper), -the Thursday before Easter. Maundy Thursday is sometimes known as -_Sheer_ or _Chare_ Thursday, either in allusion, it is thought, to the -"shearing" of heads and beards in preparation for Easter, or more -probably in the word's Middle English sense of "pure," in allusion to -the ablutions of the day. The chief ceremony, as kept from the early -middle ages onwards--the washing of the feet of twelve or more poor men -or beggars--was in the early Church almost unknown. Of Chrysostom and St -Augustine, who both speak of Maundy Thursday as being marked by a -solemn celebration of the Sacrament, the former does not mention the -foot-washing, and the latter merely alludes to it. Perhaps an indication -of it may be discerned as early as the 4th century in a custom, current -in Spain, northern Italy and elsewhere, of washing the feet of the -catechumens towards the end of Lent before their baptism. It was not, -however, universal, and in the 48th canon of the synod of Elvira (A.D. -306) it is expressly prohibited (cf. _Corp. Jur. Can._, c. 104, _caus._ -i. _qu._ 1). From the 4th century ceremonial foot-washing became yearly -more common, till it was regarded as a necessary rite, to be performed -by the pope, all Catholic sovereigns, prelates, priests and nobles. In -England the king washed the feet of as many poor men as he was years -old, and then distributed to them meat, money and clothes. At Durham -Cathedral, until the 16th century, every charity-boy had a monk to wash -his feet. At Peterborough Abbey, in 1530, Wolsey made "his maund in Our -Lady's Chapel, having fifty-nine poor men whose feet he washed and -kissed; and after he had wiped them he gave every of the said poor men -twelve pence in money, three ells of good canvas to make them shirts, a -pair of new shoes, a cast of red herrings and three white herrings." -Queen Elizabeth performed the ceremony, the paupers' feet, however, -being first washed by the yeomen of the laundry with warm water and -sweet herbs. James II. was the last English monarch to perform the rite. -William III. delegated the washing to his almoner, and this was usual -until the middle of the 18th century. Since 1754 the foot-washing has -been abandoned, and the ceremony now consists of the presentation of -Maundy money, officially called Maundy Pennies. These were first coined -in the reign of Charles II. They come straight from the Mint, and have -their edges unmilled. The service which formerly took place in the -Chapel Royal, Whitehall, is now held in Westminster Abbey. A procession -is formed in the nave, consisting of the lord high almoner representing -the sovereign, the clergy and the yeomen of the guard, the latter -carrying white and red purses in baskets. The clothes formerly given are -now commuted for in cash. The full ritual is gone through by the Roman -Catholic archbishop of Westminster, and abroad it survives in all -Catholic countries, a notable example being that of the Austrian -emperor. In the Greek Church the rite survives notably at Moscow, St -Petersburg and Constantinople. It is on Maundy Thursday that in the -Church of Rome the sacred oil is blessed, and the chrism prepared -according to an elaborate ritual which is given in the _Pontificale_. - - - - -MAUPASSANT, HENRI RENE ALBERT GUY DE (1850-1893), French novelist and -poet, was born at the Chateau of Miromesnil in the department of -Seine-Inferieure on the 5th August 1850. His grandfather, a landed -proprietor of a good Lorraine family, owned an estate at -Neuville-Champ-d'Oisel near Rouen, and bequeathed a moderate fortune to -his son, a Paris stockbroker, who married Mademoiselle Laure Lepoitevin. -Maupassant was educated at Yvetot and at the Rouen lycee. A copy of -verses entitled _Le Dieu createur_, written during his year of -philosophy, has been preserved and printed. He entered the ministry of -marine, and was promoted by M. Bardoux to the Cabinet de l'Instruction -publique. A pleasant legend says that, in a report by his official -chief, Maupassant is mentioned as not reaching the standard of the -department in the matter of style. He may very well have been an -unsatisfactory clerk, as he divided his time between rowing expeditions -and attending the literary gatherings at the house of Gustave Flaubert, -who was not, as he is often alleged to be, connected with Maupassant by -any blood tie. Flaubert was not his uncle, nor his cousin, nor even his -godfather, but merely an old friend of Madame de Maupassant, whom he had -known from childhood. At the literary meetings Maupassant seldom shared -in the conversation. Upon those who met him--Tourgenieff, Alphonse -Daudet, Catulle Mendes, Jose-Maria de Heredia and Emile Zola--he left -the impression of a simple young athlete. Even Flaubert, to whom -Maupassant submitted some sketches, was not greatly struck by their -talent, though he encouraged the youth to persevere. Maupassant's first -essay was a dramatic piece twice given at Etretat in 1873 before an -audience which included Tourgenieff, Flaubert and Meilhac. In this -indecorous performance, of which nothing more is heard, Maupassant -played the part of a woman. During the next seven years he served a -severe apprenticeship to Flaubert, who by this time realized his pupil's -exceptional gifts. In 1880 Maupassant published a volume of poems, _Des -Vers_, against which the public prosecutor of Etampes took proceedings -that were finally withdrawn through the influence of the senator -Cordier. From Flaubert, who had himself been prosecuted for his first -book, _Madame Bovary_, there came a letter congratulating the poet on -the similarity between their first literary experiences. _Des Vers_ is -an extremely interesting experiment, which shows Maupassant to us still -hesitating in his choice of a medium; but he recognized that it was not -wholly satisfactory, and that its chief deficiency--the absence of -verbal melody--was fatal. Later in the same year he contributed to the -_Soirees de Medan_, a collection of short stories by MM. Zola, J.-K. -Huysmans, Henry Ceard, Leon Hennique and Paul Alexis; and in _Boule de -suif_ the young unknown author revealed himself to his amazed -collaborators and to the public as an admirable writer of prose and a -consummate master of the _conte_. There is perhaps no other instance in -modern literary history of a writer beginning, as a fully equipped -artist, with a genuine masterpiece. This early success was quickly -followed by another. The volume entitled _La Maison Tellier_ (1881) -confirmed the first impression, and vanquished even those who were -repelled by the author's choice of subjects. In _Mademoiselle Fifi_ -(1883) he repeated his previous triumphs as a _conteur_, and in this -same year he, for the first time, attempted to write on a larger scale. -Choosing to portray the life of a blameless girl, unfortunate in her -marriage, unfortunate in her son, consistently unfortunate in every -circumstance of existence, he leaves her, ruined and prematurely old, -clinging to the tragic hope, which time, as one feels, will belie, that -she may find happiness in her grandson. This picture of an average woman -undergoing the constant agony of disillusion Maupassant calls _Une Vie_ -(1883), and as in modern literature there is no finer example of cruel -observation, so there is no sadder book than this, while the effect of -extreme truthfulness which it conveys justifies its sub-title--_L'Humble -verite_. Certain passages of _Une Vie_ are of such a character that the -sale of the volume at railway bookstalls was forbidden throughout -France. The matter was brought before the chamber of deputies, with the -result of drawing still more attention to the book, and of advertising -the _Contes de la becasse_ (1883), a collection of stories as improper -as they are clever. _Au soleil_ (1884), a book of travels which has the -eminent qualities of lucid observation and exact description, was less -read than _Clair de lune_, _Miss Harriet_, _Les Soeurs Rondoli_ and -_Yvette_, all published in 1883-1884 when Maupassant's powers were at -their highest level. Three further collections of short tales, entitled -_Contes et nouvelles_, _Monsieur Parent_, and _Contes du jour et de la -nuit_, issued in 1885, proved that while the author's vision was as -incomparable as ever, his fecundity had not improved his impeccable -form. To 1885 also belongs an elaborate novel, _Bel-ami_, the cynical -history of a particularly detestable, brutal scoundrel who makes his way -in the world by means of his handsome face. Maupassant is here no less -vivid in realizing his literary men, financiers and frivolous women than -in dealing with his favourite peasants, boors and servants, to whom he -returned in _Toine_ (1886) and in _La Petite roque_ (1886). About this -time appeared the first symptoms of the malady which destroyed him; he -wrote less, and though the novel _Mont-Oriol_ (1887) shows him -apparently in undiminished possession of his faculty, _Le Horla_ (1887) -suggests that he was already subject to alarming hallucinations. -Restored to some extent by a sea-voyage, recorded in _Sur l'eau_ (1888), -he went back to short stories in _Le Rosier de Madame Husson_ (1888), a -burst of Rabelaisian humour equal to anything he had ever written. His -novels _Pierre et Jean_ (1888), _Fort comme la mort_ (1889), and _Notre -coeur_ (1890) are penetrating studies touched with a profounder sympathy -than had hitherto distinguished him; and this softening into pity for -the tragedy of life is deepened in some of the tales included in -_Inutile beaute_ (1890). One of these, _Le Champ d'Oliviers_, is an -unsurpassable example of poignant, emotional narrative. With _La Vie -errante_ (1890), a volume of travels, Maupassant's career practically -closed. _Musotte_, a theatrical piece written in collaboration with M. -Jacques Normand, was published in 1891. By this time inherited nervous -maladies, aggravated by excessive physical exercises and by the -imprudent use of drugs, had undermined his constitution. He began to -take an interest in religious problems, and for a while made the -_Imitation_ his handbook; but his misanthropy deepened, and he suffered -from curious delusions as to his wealth and rank. A victim of general -paralysis, of which _La Folie des grandeurs_ was one of the symptoms, he -drank the waters at Aix-les-Bains during the summer of 1891, and retired -to Cannes, where he purposed passing the winter. The singularities of -conduct which had been observed at Aix-les-Bains grew more and more -marked. Maupassant's reason slowly gave way. On the 6th of January 1892 -he attempted suicide, and was removed to Paris, where he died in the -most painful circumstances on the 6th of July 1893. He is buried in the -cemetery of Montparnasse. The opening chapters of two projected novels, -_L'Angelus_ and _L'Ame etrangere_, were found among his papers; these, -with _La Paix du menage_, a comedy in two acts, and two collections of -tales, _Le Pere Milon_ (1898) and _Le Colporteur_ (1899), have been -published posthumously. A correspondence, called _Amitie amoureuse_ -(1897), and dedicated to his mother, is probably unauthentic. Among the -prefaces which he wrote for the works of others, only one--an -introduction to a French prose version of Mr Swinburne's _Poems and -Ballads_--is likely to interest English readers. - -Maupassant began as a follower of Flaubert and of M. Zola, but, whatever -the masters may have called themselves, they both remained essentially -_romantiques_. The pupil is the last of the "naturalists": he even -destroyed naturalism, since he did all that can be done in that -direction. He had no psychology, no theories of art, no moral or strong -social prejudices, no disturbing imagination, no wealth of perplexing -ideas. It is no paradox to say that his marked limitations made him the -incomparable artist that he was. Undisturbed by any external influence, -his marvellous vision enabled him to become a supreme observer, and, -given his literary sense, the rest was simple. He prided himself in -having no invention; he described nothing that he had not seen. The -peasants whom he had known as a boy figure in a score of tales; what he -saw in Government offices is set down in _L'Heritage_; from Algiers he -gathers the material for Maroca; he drinks the waters and builds up -_Mont-Oriol_; he enters journalism, constructs _Bel-ami_, and, for the -sake of precision, makes his brother, Herve de Maupassant, sit for the -infamous hero's portrait; he sees fashionable society, and, though it -wearied him intensely, he transcribes its life in _Fort comme la mort_ -and _Notre coeur_. Fundamentally he finds all men alike. In every grade -he finds the same ferocious, cunning, animal instincts at work: it is -not a gay world, but he knows no other; he is possessed by the dread of -growing old, of ceasing to enjoy; the horror of death haunts him like a -spectre. It is an extremely simple outlook. Maupassant does not prefer -good to bad, one man to another; he never pauses to argue about the -meaning of life, a senseless thing which has the one advantage of -yielding materials for art; his one aim is to discover the hidden aspect -of visible things, to relate what he has observed, to give an objective -rendering of it, and he has seen so intensely and so serenely that he is -the most exact transcriber in literature. And as the substance is, so is -the form: his style is exceedingly simple and exceedingly strong; he -uses no rare or superfluous word, and is content to use the humblest -word if only it conveys the exact picture of the thing seen. In ten -years he produced some thirty volumes. With the exception of _Pierre et -Jean_, his novels, excellent as they are, scarcely represent him at his -best, and of over two hundred _contes_ a proportion must be rejected. -But enough will remain to vindicate his claim to a permanent place in -literature as an unmatched observer and the most perfect master of the -short story. - - See also F. Brunetiere, _Le Roman naturaliste_ (1883); T. Lemaitre, - _Les Contemporains_ (vols. i. v. vi.); R. Doumic, _Ecrivains - d'aujourd'hui_ (1894); an introduction by Henry James to _The Odd - Number_ ... (1891); a critical preface by the earl of Crewe to _Pierre - and Jean_ (1902); A. Symons, _Studies in Prose and Verse_ (1904). - There are many references to Maupassant in the _Journal des Goncourt_, - and some correspondence with Marie Bashkirtseff was printed with - _Further Memoirs_ of that lady in 1901. (J. F. K.) - - - - -MAUPEOU, RENE NICOLAS CHARLES AUGUSTIN (1714-1792), chancellor of -France, was born on the 25th of February 1714, being the eldest son of -Rene Charles de Maupeou (1688-1775), who was president of the parlement -of Paris from 1743 to 1757. He married in 1744 a rich heiress, Anne de -Roncherolles, a cousin of Madame d'Epinay. Entering public life, he was -his father's right hand in the conflicts between the parlement and -Christophe de Beaumont, archbishop of Paris, who was supported by the -court. Between 1763 and 1768, dates which cover the revision of the case -of Jean Calas and the trial of the comte de Lally, Maupeou was himself -president of the parlement. In 1768, through the protection of Choiseul, -whose fall two years later was in large measure his work, he became -chancellor in succession to his father, who had held the office for a -few days only. He determined to support the royal authority against the -parlement, which in league with the provincial magistratures was seeking -to arrogate to itself the functions of the states-general. He allied -himself with the duc d'Aiguillon and Madame du Barry, and secured for a -creature of his own, the Abbe Terrai, the office of comptroller-general. -The struggle came over the trial of the case of the duc d'Aiguillon, -ex-governor of Brittany, and of La Chalotais, procureur-general of the -province, who had been imprisoned by the governor for accusations -against his administration. When the parlement showed signs of hostility -against Aiguillon, Maupeou read letters patent from Louis XV. annulling -the proceedings. Louis replied to remonstrances from the parlement by a -_lit de justice_, in which he demanded the surrender of the minutes of -procedure. On the 27th of November 1770 appeared the _Edit de reglement -et de discipline_, which was promulgated by the chancellor, forbidding -the union of the various branches of the parlement and correspondence -with the provincial magistratures. It also made a strike on the part of -the parlement punishable by confiscation of goods, and forbade further -obstruction to the registration of royal decrees after the royal reply -had been given to a first remonstrance. This edict the magistrates -refused to register, and it was registered in a _lit de justice_ held at -Versailles on the 7th of December, whereupon the parlement suspended its -functions. After five summonses to return to their duties, the -magistrates were surprised individually on the night of the 19th of -January 1771 by musketeers, who required them to sign yes or no to a -further request to return. Thirty-eight magistrates gave an affirmative -answer, but on the exile of their former colleagues by _lettres de -cachet_ they retracted, and were also exiled. Maupeou installed the -council of state to administer justice pending the establishment of six -superior courts in the provinces, and of a new parlement in Paris. The -_cour des aides_ was next suppressed. - -Voltaire praised this revolution, applauding the suppression of the old -hereditary magistrature, but in general Maupeou's policy was regarded as -the triumph of tyranny. The remonstrances of the princes, of the nobles, -and of the minor courts, were met by exile and suppression, but by the -end of 1771 the new system was established, and the Bar, which had -offered a passive resistance, recommenced to plead. But the death of -Louis XV. in May 1774 ruined the chancellor. The restoration of the -parlements was followed by a renewal of the quarrels between the new -king and the magistrature. Maupeou and Terrai were replaced by -Malesherbes and Turgot. Maupeou lived in retreat until his death at -Thuit on the 29th of July 1792, having lived to see the overthrow of the -_ancien regime_. His work, in so far as it was directed towards the -separation of the judicial and political functions and to the reform of -the abuses attaching to a hereditary magistrature, was subsequently -endorsed by the Revolution; but no justification of his violent methods -or defence of his intriguing and avaricious character is possible. He -aimed at securing absolute power for Louis XV., but his action was in -reality a serious blow to the monarchy. - - The chief authority for the administration of Maupeou is the _compte - rendu_ in his own justification presented by him to Louis XVI. in - 1789, which included a dossier of his speeches and edicts, and is - preserved in the Bibliotheque nationale. These documents, in the hands - of his former secretary, C. F. Lebrun, duc de Plaisance, formed the - basis of the judicial system of France as established under the - consulate (cf. C. F. Lebrun, _Opinions, rapports et choix d'ecrits - politiques_, published posthumously in 1829). See further _Maupeouana_ - (6 vols., Paris, 1775), which contains the pamphlets directed against - him; _Journal hist. de la revolution operee ... par M. de Maupeou_ (7 - vols., 1775); the official correspondence of Mercy-Argenteau, the - letters of Mme d'Epinay; and Jules Flammermont, _Le Chancelier Maupeou - et les parlements_ (1883). - - - - -MAUPERTUIS, PIERRE LOUIS MOREAU DE (1698-1759), French mathematician and -astronomer, was born at St Malo on the 17th of July 1698. When twenty -years of age he entered the army, becoming lieutenant in a regiment of -cavalry, and employing his leisure on mathematical studies. After five -years he quitted the army and was admitted in 1723 a member of the -Academy of Sciences. In 1728 he visited London, and was elected a fellow -of the Royal Society. In 1736 he acted as chief of the expedition sent -by Louis XV. into Lapland to measure the length of a degree of the -meridian (see EARTH, FIGURE OF), and on his return home he became a -member of almost all the scientific societies of Europe. In 1740 -Maupertuis went to Berlin on the invitation of the king of Prussia, and -took part in the battle of Mollwitz, where he was taken prisoner by the -Austrians. On his release he returned to Berlin, and thence to Paris, -where he was elected director of the Academy of Sciences in 1742, and in -the following year was admitted into the Academy. Returning to Berlin in -1744, at the desire of Frederick II., he was chosen president of the -Royal Academy of Sciences in 1746. Finding his health declining, he -repaired in 1757 to the south of France, but went in 1758 to Basel, -where he died on the 27th of July 1759. Maupertuis was unquestionably a -man of considerable ability as a mathematician, but his restless, gloomy -disposition involved him in constant quarrels, of which his -controversies with Konig and Voltaire during the latter part of his life -furnish examples. - - The following are his most important works: _Sur la figure de la - terre_ (Paris, 1738); _Discours sur la parallaxe de la lune_ (Paris, - 1741); _Discours sur la figure des astres_ (Paris, 1742); _Elements de - la geographie_ (Paris, 1742); _Lettre sur la comete de 1742_ (Paris, - 1742); _Astronomie nautique_ (Paris, 1745 and 1746); _Venus physique_ - (Paris, 1745); _Essai de cosmologie_ (Amsterdam, 1750). His _Oeuvres_ - were published in 1752 at Dresden and in 1756 at Lyons. - - - - -MAU RANIPUR, a town of British India in Jahnsi district, in the United -Provinces. Pop. (1901), 17,231. It contains a large community of wealthy -merchants and bankers. A special variety of red cotton cloth, known as -_kharua_, is manufactured and exported to all parts of India. Trees line -many of the streets, and handsome temples ornament the town. - - - - -MAUREL, ABDIAS (d. 1705), Camisard leader, became a cavalry officer in -the French army and gained distinction in Italy; here he served under -Marshal Catinat, and on this account he himself is sometimes known as -Catinat. In 1702, when the revolt in the Cevennes broke out, he became -one of the Camisard leaders, and in this capacity his name was soon -known and feared. He refused to accept the peace made by Jean Cavalier -in 1704, and after passing a few weeks in Switzerland he returned to -France and became one of the chiefs of those Camisards who were still in -arms. He was deeply concerned in a plot to capture some French towns, a -scheme which, it was hoped, would be helped by England and Holland. But -it failed; Maurel was betrayed, and with three other leaders of the -movement was burned to death at Nimes on the 22nd of April 1705. He was -a man of great physical strength; but he was very cruel, and boasted he -had killed 200 Roman Catholics with his own hands. - - - - -MAUREL, VICTOR (1848- ), French singer, was born at Marseilles, and -educated in music at the Paris Conservatoire. He made his debut in opera -at Paris in 1868, and in London in 1873, and from that time onwards his -admirable acting and vocal method established his reputation as one of -the finest of operatic baritones. He created the leading part in Verdi's -_Otello_, and was equally fine in Wagnerian and Italian opera. - - - - -MAURENBRECHER, KARL PETER WILHELM (1838-1892), German historian, was -born at Bonn on the 21st of December, 1838, and studied in Berlin and -Munich under Ranke and Von Sybel, being especially influenced by the -latter historian. After doing some research work at Simancas in Spain, -he became professor of history at the university of Dorpat in 1867; and -was then in turn professor at Konigsberg, Bonn and Leipzig. He died at -Leipzig on the 6th of November, 1892. - - Many of Maurenbrecher's works are concerned with the Reformation, - among them being _England im Reformationszeitalter_ (Dusseldorf, - 1866); _Karl V. und die deutschen Protestanten_ (Dusseldorf, 1865); - _Studien und Skizzen zur Geschichte der Reformationszeit_ (Leipzig, - 1874); and the incomplete _Geschichte der Katholischen Reformation_ - (Nordlingen, 1880). He also wrote _Don Karlos_ (Berlin, 1876); - _Grundung des deutschen Reiches 1859-1871_ (Leipzig, 1892, and again - 1902); and _Geschichte der deutschen Konigswahlen_ (Leipzig, 1889). - See G. Wolf, _Wilhelm Maurenbrecher_ (Berlin, 1893). - - - - -MAUREPAS, JEAN FREDERIC PHELYPEAUX, COMTE DE (1701-1781), French -statesman, was born on the 9th of July 1701 at Versailles, being the son -of Jerome de Pontchartrain, secretary of state for the marine and the -royal household. Maurepas succeeded to his father's charge at fourteen, -and began his functions in the royal household at seventeen, while in -1725 he undertook the actual administration of the navy. Although -essentially light and frivolous in character, Maurepas was seriously -interested in scientific matters, and he used the best brains of France -to apply science to questions of navigation and of naval construction. -He was disgraced in 1749, and exiled from Paris for an epigram against -Madame de Pompadour. On the accession of Louis XVI., twenty-five years -later, he became a minister of state and Louis XVI.'s chief adviser. He -gave Turgot the direction of finance, placed Lamoignon-Malesherbes over -the royal household and made Vergennes minister for foreign affairs. At -the outset of his new career he showed his weakness by recalling to -their functions, in deference to popular clamour, the members of the old -parlement ousted by Maupeou, thus reconstituting the most dangerous -enemy of the royal power. This step, and his intervention on behalf of -the American states, helped to pave the way for the French revolution. -Jealous of his personal ascendancy over Louis XVI., he intrigued against -Turgot, whose disgrace in 1776 was followed after six months of disorder -by the appointment of Necker. In 1781 Maurepas deserted Necker as he had -done Turgot, and he died at Versailles on the 21st of November 1781. - - Maurepas is credited with contributions to the collection of facetiae - known as the _Etrennes de la Saint Jean_ (2nd ed., 1742). Four volumes - of _Memoires de Maurepas_, purporting to be collected by his secretary - and edited by J. L. G. Soulavie in 1792, must be regarded as - apocryphal. Some of his letters were published in 1896 by the _Soc. de - l'hist. de Paris_. His _eloge_ in the Academy of Sciences was - pronounced by Condorcet. - - - - -MAURER, GEORG LUDWIG VON (1790-1872), German statesman and historian, -son of a Protestant pastor, was born at Erpolzheim, near Durkheim, in -the Rhenish Palatinate, on the 2nd of November 1790. Educated at -Heidelberg, he went in 1812 to reside in Paris, where he entered upon a -systematic study of the ancient legal institutions of the Germans. -Returning to Germany in 1814, he received an appointment under the -Bavarian government, and afterwards filled several important official -positions. In 1824 he published at Heidelberg his _Geschichte des -altgermanischen und namentlich altbayrischen offentlich-mundlichen -Gerichtsverfahrens_, which obtained the first prize of the academy of -Munich, and in 1826 he became professor in the university of Munich. In -1829 he returned to official life, and was soon offered an important -post. In 1832, when Otto (Otho), son of Louis I., king of Bavaria, was -chosen to fill the throne of Greece, a council of regency was nominated -during his minority, and Maurer was appointed a member. He applied -himself energetically to the task of creating institutions adapted to -the requirements of a modern civilized community; but grave difficulties -soon arose and Maurer was recalled in 1834, when he returned to Munich. -This loss was a serious one for Greece. Maurer was the ablest, most -energetic and most liberal-minded member of the council, and it was -through his enlightened efforts that Greece obtained a revised penal -code, regular tribunals and an improved system of civil procedure. Soon -after his recall he published _Das griechische Volk in offentlicher, -kirchlicher, und privatrechtlicher Beziehung vor und nach dem -Freiheitskampfe bis zum 31 Juli 1834_ (Heidelberg, 1835-1836), a useful -source of information for the history of Greece before Otto ascended the -throne, and also for the labours of the council of regency to the time -of the author's recall. After the fall of the ministry of Karl von Abel -(1788-1859) in 1847, he became chief Bavarian minister and head of the -departments of foreign affairs and of justice, but was overthrown in the -same year. He died at Munich on the 9th of May 1872. His only son, -Conrad von Maurer (1823-1902), was a Scandinavian scholar of some -repute, and like his father was a professor at the university of Munich. - - Maurer's most important contribution to history is a series of books - on the early institutions of the Germans. These are: _Einleitung zur - Geschichte der Mark-, Hof-, Dorf-, und Stadtverfassung und der - offentlichen Gewalt_ (Munich, 1854); _Geschichte der Markenverfassung - in Deutschland_ (Erlangen, 1856); _Geschichte der Fronhofe, der - Bauernhofe, und der Hofverfassung in Deutschland_ (Erlangen, - 1862-1863); _Geschichte der Dorfverfassung in Deutschland_ (Erlangen, - 1865-1866); and _Geschichte der Sladteverfassung in Deutschland_ - (Erlangen, 1869-1871). These works are still important authorities for - the early history of the Germans. Among other works are, _Das Stadt- - und Landrechtsbuch Ruprechts von Freising, ein Beitrag zur Geschichte - des Schwabenspiegels_ (Stuttgart, 1839); _Uber die Freipflege (plegium - liberale), und die Entstehung der grossen und kleinen Jury in England_ - (Munich, 1848); and _Uber die deutsche Reichsterritorial- und - Rechtsgeschichte_ (1830). - - Sec K. T. von Heigel, _Denkwurdigkeiten des bayrischen Staatsrats G. - L. von Maurer_ (Munich, 1903). - - - - -MAURETANIA, the ancient name of the north-western angle of the African -continent, and under the Roman Empire also of a large territory eastward -of that angle. The name had different significations at different times; -but before the Roman occupation, Mauretania comprised a considerable -part of the modern Morocco i.e. the northern portion bounded on the east -by Algiers. Towards the south we may suppose it bounded by the Atlas -range, and it seems to have been regarded by geographers as extending -along the coast to the Atlantic as far as the point where that chain -descends to the sea, in about 30 N. lat. (Strabo, p. 825). The -magnificent plateau in which the city of Morocco is situated seems to -have been unknown to ancient geographers, and was certainly never -included in the Roman Empire. On the other hand, the Gaetulians to the -south of the Atlas range, on the date-producing slopes towards the -Sahara, seem to have owned a precarious subjection to the kings of -Mauretania, as afterwards to the Roman government. A large part of the -country is of great natural fertility, and in ancient times produced -large quantities of corn, while the slopes of Atlas were clothed with -forests, which, besides other kinds of timber, produced the celebrated -ornamental wood called _citrum_ (Plin. _Hist. Nat._ 13-96), for tables -of which the Romans gave fabulous prices. (For physical geography, see -MOROCCO.) - - Mauretania, or Maurusia as it was called by Greek writers, signified - the land of the Mauri, a term still retained in the modern name of - Moors (q.v.). The origin and ethnical affinities of the race are - uncertain; but it is probable that all the inhabitants of this - northern tract of Africa were kindred races belonging to the great - Berber family, possibly with an intermingled fair-skinned race from - Europe (see Tissot, _Geographie comparee de la province romaine - d'Afrique_, i. 400 seq.; also BERBERS). They first appear in history - at the time of the Jugurthine War (110-106 B.C.), when Mauretania was - under the government of Bocchus and seems to have been recognized as - organized state (Sallust, _Jugurtha_, 19). To this Bocchus was given, - after the war, the western part of Jugurtha's kingdom of Numidia, - perhaps as far east as Saldae (Bougie). Sixty years later, at the time - of the dictator Caesar, we find two Mauretanian kingdoms, one to the - west of the river Mulucha under Bogud, and the other to the east under - a Bocchus; as to the date or cause of the division we are ignorant. - Both these kings took Caesar's part in the civil wars, and had their - territory enlarged by him (Appian, B.C. 4, 54). In 25 B.C., after - their deaths, Augustus gave the two kingdoms to Juba II. of Numidia - (see under JUBA), with the river Ampsaga as the eastern frontier - (Plin. 5. 22; Ptol. 4. 3. 1). Juba and his son Ptolemaeus after him - reigned till A.D. 40, when the latter was put to death by Caligula, - and shortly afterwards Claudius incorporated the kingdom into the - Roman state as two provinces, viz. Mauretania Tingitana to the west - of the Mulucha and M. Caesariensis to the east of that river, the - latter taking its name from the city Caesarea (formerly Iol), which - Juba had thus named and adopted as his capital. Thus the dividing line - between the two provinces was the same as that which had originally - separated Mauretania from Numidia (q.v.). These provinces were - governed until the time of Diocletian by imperial procurators, and - were occasionally united for military purposes. Under and after - Diocletian M. Tingitana was attached administratively to the - _dioicesis_ of Spain, with which it was in all respects closely - connected; while M. Caesariensis was divided by making its eastern - part into a separate government, which was called M. Sitifensis from - the Roman colony Sitifis. - - In the two provinces of Mauretania there were at the time of Pliny a - number of towns, including seven (possibly eight) Roman colonies in M. - Tingitana and eleven in M. Caesariensis; others were added later. - These were mostly military foundations, and served the purpose of - securing civilization against the inroads of the natives, who were not - in a condition to be used as material for town-life as in Gaul and - Spain, but were under the immediate government of the procurators, - retaining their own clan organization. Of these colonies the most - important, beginning from the west, were Lixus on the Atlantic, Tingis - (Tangier), Rusaddir (Melila, Melilla), Cartenna (Tenes), Iol or - Caesarea (Cherchel), Icosium (Algiers), Saldae (Bougie), Igilgili - (Jijelli) and Sitifis (Setif). All these were on the coast but the - last, which was some distance inland. Besides these there were many - municipia or _oppida civium romanorum_ (Plin. 5. 19 seq.), but, as has - been made clear by French archaeologists who have explored these - regions, Roman settlements are less frequent the farther we go west, - and M. Tingitana has as yet yielded but scanty evidence of Roman - civilization. On the whole Mauretania was in a flourishing condition - down to the irruption of the Vandals in A.D. 429; in the _Notitia_ - nearly a hundred and seventy episcopal sees are enumerated here, but - we must remember that numbers of these were mere villages. - - In 1904 the term Mauretania was revived as an official designation by - the French government, and applied to the territory north of the lower - Senegal under French protection (see SENEGAL). - - To the authorities quoted under AFRICA, ROMAN, may be added here - Gobel, _Die West-kuste Afrikas im Alterthum_. (W. W. F.*) - - - - -MAURIAC, a town of central France, capital of an arrondissement in the -department of Cantal, 39 m. N.N.W. of Aurillac by rail. Pop. (1906), -2558. Mauriac, built on the slope of a volcanic hill, has a church of -the 12th century, and the buildings of an old abbey now used as public -offices and dwellings; the town owes its origin to the abbey, founded -during the 6th century. It is the seat of a sub-prefect and has a -tribunal of first instance and a communal college. There are marble -quarries in the vicinity. - - - - -MAURICE [or MAURITIUS], ST (d. c. 286), an early Christian martyr, who, -with his companions, is commemorated by the Roman Catholic Church on the -22nd of September. The oldest form of his story is found in the _Passio_ -ascribed to Eucherius, bishop of Lyons, c. 450, who relates how the -"Theban" legion commanded by Mauritius was sent to north Italy to -reinforce the army of Maximinian. Maximinian wished to use them in -persecuting the Christians, but as they themselves were of this faith, -they refused, and for this, after having been twice decimated, the -legion was exterminated at Octodurum (Martigny) near Geneva. In late -versions this legend was expanded and varied, the martyrdom was -connected with a refusal to take part in a great sacrifice ordered at -Octodurum and the name of Exsuperius was added to that of Mauritius. -Gregory of Tours (c. 539-593) speaks of a company of the same legion -which suffered at Cologne. - - The _Magdeburg Centuries_, in spite of Mauritius being the patron - saint of Magdeburg, declared the whole legend fictitious; J. A. du - Bordien _La Legion thebeenne_ (Amsterdam, 1705); J. J. Hottinger in - _Helvetische Kirchengeschichte_ (Zurich, 1708); and F. W. Rettberg, - _Kirchengeschichte Deutschlands_ (Gottingen, 1845-1848) have also - demonstrated its untrustworthiness, while the Bollandists, De Rivaz - and Joh. Friedrich uphold it. Apart from the a priori improbability of - a whole legion being martyred, the difficulties are that in 286 - Christians everywhere throughout the empire were not molested, that at - no later date have we evidence of the presence of Maximinian in the - Valais, and that none of the writers nearest to the event (Eusebius, - Lactantius, Orosius, Sulpicius Severus) know anything of it. It is of - course quite possible that isolated cases of officers being put to - death for their faith occurred during Maximinian's reign, and on some - such cases the legend may have grown up during the century and a half - between Maximinian and Eucherius. The cult of St Maurice and the - Theban legion is found in Switzerland (where two places bear the name - in Valais, besides St Moritz in Grisons), along the Rhine, and in - north Italy. The foundation of the abbey of St Maurice (Agaunum) in - the Valais is usually ascribed to Sigismund of Burgundy (515). Relics - of the saint are preserved here and at Brieg and Turin. - - - - -MAURICE (MAURICIUS FLAVIUS TIBERIUS) (c. 539-602), East Roman emperor -from 582 to 602, was of Roman descent, but a native of Arabissus in -Cappadocia. He spent his youth at the court of Justin II., and, having -joined the army, fought with distinction in the Persian War (578-581). -At the age of forty-three he was declared Caesar by the dying emperor -Tiberius II., who bestowed upon him the hand of his daughter -Constantina. Maurice brought the Persian War to a successful close by -the restoration of Chosroes II. to the throne (591). On the northern -frontier he at first bought off the Avars by payments which compelled -him to exercise strict economy in his general administration, but after -595 inflicted several defeats upon them through his general Crispus. By -his strict discipline and his refusal to ransom a captive corps he -provoked to mutiny the army on the Danube. The revolt spread to the -popular factions in Constantinople, and Maurice consented to abdicate. -He withdrew to Chalcedon, but was hunted down and put to death after -witnessing the slaughter of his five sons. - - The work on military art ([Greek: strategika]) ascribed to him is a - contemporary work of unknown authorship (ed. Scheffer, _Arriani - tactica et Mauricii ars militaris_, Upsala, 1664; see Max Jahns, - _Gesch. d. Kriegswissensch._, i. 152-156). - - See Theophylactus Simocatta, _Vita Mauricii_ (ed. de Boor, 1887); E. - Gibbon, _The Decline and Fall of the Roman Empire_ (ed. Bury, London, - 1896, v. 19-21, 57); J. B. Bury, _The Later Roman Empire_ (London, - 1889, ii. 83-94); G. Finlay, _History of Greece_ (ed. 1877, Oxford, i. - 299-306). - - - - -MAURICE (1521-1553), elector of Saxony, elder son of Henry, duke of -Saxony, belonging to the Albertine branch of the Wettin family, was born -at Freiberg on the 21st of March 1521. In January 1541 he married Agnes, -daughter of Philip, landgrave of Hesse. In that year he became duke of -Saxony by his father's death, and he continued Henry's work in -forwarding the progress of the Reformation. Duke Henry had decreed that -his lands should be divided between his two sons, but as a partition was -regarded as undesirable the whole of the duchy came to his elder son. -Maurice, however, made generous provision for his brother Augustus, and -the desire to compensate him still further was one of the minor threads -of his subsequent policy. In 1542 he assisted the emperor Charles V. -against the Turks, in 1543 against William, duke of Cleves, and in 1544 -against the French; but his ambition soon took a wider range. The -harmonious relations which subsisted between the two branches of the -Wettins were disturbed by the interference of Maurice in Cleves, a -proceeding distasteful to the Saxon elector, John Frederick; and a -dispute over the bishopric of Meissen having widened the breach, war was -only averted by the mediation of Philip of Hesse and Luther. About this -time Maurice seized the idea of securing for himself the electoral -dignity held by John Frederick, and his opportunity came when Charles -was preparing to attack the league of Schmalkalden. Although educated as -a Lutheran, religious questions had never seriously appealed to Maurice. -As a youth he had joined the league of Schmalkalden, but this adhesion, -as well as his subsequent declaration to stand by the confession of -Augsburg, cannot be regarded as the decision of his maturer years. In -June 1546 he took a decided step by making a secret agreement with -Charles at Regensburg. Maurice was promised some rights over the -archbishopric of Magdeburg and the bishopric of Halberstadt; immunity, -in part at least, for his subjects from the Tridentine decrees; and the -question of transferring the electoral dignity was discussed. In return -the duke probably agreed to aid Charles in his proposed attack on the -league as soon as he could gain the consent of the Saxon estates, or at -all events to remain neutral during the impending war. The struggle -began in July 1546, and in October Maurice declared war against John -Frederick. He secured the formal consent of Charles to the transfer of -the electoral dignity and took the field in November. He had gained a -few successes when John Frederick hastened from south Germany to defend -his dominions. Maurice's ally, Albert Alcibiades, prince of Bayreuth, -was taken prisoner at Rochlitz; and the duke, driven from electoral -Saxony, was unable to prevent his own lands from being overrun. -Salvation, however, was at hand. Marching against John Frederick, -Charles V., aided by Maurice, gained a decisive victory at Muhlberg in -April 1547, after which by the capitulation of Wittenberg John Frederick -renounced the electoral dignity in favour of Maurice, who also obtained -a large part of his kinsman's lands. The formal investiture of the new -elector took place at Augsburg in February 1548. - -The plans of Maurice soon took a form less agreeable to the emperor. The -continued imprisonment of his father-in-law, Philip of Hesse, whom he -had induced to surrender to Charles and whose freedom he had guaranteed, -was neither his greatest nor his only cause of complaint. The emperor -had refused to complete the humiliation of the family of John Frederick; -he had embarked upon a course of action which boded danger to the -elector's Lutheran subjects, and his increased power was a menace to the -position of Maurice. Assuring Charles of his continued loyalty, the -elector entered into negotiations with the discontented Protestant -princes. An event happened which gave him a base of operations, and -enabled him to mask his schemes against the emperor. In 1550 he had been -entrusted with the execution of the imperial ban against the city of -Magdeburg, and under cover of these operations he was able to collect -troops and to concert measures with his allies. Favourable terms were -granted to Magdeburg, which surrendered and remained in the power of -Maurice, and in January 1552 a treaty was concluded with Henry II. of -France at Chambord. Meanwhile Maurice had refused to recognize the -_Interim_ issued from Augsburg in May 1548 as binding on Saxony; but a -compromise was arranged on the basis of which the Leipzig _Interim_ was -drawn up for his lands. It is uncertain how far Charles was ignorant of -the elector's preparations, but certainly he was unprepared for the -attack made by Maurice and his allies in March 1552. Augsburg was taken, -the pass of Ehrenberg was forced, and in a few days the emperor left -Innsbruck as a fugitive. Ferdinand undertook to make peace, and the -Treaty of Passau, signed in August 1552, was the result. Maurice -obtained a general amnesty and freedom for Philip of Hesse, but was -unable to obtain a perpetual religious peace for the Lutherans. Charles -stubbornly insisted that this question must be referred to the Diet, and -Maurice was obliged to give way. He then fought against the Turks, and -renewed his communications with Henry of France. Returning from Hungary -the elector placed himself at the head of the princes who were seeking -to check the career of his former ally, Albert Alcibiades, whose -depredations were making him a curse to Germany. The rival armies met at -Sievershausen on the 9th of July 1553, where after a fierce encounter -Albert was defeated. The victor, however, was wounded during the fight -and died two days later. - -Maurice was a friend to learning, and devoted some of the secularized -church property to the advancement of education. Very different -estimates have been formed of his character. He has been represented as -the saviour of German Protestantism on the one hand, and on the other as -a traitor to his faith and country. In all probability he was neither -the one nor the other, but a man of great ambition who, indifferent to -religious considerations, made good use of the exigencies of the time. -He was generous and enlightened, a good soldier and a clever -diplomatist. He left an only daughter Anna (d. 1577), who became the -second wife of William the Silent, prince of Orange. - - The elector's _Politische Korrespondenz_ has been edited by E. - Brandenburg (Leipzig, 1900-1904); and a sketch of him is given by - Roger Ascham in _A Report and Discourse of the Affairs and State of - Germany_ (London, 1864-1865). See also F. A. von Langenn, _Moritz - Herzog und Churfurst zu Sachsen_ (Leipzig, 1841); G. Voigt, _Moritz - von Sachsen_ (Leipzig, 1876); E. Brandenburg, _Moritz von Sachsen_ - (Leipzig, 1898); S. Issleib, _Moritz von Sachsen als protestantischer - Furst_ (Hamburg, 1898); J. Witter, _Die Beziehung und der Verkehr des - Kurfursten Moritz mit Konig Ferdinand_ (Jena, 1886); L. von Ranke, - _Deutsche Geschichte im Zeitalter der Reformation_, Bde. IV. and V. - (Leipzig, 1882); and W. Maurenbrecher in the _Allgemeine deutsche - Biographie_, Bd. XXII. (Leipzig, 1885). For bibliography see - Maurenbrecher; and _The Cambridge Modern History_, vol. ii. - (Cambridge, 1903). - - - - -MAURICE, JOHN FREDERICK DENISON (1805-1872), English theologian, was -born at Normanston, Suffolk, on the 29th of August, 1805. He was the son -of a Unitarian minister, and entered Trinity College, Cambridge, in -1823, though it was then impossible for any but members of the -Established Church to obtain a degree. Together with John Sterling (with -whom he founded the Apostles' Club) he migrated to Trinity Hall, whence -he obtained a first class in civil law in 1827; he then came to London, -and gave himself to literary work, writing a novel, _Eustace Conyers_, -and editing the _London Literary Chronicle_ until 1830, and also for a -short time the _Athenaeum_. At this time he was much perplexed as to his -religious opinions, and he ultimately found relief in a decision to take -a further university course and to seek Anglican orders. Entering Exeter -College, Oxford, he took a second class in classics in 1831. He was -ordained in 1834, and after a short curacy at Bubbenhall in Warwickshire -was appointed chaplain of Guy's Hospital, and became thenceforward a -sensible factor in the intellectual and social life of London. From 1839 -to 1841 Maurice was editor of the _Education Magazine_. In 1840 he was -appointed professor of English history and literature in King's College, -and to this post in 1846 was added the chair of divinity. In 1845 he was -Boyle lecturer and Warburton lecturer. These chairs he held till 1853. -In that year he published _Theological Essays_, wherein were stated -opinions which savoured to the principal, Dr R. W. Jelf, and to the -council, of unsound theology in regard to eternal punishment. He had -previously been called on to clear himself from charges of heterodoxy -brought against him in the _Quarterly Review_ (1851), and had been -acquitted by a committee of inquiry. Now again he maintained with great -warmth of conviction that his views were in close accordance with -Scripture and the Anglican standards, but the council, without -specifying any distinct "heresy" and declining to submit the case to the -judgment of competent theologians, ruled otherwise, and he was deprived -of his professorships. He held at the same time the chaplaincy of -Lincoln's Inn, for which he had resigned Guy's (1846-1860), but when he -offered to resign this the benchers refused. Nor was he assailed in the -incumbency of St. Peter's, Vere Street, which he held for nine years -(1860-1869), and where he drew round him a circle of thoughtful people. -During the early years of this period he was engaged in a hot and bitter -controversy with H. L. Mansel (afterwards dean of St Paul's), arising -out of the latter's Bampton lecture upon reason and revelation. - -During his residence in London Maurice was specially identified with two -important movements for education. He helped to found Queen's College -for the education of women (1848), and the Working Men's College (1854), -of which he was the first principal. He strongly advocated the abolition -of university tests (1853), and threw himself with great energy into all -that affected the social life of the people. Certain abortive attempts -at co-operation among working men, and the movement known as Christian -Socialism, were the immediate outcome of his teaching. In 1866 Maurice -was appointed professor of moral philosophy at Cambridge, and from 1870 -to 1872 was incumbent of St Edward's in that city. He died on the 1st of -April 1872. - -He was twice married, first to Anna Barton, a sister of John Sterling's -wife, secondly to a half-sister of his friend Archdeacon Hare. His son -Major-General Sir J. Frederick Maurice (b. 1841), became a distinguished -soldier and one of the most prominent military writers of his time. - -Those who knew Maurice best were deeply impressed with the spirituality -of his character. "Whenever he woke in the night," says his wife, "he -was always praying." Charles Kingsley called him "the most beautiful -human soul whom God has ever allowed me to meet with." As regards his -intellectual attainments we may set Julius Hare's verdict "the greatest -mind since Plato" over against Ruskin's "by nature puzzle-headed and -indeed wrong-headed." Such contradictory impressions bespeak a life made -up of contradictory elements. Maurice was a man of peace, yet his life -was spent in a series of conflicts; of deep humility, yet so polemical -that he often seemed biased; of large charity, yet bitter in his attack -upon the religious press of his time; a loyal churchman who detested the -label "Broad," yet poured out criticism upon the leaders of the Church. -With an intense capacity for visualizing the unseen, and a kindly -dignity, he combined a large sense of humour. While most of the "Broad -Churchmen" were influenced by ethical and emotional considerations in -their repudiation of the dogma of everlasting torment, he was swayed by -purely intellectual and theological arguments, and in questions of a -more general liberty he often opposed the proposed Liberal theologians, -though he as often took their side if he saw them hard pressed. He had a -wide metaphysical and philosophical knowledge which he applied to the -history of theology. He was a strenuous advocate of ecclesiastical -control in elementary education, and an opponent of the new school of -higher biblical criticism, though so far an evolutionist as to believe -in growth and development as applied to the history of nations. - - As a preacher, his message was apparently simple; his two great - convictions were the fatherhood of God, and that all religious systems - which had any stability lasted because of a portion of truth which had - to be disentangled from the error differentiating them from the - doctrines of the Church of England as understood by himself. His love - to God as his Father was a passionate adoration which filled his whole - heart. The prophetic, even apocalyptic, note of his preaching was - particularly impressive. He prophesied in London as Isaiah prophesied - to the little towns of Palestine and Syria, "often with dark - foreboding, but seeing through all unrest and convulsion the working - out of a sure divine purpose." Both at King's College and at Cambridge - Maurice gathered round him a band of earnest students, to whom he - directly taught much that was valuable drawn from wide stores of his - own reading, wide rather than deep, for he never was, strictly - speaking, a learned man. Still more did he encourage the habit of - inquiry and research, more valuable than his direct teaching. In his - Socratic power of convincing his pupils of their ignorance he did more - than perhaps any other man of his time to awaken in those who came - under his sway the desire for knowledge and the process of independent - thought. - - As a social reformer, Maurice was before his time, and gave his eager - support to schemes for which the world was not ready. From an early - period of his life in London the condition of the poor pressed upon - him with consuming force; the enormous magnitude of the social - questions involved was a burden which he could hardly bear. For many - years he was the clergyman whom working men of all opinions seemed to - trust even if their faith in other religious men and all religious - systems had faded, and he had a marvellous power of attracting the - zealot and the outcast. - - His works cover nearly 40 volumes, often obscure, often tautological, - and with no great distinction of style. But their high purpose and - philosophical outlook give his writings a permanent place in the - history of the thought of his time. The following are the more - important works--some of them were rewritten and in a measure recast, - and the date given is not necessarily that of the first appearance of - the book, but of its more complete and abiding form: _Eustace Conway, - or the Brother and Sister_, a novel (1834); _The Kingdom of Christ_ - (1842); _Christmas Day and Other Sermons_ (1843); _The Unity of the - New Testament_ (1844); _The Epistle to the Hebrews_ (1846); _The - Religions of the World_ (1847); _Moral and Metaphysical Philosophy_ - (at first an article in the _Encyclopaedia Metropolitana_, 1848); _The - Church a Family_ (1850); _The Old Testament_ (1851); _Theological - Essays_ (1853); _The Prophets and Kings of the Old Testament_ (1853); - _Lectures on Ecclesiastical History_ (1854); _The Doctrine of - Sacrifice_ (1854); _The Patriarchs and Lawgivers of the Old Testament_ - (1855); _The Epistles of St John_ (1857); _The Commandments as - Instruments of National Reformation_ (1866); _On the Gospel of St - Luke_ (1868); _The Conscience: Lectures on Casuistry_ (1868); _The - Lord's Prayer, a Manual_ (1870). The greater part of these works were - first delivered as sermons or lectures. Maurice also contributed many - prefaces and introductions to the works of friends, as to Archdeacon - Hare's _Charges_, Kingsley's _Saint's Tragedy_, &c. - - See _Life_ by his son (2 vols., London, 1884), and a monograph by C. - F. G. Masterman (1907) in "Leader of the Church" series; W. E. Collins - in _Typical English Churchmen_, pp. 327-360 (1902), and T. Hughes in - _The Friendship of Books_ (1873). - - - - -MAURICE OF NASSAU, prince of Orange (1567-1625), the second son of -William the Silent, by Anna, only daughter of the famous Maurice, -elector of Saxony, was born at Dillenburg. At the time of his father's -assassination in 1584 he was being educated at the university of Leiden, -at the expense of the states of Holland and Zeeland. Despite his youth -he was made stadtholder of those two provinces and president of the -council of state. During the period of Leicester's governorship he -remained in the background, engaged in acquiring a thorough knowledge of -the military art, and in 1586 the States of Holland conferred upon him -the title of prince. On the withdrawal of Leicester from the Netherlands -in August 1587, Johan van Oldenbarneveldt, the advocate of Holland, -became the leading statesman of the country, a position which he -retained for upwards of thirty years. He had been a devoted adherent of -William the Silent and he now used his influence to forward the -interests of Maurice. In 1588 he was appointed by the States-General -captain and admiral-general of the Union, in 1590 he was elected -stadtholder of Utrecht and Overysel, and in 1591 of Gelderland. From -this time forward, Oldenbarneveldt at the head of the civil government -and Maurice in command of the armed forces of the republic worked -together in the task of rescuing the United Netherlands from Spanish -domination (for details see HOLLAND). Maurice soon showed himself to be -a general second in skill to none of his contemporaries. He was -especially famed for his consummate knowledge of the science of sieges. -The twelve years' truce on the 9th of April 1609 brought to an end the -cordial relations between Maurice and Oldenbarneveldt. Maurice was -opposed to the truce, but the advocate's policy triumphed and -henceforward there was enmity between them. The theological disputes -between the Remonstrants and contra-Remonstrants found them on different -sides; and the theological quarrel soon became a political one. -Oldenbarneveldt, supported by the states of Holland, came forward as the -champion of provincial sovereignty against that of the states-general; -Maurice threw the weight of his sword on the side of the union. The -struggle was a short one, for the army obeyed the general who had so -often led them to victory. Oldenbarneveldt perished on the scaffold, and -the share which Maurice had in securing the illegal condemnation by a -packed court of judges of the aged patriot must ever remain a stain upon -his memory. - -Maurice, who had on the death of his elder brother Philip William, in -February 1618, become prince of Orange, was now supreme in the state, -but during the remainder of his life he sorely missed the wise counsels -of the experienced Oldenbarneveldt. War broke out again in 1621, but -success had ceased to accompany him on his campaigns. His health gave -way, and he died, a prematurely aged man, at the Hague on the 4th of -April 1625. He was buried by his father's side at Delft. - - BIBLIOGRAPHY.--I. Commelin, _Wilhelm en Maurits v. Nassau, pr. v. - Orangien, haer leven en bedrijf_ (Amsterdam, 1651); G. Groen van - Prinsterer, _Archives ou correspondance de la maison d'Orange-Nassau_, - 1^e serie, 9 vols. (Leiden, 1841-1861); G. Groen van Prinsterer, - _Maurice et Barneveldt_ (Utrecht, 1875); J. L. Motley, _Life and Death - of John of Barneveldt_ (2 vols., The Hague, 1894); C. M. Kemp, v.d. - _Maurits v. Nassau, prins v. Oranje in zijn leven en verdiensten_ (4 - vols., Rotterdam, 1845); M. O. Nutting, _The Days of Prince Maurice_ - (Boston and Chicago, 1894). - - - - -MAURISTS, a congregation of French Benedictines called after St Maurus -(d. 565), a disciple of St Benedict and the legendary introducer of the -Benedictine rule and life into Gaul.[1] At the end of the 16th century -the Benedictine monasteries of France had fallen into a state of -disorganization and relaxation. In the abbey of St Vaune near Verdun a -reform was initiated by Dom Didier de la Cour, which spread to other -houses in Lorraine, and in 1604 the reformed congregation of St Vaune -was established, the most distinguished members of which were Ceillier -and Calmet. A number of French houses joined the new congregation; but -as Lorraine was still independent of the French crown, it was considered -desirable to form on the same lines a separate congregation for France. -Thus in 1621 was established the famous French congregation of St Maur. -Most of the Benedictine monasteries of France, except those belonging to -Cluny, gradually joined the new congregation, which eventually embraced -nearly two hundred houses. The chief house was Saint-Germain-des-Pres, -Paris, the residence of the superior-general and centre of the literary -activity of the congregation. The primary idea of the movement was not -the undertaking of literary and historical work, but the return to a -strict monastic regime and the faithful carrying out of Benedictine -life; and throughout the most glorious period of Maurist history the -literary work was not allowed to interfere with the due performance of -the choral office and the other duties of the monastic life. Towards the -end of the 18th century a tendency crept in, in some quarters, to relax -the monastic observances in favour of study; but the constitutions of -1770 show that a strict monastic regime was maintained until the end. -The course of Maurist history and work was checkered by the -ecclesiastical controversies that distracted the French Church during -the 17th and 18th centuries. Some of the members identified themselves -with the Jansenist cause; but the bulk, including nearly all the -greatest names, pursued a middle path, opposing the lax moral theology -condemned in 1679 by Pope Innocent XI., and adhering to those strong -views on grace and predestination associated with the Augustinian and -Thomist schools of Catholic theology; and like all the theological -faculties and schools on French soil, they were bound to teach the four -Gallican articles. It seems that towards the end of the 18th century a -rationalistic and free-thinking spirit invaded some of the houses. The -congregation was suppressed and the monks scattered at the revolution, -the last superior-general with forty of his monks dying on the scaffold -in Paris. The present French congregation of Benedictines initiated by -Dom Gueranger in 1833 is a new creation and has no continuity with the -congregation of St Maur. - -The great claim of the Maurists to the gratitude and admiration of -posterity is their historical and critical school, which stands quite -alone in history, and produced an extraordinary number of colossal works -of erudition which still are of permanent value. The foundations of this -school were laid by Dom Tarisse, the first superior-general, who in 1632 -issued instructions to the superiors of the monasteries to train the -young monks in the habits of research and of organized work. The -pioneers in production were Menard and d'Achery. - - The following tables give, divided into groups, the most important - Maurist works, along with such information as may be useful to - students. All works are folio when not otherwise noted:-- - - I.--THE EDITIONS OF THE FATHERS - - Epistle of Barnabas Menard 1645 1 in 4^to - (editio princeps) - Lanfranc d'Achery 1648 1 - Guibert of Nogent d'Achery 1651 1 - Robert Pulleyn and Peter - of Poitiers Mathou 1655 1 - Bernard Mabillon 1667 2 - Anselm Gerberon 1675 1 - Cassiodorus Garet 1679 1 - Augustine (see Kukula, Delfau, Blampin, - _Die Mauriner-Ausgabe Coustant, Guesnie 1681-1700 11 - des Augustinus_, 1898) - Ambrose du Frische 1686-1690 2 - Acta martyrum sincera Ruinart 1689 1 - Hilary Coustant 1693 1 - Jerome Martianay 1693-1706 5 - Athanasius Loppin and Mont- - faucon 1698 3 - Gregory of Tours Ruinart 1699 1 - Gregory the Great Sainte-Marthe 1705 4 - Hildebert of Tours Beaugendre 1708 1 - Irenaeus Massuet 1710 1 - Chrysostom Montfaucon 1718-1738 13 - Cyril of Jerusalem Touttee and Maran 1720 1 - Epistolae romanorum Coustant 1721 1 - pontificum[2] - Basil Garnier and Maran 1721-1730 3 - Cyprian (Baluze, not a - Maurist) finished - by Maran 1726 1 - Origen Ch. de la Rue (1, 2, - 3) V. de la Rue (4) 1733-1759 4 - Justin and the Apologists Maran 1742 1 - Gregory Nazianzen[3] Maran and Clemencet 1778 1 - - II.--BIBLICAL WORKS - - St Jerome's Latin Bible Martianay 1693 1 - Origen's Hexapla Montfaucon 1713 2 - Old Latin versions Sabbathier 1743-1749 3 - - III.--GREAT COLLECTIONS OF DOCUMENTS - - Spicilegium d'Achery 1655-1677 13 in 4^to - Veterae analecta Mabillon 1675-1685 4 in 8^vo - Musaeum italicum Mabillon 1687-1689 2 in 4^to - Collectio nova patrum Montfaucon 1706 2 - graecorum - Thesaurus novus Martene and Durand 1717 5 - anecdotorum - Veterum scriptorum Martene and Durand 1724-1733 9 - collectio - De antiquis Martene 1690-1706 - ecclesiaeritibus (Final form) 1736-1738 4 - - IV.--MONASTIC HISTORY - - Acta of the Benedictine d'Achery, Mabillon - Saints and Ruinart 1668-1701 9 - Benedictine Annals (to Mabillon (1-4), - 1157) Massuet (5), - Martene (6) 1703-1739 6 - - V.--ECCLESIASTICAL HISTORY AND ANTIQUITIES OF FRANCE - - A.--_General._ - - Gallia Christiana (3 other Sainte-Marthe - vols. were published (1, 2, 3) 1715-1785 13 - 1856-1865) - Monuments de la monarchie Montfaucon 1729-1733 5 - francaise - Histoire litteraire de la Rivet, Clemencet, - France (16 other vols. Clement 1733-1763 12 in 4^to - were published 1814-1881) - Recueil des historiens de Bouquet (1-8), Brial - la France (4 other vols. (12-19) 1738-1833 19 - were published 1840-1876) - Concilia Galliae (the Labbat 1789 1 - printing of vol. ii. was - interrupted by the - Revolution; there were - to have been 8 vols.) - - B.--HISTORIES OF THE PROVINCES. - - Bretagne Lobineau 1707 2 - Paris Felibien and - Lobineau 1725 5 - Languedoc Vaissette and de Vic 1730-1745 5 - Bourgogne Plancher (1-3), 1739-1748 4 - Merle (4) 1781 - Bretagne Morice 1742-1756 5 - - VI.--MISCELLANEOUS WORKS OF TECHNICAL ERUDITION - - De re diplomatica Mabillon 1681 1 - Ditto Supplement Mabillon 1704 1 - Nouveau traite de Toustain and Tassin 1750-1765 6 in 4^to - diplomatique - Paleographia graeca Montfaucon 1708 1 - Bibliotheca coisliniana Montfaucon 1715 1 - Bibliotheca bibliothecarum Montfaucon 1739 2 - manuscriptorum nova - L'Antiquite explique Montfaucon 1719-1724 15 - New ed. of Du Cange's Dantine and - glossarium Carpentier 1733-1736 6 - Ditto Supplement Carpentier 1766 4 - Apparatus ad bibliothecam le Nourry 1703 2 - maximam patrum - L'Art de verifier les Dantine, Durand, - dates Clemencet 1750 1 in 4^to - Ed. 2 Clement 1770 1 - Ed. 3 Clement 1783-1787 3 - - The 58 works in the above list comprise 199 great folio volumes and 39 - in 4^to or 8^vo. The full Maurist bibliography contains the names of - some 220 writers and more than 700 works. The lesser works in large - measure cover the same fields as those in the list, but the number of - works of purely religious character, of piety, devotion and - edification, is very striking. Perhaps the most wonderful phenomenon - of Maurist work is that what was produced was only a portion of what - was contemplated and prepared for. The French Revolution cut short - many gigantic undertakings, the collected materials for which fill - hundreds of manuscript volumes in the Bibliotheque nationale of Paris - and other libraries of France. There are at Paris 31 volumes of - Berthereau's materials for the Historians of the Crusades, not only in - Latin and Greek, but in the oriental tongues; from them have been - taken in great measure the _Recueil des historiens des croisades_, - whereof 15 folio volumes have been published by the Academie des - Inscriptions. There exist also the preparations for an edition of - Rufinus and one of Eusebius, and for the continuation of the Papal - Letters and of the Concilia Galliae. Dom Caffiaux and Dom Villevielle - left 236 volumes of materials for a _Tresor genealogique_. There are - Benedictine Antiquities (37 vols.), a Monasticon Gallicanum and a - Monasticon Benedictinum (54 vols.). Of the Histories of the Provinces - of France barely half a dozen were printed, but all were in hand, and - the collections for the others fill 800 volumes of MSS. The materials - for a geography of Gaul and France in 50 volumes perished in a fire - during the Revolution. - - When these figures were considered, and when one contemplates the - vastness of the works in progress during any decade of the century - 1680-1780; and still more, when not only the quantity but the quality - of the work, and the abiding value of most of it is realized, it will - be recognized that the output was prodigious and unique in the history - of letters, as coming from a single society. The qualities that have - made Maurist work proverbial for sound learning are its fine critical - tact and its thoroughness. - - The chief source of information on the Maurists and their work is Dom - Tassin's _Histoire litteraire de la congregation de Saint-Maur_ - (1770); it has been reduced to a bare bibliography and completed by de - Lama, _Bibliotheque des ecrivains de la congr. de S.-M._ (1882). The - two works of de Broglie, _Mabillon_ (2 vols., 1888) and _Montfaucon_ - (2 vols., 1891), give a charming picture of the inner life of the - great Maurists of the earlier generation in the midst of their work - and their friends. Sketches of the lives of a few of the chief - Maurists will be found in McCarthy's _Principal Writers of the Congr. - of S. M._ (1868). Useful information about their literary undertakings - will be found in De Lisle's _Cabinet des MSS. de la Bibl. Nat. Fonds - St Germain-des-Pres_. General information will be found in the - standard authorities: Helyot, _Hist. des ordres religieux_ (1718), vi. - c. 37; Heimbucher, _Orden und Kongregationen_ (1907) i. S 36; Wetzer - und Welte, Kirchenlexicon (ed. 2) and Herzog-Hauck's - _Realencyklopadie_ (ed. 3), the latter an interesting appreciation by - the Protestant historian Otto Zockler of the spirit and the merits of - the work of the Maurists. (E. C. B.) - - -FOOTNOTES: - - [1] His festival is kept on the 15th of January. He founded the - monastery of Glanfeuil or St Maur-sur-Loire. - - [2] 14 vols. of materials collected for the continuation are at - Paris. - - [3] The printing of vol. ii. was impeded by the Revolution. - - - - -MAURITIUS, an island and British colony in the Indian Ocean (known -whilst a French possession as the _Ile de France_). It lies between 57 -deg. 18' and 57 deg. 49' E., and 19 deg. 58' and 20 deg. 32' S., 550 m. -E. of Madagascar, 2300 m. from the Cape of Good Hope, and 9500 m. from -England via Suez. The island is irregularly elliptical--somewhat -triangular--in shape, and is 36 m. long from N.N.E. to S.S.W., and about -23 m. broad. It is 130 m. in circumference, and its total area is about -710 sq. m. (For map see MADAGASCAR.) The island is surrounded by coral -reefs, so that the ports are difficult of access. - -From its mountainous character Mauritius is a most picturesque island, -and its scenery is very varied and beautiful. It has been admirably -described by Bernardin de St Pierre, who lived in the island towards the -close of the 18th century, in _Paul et Virginie_. The most level -portions of the coast districts are the north and north-east, all the -rest being broken by hills, which vary from 500 to 2700 ft. in height. -The principal mountain masses are the north-western or Pouce range, in -the district of Port Louis; the south-western, in the districts of -Riviere Noire and Savanne; and the south-eastern range, in the Grand -Port district. In the first of these, which consists of one principal -ridge with several lateral spurs, overlooking Port Louis, are the -singular peak of the Pouce (2650 ft.), so called from its supposed -resemblance to the human thumb; and the still loftier Pieter Botte (2685 -ft.), a tall obelisk of bare rock, crowned with a globular mass of -stone. The highest summit in the island is in the south-western mass of -hills, the Piton de la Riviere Noire, which is 2711 ft. above the sea. -The south-eastern group of hills consists of the Montagne du Bambou, -with several spurs running down to the sea. In the interior are -extensive fertile plains, some 1200 ft. in height, forming the districts -of Moka, Vacois, and Plaines Wilhelms; and from nearly the centre of the -island an abrupt peak, the Piton du Milieu de l'Ile rises to a height of -1932 ft. Other prominent summits are the Trois Mamelles, the Montagne du -Corps de Garde, the Signal Mountain, near Port Louis, and the Morne -Brabant, at the south-west corner of the island. - -The rivers are small, and none is navigable beyond a few hundred yards -from the sea. In the dry season little more than brooks, they become -raging torrents in the wet season. The principal stream is the Grande -Riviere, with a course of about 10 m. There is a remarkable and very -deep lake, called Grand Bassin, in the south of the island, it is -probably the extinct crater of an ancient volcano; similar lakes are the -Mare aux Vacois and the Mare aux Joncs, and there are other deep hollows -which have a like origin. - - _Geology._--The island is of volcanic origin, but has ceased to show - signs of volcanic activity. All the rocks are of basalt and - greyish-tinted lavas, excepting some beds of upraised coral. Columnar - basalt is seen in several places. The remains of ancient craters can - be distinguished, but their outlines have been greatly destroyed by - denudation. There are many caverns and steep ravines, and from the - character of the rocks the ascents are rugged and precipitous. The - island has few minerals, although iron, lead and copper in very small - quantities have in former times been obtained. The greater part of the - surface is composed of a volcanic breccia, with here and there - lava-streams exposed in ravines, and sometimes on the surface. The - commonest lavas are dolerites. In at least two places sedimentary - rocks are found at considerable elevations. In the Black River - Mountains, at a height of about 1200 ft., there is a clay-slate; and - near Midlands, in the Grand Port group of mountains, a chloritic - schist occurs about 1700 ft. above the sea, forming the hill of La - Selle. This schist is much contorted, but seems to have a general dip - to the south or south-east. Evidence of recent elevation of the island - is furnished by masses of coral reef and beach coral rock standing at - heights of 40 ft. above sea-level in the south, 12 ft. in the north - and 7 ft. on the islands situated on the bank extending to the - north-east.[1] - - _Climate._--The climate is pleasant during the cool season of the - year, but oppressively hot in summer (December to April), except in - the elevated plains of the interior, where the thermometer ranges from - 70 deg. to 80 deg. F., while in Port Louis and on the coast generally - it ranges from 90 deg. to 96 deg. The mean temperature for the year at - Port Louis is 78.6 deg. There are two seasons, the cool and - comparatively dry season, from April to November, and the hotter - season, during the rest of the year. The climate is now less healthy - than it was, severe epidemics of malarial fever having frequently - occurred, so that malaria now appears to be endemic among the - non-European population. The rainfall varies greatly in different - parts of the island. Cluny in the Grand Port (south-eastern) district - has a mean annual rainfall of 145 in.; Albion on the west coast is the - driest station, with a mean annual rainfall of 31 in. The mean monthly - rainfall for the whole island varies from 12 in. in March to 2.6 in. - in September and October. The Royal Alfred Observatory is situated at - Pamplemousses, on the north-west or dry side of the island. From - January to the middle of April, Mauritius, in common with the - neighbouring islands and the surrounding ocean from 8 deg. to 30 deg. - of southern latitude is subject to severe cyclones, accompanied by - torrents of rain, which often cause great destruction to houses and - plantations. These hurricanes generally last about eight hours, but - they appear to be less frequent and violent than in former times, - owing, it is thought, to the destruction of the ancient forests and - the consequent drier condition of the atmosphere. - - _Fauna and Flora._--Mauritius being an oceanic island of small size, - its present fauna is very limited in extent. When first seen by - Europeans it contained no mammals except a large fruit-eating bat - (_Pteropus vulgaris_), which is plentiful in the woods; but several - mammals have been introduced, and are now numerous in the uncultivated - region. Among these are two monkeys of the genera _Macacus_ and - _Cercopithecus_, a stag (_Cervus hippelaphus_), a small hare, a - shrew-mouse, and the ubiquitous rat. A lemur and one of the curious - hedgehog-like _Insectivora_ of Madagascar (_Centetes ecaudatus_) have - probably both been brought from the larger island. The avifauna - resembles that of Madagascar; there are species of a peculiar genus of - caterpillar shrikes (_Campephagidae_), as well as of the genera - _Pratincola_, _Hypsipetes_, _Phedina_, _Tchitrea_, _Zosterops_, - _Foudia_, _Collocalia_ and _Coracopsis_, and peculiar forms of doves - and parakeets. The living reptiles are small and few in number. The - surrounding seas contain great numbers of fish; the coral reefs abound - with a great variety of molluscs; and there are numerous land-shells. - The extinct fauna of Mauritius has considerable interest. In common - with the other Mascarene islands, it was the home of the dodo (_Didus - ineptus_); there were also _Aphanapteryx_, a species of rail, and a - short-winged heron (_Ardea megacephala_), which probably seldom flew. - The defenceless condition of these birds led to their extinction after - the island was colonized. Considerable quantities of the bones of the - dodo and other extinct birds--a rail (_Aphanapteryx_), and a - short-winged heron--have been discovered in the beds of some of the - ancient lakes (see DODO). Several species of large fossil tortoises - have also been discovered; they are quite different from the living - ones of Aldabra, in the same zoological region. - - Owing to the destruction of the primeval forests for the formation of - sugar plantations, the indigenous flora is only seen in parts of the - interior plains, in the river valleys and on the hills; and it is not - now easy to distinguish between what is native and what has come from - abroad. The principal timber tree is the ebony (_Diospyros ebeneum_), - which grows to a considerable size. Besides this there are bois de - cannelle, olive-tree, benzoin (_Croton Benzoe_), colophane - (_Colophonia_), and iron-wood, all of which arc useful in carpentry; - the coco-nut palm, an importation, but a tree which has been so - extensively planted during the last hundred years that it is extremely - plentiful; the palmiste (_Palma dactylifera latifolia_), the latanier - (_Corypha umbraculifera_) and the date-palm. The vacoa or vacois, - (_Pandanus utilis_) is largely grown, the long tough leaves being - manufactured into bags for the export of sugar, and the roots being - also made of use; and in the few remnants of the original forests the - traveller's tree (_Urania speciosa_), grows abundantly. A species of - bamboo is very plentiful in the river valleys and in marshy - situations. A large variety of fruit is produced, including the - tamarind, mango, banana, pine-apple, guava, shaddock, fig, - avocado-pear, litchi, custard-apple and the mabolo (_Diospyros - discolor_), a fruit of exquisite flavour, but very disagreeable odour. - Many of the roots and vegetables of Europe have been introduced, as - well as some of those peculiar to the tropics, including maize, - millet, yams, manioc, dhol, gram, &c. Small quantities of tea, rice - and sago, have been grown, as well as many of the spices (cloves, - nutmeg, ginger, pepper and allspice), and also cotton, indigo, betel, - camphor, turmeric and vanilla. The Royal Botanical Gardens at - Pamplemousses, which date from the French occupation of the island, - contain a rich collection of tropical and extra-tropical species. - -_Inhabitants._--The inhabitants consist of two great divisions, those of -European blood, chiefly French and British, together with numerous -half-caste people, and those of Asiatic or African blood. The population -of European blood, which calls itself Creole, is greater than that of -any other tropical colony; many of the inhabitants trace their descent -from ancient French families, and the higher and middle classes are -distinguished for their intellectual culture. French is more commonly -spoken than English. The Creole class is, however, diminishing, though -slowly, and the most numerous section of the population is of Indian -blood. - - The introduction of Indian coolies to work the sugar plantations dates - from the period of the emancipation of the slaves in 1834-1839. At - that time the negroes who showed great unwillingness to work on their - late masters' estates, numbered about 66,000. Immigration from India - began in 1834, and at a census taken in 1846, when the total - population was 158,462, there were already 56,245 Indians in the - island. In 1851 the total population had increased to 180,823, while - in 1861 it was 310,050. This great increase was almost entirely due to - Indian immigration, the Indian population, 77,996 in 1851, being - 192,634 in 1861. From that year the increase in the Indian population - has been more gradual but steady, while the non-Indian population has - decreased. From 102,827 in 1851 it rose to 117,416 in 1861 to sink to - 99,784 in 1871. The figures for the three following census years - were:-- - - 1881. 1891. 1901. - - Indians 248,993 255,920 259,086 - Others 110,881 114,668 111,937 - ------- ------- ------- - Total 359,874 370,588 371,023 - ------- ------- ------- - - Including the military and crews of ships in harbour, the total - population in 1901 was 373,336.[2] This total included 198,958 - Indo-Mauritians, i.e. persons of Indian descent born in Mauritius, and - 62,022 other Indians. There were 3,509 Chinese, while the remaining - 108,847 included persons of European, African or mixed descent, - Malagasy, Malays and Sinhalese. The Indian female population increased - from 51,019 in 1861 to 115,986 in 1901. In the same period the - non-Indian female population but slightly varied, being 56,070 in 1861 - and 55,485 in 1901. The Indo-Mauritians are now dominant in - commercial, agricultural and domestic callings, and much town and - agricultural land has been transferred from the Creole planters to - Indians and Chinese. The tendency to an Indian peasant proprietorship - is marked. Since 1864 real property to the value of over L1,250,000 - has been acquired by Asiatics. Between 1881 and 1901 the number of - sugar estates decreased from 171 to 115, those sold being held in - small parcels by Indians. The average death-rate for the period - 1873-1901 was 32.6 per 1000. The average birth-rate in the Indian - community is 37 per 1000; in the non-Indian community 34 per 1000. - Many Mauritian Creoles have emigrated to South Africa. The great - increase in the population since 1851 has made Mauritius one of the - most densely peopled regions of the world, having over 520 persons per - square mile. - - _Chief Towns._--The capital and seat of government, the city of Port - Louis, is on the north-western side of the island, in 20 deg. 10' S., - 57 deg. 30' E. at the head of an excellent harbour, a deep inlet about - a mile long, available for ships of the deepest draught. This is - protected by Fort William and Fort George, as well as by the citadel - (Fort Adelaide), and it has three graving-docks connected with the - inner harbour, the depths alongside quays and berths being from 12 to - 28 ft. The trade of the island passes almost entirely through the - port. Government House is a three-storeyed structure with broad - verandas, of no particular style of architecture, while the - Protestant cathedral was formerly a powder magazine, to which a tower - and spire have been added. The Roman Catholic cathedral is more - pretentious in style, but is tawdry in its interior. There are, - besides the town-hall, Royal College, public offices and theatre, - large barracks and military stores. Port Louis, which is governed by - an elective municipal council, is surrounded by lofty hills and its - unhealthy situation is aggravated by the difficulty of effective - drainage owing to the small amount of tide in the harbour. Though much - has been done to make the town sanitary, including the provision of a - good water-supply, the death-rate is generally over 44 per 1000. - Consequently all those who can make their homes in the cooler uplands - of the interior. As a result the population of the city decreased from - about 70,000 in 1891 to 53,000 in 1901. The favourite residential town - is Curepipe, where the climate resembles that of the south of France. - It is built on the central plateau about 20 m. distant from Port Louis - by rail and 1800 ft. above the sea. Curepipe was incorporated in 1888 - and had a population (1901) of 13,000. On the railway between Port - Louis and Curepipe are other residential towns--Beau Bassin, Rose Hill - and Quatre Bornes. Mahebourg, pop. (1901), 4810, is a town on the - shores of Grand Port on the south-east side of the island, Souillac a - small town on the south coast. - - _Industries.--The Sugar Plantations:_ The soil of the island is of - considerable fertility; it is a ferruginous red clay, but so largely - mingled with stones of all sizes that no plough can be used, and the - hoe has to be employed to prepare the ground for cultivation. The - greater portion of the plains is now a vast sugar plantation. The - bright green of the sugar fields is a striking feature in a view of - Mauritius from the sea, and gives a peculiar beauty and freshness to - the prospect. The soil is suitable for the cultivation of almost all - kinds of tropical produce, and it is to be regretted that the - prosperity of the colony depends almost entirely on one article of - production, for the consequences are serious when there is a failure, - more or less, of the sugar crop. Guano is extensively imported as a - manure, and by its use the natural fertility of the soil has been - increased to a wonderful extent. Since the beginning of the 20th - century some attention has been paid to the cultivation of tea and - cotton, with encouraging results. Of the exports, sugar amounts on an - average to about 95% of the total. The quantity of sugar exported rose - from 102,000 tons in 1854 to 189,164 tons in 1877. The competition of - beet-sugar and the effect of bounties granted by various countries - then began to tell on the production in Mauritius, the average crop - for the seven years ending 1900-1901 being only 150,449 tons. The - Brussels Sugar Convention of 1902 led to an increase in production, - the average annual weight of sugar exported for the three years - 1904-1906 being 182,000 tons. The value of the crop was likewise - seriously affected by the causes mentioned, and by various diseases - which attacked the canes. Thus in 1878 the value of the sugar exported - was L3,408,000; in 1888 it had sunk to L1,911,000, and in 1898 to - L1,632,000. In 1900 the value was L1,922,000, and in 1905 it had risen - to L2,172,000. India and the South African colonies between them take - some two-thirds of the total produce. The remainder is taken chiefly - by Great Britain, Canada and Hong-Kong. Next to sugar, aloe-fibre is - the most important export, the average annual export for the five - years ending 1906 being 1840 tons. In addition, a considerable - quantity of molasses and smaller quantities of rum, vanilla and - coco-nut oil are exported. The imports are mainly rice, wheat, cotton - goods, wine, coal, hardware and haberdashery, and guano. The rice - comes principally from India and Madagascar; cattle are imported from - Madagascar, sheep from South Africa and Australia, and frozen meat - from Australia. The average annual value of the exports for the ten - years 1896-1905 was L2,153,159; the average annual value of the - imports for the same period L1,453,089. These figures when compared - with those in years before the beet and bounty-fed sugar had entered - into severe competition with cane sugar, show how greatly the island - had thereby suffered. In 1864 the exports were valued at L2,249,000; - in 1868 at L2,339,000; in 1877 at L4,201,000 and in 1880 at - L3,634,000. And in each of the years named the imports exceeded - L2,000,000 in value. Nearly all the aloe-fibre exported is taken by - Great Britain, and France, while the molasses goes to India. Among the - minor exports is that of _bambara_ or sea-slugs, which are sent to - Hong-Kong and Singapore. This industry is chiefly in Chinese hands. - The great majority of the imports are from Great Britain or British - possessions. - - The currency of Mauritius is rupees and cents of a rupee, the Indian - rupee (= 16d.) being the standard unit. The metric system of weights - and measures has been in force since 1878. - - _Communications._--There is a regular fortnightly steamship service - between Marseilles and Port Louis by the Messageries Maritimes, a - four-weekly service with Southampton via Cape Town by the Union - Castle, and a four-weekly service with Colombo direct by the British - India Co.'s boats. There is also frequent communication with - Madagascar, Reunion and Natal. The average annual tonnage of ships - entering Port Louis is about 750,000 of which five-sevenths is - British. Cable communication with Europe, via the Seychelles, Zanzibar - and Aden, was established in 1893, and the Mauritius section of the - Cape-Australian cable, via Rodriguez, was completed in 1902. - - Railways connect all the principal places and sugar estates on the - island, that known as the Midland line, 36 miles long, beginning at - Port Louis crosses the island to Mahebourg, passing through Curepipe, - where it is 1822 ft. above the sea. There are in all over 120 miles of - railway, all owned and worked by the government. The first railway was - opened in 1864. The roads are well kept and there is an extensive - system of tramways for bringing produce from the sugar estates to the - railway lines. Traction engines are also largely used. There is a - complete telegraphic and telephonic service. - -_Government and Revenue._--Mauritius is a crown colony. The governor is -assisted by an executive council of five official and two elected -members, and a legislative council of 27 members, 8 sitting _ex -officio_, 9 being nominated by the governor and 10 elected on a moderate -franchise. Two of the elected members represent St Louis, the 8 rural -districts into which the island is divided electing each one member. At -least one-third of the nominated members must be persons not holding any -public office. The number of registered electors in 1908 was 6186. The -legislative session usually lasts from April to December. Members may -speak either in French or English. The average annual revenue of the -colony for the ten years 1896-1905, was L608,245, the average annual -expenditure during the same period L663,606. Up to 1854 there was a -surplus in hand, but since that time expenditure has on many occasions -exceeded income, and the public debt in 1908 was L1,305,000, mainly -incurred however on reproductive works. - -The island has largely retained the old French laws, the _codes civil_, -_de procedure_, _du commerce_, and _d'instruction criminelle_ being -still in force, except so far as altered by colonial ordinances. A -supreme court of civil and criminal justice was established in 1831 -under a chief judge and three puisne judges. - - _Religion and Education._--The majority of the European inhabitants - belong to the Roman Catholic faith. They numbered at the 1901 census - 117,102, and the Protestants 6644. Anglicans, Roman Catholics and the - Church of Scotland are helped by state grants. At the head of the - Anglican community is the bishop of Mauritius; the chief Romanist - dignitary is styled bishop of Port Louis. The Mahommedans number over - 30,000, but the majority of the Indian coolies are Hindus. - - The educational system, as brought into force in 1900, is under a - director of public instruction assisted by an advisory committee, and - consists of two branches (1) superior or secondary instruction, (2) - primary instruction. For primary instruction there are government - schools and schools maintained by the Roman Catholics, Protestants and - other faiths, to which the government gives grants in aid. In 1908 - there were 67 government schools with 8400 scholars and 90 grant - schools with 10,200 scholars, besides Hindu schools receiving no - grant. The Roman Catholic scholars number 67.72%; the Protestants - 3.80%; Mahommedans 8.37%; and Hindus and others 20.11%. Secondary and - higher education is given in the Royal College and associated schools - at Port Louis and Curepipe. - - _Defence._--Mauritius occupies an important strategic position on the - route between South Africa and India and in relation to Madagascar and - East Africa, while in Port Louis it possesses one of the finest - harbours in the Indian Ocean. A permanent garrison of some 3000 men is - maintained in the island at a cost of about L180,000 per annum. To the - cost of the troops Mauritius contributes 5(1/2)% of its annual - revenue--about L30,000. - -_History._--Mauritius appears to have been unknown to European nations, -if not to all other peoples, until the year 1505, when it was discovered -by Mascarenhas, a Portuguese navigator. It had then no inhabitants, and -there seem to be no traces of a previous occupation by any people. The -island was retained for most of the 16th century by its discoverers, but -they made no settlements in it. In 1598 the Dutch took possession, and -named the island "Mauritius," in honour of their stadtholder, Count -Maurice of Nassau. It had been previously called by the Portuguese "Ilha -do Cerne," from the belief that it was the island so named by Pliny. But -though the Dutch built a fort at Grand Port and introduced a number of -slaves and convicts, they made no permanent settlement in Mauritius, -finally abandoning the island in 1710. From 1715 to 1767 (when the -French government assumed direct control) the island was held by agents -of the French East India Company, by whom its name was again changed to -"Ile de France." The Company was fortunate in having several able men as -governors of its colony, especially the celebrated Mahe de Labourdonnais -(q.v.), who made sugar planting the main industry of the -inhabitants.[3] Under his direction roads were made, forts built, and -considerable portions of the forest were cleared, and the present -capital, Port Louis, was founded. Labourdonnais also promoted the -planting of cotton and indigo, and is remembered as the most enlightened -and best of all the French governors. He also put down the maroons or -runaway slaves who had long been the pest of the island. The colony -continued to rise in value during the time it was held by the French -crown, and to one of the intendants,[4] Pierre Poivre, was due the -introduction of the clove, nutmeg and other spices. Another governor was -D'Entrecasteaux, whose name is kept in remembrance by a group of islands -east of New Guinea. - -During the long war between France and England, at the commencement of -the 19th century, Mauritius was a continual source of much mischief to -English Indiamen and other merchant vessels; and at length the British -government determined upon an expedition for its capture. This was -effected in 1810; and upon the restoration of peace in 1814 the -possession of the island was confirmed to Britain by the Treaty of -Paris. By the eighth article of capitulation it was agreed that the -inhabitants should retain their own laws, customs, and religion; and -thus the island is still largely French in language, habits, and -predilections; but its name has again been changed to that given by the -Dutch. One of the most distinguished of the British governors was Sir -Robert Farquhar (1810-1823), who did much to abolish the Malagasy slave -trade and to establish friendly relations with the rising power of the -Hova sovereign of Madagascar. Later governors of note were Sir Henry -Barkly (1863-1871), and Sir J. Pope Hennessy (1883-1886 and 1888). - -The history of the colony since its acquisition by Great Britain has -been one of social and political evolution. At first all power was -concentrated in the hands of the governor, but in 1832 a legislative -council was constituted on which non-official nominated members served. -In 1884-1885 this council was transformed into a partly elected body. Of -more importance than the constitutional changes were the economic -results which followed the freeing of the slaves (1834-1839)--for the -loss of whose labour the planters received over L2,000,000 compensation. -Coolies were introduced to supply the place of the negroes, immigration -being definitely sanctioned by the government of India in 1842. Though -under government control the system of coolie labour led to many abuses. -A royal commission investigated the matter in 1871 and since that time -the evils which were attendant on the system have been gradually -remedied. One result of the introduction of free labour has been to -reduce the descendants of the slave population to a small and -unimportant class--Mauritius in this respect offering a striking -contrast to the British colonies in the West Indies. The last half of -the 19th century was, however, chiefly notable in Mauritius for the -number of calamities which overtook the island. In 1854 cholera caused -the death of 17,000 persons; in 1867 over 30,000 people died of malarial -fever; in 1892 a hurricane of terrific violence caused immense -destruction of property and serious loss of life; in 1893 a great part -of Port Louis was destroyed by fire. There were in addition several -epidemics of small-pox and plague, and from about 1880 onward the -continual decline in the price of sugar seriously affected the -islanders, especially the Creole population. During 1902-1905 an -outbreak of surra, which caused great mortality among draught animals, -further tried the sugar planters and necessitated government help. -Notwithstanding all these calamities the Mauritians, especially the -Indo-Mauritians, have succeeded in maintaining the position of the -colony as an important sugar-producing country. - - _Dependencies._--Dependent upon Mauritius and forming part of the - colony are a number of small islands scattered over a large extent of - the Indian Ocean. Of these the chief is Rodriguez (q.v.), 375 m. east - of Mauritius. Considerably north-east of Rodriguez lie the Oil Islands - or Chagos archipelago, of which the chief is Diego Garcia (see - CHAGOS). The Cargados, Carayos or St Brandon islets, deeps and shoals, - lie at the south end of the Nazareth Bank about 250 m. N.N.E. of - Mauritius. Until 1903 the Seychelles, Amirantes, Aldabra and other - islands lying north of Madagascar were also part of the colony of - Mauritius. In the year named they were formed into a separate colony - (see SEYCHELLES). Two islands, Farquhar and Coetivy, though - geographically within the Seychelles area, remained dependent on - Mauritius, being owned by residents in that island. In 1908, however, - Coetivy was transferred to the Seychelles administration. Amsterdam - and St Paul, uninhabited islands in the South Indian Ocean, included - in an official list of the dependencies of Mauritius drawn up in 1880, - were in 1893 annexed by France. The total population of the - dependencies of Mauritius was estimated in 1905 at 5400. - - AUTHORITIES.--F. Leguat, _Voyages et aventures en deux isles desertes - des Indes orientales_ (Eng. trans., _A New Voyage to the East Indies_; - London, 1708); Prudham, "England's Colonial Empire," vol. i., _The - Mauritius and its Dependencies_ (1846); C. P. Lucas, _A Historical - Geography of the British Colonies_, vol. i. (Oxford, 1888); Ch. Grant, - _History of Mauritius, or the Isle of France and Neighbouring Islands_ - (1801); J. Milbert, _Voyage pittoresque a l'Ile-de-France, &c._, 4 - vols. (1812); Aug. Billiard, _Voyage aux colonies orientales_ (1822); - P. Beaton, _Creoles and Coolies, or Five Years in Mauritius_ (1859); - Paul Chasteau, _Histoire et description de l'ile Maurice_ (1860); F. - P. Flemyng, _Mauritius, or the Isle of France_ (1862); Ch. J. Boyle, - _Far Away, or Sketches of Scenery and Society in Mauritius_ (1867); L. - Simonin, _Les Pays lointains, notes de voyage (Maurice, &c.)_ (1867); - N. Pike, _Sub-Tropical Rambles in the Land of the Aphanapteryx_ - (1873); A. R. Wallace. "The Mascarene Islands," in ch. xi. vol. i. of - _The Geographical Distribution of Animals_ (1876); K. Mobius, F. - Richter and E. von Martens, _Beitrage zur Meeresfauna der Insel - Mauritius und der Seychellen_ (Berlin, 1880); G. Clark, _A Brief - Notice of the Fauna of Mauritius_ (1881); A. d'Epinay, _Renseignements - pour servir a l'histoire de l'Ile de France jusqu'a 1810_ (Mauritius, - 1890); N. Decotter, _Geography of Mauritius and its Dependencies_ - (Mauritius, 1892); H. de Haga Haig, "The Physical Features and Geology - of Mauritius" in vol. li., _Q. J. Geol. Soc._ (1895); the Annual - Reports on Mauritius issued by the Colonial Office, London; _The - Mauritius Almanack_ published yearly at Port Louis. A map of the - island in six sheets on the scale of one inch to a mile was issued by - the War Office in 1905. (J. Si.*) - - -FOOTNOTES: - - [1] See _Geog. Journ._ (June 1895), p. 597. - - [2] The total population of the colony (including dependencies) on - the 1st of January 1907 was estimated at 383,206. - - [3] Labourdonnais is credited by several writers with the - introduction of the sugar cane into the island. Leguat, however, - mentions it as being cultivated during the Dutch occupation. - - [4] The regime introduced in 1767 divided the administration between - a governor, primarily charged with military matters, and an - intendant. - - - - -MAURY, JEAN SIFFREIN (1746-1817), French cardinal and archbishop of -Paris, the son of a poor cobbler, was born on the 26th of June 1746 at -Valreas in the Comtat-Venaissin, the district in France which belonged -to the pope. His acuteness was observed by the priests of the seminary -at Avignon, where he was educated and took orders. He tried his fortune -by writing _eloges_ of famous persons, then a favourite practice; and in -1771 his _eloge_ on Fenelon was pronounced next best to Laharpe's by the -Academy. The real foundation of his fortunes was the success of a -panegyric on St Louis delivered before the Academy in 1772, which caused -him to be recommended for an abbacy. In 1777 he published under the -title of _Discours choisis_ his panegyrics on Saint Louis, Saint -Augustine and Fenelon, his remarks on Bossuet and his _Essai sur -l'eloquence de la chaire_, a volume which contains much good criticism, -and remains a French classic. The book was often reprinted as _Principes -de l'eloquence_. He became a favourite preacher in Paris, and was Lent -preacher at court in 1781, when King Louis XVI. said of his sermon: "If -the abbe had only said a few words on religion he would have discussed -every possible subject." In 1781 he obtained the rich priory of Lyons, -near Peronne, and in 1785 he was elected to the Academy, as successor of -Lefranc de Pompignan. His morals were as loose as those of his great -rival Mirabeau, but he was famed in Paris for his wit and gaiety. In -1789 he was elected a member of the states-general by the clergy of the -bailliage of Peronne, and from the first proved to be the most able and -persevering defender of the _ancien regime_, although he had drawn up -the greater part of the _cahier_ of the clergy of Peronne, which -contained a considerable programme of reform. It is said that he -attempted to emigrate both in July and in October 1789; but after that -time he held firmly to his place, when almost universally deserted by -his friends. In the Constituent Assembly he took an active part in every -important debate, combating with especial vigour the alienation of the -property of the clergy. His life was often in danger, but his ready wit -always saved it, and it was said that one _bon mot_ would preserve him -for a month. When he did emigrate in 1792 he found himself regarded as -a martyr to the church and the king, and was at once named archbishop -_in partibus_, and extra nuncio to the diet at Frankfort, and in 1794 -cardinal. He was finally made bishop of Montefiascone, and settled down -in that little Italian town--but not for long, for in 1798 the French -drove him from his retreat, and he sought refuge in Venice and St -Petersburg. Next year he returned to Rome as ambassador of the exiled -Louis XVIII. at the papal court. In 1804 he began to prepare his return -to France by a well-turned letter to Napoleon, congratulating him on -restoring religion to France once more. In 1806 he did return; in 1807 -he was again received into the Academy; and in 1810, on the refusal of -Cardinal Fesch, was made archbishop of Paris. He was presently ordered -by the pope to surrender his functions as archbishop of Paris. This he -refused to do. On the restoration of the Bourbons he was summarily -expelled from the Academy and from the archiepiscopal palace. He retired -to Rome, where he was imprisoned in the castle of St Angelo for six -months for his disobedience to the papal orders, and died in 1817, a -year or two after his release, of disease contracted in prison and of -chagrin. As a critic he was a very able writer, and Sainte-Beuve gives -him the credit of discovering Father Jacques Bridayne, and of giving -Bossuet his rightful place as a preacher above Massillon; as a -politician, his wit and eloquence make him a worthy rival of Mirabeau. -He sacrificed too much to personal ambition, yet it would have been a -graceful act if Louis XVIII. had remembered the courageous supporter of -Louis XVI., and the pope the one intrepid defender of the Church in the -states-general. - - The _Oeuvres choisies du Cardinal Maury_ (5 vols., 1827) contain what - is worth preserving. Mgr Ricard has published Maury's _Correspondance - diplomatique_ (2 vols., Lille, 1891). For his life and character see - _Vie du Cardinal Maury_, by Louis Siffrein Maury, his nephew (1828); - J. J. F. Poujoulat, _Cardinal Maury, sa vie et ses oeuvres_ (1855); - Sainte-Beuve, _Causeries du lundi_ (vol. iv.); Mgr Ricard, _L'Abbe - Maury_ (1746-1791), _L'Abbe Maury avant 1789, L'Abbe Maury et - Mirabeau_ (1887); G. Bonet-Maury, _Le Cardinal Maury d'apres ses - memoires et sa correspondance inedits_ (Paris, 1892); A. Aulard, _Les - Orateurs de la constituante_ (Paris, 1882). Of the many libels written - against him during the Revolution the most noteworthy are the _Petit - careme de l'abbe Maury_, with a supplement called the _Seconde annee_ - (1790), and the _Vie privee de l'abbe Maury_ (1790), claimed by J. R. - Hebert, but attributed by some writers to Restif de la Bretonne. For - further bibliographical details see J. M. Querard, _La France - litteraire_, vol. v. (1833). - - - - -MAURY, LOUIS FERDINAND ALFRED (1817-1892), French scholar, was born at -Meaux on the 23rd of March 1817. In 1836, having completed his -education, he entered the Bibliotheque Nationale, and afterwards the -Bibliotheque de l'Institut (1844), where he devoted himself to the study -of archaeology, ancient and modern languages, medicine and law. Gifted -with a great capacity for work, a remarkable memory and an unbiassed and -critical mind, he produced without great effort a number of learned -pamphlets and books on the most varied subjects. He rendered great -service to the Academie des Inscriptions et Belles Lettres, of which he -had been elected a member in 1857. Napoleon III. employed him in -research work connected with the _Histoire de Cesar_, and he was -rewarded, proportionately to his active, if modest, part in this work, -with the positions of librarian of the Tuileries (1860), professor at -the College of France (1862) and director-general of the Archives -(1868). It was not, however, to the imperial favour that he owed these -high positions. He used his influence for the advancement of science and -higher education, and with Victor Duruy was one of the founders of the -Ecole des Hautes Etudes. He died at Paris four years after his -retirement from the last post, on the 11th of February 1892. - - BIBLIOGRAPHY.--His works are numerous: _Les Fees au moyen age_ and - _Histoire des legendes pieuses au moyen age_; two books filled with - ingenious ideas, which were published in 1843, and reprinted after the - death of the author, with numerous additions under the title - _Croyances et legendes du moyen age_ (1896); _Histoire des grandes - forets de la Gaule et de l'ancienne France_ (1850, a 3rd ed. revised - appeared in 1867 under the title _Les Forets de la Gaule et de - l'ancienne France); La Terre et l'homme_, a general historical sketch - of geology, geography and ethnology, being the introduction to the - _Histoire universelle_, by Victor Duruy (1854); _Histoire des - religions de la_ _Grece antique_, (3 vols., 1857-1859); _La Magie et - l'astrologie dans l'antiquite et dans le moyen age_ (1863); _Histoire - de l'ancienne academie des sciences_ (1864); _Histoire de l'Academie - des Inscriptions et Belles Lettres_ (1865); a learned paper on the - reports of French archaeology, written on the occasion of the - universal exhibition (1867); a number of articles in the _Encyclopedie - moderne_ (1846-1851), in Michaud's _Biographie universelle_ (1858 and - seq.), in the _Journal des savants_ in the _Revue des deux mondes_ - (1873, 1877, 1879-1880, &c.). A detailed bibliography of his works has - been placed by Auguste Longnon at the beginning of the volume _Les - Croyances et legendes du moyen age_. - - - - -MAURY, MATTHEW FONTAINE (1806-1873), American naval officer and -hydrographer, was born near Fredericksburg in Spottsylvania county, -Virginia, on the 24th of January 1806. He was educated at Harpeth -academy, and in 1825 entered the navy as midshipman, circumnavigating -the globe in the "Vincennes," during a cruise of four years (1826-1830). -In 1831 he was appointed master of the sloop "Falmouth" on the Pacific -station, and subsequently served in other vessels before returning home -in 1834, when he married his cousin, Ann Herndon. In 1835-1836 he was -actively engaged in producing for publication a treatise on navigation, -a remarkable achievement at so early a stage in his career; he was at -this time made lieutenant, and gazetted astronomer to a South Sea -exploring expedition, but resigned this position and was appointed to -the survey of southern harbours. In 1839 he met with an accident which -resulted in permanent lameness, and unfitted him for active service. In -the same year, however, he began to write a series of articles on naval -reform and other subjects, under the title of _Scraps from the -Lucky-Bag_, which attracted much attention; and in 1841 he was placed in -charge of the Depot of Charts and Instruments, out of which grew the -United States Naval Observatory and the Hydrographie Office. He laboured -assiduously to obtain observations as to the winds and currents by -distributing to captains of vessels specially prepared log-books; and in -the course of nine years he had collected a sufficient number of logs to -make two hundred manuscript volumes, each with about two thousand five -hundred days' observations. One result was to show the necessity for -combined action on the part of maritime nations in regard to ocean -meteorology. This led to an international conference at Brussels in -1853, which produced the greatest benefit to navigation as well as -indirectly to meteorology. Maury attempted to organize co-operative -meteorological work on land, but the government did not at this time -take any steps in this direction. His oceanographical work, however, -received recognition in all parts of the civilized world, and in 1855 it -was proposed in the senate to remunerate him, but in the same year the -Naval Retiring Board, erected under an act to promote the efficiency of -the navy, placed him on the retired list. This action aroused wide -opposition, and in 1858 he was reinstated with the rank of commander as -from 1855. In 1853 Maury had published his _Letters on the Amazon and -Atlantic Slopes of South America_, and the most widely popular of his -works, the _Physical Geography of the Sea_, was published in London in -1855, and in New York in 1856; it was translated into several European -languages. On the outbreak of the American Civil War in 1861, Maury -threw in his lot with the South, and became head of coast, harbour and -river defences. He invented an electric torpedo for harbour defence, and -in 1862 was ordered to England to purchase torpedo material, &c. Here he -took active part in organizing a petition for peace to the American -people, which was unsuccessful. Afterwards he became imperial -commissioner of emigration to the emperor Maximilian of Mexico, and -attempted to form a Virginian colony in that country. Incidentally he -introduced there the cultivation of cinchona. The scheme of colonization -was abandoned by the emperor (1866), and Maury, who had lost nearly his -all during the war, settled for a while in England, where he was -presented with a testimonial raised by public subscription, and among -other honours received the degree of LL.D. of Cambridge University -(1868). In the same year, a general amnesty admitting of his return to -America, he accepted the professorship of meteorology in the Virginia -Military Institute, and settled at Lexington, Virginia, where he died on -the 1st of February 1873. - - Among works published by Maury, in addition to those mentioned, are - the papers contributed by him to the _Astronomical Observations_ of - the United States Observatory, _Letter concerning Lanes for Steamers - crossing the Atlantic_ (1855); _Physical Geography_ (1864) and _Manual - of Geography_ (1871). In 1859 he began the publication of a series of - _Nautical Monographs_. - - See Diana Fontaine Maury Corbin (his daughter), _Life of Matthew - Fontaine Maury_ (London, 1888). - - - - -MAUSOLEUM, the term given to a monument erected to receive the remains -of a deceased person, which may sometimes take the form of a sepulchral -chapel. The term _cenotaph_ ([Greek: kenos], empty, [Greek: taphos], -tomb) is employed for a similar monument where the body is not buried in -the structure. The term "mausoleum" originated with the magnificent -monument erected by Queen Artemisia in 353 B.C. in memory of her husband -King Mausolus, of which the remains were brought to England in 1859 by -Sir Charles Newton and placed in the British Museum. The tombs of -Augustus and of Hadrian in Rome are perhaps the largest monuments of the -kind ever erected. - - - - -MAUSOLUS (more correctly MAUSSOLLUS), satrap and practically ruler of -Caria (377-353 B.C.). The part he took in the revolt against Artaxerxes -Mnemon, his conquest of a great part of Lycia, Ionia and of several of -the Greek islands, his co-operation with the Rhodians and their allies -in the war against Athens, and the removal of his capital from Mylasa, -the ancient seat of the Carian kings, to Halicarnassus are the leading -facts of his history. He is best known from the tomb erected for him by -his widow Artemisia. The architects Satyrus and Pythis, and the -sculptors Scopas, Leochares, Bryaxis and Timotheus, finished the work -after her death. (See HALICARNASSUS.) An inscription discovered at -Mylasa (Bockh, _Inscr. gr._ ii. 2691 _c._) details the punishment of -certain conspirators who had made an attempt upon his life at a festival -in a temple at Labranda in 353. - - See Diod. Sic. xv. 90, 3, xvi. 7, 4, 36, 2; Demosthenes, _De Rhodiorum - libertate_; J. B. Bury, _Hist. of Greece_ (1902), ii. 271; W. Judeich, - _Kleinasiatische Studien_ (Marburg, 1892), pp. 226-256, and - authorities under HALICARNASSUS. - - - - -MAUVE, ANTON (1838-1888), Dutch landscape painter, was born at Zaandam, -the son of a Baptist minister. Much against the wish of his parents he -took up the study of art and entered the studio of Van Os, whose dry -academic manner had, however, but little attraction for him. He -benefited far more by his intimacy with his friends Jozef Israels and W. -Maris. Encouraged by their example he abandoned his early tight and -highly finished manner for a freer, looser method of painting, and the -brilliant palette of his youthful work for a tender lyric harmony which -is generally restricted to delicate greys, greens, and light blue. He -excelled in rendering the soft hazy atmosphere that lingers over the -green meadows of Holland, and devoted himself almost exclusively to -depicting the peaceful rural life of the fields and country lanes of -Holland--especially of the districts near Oosterbeck and Wolfhezen, the -sand dunes of the coast at Scheveningen, and the country near Laren, -where he spent the last years of his life. A little sad and melancholy, -his pastoral scenes are nevertheless conceived in a peaceful soothing -lyrical mood, which is in marked contrast to the epic power and almost -tragic intensity of J. F. Millet. There are fourteen of Mauve's pictures -at the Mesdag Museum at the Hague, and two ("Milking Time" and "A -Fishing Boat putting to Sea") at the Ryks Museum in Amsterdam. The -Glasgow Corporation Gallery owns his painting of "A Flock of Sheep." The -finest and most representative private collection of pictures by Mauve -was made by Mr J. C. J. Drucker, London. - - - - -MAVROCORDATO, MAVROCORDAT or MAVROGORDATO, the name of a family of -Phanariot Greeks, distinguished in the history of Turkey, Rumania and -modern Greece. The family was founded by a merchant of Chios, whose son -Alexander Mavrocordato (c. 1636-1709), a doctor of philosophy and -medicine of Bologna, became dragoman to the sultan in 1673, and was much -employed in negotiations with Austria. It was he who drew up the treaty -of Karlowitz (1699). He became a secretary of state, and was created a -count of the Holy Roman Empire. His authority, with that of Hussein -Kupruli and Rami Pasha, was supreme at the court of Mustapha II., and he -did much to ameliorate the condition of the Christians in Turkey. He -was disgraced in 1703, but was recalled to court by Sultan Ahmed III. He -left some historical, grammatical, &c. treatises of little value. - -His son NICHOLAS MAVROCORDATO (1670-1730) was grand dragoman to the -Divan (1697), and in 1708 was appointed hospodar (prince) of Moldavia. -Deposed, owing to the sultan's suspicions, in favour of Demetrius -Cantacuzene, he was restored in 1711, and soon afterwards became -hospodar of Walachia. In 1716 he was deposed by the Austrians, but was -restored after the peace of Passarowitz. He was the first Greek set to -rule the Danubian principalities, and was responsible for establishing -the system which for a hundred years was to make the name of Greek -hateful to the Rumanians. He introduced Greek manners, the Greek -language and Greek costume, and set up a splendid court on the Byzantine -model. For the rest he was a man of enlightenment, founded libraries and -was himself the author of a curious work entitled [Greek: Peri -kathekonton] (Bucharest, 1719). He was succeeded as grand dragoman -(1709) by his son John (Ioannes), who was for a short while hospodar of -Moldavia, and died in 1720. - -Nicholas Mavrocordato was succeeded as prince of Walachia in 1730 by his -son Constantine. He was deprived in the same year, but again ruled the -principality from 1735 to 1741 and from 1744 to 1748; he was prince of -Moldavia from 1741 to 1744 and from 1748 to 1749. His rule was -distinguished by numerous tentative reforms in the fiscal and -administrative systems. He was wounded and taken prisoner in the affair -of Galati during the Russo-Turkish War, on the 5th of November 1769, and -died in captivity. - -PRINCE ALEXANDER MAVROCORDATO (1791-1865), Greek statesman, a descendant -of the hospodars, was born at Constantinople on the 11th of February -1791. In 1812 he went to the court of his uncle Ioannes Caradja, -hospodar of Walachia, with whom he passed into exile in Russia and Italy -(1817). He was a member of the Hetairia Philike and was among the -Phanariot Greeks who hastened to the Morea on the outbreak of the War of -Independence in 1821. He was active in endeavouring to establish a -regular government, and in January 1822 presided over the first Greek -national assembly at Epidaurus. He commanded the advance of the Greeks -into western Hellas the same year, and suffered a defeat at Peta on the -16th of July, but retrieved this disaster somewhat by his successful -resistance to the first siege of Missolonghi (Nov. 1822 to Jan. 1823). -His English sympathies brought him, in the subsequent strife of -factions, into opposition to the "Russian" party headed by Demetrius -Ypsilanti and Kolokotrones; and though he held the portfolio of foreign -affairs for a short while under the presidency of Petrobey (Petros -Mavromichales), he was compelled to withdraw from affairs until February -1825, when he again became a secretary of state. The landing of Ibrahim -Pasha followed, and Mavrocordato again joined the army, only escaping -capture in the disaster at Sphagia (Spakteria), on the 9th of May 1815, -by swimming to Navarino. After the fall of Missolonghi (April 22, 1826) -he went into retirement, until President Capo d'Istria made him a member -of the committee for the administration of war material, a position he -resigned in 1828. After Capo d'Istria's murder (Oct. 9, 1831) and the -resignation of his brother and successor, Agostino Capo d'Istria (April -13, 1832), Mavrocordato became minister of finance. He was -vice-president of the National Assembly at Argos (July, 1832), and was -appointed by King Otto minister of finance, and in 1833 premier. From -1834 onwards he was Greek envoy at Munich, Berlin, London and--after a -short interlude as premier in Greece in 1841--Constantinople. In 1843, -after the revolution of September, he returned to Athens as minister -without portfolio in the Metaxas cabinet, and from April to August 1844 -was head of the government formed after the fall of the "Russian" party. -Going into opposition, he distinguished himself by his violent attacks -on the Kolettis government. In 1854-1855 he was again head of the -government for a few months. He died in Aegina on the 18th of August -1865. - - See E. Legrand, _Genealogie des Mavrocordato_ (Paris, 1886). - - - - -MAWKMAI (Burmese _Maukme_), one of the largest states in the eastern -division of the southern Shan States of Burma. It lies approximately -between 19 deg. 30' and 20 deg. 30' N. and 97 deg. 30' and 98 deg. 15' -E., and has an area of 2,787 sq. m. The central portion of the state -consists of a wide plain well watered and under rice cultivation. The -rest is chiefly hills in ranges running north and south. There is a good -deal of teak in the state, but it has been ruinously worked. The sawbwa -now works as contractor for government, which takes one-third of the net -profits. Rice is the chief crop, but much tobacco of good quality is -grown in the Langko district on the Teng river. There is also a great -deal of cattle-breeding. The population in 1901 was 29,454, over -two-thirds of whom were Shans and the remainder Taungthu, Burmese, -Yangsek and Red Karens. The capital, MAWKMAI, stands in a fine rice -plain in 20 deg. 9' N. and 97 deg. 25' E. It had about 150 houses when -it first submitted in 1887, but was burnt out by the Red Karens in the -following year. It has since recovered. There are very fine orange -groves a few miles south of the town at Kantu-awn, called Kadugate by -the Burmese. - - - - -MAXENTIUS, MARCUS AURELIUS VALERIUS, Roman emperor from A.D. 306 to 312, -was the son of Maximianus Herculius, and the son-in-law of Galerius. -Owing to his vices and incapacity he was left out of account in the -division of the empire which took place in 305. A variety of causes, -however, had produced strong dissatisfaction at Rome with many of the -arrangements established by Diocletian, and on the 28th of October 306, -the public discontent found expression in the massacre of those -magistrates who remained loyal to Flavius Valerius Severus and in the -election of Maxentius to the imperial dignity. With the help of his -father, Maxentius was enabled to put Severus to death and to repel the -invasion of Galerius; his next steps were first to banish Maximianus, -and then, after achieving a military success in Africa against the -rebellious governor, L. Domitius Alexander, to declare war against -Constantine as having brought about the death of his father Maximianus. -His intention of carrying the war into Gaul was anticipated by -Constantine, who marched into Italy. Maxentius was defeated at Saxa -Rubra near Rome and drowned in the Tiber while attempting to make his -way across the Milvian bridge into Rome. He was a man of brutal and -worthless character; but although Gibbon's statement that he was "just, -humane and even partial towards the afflicted Christians" may be -exaggerated, it is probable that he never exhibited any special -hostility towards them. - - See De Broglie, _L'Eglise et l'empire Romain au quatrieme siecle_ - (1856-1866), and on the attitude of the Romans towards Christianity - generally, app. 8 in vol. ii. of J. B. Bury's edition of Gibbon - (Zosimus ii. 9-18; Zonaras xii. 33, xiii. 1; Aurelius Victor, _Epit._ - 40; Eutropius, x. 2). - - - - -MAXIM, SIR HIRAM STEVENS (1840- ), Anglo-American engineer and -inventor, was born at Sangerville, Maine, U.S.A., on the 5th of February -1840. After serving an apprenticeship with a coachbuilder, he entered -the machine works of his uncle, Levi Stevens, at Fitchburg, -Massachusetts, in 1864, and four years later he became a draughtsman in -the Novelty Iron Works and Shipbuilding Company in New York City. About -this period he produced several inventions connected with illumination -by gas; and from 1877 he was one of the numerous inventors who were -trying to solve the problem of making an efficient and durable -incandescent electric lamp, in this connexion introducing the -widely-used process of treating the carbon filaments by heating them in -an atmosphere of hydrocarbon vapour. In 1880 he came to Europe, and soon -began to devote himself to the construction of a machine-gun which -should be automatically loaded and fired by the energy of the recoil -(see MACHINE-GUN). In order to realize the full usefulness of the -weapon, which was first exhibited in an underground range at Hatton -Garden, London, in 1884, he felt the necessity of employing a smokeless -powder, and accordingly he devised maximite, a mixture of -trinitrocellulose, nitroglycerine and castor oil, which was patented in -1889. He also undertook to make a flying machine, and after numerous -preliminary experiments constructed an apparatus which was tried at -Bexley Heath, Kent, in 1894. (See FLIGHT.) Having been naturalized as a -British subject, he was knighted in 1901. His younger brother, Hudson -Maxim (b. 1853), took out numerous patents in connexion with explosives. - - - - -MAXIMA AND MINIMA, in mathematics. By the _maximum_ or _minimum_ value -of an expression or quantity is meant primarily the "greatest" or -"least" value that it can receive. In general, however, there are points -at which its value ceases to increase and begins to decrease; its value -at such a point is called a maximum. So there are points at which its -value ceases to decrease and begins to increase; such a value is called -a minimum. There may be several maxima or minima, and a minimum is not -necessarily less than a maximum. For instance, the expression (x^2 + x + -2)/(x - 1) can take all values from -[oo] to -1 and from +7 to +[oo], -but has, so long as x is real, no value between -1 and +7. Here -1 is a -maximum value, and +7 is a minimum value of the expression, though it -can be made greater or less than any assignable quantity. - -The first general method of investigating maxima and minima seems to -have been published in A.D. 1629 by Pierre Fermat. Particular cases had -been discussed. Thus Euclid in book III. of the _Elements_ finds the -greatest and least straight lines that can be drawn from a point to the -circumference of a circle, and in book VI. (in a proposition generally -omitted from editions of his works) finds the parallelogram of greatest -area with a given perimeter. Apollonius investigated the greatest and -least distances of a point from the perimeter of a conic section, and -discovered them to be the normals, and that their feet were the -intersections of the conic with a rectangular hyperbola. Some remarkable -theorems on maximum areas are attributed to Zenodorus, and preserved by -Pappus and Theon of Alexandria. The most noteworthy of them are the -following:-- - - 1. Of polygons of n sides with a given perimeter the regular polygon - encloses the greatest area. - - 2. Of two regular polygons of the same perimeter, that with the - greater number of sides encloses the greater area. - - 3. The circle encloses a greater area than any polygon of the same - perimeter. - - 4. The sum of the areas of two isosceles triangles on given bases, the - sum of whose perimeters is given, is greatest when the triangles are - similar. - - 5. Of segments of a circle of given perimeter, the semicircle encloses - the greatest area. - - 6. The sphere is the surface of given area which encloses the greatest - volume. - -Serenus of Antissa investigated the somewhat trifling problem of finding -the triangle of greatest area whose sides are formed by the -intersections with the base and curved surface of a right circular cone -of a plane drawn through its vertex. - -The next problem on maxima and minima of which there appears to be any -record occurs in a letter from Regiomontanus to Roder (July 4, 1471), -and is a particular numerical example of the problem of finding the -point on a given straight line at which two given points subtend a -maximum angle. N. Tartaglia in his _General trattato de numeri et -mesuri_ (c. 1556) gives, without proof, a rule for dividing a number -into two parts such that the continued product of the numbers and their -difference is a maximum. - -Fermat investigated maxima and minima by means of the principle that in -the neighbourhood of a maximum or minimum the differences of the values -of a function are insensible, a method virtually the same as that of the -differential calculus, and of great use in dealing with geometrical -maxima and minima. His method was developed by Huygens, Leibnitz, Newton -and others, and in particular by John Hudde, who investigated maxima and -minima of functions of more than one independent variable, and made some -attempt to discriminate between maxima and minima, a question first -definitely settled, so far as one variable is concerned, by Colin -Maclaurin in his _Treatise on Fluxions_ (1742). The method of the -differential calculus was perfected by Euler and Lagrange. - -John Bernoulli's famous problem of the "brachistochrone," or curve of -quickest descent from one point to another under the action of gravity, -proposed in 1696, gave rise to a new kind of maximum and minimum problem -in which we have to find a curve and not points on a given curve. From -these problems arose the "Calculus of Variations." (See VARIATIONS, -CALCULUS OF.) - -The only general methods of attacking problems on maxima and minima are -those of the differential calculus or, in geometrical problems, what is -practically Fermat's method. Some problems may be solved by algebra; -thus if y = f(x) / [phi](x), where f(x) and [phi](x) are polynomials in -x, the limits to the values of y[phi] may be found from the -consideration that the equation y[phi](x) - f(x) = 0 must have real -roots. This is a useful method in the case in which [phi](x) and f(x) -are quadratics, but scarcely ever in any other case. The problem of -finding the maximum product of n positive quantities whose sum is given -may also be found, algebraically, thus. If a and b are any two real -unequal quantities whatever {(1/2)(a + b)}^2 > ab, so that we can -increase the product leaving the sum unaltered by replacing any two -terms by half their sum, and so long as any two of the quantities are -unequal we can increase the product. Now, the quantities being all -positive, the product cannot be increased without limit and must -somewhere attain a maximum, and no other form of the product than that -in which they are all equal can be the maximum, so that the product is a -maximum when they are all equal. Its minimum value is obviously zero. If -the restriction that all the quantities shall be positive is removed, -the product can be made equal to any quantity, positive or negative. So -other theorems of algebra, which are stated as theorems on inequalities, -may be regarded as algebraic solutions of problems on maxima and minima. - -For purely geometrical questions the only general method available is -practically that employed by Fermat. If a quantity depends on the -position of some point P on a curve, and if its value is equal at two -neighbouring points P and P', then at some position between P and P' it -attains a maximum or minimum, and this position may be found by making P -and P' approach each other indefinitely. Take for instance the problem -of Regiomontanus "to find a point on a given straight line which -subtends a maximum angle at two given points A and B." Let P and P' be -two near points on the given straight line such that the angles APB and -AP'B are equal. Then ABPP' lie on a circle. By making P and P' approach -each other we see that for a maximum or minimum value of the angle APB, -P is a point in which a circle drawn through AB touches the given -straight line. There are two such points, and unless the given straight -line is at right angles to AB the two angles obtained are not the same. -It is easily seen that both angles are maxima, one for points on the -given straight line on one side of its intersection with AB, the other -for points on the other side. For further examples of this method -together with most other geometrical problems on maxima and minima of -any interest or importance the reader may consult such a book as J. W. -Russell's _A Sequel lo Elementary Geometry_ (Oxford, 1907). - - The method of the differential calculus is theoretically very simple. - Let u be a function of several variables x1, x2, x3 ... x_n, supposed - for the present independent; if u is a maximum or minimum for the set - of values x1, x2, x3, ... x_n, and u becomes u + [delta]u, when x1, - x2, x3 ... x_n receive small increments [delta]x1, [delta]x2, ... - [delta]x_n; then [delta]u must have the same sign for all possible - values of [delta]x1, [delta]2 ... [delta]x_n. - - Now - _ _ - __ [delta]u | __ [delta]^2u __ [delta]^3u | - [delta]u = \ --------- [delta]x1 + (1/2) | \ ----------- + 2 \ ------------------- [delta]x1 [delta]x2 ... | + ... - /__ [delta]x1 |_ /__ [delta]x1^2 /__ [delta]x1 [delta]x2 _| - - The sign of this expression in general is that of - [Sigma]([delta]u/[delta]x1)[delta]x1, which cannot be one-signed when - x1, x2, ... x_n can take all possible values, for a set of increments - [delta]x1, [delta]x2 ... [delta]x_n, will give an opposite sign to the - set -[delta]x1, -[delta]x2, ... -[delta]x_n. Hence - [Sigma]([delta]u/[delta]x1)[delta]x1 must vanish for all sets of - increments [delta]x1, ... [delta]x_n, and since these are independent, - we must have [delta]u/[delta]x1 = 0, [delta]u/[delta]x2 = 0, ... - [delta]u/[delta]x_n = 0. A value of u given by a set of solutions of - these equations is called a "critical value" of u. The value of - [delta]u now becomes - _ _ - | __ [delta]^2u __ [delta]^2u | - (1/2) | \ --------- [delta]x1^2 + 2 \ ------------------- [delta]x1 [delta]x2 + ... |; - |_ /__ [delta]x1^2 /__ [delta]x1 [delta]x2 _| - - for u to be a maximum or minimum this must have always the same sign. - For the case of a single variable x, corresponding to a value of x - given by the equation du/dx = 0, u is a maximum or minimum as d^2u/dx^2 - is negative or positive. If d^2u/dx^2 vanishes, then there is no maximum - or minimum unless d^2u/dx^2 vanishes, and there is a maximum or minimum - according as d^4u/dx^4 is negative or positive. Generally, if the - first differential coefficient which does not vanish is even, there is - a maximum or minimum according as this is negative or positive. If it - is odd, there is no maximum or minimum. - - In the case of several variables, the quadratic - - __ [delta]^2u __ [delta]^2u - \ ---------- [delta]x1^2 + 2 \ ------------------- + ... - /__ [delta]x1^2 /__ [delta]x1 [delta]x2 - - must be one-signed. The condition for this is that the series of - discriminants - - a11 , | a11 a12 | , | a11 a12 a13 | , ... - | a21 a22 | | a21 a22 a23 | - | a31 a32 a33 | - - where a_pq denotes [delta]^2u/[delta]a_p[delta]a_q should be all - positive, if the quadratic is always positive, and alternately - negative and positive, if the quadratic is always negative. If the - first condition is satisfied the critical value is a minimum, if the - second it is a maximum. For the case of two variables the conditions - are - - [delta]^2u [delta]^2u / [delta]^2 \^2 - ----------- . ----------- > ( ------------------- ) - [delta]x1^2 [delta]x2^2 \ [delta]x1 [delta]x2 / - - for a maximum or minimum at all and [delta]^2u/[delta]x1^2 and - [delta]^2u/[delta]x2^2 both negative for a maximum, and both positive - for a minimum. It is important to notice that by the quadratic being - one-signed is meant that it cannot be made to vanish except when - [delta]x1, [delta]x2, ... [delta]x_n all vanish. If, in the case of - two variables, - - [delta]^2u [delta]^2u / [delta]^2u \^2 - ----------- . ----------- = ( ------------------- ) - [delta]x1^2 [delta]x2^2 \ [delta]x1 [delta]x2 / - - then the quadratic is one-signed unless it vanishes, but the value of - u is not necessarily a maximum or minimum, and the terms of the third - and possibly fourth order must be taken account of. - - Take for instance the function u = x^2 - xy^2 + y^2. Here the values x - = 0, y = 0 satisfy the equations [delta]u/[delta]x = 0, - [delta]u/[delta]y = 0, so that zero is a critical value of u, but it - is neither a maximum nor a minimum although the terms of the second - order are ([delta]x)^2, and are never negative. Here [delta]u = - [delta]x^2 - [delta]x[delta]y^2 + [delta]y^2, and by putting [delta]x - = 0 or an infinitesimal of the same order as [delta]y^2, we can make - the sign of [delta]u depend on that of [delta]y^2, and so be positive - or negative as we please. On the other hand, if we take the function u - = x^2 - xy^2 + y^4, x = 0, y = 0 make zero a critical value of u, and - here [delta]u = [delta]x^2 - [delta]x[delta]y^2 + [delta]y^4, which is - always positive, because we can write it as the sum of two squares, - viz. ([delta]x - (1/2)[delta]y^2)^2 + (3/4)[delta]y^4; so that in this - case zero is a minimum value of u. - - A critical value usually gives a maximum or minimum in the case of a - function of one variable, and often in the case of several independent - variables, but all maxima and minima, particularly absolutely greatest - and least values, are not necessarily critical values. If, for - example, x is restricted to lie between the values a and b and - [phi]'(x) = 0 has no roots in this interval, it follows that [phi]'(x) - is one-signed as x increases from a to b, so that [phi](x) is - increasing or diminishing all the time, and the greatest and least - values of [phi](x) are [phi](a) and [phi](b), though neither of them - is a critical value. Consider the following example: A person in a - boat a miles from the nearest point of the beach wishes to reach as - quickly as possible a point b miles from that point along the shore. - The ratio of his rate of walking to his rate of rowing is cosec - [alpha]. Where should he land? - - Here let AB be the direction of the beach, A the nearest point to the - boat O, and B the point he wishes to reach. Clearly he must land, if - at all, between A and B. Suppose he lands at P. Let the angle AOP be - [theta], so that OP = a sec[theta], and PB = b - a tan [theta]. If his - rate of rowing is V miles an hour his time will be a sec [theta]/V + - (b - a tan [theta]) sin [alpha]/V hours. Call this T. Then to the - first power of [delta][theta], [delta]T = (a/V) sec^2[theta] (sin - [theta] - sin [alpha])[delta][theta], so that if AOB > [alpha], - [delta]T and [delta][theta] have opposite signs from [theta] = 0 to - [theta] = [alpha], and the same signs from [theta] = [alpha] to - [theta] = AOB. So that when AOB is > [alpha], T decreases from [theta] - = 0 to [theta] = [alpha], and then increases, so that he should land - at a point distant a tan [alpha] from A, unless a tan [alpha] > b. - When this is the case, [delta]T and [delta][theta] have opposite signs - throughout the whole range of [theta], so that T decreases as [theta] - increases, and he should row direct to B. In the first case the - minimum value of T is also a critical value; in the second case it is - not. - - The greatest and least values of the bending moments of loaded rods - are often at the extremities of the divisions of the rods and not at - points given by critical values. - - In the case of a function of several variables, X1, x2, ... x_n, not - independent but connected by m functional relations u1 = 0, u2 = 0, - ..., u_m = 0, we might proceed to eliminate m of the variables; but - Lagrange's "Method of undetermined Multipliers" is more elegant and - generally more useful. - - We have [delta]u1 = 0, [delta]u2 = 0, ..., [delta]u_m = 0. Consider - instead of [delta]u, what is the same thing, viz., [delta]u + - [lambda]1[delta]u1 + [lambda]2[delta]u2 + ... + [lambda]_m[delta]u_m, - where [lambda]1, [lambda]2, ... [lambda]_m, are arbitrary multipliers. - The terms of the first order in this expression are - - __ [delta]u __ [delta]u1 __ [delta]u_m - \ --------- [delta]x1 + [lambda]1 \ --------- [delta]x1 + ... + [lambda]_m \ ---------- [delta]x1. - /__ [delta]x1 /__ [delta]x1 /__ [delta]x1 - - We can choose [lambda]1, ... [lambda]_m, to make the coefficients of - [delta]x1, [delta]x2, ... [delta]x_m, vanish, and the remaining - [delta]x_(m+1) to [delta]x_n may be regarded as independent, so that, - when u has a critical value, their coefficients must also vanish. So - that we put - - [delta]u [delta]u1 [delta]u_m - ---------- + [lambda]1 ---------- + ... + [lambda]_m ---------- = 0 - [delta]x_r [delta]x_r [delta]x_r - - for all values of r. These equations with the equations u1 = 0, ..., - u_m = 0 are exactly enough to determine [lambda]1, ..., [lambda]_m, x1 - x2, ..., x_n, so that we find critical values of u, and examine the - terms of the second order to decide whether we obtain a maximum or - minimum. - - To take a very simple illustration; consider the problem of - determining the maximum and minimum radii vectors of the ellipsoid - x^2/a^2 + y^2/b^2 + z^2/c^2 = 1, where a^2 > b^2 > c^2. Here we - require the maximum and minimum values of x^2 + y^2 + z^2 where - x^2/a^2 + y^2/b^2 + z^2/c^2 = 1. - - We have - - / [lambda]\ / [lambda]\ / [lambda]\ - [delta]u = 2x [delta]x ( 1 + -------- ) + 2y [delta]y ( 1 + -------- ) + 2z [delta]z ( 1 + -------- ) - \ a^2 / \ b^2 / \ c^2 / - - / [lambda]\ / [lambda]\ / [lambda]\ - + [delta]x^2 ( 1 + -------- ) + [delta]y^2 ( 1 + -------- ) + [delta]z^2 ( 1 + -------- ). - \ a^2 / \ b^2 / \ c^2 / - - To make the terms of the first order disappear, we have the three - equations:-- - - x(1 + [lambda]/a^2) = 0, y(1 + [lambda]/b^2) = 0, z(1 + [lambda]/c^2) = - 0. - - These have three sets of solutions consistent with the conditions - x^2/a^2 + y^2/b^2 + z^2/c^2 = 1, a^2 > b^2 > c^2, viz.:-- - - (1) y = 0, z = 0, [lambda] = -a^2; (2) z = 0, x = 0, [lambda] = -b^2; - - (3) x = 0, y = 0, [lambda] = -c^2. - - In the case of (1) [delta]u = [delta]y^2 (1 - a^2/b^2) + [delta]z^2 (1 - - a^2/c^2), which is always negative, so that u = a^2 gives a maximum. - - In the case of (3) [delta]u = [delta]x^2 (1 - c^2/a^2) + [delta]y^2 (1 - - c^2/b^2), which is always positive, so that u = c^2 gives a minimum. - - In the case of (2) [delta]u = [delta]x^2(1 - b^2/a^2) - - [delta]z^2(b^2/c^2 - 1), which can be made either positive or - negative, or even zero if we move in the planes x^2(1 - b^2/a^2) = - z^2(b^2/c^2 - 1), which are well known to be the central planes of - circular section. So that u = b^2, though a critical value, is neither - a maximum nor minimum, and the central planes of circular section - divide the ellipsoid into four portions in two of which a^2 > r^2 > - b^2, and in the other two b^2 > r^2 > c^2. (A. E. J.) - - - - -MAXIMIANUS, a Latin elegiac poet who flourished during the 6th century -A.D. He was an Etruscan by birth, and spent his youth at Rome, where he -enjoyed a great reputation as an orator. At an advanced age he was sent -on an important mission to the East, perhaps by Theodoric, if he is the -Maximianus to whom that monarch addressed a letter preserved in -Cassiodorus (_Variarum_, i. 21). The six elegies extant under his name, -written in old age, in which he laments the loss of his youth, contain -descriptions of various amours. They show the author's familiarity with -the best writers of the Augustan age. - - Editions by J. C. Wernsdorf, _Poetae latini minores_, vi.; E. Bahrens, - _Poetae latini minores_, v.; M. Petschenig (1890), in C. F. - Ascherson's _Berliner Studien_, xi.; R. Webster (Princeton, 1901; see - _Classical Review_, Oct. 1901), with introduction and commentary; see - also Robinson Ellis in _American Journal of Philology_, v. (1884) and - Teuffel-Schwabe, _Hist. of Roman Literature_ (Eng. trans.), S 490. - There is an English version (as from Cornelius Gallus), by Hovenden - Walker (1689), under the title of _The Impotent Lover_. - - - - -MAXIMIANUS, MARCUS AURELIUS VALERIUS, surnamed Herculius, Roman emperor -from A.D. 286 to 305, was born of humble parents at Sirmium in Pannonia. -He achieved distinction during long service in the army, and having been -made Caesar by Diocletian in 285, received the title of Augustus in the -following year (April 1, 286). In 287 he suppressed the rising of the -peasants (Bagaudae) in Gaul, but in 289, after a three years' struggle, -his colleague and he were compelled to acquiesce in the assumption by -his lieutenant Carausius (who had crossed over to Britain) of the title -of Augustus. After 293 Maximianus left the care of the Rhine frontier to -Constantius Chlorus, who had been designated Caesar in that year, but in -297 his arms achieved a rapid and decisive victory over the barbarians -of Mauretania, and in 302 he shared at Rome the triumph of Diocletian, -the last pageant of the kind ever witnessed by that city. On the 1st of -May 305, the day of Diocletian's abdication, he also, but without his -colleague's sincerity, divested himself of the imperial dignity at -Mediolanum (Milan), which had been his capital, and retired to a villa -in Lucania; in the following year, however, he was induced by his son -Maxentius to reassume the purple. In 307 he brought the emperor Flavius -Valerius Severus a captive to Rome, and also compelled Galerius to -retreat, but in 308 he was himself driven by Maxentius from Italy into -Illyricum, whence again he was compelled to seek refuge at Arelate -(Arles), the court of his son-in-law, Constantine. Here a false report -was received, or invented, of the death of Constantine, at that time -absent on the Rhine. Maximianus at once grasped at the succession, but -was soon driven to Massilia (Marseilles), where, having been delivered -up to his pursuers, he strangled himself. - - See Zosimus ii. 7-11; Zonaras xii. 31-33; Eutropius ix. 20, x. 2, 3; - Aurelius Victor p. 39. For the emperor Galerius Valerius Maximianus - see GALERIUS. - - - - -MAXIMILIAN I. (1573-1651), called "the Great," elector and duke of -Bavaria, eldest son of William V. of Bavaria, was born at Munich on the -17th of April 1573. He was educated by the Jesuits at the university of -Ingolstadt, and began to take part in the government in 1591. He married -in 1595 his cousin, Elizabeth, daughter of Charles II., duke of -Lorraine, and became duke of Bavaria upon his father's abdication in -1597. He refrained from any interference in German politics until 1607, -when he was entrusted with the duty of executing the imperial ban -against the free city of Donauworth, a Protestant stronghold. In -December 1607 his troops occupied the city, and vigorous steps were -taken to restore the supremacy of the older faith. Some Protestant -princes, alarmed at this action, formed a union to defend their -interests, which was answered in 1609 by the establishment of a league, -in the formation of which Maximilian took an important part. Under his -leadership an army was set on foot, but his policy was strictly -defensive and he refused to allow the league to become a tool in the -hands of the house of Habsburg. Dissensions among his colleagues led the -duke to resign his office in 1616, but the approach of trouble brought -about his return to the league about two years later. - -Having refused to become a candidate for the imperial throne in 1619, -Maximilian was faced with the complications arising from the outbreak of -war in Bohemia. After some delay he made a treaty with the emperor -Ferdinand II. in October 1619, and in return for large concessions -placed the forces of the league at the emperor's service. Anxious to -curtail the area of the struggle, he made a treaty of neutrality with -the Protestant Union, and occupied Upper Austria as security for the -expenses of the campaign. On the 8th of November 1620 his troops under -Count Tilly defeated the forces of Frederick, king of Bohemia and count -palatine of the Rhine, at the White Hill near Prague. In spite of the -arrangement with the union Tilly then devastated the Rhenish Palatinate, -and in February 1623 Maximilian was formally invested with the electoral -dignity and the attendant office of imperial steward, which had been -enjoyed since 1356 by the counts palatine of the Rhine. After receiving -the Upper Palatinate and restoring Upper Austria to Ferdinand, -Maximilian became leader of the party which sought to bring about -Wallenstein's dismissal from the imperial service. At the diet of -Regensburg in 1630 Ferdinand was compelled to assent to this demand, but -the sequel was disastrous both for Bavaria and its ruler. Early in 1632 -the Swedes marched into the duchy and occupied Munich, and Maximilian -could only obtain the assistance of the imperialists by placing himself -under the orders of Wallenstein, now restored to the command of the -emperor's forces. The ravages of the Swedes and their French allies -induced the elector to enter into negotiations for peace with Gustavus -Adolphus and Cardinal Richelieu. He also proposed to disarm the -Protestants by modifying the Restitution edict of 1629; but these -efforts were abortive. In March 1647 he concluded an armistice with -France and Sweden at Ulm, but the entreaties of the emperor Ferdinand -III. led him to disregard his undertaking. Bavaria was again ravaged, -and the elector's forces defeated in May 1648 at Zusmarshausen. But the -peace of Westphalia soon put an end to the struggle. By this treaty it -was agreed that Maximilian should retain the electoral dignity, which -was made hereditary in his family; and the Upper Palatinate was -incorporated with Bavaria. The elector died at Ingolstadt on the 27th of -September 1651. By his second wife, Maria Anne, daughter of the emperor -Ferdinand II., he left two sons, Ferdinand Maria, who succeeded him, and -Maximilian Philip. In 1839 a statue was erected to his memory at Munich -by Louis I., king of Bavaria. Weak in health and feeble in frame, -Maximilian had high ambitions both for himself and his duchy, and was -tenacious and resourceful in prosecuting his designs. As the ablest -prince of his age he sought to prevent Germany from becoming the -battleground of Europe, and although a rigid adherent of the Catholic -faith, was not always subservient to the priest. - - See P. P. Wolf, _Geschichte Kurfurst Maximilians I. und seiner Zeit_ - (Munich, 1807-1809); C. M. Freiherr von Aretin, _Geschichte des - bayerschen Herzogs und Kurfursten Maximilian des Ersten_ (Passau, - 1842); M. Lossen, _Die Reichstadt Donauworth und Herzog Maximilian_ - (Munich, 1866); F. Stieve, _Kurfurst Maximilian I. von Bayern_ - (Munich, 1882); F. A. W. Schreiber, _Maximilian I. der Katholische - Kurfurst von Bayern, und der dreissigjahrige Krieg_ (Munich, 1868); M. - Hogl, _Die Bekehrung der Oberpfalz durch Kurfurst Maximilian I._ - (Regensburg, 1903). - - - - -MAXIMILIAN I. (MAXIMILIAN JOSEPH) (1756-1825), king of Bavaria, was the -son of the count palatine Frederick of Zweibrucken-Birkenfeld, and was -born on the 27th of May 1756. He was carefully educated under the -supervision of his uncle, Duke Christian IV. of Zweibrucken, took -service in 1777 as a colonel in the French army, and rose rapidly to the -rank of major-general. From 1782 to 1789 he was stationed at Strassburg, -but at the outbreak of the revolution he exchanged the French for the -Austrian service, taking part in the opening campaigns of the -revolutionary wars. On the 1st of April 1795 he succeeded his brother, -Charles II., as duke of Zweibrucken, and on the 16th of February 1799 -became elector of Bavaria on the extinction of the Sulzbach line with -the death of the elector Charles Theodore. - -The sympathy with France and with French ideas of enlightenment which -characterized his reign was at once manifested. In the newly organized -ministry Count Max Josef von Montgelas (q.v.), who, after falling into -disfavour with Charles Theodore, had acted for a time as Maximilian -Joseph's private secretary, was the most potent influence, an influence -wholly "enlightened" and French. Agriculture and commerce were fostered, -the laws were ameliorated, a new criminal code drawn up, taxes and -imposts equalized without regard to traditional privileges, while a -number of religious houses were suppressed and their revenues used for -educational and other useful purposes. In foreign politics Maximilian -Joseph's attitude was from the German point of view less commendable. -With the growing sentiment of German nationality he had from first to -last no sympathy, and his attitude throughout was dictated by wholly -dynastic, or at least Bavarian, considerations. Until 1813 he was the -most faithful of Napoleon's German allies, the relation being cemented -by the marriage of his daughter to Eugene Beauharnais. His reward came -with the treaty of Pressburg (Dec. 26, 1805), by the terms of which he -was to receive the royal title and important territorial acquisitions in -Swabia and Franconia to round off his kingdom. The style of king he -actually assumed on the 1st of January 1806. - -The new king of Bavaria was the most important of the princes belonging -to the Confederation of the Rhine, and remained Napoleon's ally until -the eve of the battle of Leipzig, when by the convention of Ried (Oct. -8, 1813) he made the guarantee of the integrity of his kingdom the price -of his joining the Allies. By the first treaty of Paris (June 3, 1814), -however, he ceded Tirol to Austria in exchange for the former duchy of -Wurzburg. At the congress of Vienna, too, which he attended in person, -Maximilian had to make further concessions to Austria, ceding the -quarters of the Inn and Hausruck in return for a part of the old -Palatinate. The king fought hard to maintain the contiguity of the -Bavarian territories as guaranteed at Ried; but the most he could obtain -was an assurance from Metternich in the matter of the Baden succession, -in which he was also doomed to be disappointed (see BADEN: _History_, -iii. 506). - -At Vienna and afterwards Maximilian sturdily opposed any reconstitution -of Germany which should endanger the independence of Bavaria, and it -was his insistence on the principle of full sovereignty being left to -the German reigning princes that largely contributed to the loose and -weak organization of the new German Confederation. The Federal Act of -the Vienna congress was proclaimed in Bavaria, not as a law but as an -international treaty. It was partly to secure popular support in his -resistance to any interference of the federal diet in the internal -affairs of Bavaria, partly to give unity to his somewhat heterogeneous -territories, that Maximilian on the 26th of May 1818 granted a liberal -constitution to his people. Montgelas, who had opposed this concession, -had fallen in the previous year, and Maximilian had also reversed his -ecclesiastical policy, signing on the 24th of October 1817 a concordat -with Rome by which the powers of the clergy, largely curtailed under -Montgelas's administration, were restored. The new parliament proved so -intractable that in 1819 Maximilian was driven to appeal to the powers -against his own creation; but his Bavarian "particularism" and his -genuine popular sympathies prevented him from allowing the Carlsbad -decrees to be strictly enforced within his dominions. The suspects -arrested by order of the Mainz Commission he was accustomed to examine -himself, with the result that in many cases the whole proceedings were -quashed, and in not a few the accused dismissed with a present of money. -Maximilian died on the 13th of October 1825 and was succeeded by his son -Louis I. - -In private life Maximilian was kindly and simple. He loved to play the -part of _Landesvater_, walking about the streets of his capital _en -bourgeois_ and entering into conversation with all ranks of his -subjects, by whom he was regarded with great affection. He was twice -married: (1) in 1785 to Princess Wilhelmine Auguste of Hesse-Darmstadt, -(2) in 1797 to Princess Caroline Friederike of Baden. - - See G. Freiherr von Lerchenfeld, _Gesch. Bayerns unter Konig - Maximilian Joseph I._ (Berlin, 1854); J. M. Soltl, _Max Joseph, Konig - von Bayern_ (Stuttgart, 1837); L. von Kobell, _Unter den vier ersten - Konigen Bayerns. Nach Briefen und eigenen Erinnerungen_ (Munich, - 1894). - - - - -MAXIMILIAN II. (1811-1864), king of Bavaria, son of king Louis I. and of -his consort Theresa of Saxe-Hildburghausen, was born on the 28th of -November 1811. After studying at Gottingen and Berlin and travelling in -Germany, Italy and Greece, he was introduced by his father into the -council of state (1836). From the first he showed a studious -disposition, declaring on one occasion that had he not been born in a -royal cradle his choice would have been to become a professor. As crown -prince, in the chateau of Hohenschwangau near Fussen, which he had -rebuilt with excellent taste, he gathered about him an intimate society -of artists and men of learning, and devoted his time to scientific and -historical study. When the abdication of Louis I. (March 28, 1848) -called him suddenly to the throne, his choice of ministers promised a -liberal regime. The progress of the revolution, however, gave him pause. -He strenuously opposed the unionist plans of the Frankfort parliament, -refused to recognize the imperial constitution devised by it, and -assisted Austria in restoring the federal diet and in carrying out the -federal execution in Hesse and Holstein. Although, however, from 1850 -onwards his government tended in the direction of absolutism, he refused -to become the tool of the clerical reaction, and even incurred the -bitter criticism of the Ultramontanes by inviting a number of celebrated -men of learning and science (e.g. Liebig and Sybel) to Munich, -regardless of their religious views. Finally, in 1859, he dismissed the -reactionary ministry of von der Pfordten, and met the wishes of his -people for a moderate constitutional government. In his German policy he -was guided by the desire to maintain the union of the princes, and hoped -to attain this as against the perilous rivalry of Austria and Prussia by -the creation of a league of the "middle" and small states--the so-called -Trias. In 1863, however, seeing what he thought to be a better way, he -supported the project of reform proposed by Austria at the Furstentag of -Frankfort. The failure of this proposal, and the attitude of Austria -towards the Confederation and in the Schleswig-Holstein question, -undeceived him; but before he could deal with the new situation created -by the outbreak of the war with Denmark he died suddenly at Munich, on -the 10th of March 1864. - -Maximilian was a man of amiable qualities and of intellectual -attainments far above the average, but as a king he was hampered by -constant ill-health, which compelled him to be often abroad, and when at -home to live much in the country. By his wife, Maria Hedwig, daughter of -Prince William of Prussia, whom he married in 1842, he had two sons, -Louis II., king of Bavaria, and Otto, king of Bavaria, both of whom lost -their reason. - - See J. M. Soltl, _Max der Zweite, Konig von Bayern_ (Munich, 1865); - biography by G. K. Heigel in _Allgem. Deutsche Biographie_, vol. xxi. - (Leipzig, 1885). Maximilian's correspondence with Schlegel was - published at Stuttgart in 1890. - - - - -MAXIMILIAN I. (1459-1519), Roman emperor, son of the emperor Frederick -III. and Leonora, daughter of Edward, king of Portugal, was born at -Vienna Neustadt on the 22nd of March 1459. On the 18th of August 1477, -by his marriage at Ghent to Mary, who had just inherited Burgundy and -the Netherlands from her father Charles the Bold, duke of Burgundy, he -effected a union of great importance in the history of the house of -Habsburg. He at once undertook the defence of his wife's dominions from -an attack by Louis XI., king of France, and defeated the French forces -at Guinegatte, the modern Enguinegatte, on the 7th of August 1479. But -Maximilian was regarded with suspicion by the states of Netherlands, and -after suppressing a rising in Gelderland his position was further -weakened by the death of his wife on the 27th of March 1482. He claimed -to be recognized as guardian of his young son Philip and as regent of -the Netherlands, but some of the states refused to agree to his demands -and disorder was general. Maximilian was compelled to assent to the -treaty of Arras in 1482 between the states of the Netherlands and Louis -XI. This treaty provided that Maximilian's daughter Margaret should -marry Charles, the dauphin of France, and have for her dowry Artois and -Franche-Comte, two of the provinces in dispute, while the claim of Louis -on the duchy of Burgundy was tacitly admitted. Maximilian did not, -however, abandon the struggle in the Netherlands. Having crushed a -rebellion at Utrecht, he compelled the burghers of Ghent to restore -Philip to him in 1485, and returning to Germany was chosen king of the -Romans, or German king, at Frankfort on the 16th of February 1486, and -crowned at Aix-la-Chapelle on the 9th of the following April. Again in -the Netherlands, he made a treaty with Francis II., duke of Brittany, -whose independence was threatened by the French regent, Anne of Beaujeu, -and the struggle with France was soon renewed. This war was very -unpopular with the trading cities of the Netherlands, and early in 1488 -Maximilian, having entered Bruges, was detained there as a prisoner for -nearly three months, and only set at liberty on the approach of his -father with a large force. On his release he had promised he would -maintain the treaty of Arras and withdraw from the Netherlands; but he -delayed his departure for nearly a year and took part in a punitive -campaign against his captors and their allies. On his return to Germany -he made peace with France at Frankfort in July 1489, and in October -several of the states of the Netherlands recognized him as their ruler -and as guardian of his son. In March 1490 the county of Tirol was added -to his possessions through the abdication of his kinsman, Count -Sigismund, and this district soon became his favourite residence. - -Meanwhile the king had formed an alliance with Henry VII. king of -England, and Ferdinand II., king of Aragon, to defend the possessions of -the duchess Anne, daughter and successor of Francis, duke of Brittany. -Early in 1490 he took a further step and was betrothed to the duchess, -and later in the same year the marriage was celebrated by proxy; but -Brittany was still occupied by French troops, and Maximilian was unable -to go to the assistance of his bride. The sequel was startling. In -December 1491 Anne was married to Charles VIII., king of France, and -Maximilian's daughter Margaret, who had resided in France since her -betrothal, was sent back to her father. The inaction of Maximilian at -this time is explained by the condition of affairs in Hungary, where -the death of king Matthias Corvinus had brought about a struggle for -this throne. The Roman king, who was an unsuccessful candidate, took up -arms, drove the Hungarians from Austria, and regained Vienna, which had -been in the possession of Matthias since 1485; but he was compelled by -want of money to retreat, and on the 7th of November 1491 signed the -treaty of Pressburg with Ladislaus, king of Bohemia, who had obtained -the Hungarian throne. By this treaty it was agreed that Maximilian -should succeed to the crown in case Ladislaus left no legitimate male -issue. Having defeated the invading Turks at Villach in 1492, the king -was eager to take revenge upon the king of France; but the states of the -Netherlands would afford him no assistance. The German diet was -indifferent, and in May 1493 he agreed to the peace of Senlis and -regained Artois and Franche-Comte. - -In August 1493 the death of the emperor left Maximilian sole ruler of -Germany and head of the house of Habsburg; and on the 16th of March 1494 -he married at Innsbruck Bianca Maria Sforza, daughter of Galeazzo -Sforza, duke of Milan (d. 1476). At this time Bianca's uncle, Ludovico -Sforza, was invested with the duchy of Milan in return for the -substantial dowry which his niece brought to the king. Maximilian -harboured the idea of driving the Turks from Europe; but his appeal to -all Christian sovereigns was ineffectual. In 1494 he was again in the -Netherlands, where he led an expedition against the rebels of -Gelderland, assisted Perkin Warbeck to make a descent upon England, and -formally handed over the government of the Low Countries to Philip. His -attention was next turned to Italy, and, alarmed at the progress of -Charles VIII. in the peninsula, he signed the league of Venice in March -1495, and about the same time arranged a marriage between his son Philip -and Joanna, daughter of Ferdinand and Isabella, king and queen of -Castile and Aragon. The need for help to prosecute the war in Italy -caused the king to call the diet to Worms in March 1495, when he urged -the necessity of checking the progress of Charles. As during his -father's lifetime Maximilian had favoured the reforming party among the -princes, proposals for the better government of the empire were brought -forward at Worms as a necessary preliminary to financial and military -support. Some reforms were adopted, the public peace was proclaimed -without any limitation of time and a general tax was levied. The three -succeeding years were mainly occupied with quarrels with the diet, with -two invasions of France, and a war in Gelderland against Charles, count -of Egmont, who claimed that duchy, and was supported by French troops. -The reforms of 1495 were rendered abortive by the refusal of Maximilian -to attend the diets or to take any part in the working of the new -constitution, and in 1497 he strengthened his own authority by -establishing an Aulic Council (_Reichshofrath_), which he declared was -competent to deal with all business of the empire, and about the same -time set up a court to centralize the financial administration of -Germany. - -In February 1499 the king became involved in a war with the Swiss, who -had refused to pay the imperial taxes or to furnish a contribution for -the Italian expedition. Aided by France they defeated the German troops, -and the peace of Basel in September 1499 recognized them as virtually -independent of the empire. About this time Maximilian's ally, Ludovico -of Milan, was taken prisoner by Louis XII., king of France, and -Maximilian was again compelled to ask the diet for help. An elaborate -scheme for raising an army was agreed to, and in return a council of -regency (_Reichsregiment_) was established, which amounted, in the words -of a Venetian envoy, to a deposition of the king. The relations were now -very strained between the reforming princes and Maximilian, who, unable -to raise an army, refused to attend the meetings of the council at -Nuremberg, while both parties treated for peace with France. The -hostility of the king rendered the council impotent. He was successful -in winning the support of many of the younger princes, and in -establishing a new court of justice, the members of which were named by -himself. The negotiations with France ended in the treaty of Blois, -signed in September 1504, when Maximilian's grandson Charles was -betrothed to Claude, daughter of Louis XII., and Louis, invested with -the duchy of Milan, agreed to aid the king of the Romans to secure the -imperial crown. A succession difficulty in Bavaria-Landshut was only -decided after Maximilian had taken up arms and narrowly escaped with his -life at Regensburg. In the settlement of this question, made in 1505, he -secured a considerable increase of territory, and when the king met the -diet at Cologne in 1505 he was at the height of his power. His enemies -at home were crushed, and their leader, Berthold, elector of Mainz, was -dead; while the outlook abroad was more favourable than it had been -since his accession. - -It is at this period that Ranke believes Maximilian to have entertained -the idea of a universal monarchy; but whatever hopes he may have had -were shattered by the death of his son Philip and the rupture of the -treaty of Blois. The diet of Cologne discussed the question of reform in -a halting fashion, but afforded the king supplies for an expedition into -Hungary, to aid his ally Ladislaus, and to uphold his own influence in -the East. Having established his daughter Margaret as regent for Charles -in the Netherlands, Maximilian met the diet at Constance in 1507, when -the imperial chamber (_Reichskammergericht_) was revised and took a more -permanent form, and help was granted for an expedition to Italy. The -king set out for Rome to secure his coronation, but Venice refused to -let him pass through her territories; and at Trant, on the 4th of -February 1508, he took the important step of assuming the title of Roman -Emperor Elect, to which he soon received the assent of pope Julius II. -He attacked the Venetians, but finding the war unpopular with the -trading cities of southern Germany, made a truce with the republic for -three years. The treaty of Blois had contained a secret article -providing for an attack on Venice, and this ripened into the league of -Cambray, which was joined by the emperor in December 1509. He soon took -the field, but after his failure to capture Padua the league broke up; -and his sole ally, the French king, joined him in calling a general -council at Pisa to discuss the question of Church reform. A breach with -pope Julius followed, and at this time Maximilian appears to have -entertained, perhaps quite seriously, the idea of seating himself in the -chair of St Peter. After a period of vacillation he deserted Louis and -joined the Holy League, which had been formed to expel the French from -Italy; but unable to raise troops, he served with the English forces as -a volunteer and shared in the victory gained over the French at the -battle of the Spurs near Therouanne on the 16th of August 1513. In 1500 -the diet had divided Germany into six circles, for the maintenance of -peace, to which the emperor at the diet of Cologne in 1512 added four -others. Having made an alliance with Christian II., king of Denmark, and -interfered to protect the Teutonic Order against Sigismund I., king of -Poland, Maximilian was again in Italy early in 1516 fighting the French -who had overrun Milan. His want of success compelled him on the 4th of -December 1516 to sign the treaty of Brussels, which left Milan in the -hands of the French king, while Verona was soon afterwards transferred -to Venice. He attempted in vain to secure the election of his grandson -Charles as king of the Romans, and in spite of increasing infirmity was -eager to lead the imperial troops against the Turks. At the diet of -Augsburg in 1518 the emperor heard warnings of the Reformation in the -shape of complaints against papal exactions, and a repetition of the -complaints preferred at the diet of Mainz in 1517 about the -administration of Germany. Leaving the diet, he travelled to Wels in -Upper Austria, where he died on the 12th of January 1519. He was buried -in the church of St George in Vienna Neustadt, and a superb monument, -which may still be seen, was raised to his memory at Innsbruck. - - Maximilian had many excellent personal qualities. He was not handsome, - but of a robust and well-proportioned frame. Simple in his habits, - conciliatory in his bearing, and catholic in his tastes, he enjoyed - great popularity and rarely made a personal enemy. He was a skilled - knight and a daring huntsman, and although not a great general, was - intrepid on the field of battle. His mental interests were extensive. - He knew something of six languages, and could discuss art, music, - literature or theology. He reorganized the university of Vienna and - encouraged the development of the universities of Ingolstadt and - Freiburg. He was the friend and patron of scholars, caused manuscripts - to be copied and medieval poems to be collected. He was the author of - military reforms, which included the establishment of standing troops, - called _Landsknechte_, the improvement of artillery by making cannon - portable, and some changes in the equipment of the cavalry. He was - continually devising plans for the better government of Austria, and - although they ended in failure, he established the unity of the - Austrian dominions. Maximilian has been called the second founder of - the house of Habsburg, and certainly by bringing about marriages - between Charles and Joanna and between his grandson Ferdinand and - Anna, daughter of Ladislaus, king of Hungary and Bohemia, he paved the - way for the vast empire of Charles V. and for the influence of the - Habsburgs in eastern Europe. But he had many qualities less desirable. - He was reckless and unstable, resorting often to lying and deceit, and - never pausing to count the cost of an enterprise or troubling to adapt - means to ends. For absurd and impracticable schemes in Italy and - elsewhere he neglected Germany, and sought to involve its princes in - wars undertaken solely for private aggrandizement or personal - jealousy. Ignoring his responsibilities as ruler of Germany, he only - considered the question of its government when in need of money and - support from the princes. As the "last of the knights" he could not - see that the old order of society was passing away and a new order - arising, while he was fascinated by the glitter of the medieval empire - and spent the better part of his life in vague schemes for its - revival. As "a gifted amateur in politics" he increased the disorder - of Germany and Italy and exposed himself and the empire to the jeers - of Europe. - - Maximilian was also a writer of books, and his writings display his - inordinate vanity. His _Geheimes Jagdbuch_, containing about 2500 - words, is a treatise purporting to teach his grandsons the art of - hunting. He inspired the production of _The Dangers and Adventures of - the Famous Hero and Knight Sir Teuerdank_, an allegorical poem - describing his adventures on his journey to marry Mary of Burgundy. - The emperor's share in the work is not clear, but it seems certain - that the general scheme and many of the incidents are due to him. It - was first published at Nuremberg by Melchior Pfintzing in 1517, and - was adorned with woodcuts by Hans Leonhard Schaufelein. The - _Weisskunig_ was long regarded as the work of the emperor's secretary, - Marx Treitzsaurwein, but it is now believed that the greater part of - the book at least is the work of the emperor himself. It is an - unfinished autobiography containing an account of the achievements of - Maximilian, who is called "the young white king." It was first - published at Vienna in 1775. He also is responsible for _Freydal_, an - allegorical account of the tournaments in which he took part during - his wooing of Mary of Burgundy; _Ehrenpforten_, _Triumphwagen_ and - _Der weisen konige Stammbaum_, books concerning his own history and - that of the house of Habsburg, and works on various subjects, as _Das - Stahlbuch_, _Die Baumeisterei_ and _Die Gartnerei_. These works are - all profusely illustrated, some by Albrecht Durer, and in the - preparation of the woodcuts Maximilian himself took the liveliest - interest. A facsimile of the original editions of Maximilian's - autobiographical and semi-autobiographical works has been published in - nine volumes in the _Jahrbucher der kunsthistorischen Sammlungen des - Kaiserhauses_ (Vienna, 1880-1888). For this edition S. Laschitzer - wrote an introduction to _Sir Teuerdank_, Q. von Leitner to _Freydal_, - and N. A. von Schultz to _Der Weisskunig_. The Holbein society issued - a facsimile of _Sir Teuerdank_ (London, 1884) and _Triumphwagen_ - (London, 1883). - - See _Correspondance de l'empereur Maximilien I. et de Marguerite - d'Autriche, 1507-1519_, edited by A. G. le Glay (Paris, 1839); - _Maximilians I. vertraulicher Briefwechsel mit Sigmund Pruschenk_, - edited by V. von Kraus (Innsbruck, 1875); J. Chmel, _Urkunden, Briefe - und Aktenstucke zur Geschichte Maximilians I. und seiner Zeit_. - (Stuttgart, 1845) and _Aktenstucke und Briefe zur Geschichte des - Hauses Habsburg im Zeitalter Maximilians I._ (Vienna, 1854-1858); K. - Klupfel, _Kaiser Maximilian I._ (Berlin, 1864); H. Ulmann, _Kaiser - Maximilian I._ (Stuttgart, 1884); L. P. Gachard, _Lettres inedites de - Maximilien I. sur les affaires des Pays Bas_ (Brussels, 1851-1852); L. - von Ranke, _Geschichte der romanischen und germanischen Volker, - 1494-1514_ (Leipzig, 1874); R. W. S. Watson, _Maximilian I._ (London, - 1902); A. Jager, _Uber Kaiser Maximilians I. Verhaltnis zum Papstthum_ - (Vienna, 1854); H. Ulmann, _Kaiser Maximilians I. Absichten auf das - Papstthum_ (Stuttgart, 1888), and A. Schulte, _Kaiser Maximilian I. - als Kandidat fur den papstlichen Stuhl_ (Leipzig, 1906). - (A. W. H.*) - - - - -MAXIMILIAN II. (1527-1576), Roman emperor, was the eldest son of the -emperor Ferdinand I. by his wife Anne, daughter of Ladislaus, king of -Hungary and Bohemia, and was born in Vienna on the 31st of July 1527. -Educated principally in Spain, he gained some experience of warfare -during the campaign of Charles V. against France in 1544, and also -during the war of the league of Schmalkalden, and soon began to take -part in imperial business. Having in September 1548 married his cousin -Maria, daughter of Charles V., he acted as the emperor's representative -in Spain from 1548 to 1550, returning to Germany in December 1550 in -order to take part in the discussion over the imperial succession. -Charles V. wished his son Philip (afterwards king of Spain) to succeed -him as emperor, but his brother Ferdinand, who had already been -designated as the next occupant of the imperial throne, and Maximilian -objected to this proposal. At length a compromise was reached. Philip -was to succeed Ferdinand, but during the former's reign Maximilian, as -king of the Romans, was to govern Germany. This arrangement was not -carried out, and is only important because the insistence of the emperor -seriously disturbed the harmonious relations which had hitherto existed -between the two branches of the Habsburg family; and the estrangement -went so far that an illness which befell Maximilian in 1552 was -attributed to poison given to him in the interests of his cousin and -brother-in-law, Philip of Spain. About this time he took up his -residence in Vienna, and was engaged mainly in the government of the -Austrian dominions and in defending them against the Turks. The -religious views of the king of Bohemia, as Maximilian had been called -since his recognition as the future ruler of that country in 1549, had -always been somewhat uncertain, and he had probably learned something of -Lutheranism in his youth; but his amicable relations with several -Protestant princes, which began about the time of the discussion over -the succession, were probably due more to political than to religious -considerations. However, in Vienna he became very intimate with -Sebastian Pfauser (1520-1569), a court preacher with strong leanings -towards Lutheranism, and his religious attitude caused some uneasiness -to his father. Fears were freely expressed that he would definitely -leave the Catholic Church, and when Ferdinand became emperor in 1558 he -was prepared to assure Pope Paul IV. that his son should not succeed him -if he took this step. Eventually Maximilian remained nominally an -adherent of the older faith, although his views were tinged with -Lutheranism until the end of his life. After several refusals he -consented in 1560 to the banishment of Pfauser, and began again to -attend the services of the Catholic Church. This uneasiness having been -dispelled, in November 1562 Maximilian was chosen king of the Romans, or -German king, at Frankfort, where he was crowned a few days later, after -assuring the Catholic electors of his fidelity to their faith, and -promising the Protestant electors that he would publicly accept the -confession of Augsburg when he became emperor. He also took the usual -oath to protect the Church, and his election was afterwards confirmed by -the papacy. In September 1563 he was crowned king of Hungary, and on his -father's death, in July 1564, succeeded to the empire and to the -kingdoms of Hungary and Bohemia. - -The new emperor had already shown that he believed in the necessity for -a thorough reform of the Church. He was unable, however, to obtain the -consent of Pope Pius IV. to the marriage of the clergy, and in 1568 the -concession of communion in both kinds to the laity was withdrawn. On his -part Maximilian granted religious liberty to the Lutheran nobles and -knights in Austria, and refused to allow the publication of the decrees -of the council of Trent. Amid general expectations on the part of the -Protestants he met his first Diet at Augsburg in March 1566. He refused -to accede to the demands of the Lutheran princes; on the other hand, -although the increase of sectarianism was discussed, no decisive steps -were taken to suppress it, and the only result of the meeting was a -grant of assistance for the Turkish War, which had just been renewed. -Collecting a large and splendid army Maximilian marched to defend his -territories; but no decisive engagement had taken place when a truce was -made in 1568, and the emperor continued to pay tribute to the sultan for -Hungary. Meanwhile the relations between Maximilian and Philip of Spain -had improved; and the emperor's increasingly cautious and moderate -attitude in religious matters was doubtless due to the fact that the -death of Philip's son, Don Carlos, had opened the way for the succession -of Maximilian, or of one of his sons, to the Spanish throne. Evidence -of this friendly feeling was given in 1570, when the emperor's daughter, -Anne, became the fourth wife of Philip; but Maximilian was unable to -moderate the harsh proceedings of the Spanish king against the revolting -inhabitants of the Netherlands. In 1570 the emperor met the diet at -Spires and asked for aid to place his eastern borders in a state of -defence, and also for power to repress the disorder caused by troops in -the service of foreign powers passing through Germany. He proposed that -his consent should be necessary before any soldiers for foreign service -were recruited in the empire; but the estates were unwilling to -strengthen the imperial authority, the Protestant princes regarded the -suggestion as an attempt to prevent them from assisting their -coreligionists in France and the Netherlands, and nothing was done in -this direction, although some assistance was voted for the defence of -Austria. The religious demands of the Protestants were still -unsatisfied, while the policy of toleration had failed to give peace to -Austria. Maximilian's power was very limited; it was inability rather -than unwillingness that prevented him from yielding to the entreaties of -Pope Pius V. to join in an attack on the Turks both before and after the -victory of Lepanto in 1571; and he remained inert while the authority of -the empire in north-eastern Europe was threatened. His last important -act was to make a bid for the throne of Poland, either for himself or -for his son Ernest. In December 1575 he was elected by a powerful -faction, but the diet which met at Regensburg was loath to assist; and -on the 12th of October 1576 the emperor died, refusing on his deathbed -to receive the last sacraments of the Church. - -By his wife Maria he had a family of nine sons and six daughters. He was -succeeded by his eldest surviving son, Rudolph, who had been chosen king -of the Romans in October 1575. Another of his sons, Matthias, also -became emperor; three others, Ernest, Albert and Maximilian, took some -part in the government of the Habsburg territories or of the -Netherlands, and a daughter, Elizabeth, married Charles IX. king of -France. - - The religious attitude of Maximilian has given rise to much - discussion, and on this subject the writings of W. Maurenbrecher, W. - Goetz and E. Reimann in the _Historische Zeitschrift_, Bande VII., - XV., XXXII. and LXXVII. (Munich, 1870 fol.) should be consulted, and - also O. H. Hopfen, _Maximilian II. und der Kompromisskatholizismus_ - (Munich, 1895); C. Haupt, _Melanchthons und seiner Lehrer Einfluss auf - Maximilian II._ (Wittenberg, 1897); F. Walter, _Die Wahl Maximilians - II._ (Heidelberg, 1892); W. Goetz, _Maximilians II. Wahl zum romischen - Konige_ (Wurzburg, 1891), and T. J. Scherg, _Uber die religiose - Entwickelung Kaiser Maximilians II. bis zu seiner Wahl zum romischen - Konige_ (Wurzburg, 1903). For a more general account of his life and - work see _Briefe und Akten zur Geschichte Maximilians II._, edited by - W. E. Schwarz (Paderborn, 1889-1891); M. Koch, _Quellen zur Geschichte - des Kaisers Maximilian II. in Archiven gesammelt_ (Leipzig, - 1857-1861); R. Holtzmann, _Kaiser Maximilian II. bis zu seiner - Thronbesteigung_ (Berlin, 1903); E. Wertheimer, _Zur Geschichte der - Turkenkriege Maximilians II._ (Vienna, 1875); L. von Ranke, _Uber die - Zeiten Ferdinands I. und Maximilians II._ in Band VII. of his - _Sammtliche Werke_ (Leipzig, 1874), and J. Janssen, _Geschichte des - deutschen Volkes seit dem Ausgang des Mittelalters,_ Bande IV. to - VIII. (Freiburg, 1885-1894), English translation by M. A. Mitchell and - A. M. Christie (London, 1896 fol.). - - - - -MAXIMILIAN (1832-1867), emperor of Mexico, second son of the archduke -Francis Charles of Austria, was born in the palace of Schonbrunn, on the -6th of July 1832. He was a particularly clever boy, showed considerable -taste for the arts, and early displayed an interest in science, -especially botany. He was trained for the navy, and threw himself into -this career with so much zeal that he quickly rose to high command, and -was mainly instrumental in creating the naval port of Trieste and the -fleet with which Tegethoff won his victories in the Italian War. He had -some reputation as a Liberal, and this led, in February 1857, to his -appointment as viceroy of the Lombardo-Venetian kingdom; in the same -year he married the Princess Charlotte, daughter of Leopold I., king of -the Belgians. On the outbreak of the war of 1859 he retired into private -life, chiefly at Trieste, near which he built the beautiful chateau of -Miramar. In this same year he was first approached by Mexican exiles -with the proposal to become the candidate for the throne of Mexico. He -did not at first accept, but sought to satisfy his restless desire for -adventure by a botanical expedition to the tropical forests of Brazil. -In 1863, however, under pressure from Napoleon III., and after General -Forey's capture of the city of Mexico and the plebiscite which confirmed -his proclamation of the empire, he consented to accept the crown. This -decision was contrary to the advice of his brother, the emperor Francis -Joseph, and involved the loss of all his rights in Austria. Maximilian -landed at Vera Cruz on the 28th of May 1864; but from the very outset he -found himself involved in difficulties of the most serious kind, which -in 1866 made apparent to almost every one outside of Mexico the -necessity for his abdicating. Though urged to this course by Napoleon -himself, whose withdrawal from Mexico was the final blow to his cause, -Maximilian refused to desert his followers. Withdrawing, in February -1867, to Queretaro, he there sustained a siege for several weeks, but on -the 15th of May resolved to attempt an escape through the enemy's lines. -He was, however, arrested before he could carry out this resolution, and -after trial by court-martial was condemned to death. The sentence was -carried out on the 19th of June 1867. His remains were conveyed to -Vienna, where they were buried in the imperial vault early in the -following year. (See MEXICO.) - - Maximilian's papers were published at Leipzig in 1867, in seven - volumes, under the title _Aus meinem Leben, Reiseskizzen, Aphorismen, - Gedichte._ See Pierre de la Gorce, _Hist. du Second Empire_, IV., liv. - xxv. ii. (Paris, 1904); article by von Hoffinger in _Allgemeine - Deutsche Biographie_, xxi. 70, where authorities are cited. - - - - -MAXIMINUS, GAIUS JULIUS VERUS, Roman emperor from A.D. 235 to 238, was -born in a village on the confines of Thrace. He was of barbarian -parentage and was brought up as a shepherd. His immense stature and -enormous feats of strength attracted the attention of the emperor -Septimius Severus. He entered the army, and under Caracalla rose to the -rank of centurion. He carefully absented himself from court during the -reign of Heliogabalus, but under his successor Alexander Severus, was -appointed supreme commander of the Roman armies. After the murder of -Alexander in Gaul, hastened, it is said, by his instigation, Maximinus -was proclaimed emperor by the soldiers on the 19th of March 235. The -three years of his reign, which were spent wholly in the camp, were -marked by great cruelty and oppression; the widespread discontent thus -produced culminated in a revolt in Africa and the assumption of the -purple by Gordian (q.v.). Maximinus, who was in Pannonia at the time, -marched against Rome, and passing over the Julian Alps descended on -Aquileia; while detained before that city he and his son were murdered -in their tent by a body of praetorians. Their heads were cut off and -despatched to Rome, where they were burnt on the Campus Martius by the -exultant crowd. - - Capitolinus, _Maximini duo_; Herodian vi. 8, vii., viii. 1-5; Zosimus - i. 13-15. - - - - -MAXIMINUS [MAXIMIN], GALERIUS VALERIUS, Roman emperor from A.D. 308 to -314, was originally an Illyrian shepherd named Daia. He rose to high -distinction after he had joined the army, and in 305 he was raised by -his uncle, Galerius, to the rank of Caesar, with the government of Syria -and Egypt. In 308, after the elevation of Licinius, he insisted on -receiving the title of Augustus; on the death of Galerius, in 311, he -succeeded to the supreme command of the provinces of Asia, and when -Licinius and Constantine began to make common cause with one another -Maximinus entered into a secret alliance with Maxentius. He came to an -open rupture with Licinius in 313, sustained a crushing defeat in the -neighbourhood of Heraclea Pontica on the 30th of April, and fled, first -to Nicomedia and afterwards to Tarsus, where he died in August -following. His death was variously ascribed "to despair, to poison, and -to the divine justice." Maximinus has a bad name in Christian annals, as -having renewed persecution after the publication of the toleration edict -of Galerius, but it is probable that he has been judged too harshly. - - See MAXENTIUS; Zosimus ii. 8; Aurelius Victor, _Epit_. 40. - - - - -MAXIMS, LEGAL. A maxim is an established principle or proposition. The -Latin term _maxima_ is not to be found in Roman law with any meaning -exactly analogous to that of a legal maxim in the modern sense of the -word, but the treatises of many of the Roman jurists on _Regulae -definitiones_, and _Sententiae juris_ are, in some measure, collections -of maxims (see an article on "Latin Maxims in English Law" in _Law Mag. -and Rev._ xx. 285); Fortescue (_De laudibus_, c. 8) and Du Cange treat -_maxima_ and _regula_ as identical. The attitude of early English -commentators towards the maxims of the law was one of unmingled -adulation. In _Doctor and Student_ (p. 26) they are described as "of the -same strength and effect in the law as statutes be." Coke (Co. _Litt._ -11 A) says that a maxim is so called "Quia maxima est ejus dignitas et -certissima auctoritas, atque quod maxime omnibus probetur." "Not only," -observes Bacon in the Preface to his _Collection of Maxims_, "will the -use of maxims be in deciding doubt and helping soundness of judgment, -but, further, in gracing argument, in correcting unprofitable subtlety, -and reducing the same to a more sound and substantial sense of law, in -reclaiming vulgar errors, and, generally, in the amendment in some -measure of the very nature and complexion of the whole law." A similar -note was sounded in Scotland; and it has been well observed that "a -glance at the pages of Morrison's _Dictionary_ or at other early reports -will show how frequently in the older Scots law questions respecting the -rights, remedies and liabilities of individuals were determined by an -immediate reference to legal maxims" (J. M. Irving, _Encyclo. Scots -Law_, s.v. "Maxims"). In later times less value has been attached to the -maxims of the law, as the development of civilization and the increasing -complexity of business relations have shown the necessity of qualifying -the propositions which they enunciate (see Stephen, _Hist. Crim. Law_, -ii. 94 _n: Yarmouth_ v. _France_, 1887, 19 Q.B.D., per Lord Esher, at p. -653, and American authorities collected in Bouvier's _Law Dict._ s.v. -"Maxim"). But both historically and practically they must always possess -interest and value. - - A brief reference need only be made here, with examples by way of - illustration, to the field which the maxims of the law cover. - - Commencing with rules founded on public policy, we may note the famous - principle--_Salus populi suprema lex_ (xii. Tables: Bacon, _Maxims_, - reg. 12)--"the public welfare is the highest law." It is on this maxim - that the coercive action of the State towards individual liberty in a - hundred matters is based. To the same category belong the - maxims--_Summa ratio est quae pro religione facit_ (Co. _Litt._ 341 - a)--"the best rule is that which advances religion"--a maxim which - finds its application when the enforcement of foreign laws or - judgments supposed to violate our own laws or the principles of - natural justice is in question; and _Dies dominicus non est - juridicus_, which exempts Sunday from the lawful days for juridical - acts. Among the maxims relating to the crown, the most important are - _Rex non potest peccare_ (2 Rolle R. 304)--"The King can do no - wrong"--which enshrines the principle of ministerial responsibility, - and _Nullum tempus occurrit regi_ (2 Co. Inst. 273)--"lapse of time - does not bar the crown," a maxim qualified by various enactments in - modern times. Passing to the judicial office and the administration of - justice, we may refer to the rules--_Audi alteram partem_--a - proposition too familiar to need either translation or comment; _Nemo - debet esse judex in propria sua causa_ (12 Co. _Rep._ 114)--"no man - ought to be judge in his own cause"--a maxim which French law, and the - legal systems based upon or allied to it, have embodied in an - elaborate network of rules for judicial challenge; and the maxim which - defines the relative functions of judge and jury, _Ad quaestionem - facti non respondent judices, ad quaestionem legis non respondent - juratores_ (8 Co. _Rep._ 155). The maxim _Boni judicis est ampliare - jurisdictionem_ (Ch. Prec. 329) is certainly erroneous as it stands, - as a judge has no right to "extend his jurisdiction." If _justitiam_ - is substituted for _jurisdictionem_, as Lord Mansfield said it should - be (1 Burr. 304), the maxim is near the truth. A group of maxims - supposed to embody certain fundamental principles of legal right and - obligations may next be referred to: (a) _Ubi jus ibi remedium_ (see - Co. _Litt._ 197 b)--a maxim to which the evolution of the flexible - "action on the case," by which wrongs unknown to the "original writs" - were dealt with, was historically due, but which must be taken with - the gloss _Damnum absque injuria_--"there are forms of actual damage - which do not constitute legal injury" for which the law supplies no - remedy; (b) _Actus Dei nemini facit injuriam_ (2 Blackstone, 122)--and - its allied maxim, _Lex non cogit ad impossibilia_ (Co. _Litt._ 231 - b)--on which the whole doctrine of _vis major_ (_force majeure_) and - impossible conditions in the law of contract has been built up. In - this category may also be classed _Volenti non fit injuria_ (Wingate, - _Maxims_), out of which sprang the theory--now profoundly modified by - statute--of "common employment" in the law of employers' liability; - see _Smith_ v. _Baker_, 1891, A.C. 325. Other maxims deal with rights - of property--_Qui prior est tempore, potior est jure_ (Co. _Litt._ 14 - a), which consecrates the position of the _beati possidentes_ alike in - municipal and in international law; _Sic utere tuo ut alienum non - laedas_ (9 Co. _Rep._ 59), which has played its part in the - determination of the rights of adjacent owners; and _Domus sua cuique - est tutissimum refugium_ (5 Co. _Rep._ 92)--"a man's house is his - castle," a doctrine which has imposed limitations on the rights of - execution creditors (see EXECUTION). In the laws of family relations - there are the maxims _Consensus non concubitus facit matrimonium_ (Co. - _Litt._ 33 a)--the canon law of Europe prior to the council of Trent, - and still law in Scotland, though modified by legislation in England; - and _Pater is est quem nuptiae demonstrant_ (see Co. _Litt._ 7 b), on - which, in most civilized countries, the presumption of legitimacy - depends. In the interpretation of written instruments, the maxim - _Noscitur a sociis_ (3 _Term Reports_, 87), which proclaims the - importance of the context, still applies. So do the rules _Expressio - unius est exclusio alterius_ (Co. _Litt._ 210 a), and _Contemporanea - expositio est optima et fortissima in lege_ (2 Co. _Inst._ 11), which - lets in evidence of contemporaneous user as an aid to the - interpretation of statutes or documents; see _Van Diemen's Land Co._ - v. _Table Cape Marine Board_, 1906, A.C. 92, 98. We may conclude this - sketch with a miscellaneous summary: _Caveat emptor_ (Hob. 99)--"let - the purchaser beware"; _Qui facit per alium facile per se_, which - affirms the principal's liability for the acts of his agent; - _Ignorantia juris neminem excusat_, on which rests the ordinary - citizen's obligation to know the law; and _Vigilantibus non - dormientibus jura subveniunt_ (2 Co. _Inst._ 690), one of the maxims - in accordance with which courts of equity administer relief. Among - other "maxims of equity" come the rules that "he that seeks equity - must do equity," i.e. must act fairly, and that "equity looks upon - that as done which ought to be done"--a principle from which the - "conversion" into money of land directed to be sold, and of money - directed to be invested in the purchase of land, is derived. - - The principal collections of legal maxims are: _English Law_: Bacon, - _Collection of Some Principal Rules and Maxims of the Common Law_ - (1630); Noy, _Treatise of the principal Grounds and Maxims of the Law - of England_ (1641, 8th ed., 1824); Wingate, _Maxims of Reason_ (1728); - Francis, _Grounds and Rudiments of Law and Equity_ (2nd ed. 1751); - Lofft (annexed to his Reports, 1776); Broom, _Legal Maxims_ (7th ed. - London, 1900). _Scots Law_: Lord Trayner, _Latin Maxims and Phrases_ - (2nd ed., 1876); Stair, _Institutions of the Law of Scotland_, with - Index by More (Edinburgh, 1832). _American Treatises_: A. I. Morgan, - _English Version of Legal Maxims_ (Cincinnati, 1878); S. S. Peloubet, - _Legal Maxims in Law and Equity_ (New York, 1880). (A. W. R.) - - - - -MAXIMUS, the name of four Roman emperors. - -I. M. CLODIUS PUPIENUS MAXIMUS, joint emperor with D. Caelius Calvinus -Balbinus during a few months of the year A.D. 238. Pupienus was a -distinguished soldier, who had been proconsul of Bithynia, Achaea, and -Gallia Narbonensis. At the advanced age of seventy-four, he was chosen by -the senate with Balbinus to resist the barbarian Maximinus. Their complete -equality is shown by the fact that each assumed the titles of pontifex -maximus and princeps senatus. It was arranged that Pupienus should take -the field against Maximinus, while Balbinus remained at Rome to maintain -order, a task in which he signally failed. A revolt of the praetorians was -not repressed till much blood had been shed and a considerable part of the -city reduced to ashes. On his march, Pupienus, having received the news -that Maximinus had been assassinated by his own troops, returned in -triumph to Rome. Shortly afterwards, when both emperors were on the point -of leaving the city on an expedition--Pupienus against the Persians and -Balbinus against the Goths--the praetorians, who had always resented the -appointment of the senatorial emperors and cherished the memory of the -soldier-emperor Maximinus, seized the opportunity of revenge. When most of -the people were at the Capitoline games, they forced their way into the -palace, dragged Balbinus and Pupienus through the streets, and put them to -death. - - See Capitolinus, _Life of Maximus and Balbinus_; Herodian vii. 10, - viii. 6; Zonaras xii. 16; Orosius vii. 19; Eutropius ix. 2; Zosimus i. - 14; Aurelius Victor, _Caesares_, 26, _epit._ 26; H. Schiller, - _Geschichte der romischen Kaiserzeit_, i. 2; Gibbon, _Decline and - Fall_, ch. 7 and (for the chronology) appendix 12 (Bury's edition). - -II. MAGNUS MAXIMUS, a native of Spain, who had accompanied Theodosius on -several expeditions and from 368 held high military rank in Britain. The -disaffected troops having proclaimed Maximus emperor, he crossed over -to Gaul, attacked Gratian (q.v.), and drove him from Paris to Lyons, -where he was murdered by a partisan of Maximus. Theodosius being unable -to avenge the death of his colleague, an agreement was made (384 or 385) -by which Maximus was recognized as Augustus and sole emperor in Gaul, -Spain and Britain, while Valentinian II. was to remain unmolested in -Italy and Illyricum, Theodosius retaining his sovereignty in the East. -In 387 Maximus crossed the Alps, Valentinian was speedily put to flight, -while the invader established himself in Milan and for the time became -master of Italy. Theodosius now took vigorous measures. Advancing with a -powerful army, he twice defeated the troops of Maximus--at Siscia on the -Save, and at Poetovio on the Danube. He then hurried on to Aquileia, -where Maximus had shut himself up, and had him beheaded. Under the name -of Maxen Wledig, Maximus appears in the list of Welsh royal heroes (see -R. Williams, _Biog. Dict. of Eminent Welshmen_, 1852; "The Dream of -Maxen Wledig," in the _Mabinogion_). - - Full account with classical references in H. Richter, _Das - westromische Reich, besonders unter den Kaisern Gratian, Valentinian - II. und Maximus_ (1865); see also H. Schiller, _Geschichte der - romischen Kaiserzeit_, ii. (1887); Gibbon, _Decline and Fall_, ch. 27; - Tillemont, _Hist. des empereurs_, v. - -III. MAXIMUS TYRANNUS, made emperor in Spain by the Roman general, -Gerontius, who had rebelled against the usurper Constantine in 408. -After the defeat of Gerontius at Arelate (Arles) and his death in 411 -Maximus renounced the imperial title and was permitted by Constantine to -retire into private life. About 418 he rebelled again, but, failing in -his attempt, was seized, carried into Italy, and put to death at Ravenna -in 422. - - See Orosius vii. 42; Zosimus vi. 5; Sozomen ix. 3; E. A. Freeman, "The - Tyrants of Britain, Gaul and Spain, A.D. 406-411," in _English - Historical Review_, i. (1886). - -IV. PETRONIUS MAXIMUS, a member of the higher Roman nobility, had held -several court and public offices, including those of _praefectus Romae_ -(420) and _Italiae_ (439-441 and 445), and consul (433, 443). He was one -of the intimate associates of Valentinian III., whom he assisted in the -palace intrigues which led to the death of Aetius in 454; but an outrage -committed on the wife of Maximus by the emperor turned his friendship -into hatred. Maximus was proclaimed emperor immediately after -Valentinian's murder (March 16, 455), but after reigning less than three -months, he was murdered by some Burgundian mercenaries as he was fleeing -before the troops of Genseric, who, invited by Eudoxia, the widow of -Valentinian, had landed at the mouth of the Tiber (May or June 455). - - See Procopius, _Vand._ i. 4; Sidonius Apollinaris, _Panegyr. Aviti_, - ep. ii. 13; the various _Chronicles_; Gibbon, _Decline and Fall_, chs. - 35, 36; Tillemont, _Hist. des empereurs_, vi. - - - - -MAXIMUS, ST (c. 580-662), abbot of Chrysopolis, known as "the Confessor" -from his orthodox zeal in the Monothelite (q.v.) controversy, or as "the -monk," was born of noble parentage at Constantinople about the year 580. -Educated with great care, he early became distinguished by his talents -and acquirements, and some time after the accession of the emperor -Heraclius in 610 was made his private secretary. In 630 he abandoned the -secular life and entered the monastery of Chrysopolis (Scutari), -actuated, it was believed, less by any longing for the life of a recluse -than by the dissatisfaction he felt with the Monothelite leanings of his -master. The date of his promotion to the abbacy is uncertain. In 633 he -was one of the party of Sophronius of Jerusalem (the chief original -opponent of the Monothelites) at the council of Alexandria; and in 645 -he was again in Africa, when he held in presence of the governor and a -number of bishops the disputation with Pyrrhus, the deposed and banished -patriarch of Constantinople, which resulted in the (temporary) -conversion of his interlocutor to the Dyothelite view. In the following -year several African synods, held under the influence of Maximus, -declared for orthodoxy. In 649, after the accession of Martin I., he -went to Rome, and did much to fan the zeal of the new pope, who in -October of that year held the (first) Lateran synod, by which not only -the Monothelite doctrine but also the moderating _ecthesis_ of Heraclius -and _typus_ of Constans II. were anathematized. About 653 Maximus, for -the part he had taken against the latter document especially, was -apprehended (together with the pope) by order of Constans and carried a -prisoner to Constantinople. In 655, after repeated examinations, in -which he maintained his theological opinions with memorable constancy, -he was banished to Byzia in Thrace, and afterwards to Perberis. In 662 -he was again brought to Constantinople and was condemned by a synod to -be scourged, to have his tongue cut out by the root, and to have his -right hand chopped off. After this sentence had been carried out he was -again banished to Lazica, where he died on the 13th of August 662. He is -venerated as a saint both in the Greek and in the Latin Churches. -Maximus was not only a leader in the Monothelite struggle but a mystic -who zealously followed and advocated the system of Pseudo-Dionysius, -while adding to it an ethical element in the conception of the freedom -of the will. His works had considerable influence in shaping the system -of John Scotus Erigena. - - The most important of the works of Maximus will be found in Migne, - _Patrologia graeca_, xc. xci., together with an anonymous life; an - exhaustive list in Wagenmann's article in vol. xii. (1903) of - Hauck-Herzog's _Realencyklopadie_ where the following classification - is adopted: (a) exegetical, (b) scholia on the Fathers, (c) dogmatic - and controversial, (d) ethical and ascetic, (e) miscellaneous. The - details of the disputation with Pyrrhus and of the martyrdom are given - very fully and clearly in Hefele's _Conciliengeschichte_, iii. For - further literature see H. Gelzer in C. Krumbacher's _Geschichte der - byzantinischen Litteratur_ (1897). - - - - -MAXIMUS OF SMYRNA, a Greek philosopher of the Neo-platonist school, who -lived towards the end of the 4th century A.D. He was perhaps the most -important of the followers of Iamblichus. He is said to have been of a -rich and noble family, and exercised great influence over the emperor -Julian, who was commended to him by Aedesius. He pandered to the -emperor's love of magic and theurgy, and by judicious administration of -the omens won a high position at court. His overbearing manner made him -numerous enemies, and, after being imprisoned on the death of Julian, he -was put to death by Valens. He is a representative of the least -attractive side of Neoplatonism. Attaching no value to logical proof and -argument, he enlarged on the wonders and mysteries of nature, and -maintained his position by the working of miracles. In logic he is -reported to have agreed with Eusebius, Iamblichus and Porphyry in -asserting the validity of the second and third figures of the syllogism. - - - - -MAXIMUS OF TYRE (CASSIUS MAXIMUS TYRIUS), a Greek rhetorician and -philosopher who flourished in the time of the Antonines and Commodus -(2nd century A.D.). After the manner of the sophists of his age, he -travelled extensively, delivering lectures on the way. His writings -contain many allusions to the history of Greece, while there is little -reference to Rome; hence it is inferred that he lived longer in Greece, -perhaps as a professor at Athens. Although nominally a Platonist, he is -really an Eclectic and one of the precursors of Neoplatonism. There are -still extant by him forty-one essays or discourses ([Greek: dialexeis]) -on theological, ethical, and other philosophical commonplaces. With him -God is the supreme being, one and indivisible though called by many -names, accessible to reason alone; but as animals form the intermediate -stage between plants and human beings, so there exist intermediaries -between God and man, viz. daemons, who dwell on the confines of heaven -and earth. The soul in many ways bears a great resemblance to the -divinity; it is partly mortal, partly immortal, and, when freed from the -fetters of the body, becomes a daemon. Life is the sleep of the soul, -from which it awakes at death. The style of Maximus is superior to that -of the ordinary sophistical rhetorician, but scholars differ widely as -to the merits of the essays themselves. - -Maximus of Tyre must be distinguished from the Stoic Maximus, tutor of -Marcus Aurelius. - - Editions by J. Davies, revised with valuable notes by J. Markland - (1740); J. J. Reiske (1774); F. Dubner (1840, with Theophrastus, &c., - in the Didot series). Monographs by R. Rohdich (Beuthen, 1879); H. - Hobein, _De Maximo Tyrio quaestiones philol._ (Jena, 1895). There is - an English translation (1804) by Thomas Taylor, the Platonist. - - - - -MAX MULLER, FRIEDRICH (1823-1900), Anglo-German orientalist and -comparative philologist, was born at Dessau on the 6th of December 1823, -being the son of Wilhelm Muller (1794-1827), the German poet, celebrated -for his phil-Hellenic lyrics, who was ducal librarian at Dessau. The -elder Muller had endeared himself to the most intellectual circles in -Germany by his amiable character and his genuine poetic gift; his songs -had been utilized by musical composers, notably Schubert; and it was his -son's good fortune to meet in his youth with a succession of eminent -friends, who, already interested in him for his father's sake, and -charmed by the qualities which they discovered in the young man himself, -powerfully aided him by advice and patronage. Mendelssohn, who was his -godfather, dissuaded him from indulging his natural bent to the study of -music; Professor Brockhaus of the University of Leipzig, where Max -Muller matriculated in 1841, induced him to take up Sanskrit; Bopp, at -the University of Berlin (1844), made the Sanskrit student a scientific -comparative philologist; Schelling at the same university, inspired him -with a love for metaphysical speculation, though failing to attract him -to his own philosophy; Burnouf, at Paris in the following year, by -teaching him Zend, started him on the track of inquiry into the science -of comparative religion, and impelled him to edit the _Rig Veda_; and -when, in 1846, Max Muller came to England upon this errand, Bunsen, in -conjunction with Professor H. H. Wilson, prevailed upon the East India -Company to undertake the expense of publication. Up to this time Max -Muller had lived the life of a poor student, supporting himself partly -by copying manuscripts, but Bunsen's introductions to Queen Victoria and -the prince consort, and to Oxford University, laid the foundation for -him of fame and fortune. In 1848 the printing of his _Rig Veda_ at the -University Press obliged him to settle in Oxford, a step which decided -his future career. He arrived at a favourable conjuncture: the -Tractarian strife, which had so long thrust learning into the -background, was just over, and Oxford was becoming accessible to modern -ideas. The young German excited curiosity and interest, and it was soon -discovered that, although a genuine scholar, he was no mere bookworm. -Part of his social success was due to his readiness to exert his musical -talents at private parties. Max Muller was speedily subjugated by the -_genius loci_. He was appointed deputy Taylorian professor of modern -languages in 1850, and the German government failed to tempt him back to -Strassburg. In the following year he was made M.A. and honorary fellow -of Christ Church, and in 1858 he was elected a fellow of All Souls. In -1854 the Crimean War gave him the opportunity of utilizing his oriental -learning in vocabularies and schemes of transliteration. In 1857 he -successfully essayed another kind of literature in his beautiful story -_Deutsche Liebe_, written both in German and English. He had by this -time become an extensive contributor to English periodical literature, -and had written several of the essays subsequently collected as _Chips -from a German Workshop_. The most important of them was the fascinating -essay on "Comparative Mythology" in the _Oxford Essays_ for 1856. His -valuable _History of Ancient Sanskrit Literature_, so far as it -illustrates the primitive religion of the Brahmans (and hence the Vedic -period only), was published in 1850. - -Though Max Muller's reputation was that of a comparative philologist and -orientalist, his professional duties at Oxford were long confined to -lecturing on modern languages, or at least their medieval forms. In 1860 -the death of Horace Hayman Wilson, professor of Sanskrit, seemed to open -a more congenial sphere to him. His claims to the succession seemed -incontestable, for his opponent, Monier Williams, though well qualified -as a Sanskritist, lacked Max Muller's brilliant versatility, and -although educated at Oxford, had held no University office. But Max -Muller was a Liberal, and the friend of Liberals in university matters, -in politics, and in theology, and this consideration united with his -foreign birth to bring the country clergy in such hosts to the poll that -the voice of resident Oxford was overborne, and Monier Williams was -elected by a large majority. It was the one great disappointment of Max -Muller's life, and made a lasting impression upon him. It was, -nevertheless, serviceable to his influence and reputation by permitting -him to enter upon a wider field of subjects than would have been -possible otherwise. Directly, Sanskrit philology received little more -from him, except in connexion with his later undertaking of _The Sacred -Books of the East_; but indirectly he exalted it more than any -predecessor by proclaiming its commanding position in the history of the -human intellect by his _Science of Language_, two courses of lectures -delivered at the Royal Institution in 1861 and 1863. Max Muller ought -not to be described as "the introducer of comparative philology into -England." Prichard had proved the Aryan affinities of the Celtic -languages by the methods of comparative philology so long before as -1831; Winning's _Manual of Comparative Philology_ had been published in -1838; the discoveries of Bopp and Pott and Pictet had been recognized in -brilliant articles in the _Quarterly Review_, and had guided the -researches of Rawlinson. But Max Muller undoubtedly did far more to -popularize the subject than had been done, or could have been done, by -any predecessor. He was on less sure ground in another department of the -study of language--the problem of its origin. He wrote upon it as a -disciple of Kant, whose _Critique of Pure Reason_ he translated. His -essays on mythology are among the most delightful of his writings, but -their value is somewhat impaired by a too uncompromising adherence to -the seductive generalization of the solar myth. - -Max Muller's studies in mythology led him to another field of activity -in which his influence was more durable and extensive, that of the -comparative science of religions. Here, so far as Great Britain is -concerned, he does deserve the fame of an originator, and his -_Introduction to the Science of Religion_ (1873: the same year in which -he lectured on the subject, at Dean Stanley's invitation, in Westminster -Abbey, this being the only occasion on which a layman had given an -address there) marks an epoch. It was followed by other works of -importance, especially the four volumes of Gifford lectures, delivered -between 1888 and 1892; but the most tangible result of the impulse he -had given was the publication under his editorship, from 1875 onwards, -of _The Sacred Books of the East_, in fifty-one volumes, including -indexes, all but three of which appeared under his superintendence -during his lifetime. These comprise translations by the most competent -scholars of all the really important non-Christian scriptures of -Oriental nations, which can now be appreciated without a knowledge of -the original languages. Max Muller also wrote on Indian philosophy in -his latter years, and his exertions to stimulate search for Oriental -manuscripts and inscriptions were rewarded with important discoveries of -early Buddhist scriptures, in their Indian form, made in Japan. He was -on particularly friendly terms with native Japanese scholars, and after -his death his library was purchased by the university of Tokyo. - -In 1868 Max Muller had been indemnified for his disappointment over the -Sanskrit professorship by the establishment of a chair of Comparative -Philology to be filled by him. He retired, however, from the actual -duties of the post in 1875, when entering upon the editorship of _The -Sacred Books of the East_. The most remarkable external events of his -latter years were his delivery of lectures at the restored university of -Strassburg in 1872, when he devoted his honorarium to the endowment of a -Sanskrit lectureship, and his presidency over the International Congress -of Orientalists in 1892. But his days, if uneventful, were busy. He -participated in every movement at Oxford of which he could approve, and -was intimate with nearly all its men of light and leading; he was a -curator of the Bodleian Library, and a delegate of the University Press. -He was acquainted with most of the crowned heads - -of Europe, and was an especial favourite with the English royal family. -His hospitality was ample, especially to visitors from India, where he -was far better known than any other European Orientalist. His -distinctions, conferred by foreign governments and learned societies, -were innumerable, and, having been naturalized shortly after his arrival -in England, he received the high honour of being made a privy -councillor. In 1898 and 1899 he published autobiographical reminiscences -under the title of _Auld Lang Syne_. He was writing a more detailed -autobiography when overtaken by death on the 28th of October 1900. Max -Muller married in 1859 Georgiana Adelaide Grenfell, sister of the wives -of Charles Kingsley and J. A. Froude. One of his daughters, Mrs -Conybeare, distinguished herself by a translation of Scherer's _History -of German Literature_. - -Though undoubtedly a great scholar, Max Muller did not so much represent -scholarship pure and simple as her hybrid types--the scholar-author and -the scholar-courtier. In the former capacity, though manifesting little -of the originality of genius, he rendered vast service by popularizing -high truths among high minds. In his public and social character he -represented Oriental studies with a brilliancy, and conferred upon them -a distinction, which they had not previously enjoyed in Great Britain. -There were drawbacks in both respects: the author was too prone to build -upon insecure foundations, and the man of the world incurred censure for -failings which may perhaps be best indicated by the remark that he -seemed too much of a diplomatist. But the sum of foibles seems -insignificant in comparison with the life of intense labour dedicated to -the service of culture and humanity. - - Max Muller's _Collected Works_ were published in 1903. (R. G.) - - - - -MAXWELL, the name of a Scottish family, members of which have held the -titles of earl of Morton, earl of Nithsdale, Lord Maxwell, and Lord -Herries. The name is taken probably from Maccuswell, or Maxwell, near -Kelso, whither the family migrated from England about 1100. Sir Herbert -Maxwell won great fame by defending his castle of Carlaverock against -Edward I. in 1300; another Sir Herbert was made a lord of the Scottish -parliament before 1445; and his great-grandson John, 3rd Lord Maxwell, -was killed at Flodden in 1513. John's son Robert, the 4th lord (d. -1546), was a member of the royal council under James V.; he was also an -extraordinary lord of session, high admiral, and warden of the west -marches, and was taken prisoner by the English at the rout of Solway -Moss in 1542. Robert's grandson John, 7th Lord Maxwell (1553-1593), was -the second son of Robert, the 5th lord (d. 1552), and his wife Beatrix, -daughter of James Douglas, 3rd earl of Morton. After the execution of -the regent Morton, the 4th earl, in 1581 this earldom was bestowed upon -Maxwell, but in 1586 the attainder of the late earl was reversed and he -was deprived of his new title. He had helped in 1585 to drive the royal -favourite James Stewart, earl of Arran, from power, and he made active -preparations to assist the invading Spaniards in 1588. His son John, the -8th lord (c. 1586-1613), was at feud with the Johnstones, who had killed -his father in a skirmish, and with the Douglases over the earldom of -Morton, which he regarded as his inheritance. After a life of -exceptional and continuous lawlessness he escaped from Scotland and in -his absence was sentenced to death; having returned to his native -country he was seized and was beheaded in Edinburgh. In 1618 John's -brother and heir Robert (d. 1646) was restored to the lordship of -Maxwell, and in 1620 was created earl of Nithsdale, surrendering at this -time his claim to the earldom of Morton. He and his son Robert, -afterwards the 2nd earl, fought under Montrose for Charles I. during the -Civil War. Robert died without sons in October 1667, when a cousin John -Maxwell, 7th Lord Herries (d. 1677), became third earl. - -William, 5th earl of Nithsdale (1676-1744), a grandson of the third -earl, was like his ancestor a Roman Catholic and was attached to the -cause of the exiled house of Stuart. In 1715 he joined the Jacobite -insurgents, being taken prisoner at the battle of Preston and sentenced -to death. He escaped, however, from the Tower of London through the -courage and devotion of his wife Winifred (d. 1749), daughter of William -Herbert, 1st marquess of Powis. He was attainted in 1716 and his titles -became extinct, but his estates passed to his son William (d. 1776), -whose descendant, William Constable-Maxwell, regained the title of Lord -Herries in 1858. The countess of Nithsdale wrote an account of her -husband's escape, which is published in vol. i. of the _Transactions of -the Society of Antiquaries of Scotland_. - - A few words may be added about other prominent members of the Maxwell - family. John Maxwell (c. 1590-1647), archbishop of Tuam, was a - Scottish ecclesiastic who took a leading part in helping Archbishop - Laud in his futile attempt to restore the liturgy in Scotland. He was - bishop of Ross from 1633 until 1638, when he was deposed by the - General Assembly; then crossing over to Ireland he was bishop of - Killala and Achonry from 1640 to 1645, and archbishop of Tuam from - 1645 until his death. James Maxwell of Kirkconnell (c. 1708-1762), the - Jacobite, wrote the _Narrative of Charles Prince of Wales's Expedition - to Scotland in 1745_, which was printed for the Maitland Club in 1841. - Robert Maxwell (1695-1765) was the author of _Select Transactions of - the Society of Improvers_ and was a great benefactor to Scottish - agriculture. Sir Murray Maxwell (1775-1831), a naval officer, gained - much fame by his conduct when his ship the "Alceste" was wrecked in - Gaspar Strait in 1817. William Hamilton Maxwell (1792-1850), the Irish - novelist, wrote, in addition to several novels, a _Life of the Duke of - Wellington_ (1839-1841 and again 1883), and a _History of the Irish - Rebellion in 1798_ (1845 and 1891). Sir Herbert Maxwell, 7th bart. (b. - 1845), member of parliament for Wigtownshire from 1880 to 1906, and - president of the Society of Antiquaries of Scotland, became well known - as a writer, his works including _Life and Times of the Right Hon. W. - H. Smith_ (1893); _Life of the Duke of Wellington_ (1899); _The House - of Douglas_ (1902); _Robert the Bruce_ (1897) and _A Duke of Britain_ - (1895). - - - - -MAXWELL, JAMES CLERK (1831-1879), British physicist, was the last -representative of a younger branch of the well-known Scottish family of -Clerk of Penicuik, and was born at Edinburgh on the 13th of November -1831. He was educated at the Edinburgh Academy (1840-1847) and the -university of Edinburgh (1847-1850). Entering at Cambridge in 1850, he -spent a term or two at Peterhouse, but afterwards migrated to Trinity. -In 1854 he took his degree as second wrangler, and was declared equal -with the senior wrangler of his year (E. J. Routh, q.v.) in the higher -ordeal of the Smith's prize examination. He held the chair of Natural -Philosophy in Marischal College, Aberdeen, from 1856 till the fusion of -the two colleges there in 1860. For eight years subsequently he held the -chair of Physics and Astronomy in King's College, London, but resigned -in 1868 and retired to his estate of Glenlair in Kirkcudbrightshire. He -was summoned from his seclusion in 1871 to become the first holder of -the newly founded professorship of Experimental Physics in Cambridge; -and it was under his direction that the plans of the Cavendish -Laboratory were prepared. He superintended every step of the progress of -the building and of the purchase of the very valuable collection of -apparatus with which it was equipped at the expense of its munificent -founder the seventh duke of Devonshire (chancellor of the university, -and one of its most distinguished alumni). He died at Cambridge on the -5th of November 1879. - -For more than half of his brief life he held a prominent position in the -very foremost rank of natural philosophers. His contributions to -scientific societies began in his fifteenth year, when Professor J. D. -Forbes communicated to the Royal Society of Edinburgh a short paper of -his on a mechanical method of tracing Cartesian ovals. In his eighteenth -year, while still a student in Edinburgh, he contributed two valuable -papers to the _Transactions_ of the same society--one of which, "On the -Equilibrium of Elastic Solids," is remarkable, not only on account of -its intrinsic power and the youth of its author, but also because in it -he laid the foundation of one of the most singular discoveries of his -later life, the temporary double refraction produced in viscous liquids -by shearing stress. Immediately after taking his degree, he read to the -Cambridge Philosophical Society a very novel memoir, "On the -Transformation of Surfaces by Bending." This is one of the few purely -mathematical papers he published, and it exhibited at once to experts -the full genius of its author. About the same time appeared his -elaborate memoir, "On Faraday's Lines of Force," in which he gave the -first indication of some of those extraordinary electrical -investigations which culminated in the greatest work of his life. He -obtained in 1859 the Adams prize in Cambridge for a very original and -powerful essay, "On the Stability of Saturn's Rings." From 1855 to 1872 -he published at intervals a series of valuable investigations connected -with the "Perception of Colour" and "Colour-Blindness," for the earlier -of which he received the Rumford medal from the Royal Society in 1860. -The instruments which he devised for these investigations were simple -and convenient, but could not have been thought of for the purpose -except by a man whose knowledge was co-extensive with his ingenuity. One -of his greatest investigations bore on the "Kinetic Theory of Gases." -Originating with D. Bernoulli, this theory was advanced by the -successive labours of John Herapath, J. P. Joule, and particularly R. -Clausius, to such an extent as to put its general accuracy beyond a -doubt; but it received enormous developments from Maxwell, who in this -field appeared as an experimenter (on the laws of gaseous friction) as -well as a mathematician. He wrote an admirable textbook of the _Theory -of Heat_ (1871), and a very excellent elementary treatise on _Matter and -Motion_ (1876). - -But the great work of his life was devoted to electricity. He began by -reading, with the most profound admiration and attention, the whole of -Faraday's extraordinary self-revelations, and proceeded to translate the -ideas of that master into the succinct and expressive notation of the -mathematicians. A considerable part of this translation was accomplished -during his career as an undergraduate in Cambridge. The writer had the -opportunity of perusing the MS. of "On Faraday's Lines of Force," in a -form little different from the final one, a year before Maxwell took his -degree. His great object, as it was also the great object of Faraday, -was to overturn the idea of action at a distance. The splendid -researches of S. D. Poisson and K. F. Gauss had shown how to reduce all -the phenomena of statical electricity to mere attractions and repulsions -exerted at a distance by particles of an imponderable on one another. -Lord Kelvin (Sir W. Thomson) had, in 1846, shown that a totally -different assumption, based upon other analogies, led (by its own -special mathematical methods) to precisely the same results. He treated -the resultant electric force at any point as analogous to the _flux of -heat_ from sources distributed in the same manner as the supposed -electric particles. This paper of Thomson's, whose ideas Maxwell -afterwards developed in an extraordinary manner, seems to have given the -first hint that there are at least two perfectly distinct methods of -arriving at the known formulae of statical electricity. The step to -magnetic phenomena was comparatively simple; but it was otherwise as -regards electro-magnetic phenomena, where current electricity is -essentially involved. An exceedingly ingenious, but highly artificial, -theory had been devised by W. E. Weber, which was found capable of -explaining all the phenomena investigated by Ampere as well as the -induction currents of Faraday. But this was based upon the assumption of -a distance-action between electric particles, the intensity of which -depended on their relative motion as well as on their position. This -was, of course, even more repugnant to Maxwell's mind than the statical -distance-action developed by Poisson. The first paper of Maxwell's in -which an attempt at an admissible physical theory of electromagnetism -was made was communicated to the Royal Society in 1867. But the theory, -in a fully developed form, first appeared in 1873 in his great treatise -on _Electricity and Magnetism_. This work was one of the most splendid -monuments ever raised by the genius of a single individual. Availing -himself of the admirable generalized co-ordinate system of Lagrange, -Maxwell showed how to reduce all electric and magnetic phenomena to -stresses and motions of a material medium, and, as one preliminary, but -excessively severe, test of the truth of his theory, he pointed out that -(if the electro-magnetic medium be that which is required for the -explanation of the phenomena of light) the velocity of light in vacuo -should be numerically the same as the ratio of the electro-magnetic and -electrostatic units. In fact, the means of the best determinations of -each of these quantities separately agree with one another more closely -than do the various values of either. - -One of Maxwell's last great contributions to science was the editing -(with copious original notes) of the _Electrical Researches of the Hon. -Henry Cavendish_, from which it appeared that Cavendish, already famous -by many other researches (such as the mean density of the earth, the -composition of water, &c.), must be looked on as, in his day, a man of -Maxwell's own stamp as a theorist and an experimenter of the very first -rank. - -In private life Clerk Maxwell was one of the most lovable of men, a -sincere and unostentatious Christian. Though perfectly free from any -trace of envy or ill-will, he yet showed on fit occasion his contempt -for that pseudo-science which seeks for the applause of the ignorant by -professing to reduce the whole system of the universe to a fortuitous -sequence of uncaused events. - - His collected works, including the series of articles on the - properties of matter, such as "Atom," "Attraction," "Capillary - Action," "Diffusion," "Ether," &c., which he contributed to the 9th - edition of this encyclopaedia, were issued in two volumes by the - Cambridge University Press in 1890; and an extended biography, by his - former schoolfellow and lifelong friend Professor Lewis Campbell, was - published in 1882. (P. G. T.) - - - - -MAXWELLTOWN, a burgh of barony and police burgh of Kirkcudbrightshire, -Scotland. Pop. (1901), 5796. It lies on the Nith, opposite to Dumfries, -with which it is connected by three bridges, being united with it for -parliamentary purposes. It has a station on the Glasgow & South-Western -line from Dumfries to Kirkcudbright. Its public buildings include a -court-house, the prison for the south-west of Scotland, and an -observatory and museum, housed in a disused windmill. The chief -manufactures are woollens and hosiery, besides dyeworks and sawmills. It -was a hamlet known as Bridgend up till 1810, in which year it was -erected into a burgh of barony under its present name. To the north-west -lies the parish of Terregles, said to be a corruption of Tir-eglwys -(_terra ecclesia_, that is, "Kirk land"). The parish contains the -beautiful ruin of Lincluden Abbey (see DUMFRIES), and Terregles House, -once the seat of William Maxwell, last earl of Nithsdale. In the parish -of Lochrutton, a few miles south-west of Maxwelltown, there is a good -example of a stone circle, the "Seven Grey Sisters," and an old -peel-tower in the Mains of Hills. - - - - -MAY, PHIL (1864-1903), English caricaturist, was born at Wortley, near -Leeds, on the 22nd of April 1864, the son of an engineer. His father -died when the child was nine years old, and at twelve he had begun to -earn his living. Before he was fifteen he had acted as time-keeper at a -foundry, had tried to become a jockey, and had been on the stage at -Scarborough and Leeds. When he was about seventeen he went to London -with a sovereign in his pocket. He suffered extreme want, sleeping out -in the parks and streets, until he obtained employment as designer to a -theatrical costumier. He also drew posters and cartoons, and for about -two years worked for the _St Stephen's Review_, until he was advised to -go to Australia for his health. During the three years he spent there he -was attached to the _Sydney Bulletin_, for which many of his best -drawings were made. On his return to Europe he went to Paris by way of -Rome, where he worked hard for some time before he appeared in 1892 in -London to resume his interrupted connexion with the _St Stephen's -Review_. His studies of the London "guttersnipe" and the coster-girl -rapidly made him famous. His overflowing sense of fun, his genuine -sympathy with his subjects, and his kindly wit were on a par with his -artistic ability. It was often said that the extraordinary economy of -line which was a characteristic feature of his drawings had been forced -upon him by the deficiencies of the printing machines of the _Sydney -Bulletin_. It was in fact the result of a laborious process which -involved a number of preliminary sketches, and of a carefully considered -system of elimination. His later work included some excellent political -portraits. He became a regular member of the staff of _Punch_ in 1896, -and in his later years his services were retained exclusively for -_Punch_ and the _Graphic_. He died on the 5th of August 1903. - - There was an exhibition of his drawings at the Fine Arts Society in - 1895, and another at the Leicester Galleries in 1903. A selection of - his drawings contributed to the periodical press and from _Phil May's - Annual_ and _Phil May's Sketch Books_, with a portrait and biography - of the artist, entitled _The Phil May Folio_, appeared in 1903. - - - - -MAY, THOMAS (1595-1650), English poet and historian, son of Sir Thomas -May of Mayfield, Sussex, was born in 1595. He entered Sidney Sussex -College, Cambridge, in 1609, and took his B.A. degree three years later. -His father having lost his fortune and sold the family estate, Thomas -May, who was hampered by an impediment in his speech, made literature -his profession. In 1620 he produced _The Heir_, an ingeniously -constructed comedy, and, probably about the same time, _The Old Couple_, -which was not printed until 1658. His other dramatic works are classical -tragedies on the subjects of Antigone, Cleopatra, and Agrippina. F. G. -Fleay has suggested that the more famous anonymous tragedy of _Nero_ -(printed 1624, reprints in A. H. Bullen's _Old English Plays_ and the -_Mermaid Series_) should also be assigned to May. But his most important -work in the department of pure literature was his translation (1627) -into heroic couplets of the _Pharsalia_ of Lucan. Its success led May to -write a continuation of Lucan's narrative down to the death of Caesar. -Charles I. became his patron, and commanded him to write metrical -histories of Henry II. and Edward III., which were completed in 1635. -When the earl of Pembroke, then lord chamberlain, broke his staff across -May's shoulders at a masque, the king took him under his protection as -"my poet," and Pembroke made him an apology accompanied with a gift of -L50. These marks of the royal favour seem to have led May to expect the -posts of poet-laureate and city chronologer when they fell vacant on the -death of Ben Jonson in 1637, but he was disappointed, and he forsook the -court and attached himself to the party of the Parliament. In 1646 he is -styled one of the "secretaries" of the Parliament, and in 1647 he -published his best known work, _The History of the Long Parliament_. In -this official apology for the moderate or Presbyterian party, he -professes to give an impartial statement of facts, unaccompanied by any -expression of party or personal opinion. If he refrained from actual -invective, he accomplished his purpose, according to Guizot, by -"omission, palliation and dissimulation." Accusations of this kind were -foreseen by May, who says in his preface that if he gives more -information about the Parliament men than their opponents it is that he -was more conversant with them and their affairs. In 1650 he followed -this with another work written with a more definite bias, a _Breviary of -the History of the Parliament of England_, in Latin and English, in -which he defended the position of the Independents. He stopped short of -the catastrophe of the king's execution, and it seems likely that his -subservience to Cromwell was not quite voluntary. In February 1650 he -was brought to London from Weymouth under a strong guard for having -spread false reports of the Parliament and of Cromwell. He died on the -13th of November in the same year, and was buried in Westminster Abbey, -but after the Restoration his remains were exhumed and buried in a pit -in the yard of St Margaret's, Westminster. May's change of side made him -many bitter enemies, and he is the object of scathing condemnation from -many of his contemporaries. - - There is a long notice of May in the _Biographia Britannica_. See also - W. J. Courthope, _Hist. of Eng. Poetry_, vol. 3; and Guizot, _Etudes - biographiques sur la revolution d'Angleterre_ (pp. 403-426, ed. 1851). - - - - -MAY, or MEY(E), WILLIAM (d. 1560), English divine, was the brother of -John May, bishop of Carlisle. He was educated at Cambridge, where he was -a fellow of Trinity Hall, and in 1537, president of Queen's College. May -heartily supported the Reformation, signed the Ten Articles in 1536, and -helped in the production of _The Institution of a Christian Man_. He had -close connexion with the diocese of Ely, being successively chancellor, -vicar-general and prebendary. In 1545 he was made a prebendary of St -Paul's, and in the following year dean. His favourable report on the -Cambridge colleges saved them from dissolution. He was dispossessed -during the reign of Mary, but restored to the deanery on Elizabeth's -accession. He died on the day of his election to the archbishopric of -York. - - - - -MAY, the fifth month of our modern year, the third of the old Roman -calendar. The origin of the name is disputed; the derivation from Maia, -the mother of Mercury, to whom the Romans were accustomed to sacrifice -on the first day of this month, is usually accepted. The ancient Romans -used on May Day to go in procession to the grotto of Egeria. From the -28th of April to the 2nd of May was kept the festival in honour of -Flora, goddess of flowers. By the Romans the month was regarded as -unlucky for marriages, owing to the celebration on the 9th, 11th and -13th of the Lemuria, the festival of the unhappy dead. This superstition -has survived to the present day. - -In medieval and Tudor England, May Day was a great public holiday. All -classes of the people, young and old alike, were up with the dawn, and -went "a-Maying" in the woods. Branches of trees and flowers were borne -back in triumph to the towns and villages, the centre of the procession -being occupied by those who shouldered the maypole, glorious with -ribbons and wreaths. The maypole was usually of birch, and set up for -the day only; but in London and the larger towns the poles were of -durable wood and permanently erected. They were special eyesores to the -Puritans. John Stubbes in his _Anatomy of Abuses_ (1583) speaks of them -as those "stinckyng idols," about which the people "leape and daunce, as -the heathen did." Maypoles were forbidden by the parliament in 1644, but -came once more into favour at the Restoration, the last to be erected in -London being that set up in 1661. This pole, which was of cedar, 134 ft. -high, was set up by twelve British sailors under the personal -supervision of James II., then duke of York and lord high admiral, in -the Strand on or about the site of the present church of St -Mary's-in-the-Strand. Taken down in 1717, it was conveyed to Wanstead -Park in Essex, where it was fixed by Sir Isaac Newton as part of the -support of a large telescope, presented to the Royal Society by a French -astronomer. - - For an account of the May Day survivals in rural England see P. H. - Ditchfield, _Old English Customs extant at Present Times_ (1897). - - - - -MAY, ISLE OF, an island belonging to Fifeshire, Scotland, at the -entrance to the Firth of Forth, 5 m. S.E. of Crail and Anstruther. It -has a N.W. to S.E. trend, is more than 1 m. long, and measures at its -widest about 1/3 m. St Adrian, who had settled here, was martyred by the -Danes about the middle of the 9th century. The ruins of the small chapel -dedicated to him, which was a favourite place of pilgrimage, still -exist. The place where the pilgrims--of whom James IV. was often -one--landed is yet known as Pilgrims' Haven, and traces may yet be seen -of the various wells of St Andrew, St John, Our Lady, and the Pilgrims, -though their waters have become brackish. In 1499 Sir Andrew Wood of -Largo, with the "Yellow Carvel" and "Mayflower," captured the English -seaman Stephen Bull, and three ships, after a fierce fight which took -place between the island and the Bass Rock. In 1636 a coal beacon was -lighted on the May and maintained by Alexander Cunningham of Barns. The -oil light substituted for it in 1816 was replaced in 1888 by an electric -light. - - - - -MAYA, an important tribe and stock of American Indians, the dominant -race of Yucatan and other states of Mexico and part of Central America -at the time of the Spanish conquest. They were then divided into many -nations, chief among them being the Maya proper, the Huastecs, the -Tzental, the Pokom, the Mame and the Cakchiquel and Quiche. They were -spread over Yucatan, Vera Cruz, Tabasco, Campeche, and Chiapas in -Mexico, and over the greater part of Guatemala and Salvador. In -civilization the Mayan peoples rivalled the Aztecs. Their traditions -give as their place of origin the extreme north; thence a migration took -place, perhaps at the beginning of the Christian era. They appear to -have reached Yucatan as early as the 5th century. From the evidence of -the Quiche chronicles, which are said to date back to about A.D. 700, -Guatemala was shortly afterwards overrun. Physically the Mayans are a -dark-skinned, round-headed, short and sturdy type. Although they were -already decadent when the Spaniards arrived they made a fierce -resistance. They still form the bulk of the inhabitants of Yucatan. For -their culture, ruined cities, &c. see CENTRAL AMERICA and MEXICO. - - - - -MAYAGUEZ, the third largest city of Porto Rico, a seaport, and the seat -of government of the department of Mayaguez, on the west coast, at the -mouth of Rio Yaguez, about 72 m. W. by S. of San Juan. Pop. of the city -(1899), 15,187, including 1381 negroes and 4711 of mixed races; (1910), -16,591; of the municipal district, 35,700 (1899), of whom 2687 were -negroes and 9933 were of mixed races. Mayaguez is connected by the -American railroad of Porto Rico with San Juan and Ponce, and it is -served regularly by steamboats from San Juan, Ponce and New York, -although its harbour is not accessible to vessels drawing more than 16 -ft. of water. It is situated at the foot of Las Mesas mountains and -commands picturesque views. The climate is healthy and good water is -obtained from the mountain region. From the shipping district along the -water-front a thoroughfare leads to the main portion of the city, about -1 m. distant. There are four public squares, in one of which is a statue -of Columbus. Prominent among the public buildings are the City Hall -(containing a public library), San Antonio Hospital, Roman Catholic -churches, a Presbyterian church, the court-house and a theatre. The -United States has an agricultural experiment station here, and the -Insular Reform School is 1 m. south of the city. Coffee, sugar-cane and -tropical fruits are grown in the surrounding country; and the business -of the city consists chiefly in their export and the import of flour. -Among the manufactures are sugar, tobacco and chocolate. Mayaguez was -founded about the middle of the 18th century on the site of a hamlet -which was first settled about 1680. It was incorporated as a town in -1836, and became a city in 1873. In 1841 it was nearly all destroyed by -fire. - - - - -MAYAVARAM, a town of British India, in the Tanjore district of Madras, -on the Cauvery river; junction on the South Indian railway, 174 m. S.W. -of Madras. Pop. (1901), 24,276. It possesses a speciality of fine cotton -and silk cloth, known as Kornad from the suburb in which the weavers -live. During October and November the town is the scene of a great -pilgrimage to the holy waters of the Cauvery. - - - - -MAYBOLE, a burgh of barony and police burgh of Ayrshire, Scotland. Pop. -(1901), 5892. It is situated 9 m. S. of Ayr and 50(1/4) m. S.W. of Glasgow -by the Glasgow & South-Western railway. It is an ancient place, having -received a charter from Duncan II. in 1193. In 1516 it was made a burgh -of regality, but for generations it remained under the subjection of the -Kennedys, afterwards earls of Cassillis and marquesses of Ailsa, the -most powerful family in Ayrshire. Of old Maybole was the capital of the -district of Carrick, and for long its characteristic feature was the -family mansions of the barons of Carrick. The castle of the earls of -Cassillis still remains. The public buildings include the town-hall, the -Ashgrove and the Lumsden fresh-air fortnightly homes, and the Maybole -combination poorhouse. The leading manufactures are of boots and shoes -and agricultural implements. Two miles to the south-west are the ruins -of Crossraguel (Cross of St Regulus) Abbey, founded about 1240. -KIRKOSWALD, where Burns spent his seventeenth year, learning -land-surveying, lies a little farther west. In the parish churchyard lie -"Tam o' Shanter" (Douglas Graham) and "Souter Johnnie" (John Davidson). -Four miles to the west of Maybole on the coast is Culzean Castle, the -chief seat of the marquess of Ailsa, dating from 1777; it stands on a -basaltic cliff, beneath which are the Coves of Culzean, once the retreat -of outlaws and a resort of the fairies. Farther south are the ruins of -Turnberry Castle, where Robert Bruce is said to have been born. A few -miles to the north of Culzean are the ruins of Dunure Castle, an ancient -stronghold of the Kennedys. - - - - -MAYEN, a town of Germany, in the Prussian Rhine province, on the -northern declivity of the Eifel range, 16 m. W. from Coblenz, on the -railway Andernach-Gerolstein. Pop. (1905), 13,435. It is still partly -surrounded by medieval walls, and the ruins of a castle rise above the -town. There are some small industries, embracing textile manufactures, -oil mills and tanneries, and a trade in wine, while near the town are -extensive quarries of basalt. Having been a Roman settlement, Mayen -became a town in 1291. In 1689 it was destroyed by the French. - - - - -MAYENNE, CHARLES OF LORRAINE, DUKE OF (1554-1611), second son of Francis -of Lorraine, second duke of Guise, was born on the 26th of March 1554. -He was absent from France at the time of the massacre of Saint -Bartholomew, but took part in the siege of La Rochelle in the following -year, when he was created duke and peer of France. He went with Henry of -Valois, duke of Anjou (afterwards Henry III.), on his election as king -of Poland, but soon returned to France to become the energetic supporter -and lieutenant of his brother, the 3rd duke of Guise. In 1577 he gained -conspicuous successes over the Huguenot forces in Poitou. As governor of -Burgundy he raised his province in the cause of the League in 1585. The -assassination of his brothers at Blois on the 23rd and 24th of December -1588 left him at the head of the Catholic party. The Venetian -ambassador, Mocenigo, states that Mayenne had warned Henry III. that -there was a plot afoot to seize his person and to send him by force to -Paris. At the time of the murder he was at Lyons, where he received a -letter from the king saying that he had acted on his warning, and -ordering him to retire to his government. Mayenne professed obedience, -but immediately made preparations for marching on Paris. After a vain -attempt to recover the persons of those of his relatives who had been -arrested at Blois he proceeded to recruit troops in his government of -Burgundy and in Champagne. Paris was devoted to the house of Guise and -had been roused to fury by the news of the murder. When Mayenne entered -the city in February 1589 he found it dominated by representatives of -the sixteen quarters of Paris, all fanatics of the League. He formed a -council general to direct the affairs of the city and to maintain -relations with the other towns faithful to the League. To this council -each quarter sent four representatives, and Mayenne added -representatives of the various trades and professions of Paris in order -to counterbalance this revolutionary element. He constituted himself -"lieutenant-general of the state and crown of France," taking his oath -before the parlement of Paris. In April he advanced on Tours. Henry III. -in his extremity sought an alliance with Henry of Navarre, and the -allied forces drove the leaguers back, and had laid siege to Paris, when -the murder of Henry III. by a Dominican fanatic changed the face of -affairs and gave new strength to the Catholic party. - -Mayenne was urged to claim the crown for himself, but he was faithful to -the official programme of the League and proclaimed Charles, cardinal of -Bourbon, at that time a prisoner in the hands of Henry IV., as Charles -X. Henry IV. retired to Dieppe, followed by Mayenne, who joined his -forces with those of his cousin Charles, duke of Aumale, and Charles de -Cosse, comte de Brissac, and engaged the royal forces in a succession of -fights in the neighbourhood of Arques (September 1589). He was defeated -and out-marched by Henry IV., who moved on Paris, but retreated before -Mayenne's forces. In 1590 Mayenne received additions to his army from -the Spanish Netherlands, and took the field again, only to suffer -complete defeat at Ivry (March 14, 1590). He then escaped to Mantes, and -in September collected a fresh army at Meaux, and with the assistance of -Alexander Farnese, prince of Parma, sent by Philip II., raised the siege -of Paris, which was about to surrender to Henry IV. Mayenne feared with -reason the designs of Philip II., and his difficulties were increased by -the death of Charles X., the "king of the league." The extreme section -of the party, represented by the Sixteen, urged him to proceed to the -election of a Catholic king and to accept the help and the claims of -their Spanish allies. But Mayenne, who had not the popular gifts of his -brother, the duke of Guise, had no sympathy with the demagogues, and -himself inclined to the moderate side of his party, which began to urge -reconciliation with Henry IV. He maintained the ancient forms of the -constitution against the revolutionary policy of the Sixteen, who during -his absence from Paris took the law into their own hands and in November -1591 executed one of the leaders of the more moderate party, Barnabe -Brisson, president of the parlement. He returned to Paris and executed -four of the chief malcontents. The power of the Sixteen diminished from -that time, but with it the strength of the League.[1] - -Mayenne entered into negotiations with Henry IV. while he was still -appearing to consider with Philip II. the succession to the French crown -of the Infanta Elizabeth, granddaughter, through her mother Elizabeth of -Valois, of Henry II. He demanded that Henry IV. should accomplish his -conversion to Catholicism before he was recognized by the leaguers. He -also desired the continuation to himself of the high offices which had -accumulated in his family and the reservation of their provinces to his -relatives among the leaguers. In 1593 he summoned the States General to -Paris and placed before them the claims of the Infanta, but they -protested against foreign intervention. Mayenne signed a truce at La -Villette on the 31st of July 1593. The internal dissensions of the -league continued to increase, and the principal chiefs submitted. -Mayenne finally made his peace only in October 1595. Henry IV. allowed -him the possession of Chalon-sur-Saone, of Seurre and Soissons for three -years, made him governor of the Isle of France and paid a large -indemnity. Mayenne died at Soissons on the 3rd of October 1611. - - A _Histoire de la vie et de la mort du duc de Mayenne_ appeared at - Lyons in 1618. See also J. B. H. Capefigue, _Hist. de la Reforme, de - la ligue et du regne de Henri IV._ (8 vols., 1834-1835) and the - literature dealing with the house of Guise (q.v.). - - -FOOTNOTE: - - [1] The estates of the League in 1593 were the occasion of the famous - _Satire Menippee_, circulated in MS. in that year, but only printed - at Tours in 1594. It was the work of a circle of men of letters who - belonged to the _politiques_ or party of the centre and ridiculed the - League. The authors were Pierre Le Roy, Jean Passerat, Florent - Chrestien, Nicolas Rapin and Pierre Pithou. It opened with "La vertu - du catholicon," in which a Spanish quack (the cardinal of Plaisance) - vaunts the virtues of his drug "catholicon compose," manufactured in - the Escurial, while a Lorrainer rival (the cardinal of Pelleve) tries - to sell a rival cure. A mock account of the estates, with harangues - delivered by Mayenne and the other chiefs of the League, followed. - Mayenne's discourse is said to have been written by the jurist - Pithou. - - - - -MAYENNE, a department of north-western France, three-fourths of which -formerly belonged to Lower Maine and the remainder to Anjou, bounded on -the N. by Manche and Orne, E. by Sarthe, S. by Maine-et-Loire and W. by -Ille-et-Vilaine. Area, 2012 sq. m. Pop. (1906), 305,457. Its ancient -geological formations connect it with Brittany. The surface is agreeably -undulating; forests are numerous, and the beauty of the cultivated -portions is enhanced by the hedgerows and lines of trees by which the -farms are divided. The highest point of the department, and indeed of -the whole north-west of France, is the Mont des Avaloirs (1368 ft.). -Hydrographically Mayenne belongs to the basins of the Loire, the Vilaine -and the Selune, the first mentioned draining by far the larger part of -the entire area. The principal stream is the Mayenne, which passes -successively from north to south through Mayenne, Laval and -Chateau-Gontier; by means of weirs and sluices it is navigable below -Mayenne, but traffic is inconsiderable. The chief affluents are the -Jouanne on the left, and on the right the Colmont, the Ernee and the -Oudon. A small area in the east of the department drains by the Erve -into the Sarthe; the Vilaine rises in the west, and in the north-west -two small rivers flow into the Selune. The climate of Mayenne is -generally healthy except in the neighbourhood of the numerous marshes. -The temperature is lower and the moisture of the atmosphere greater than -in the neighbouring departments; the rainfall (about 32 in. annually) is -above the average for France. - - Agriculture and stock-raising are prosperous. A large number of horned - cattle are reared, and in no other French department are so many - horses found within the same area; the breed, that of Craon, is famed - for its strength. Craon has also given its name to the most prized - breed of pigs in western France. Mayenne produces excellent butter and - poultry and a large quantity of honey. The cultivation of the vine is - very limited, and the most common beverage is cider. Wheat, oats, - barley and buckwheat, in the order named, are the most important - crops, and a large quantity of flax and hemp is produced. Game is - abundant. The timber grown is chiefly beech, oak, birch, elm and - chestnut. The department produces antimony, auriferous quartz and - coal. Marble, slate and other stone are quarried. There are several - chalybeate springs. The industries include flour-milling, brick and - tile making, brewing, cotton and wool spinning, and the production of - various textile fabrics (especially ticking) for which Laval and - Chateau-Gontier are the centres, agricultural implement making, wood - and marble sawing, tanning and dyeing. The exports include - agricultural produce, live-stock, stone and textiles; the chief - imports are coal, brandy, wine, furniture and clothing. The department - is served by the Western railway. It forms part of the - circumscriptions of the IV. army corps, the academie (educational - division) of Rennes, and the court of appeal of Angers. It comprises - three arrondissements (Laval, Chateau-Gontier and Mayenne), with 27 - cantons and 276 communes. Laval, the capital, is the seat of a - bishopric of the province of Tours. The other principal towns are - Chateau-Gontier and Mayenne, which are treated under separate - headings. The following places are also of interest: Evron, which has - a church of the 12th and 13th centuries; Jublains, with a Roman fort - and other Roman remains; Lassay, with a fine chateau of the 14th and - 16th centuries; and Ste Suzanne, which has remains of medieval - ramparts and a fortress with a keep of the Romanesque period. - - - - -MAYENNE, a town of north-western France, capital of an arrondissement in -the department of Mayenne, 19 m. N.N.E. of Laval by rail. Pop., town -7003, commune 10,020. Mayenne is an old feudal town, irregularly built -on hills on both sides of the river Mayenne. Of the old castle -overlooking the river several towers remain, one of which has retained -its conical roof; the vaulted chambers and chapel are ornamented in the -style of the 13th century; the building is now used as a prison. The -church of Notre-Dame, beside which there is a statue of Joan of Arc, -dates partly from the 12th century; the choir was rebuilt in the 19th -century. In the Place de Cheverus is a statue, by David of Angers, to -Cardinal Jean de Cheverus (1768-1836), who was born in Mayenne. Mayenne -has a subprefecture, tribunals of first instance and of commerce, a -chamber of arts and manufactures, and a board of trade-arbitration. -There is a school of agriculture in the vicinity. The chief industry of -the place is the manufacture of tickings, linen, handkerchiefs and -calicoes. - -Mayenne had its origin in the castle built here by Juhel, baron of -Mayenne, the son of Geoffrey of Maine, in the beginning of the 11th -century. It was taken by the English in 1424, and several times suffered -capture by the opposing parties in the wars of religion and the Vendee. -At the beginning of the 16th century the territory passed to the family -of Guise, and in 1573 was made a duchy in favour of Charles of Mayenne, -leader of the League. - - - - -MAYER, JOHANN TOBIAS (1723-1762), German astronomer, was born at -Marbach, in Wurtemberg, on the 17th of February 1723, and brought up at -Esslingen in poor circumstances. A self-taught mathematician, he had -already published two original geometrical works when, in 1746, he -entered J. B. Homann's cartographic establishment at Nuremberg. Here he -introduced many improvements in map-making, and gained a scientific -reputation which led (in 1751) to his election to the chair of economy -and mathematics in the university of Gottingen. In 1754 he became -superintendent of the observatory, where he laboured with great zeal and -success until his death, on the 20th of February 1762. His first -important astronomical work was a careful investigation of the libration -of the moon (_Kosmographische Nachrichten_, Nuremberg, 1750), and his -chart of the full moon (published in 1775) was unsurpassed for half a -century. But his fame rests chiefly on his lunar tables, communicated in -1752, with new solar tables, to the Royal Society of Gottingen, and -published in their _Transactions_ (vol. ii.). In 1755 he submitted to -the English government an amended body of MS. tables, which James -Bradley compared with the Greenwich observations, and found to be -sufficiently accurate to determine the moon's place to 75", and -consequently the longitude at sea to about half a degree. An improved -set was afterwards published in London (1770), as also the theory -(_Theoria lunae juxta systema Newtonianum_, 1767) upon which the tables -are based. His widow, by whom they were sent to England, received in -consideration from the British government a grant of L3000. Appended to -the London edition of the solar and lunar tables are two short -tracts--the one on determining longitude by lunar distances, together -with a description of the repeating circle (invented by Mayer in 1752), -the other on a formula for atmospheric refraction, which applies a -remarkably accurate correction for temperature. - -Mayer left behind him a considerable quantity of manuscript, part of -which was collected by G. C. Lichtenberg and published in one volume -(_Opera inedita_, Gottingen, 1775). It contains an easy and accurate -method for calculating eclipses; an essay on colour, in which three -primary colours are recognized; a catalogue of 998 zodiacal stars; and a -memoir, the earliest of any real value, on the proper motion of eighty -stars, originally communicated to the Gottingen Royal Society in 1760. -The manuscript residue includes papers on atmospheric refraction (dated -1755), on the motion of Mars as affected by the perturbations of Jupiter -and the Earth (1756), and on terrestrial magnetism (1760 and 1762). In -these last Mayer sought to explain the magnetic action of the earth by a -modification of Euler's hypothesis, and made the first really definite -attempt to establish a mathematical theory of magnetic action (C. -Hansteen, _Magnetismus der Erde_, i. 283). E. Klinkerfuss published in -1881 photo-lithographic reproductions of Mayer's local charts and -general map of the moon; and his star-catalogue was re-edited by F. -Baily in 1830 (_Memoirs Roy. Astr. Soc._ iv. 391) and by G. F. J. A. -Auvers in 1894. - - AUTHORITIES.--A. G. Kastner, _Elogium Tobiae Mayeri_ (Gottingen, - 1762); _Connaissance des temps, 1767_, p. 187 (J. Lalande); - _Monatliche Correspondenz_ viii. 257, ix. 45, 415, 487, xi. 462; - _Allg. Geographische Ephemeriden_ iii. 116, 1799 (portrait); _Berliner - Astr. Jahrbuch_, Suppl. Bd. iii. 209, 1797 (A. G. Kastner); J. B. J. - Delambre, _Hist. de l'Astr. au XVIII^e siecle_, p. 429; R. Grant, - _Hist. of Phys. Astr._ pp. 46, 488, 555; A. Berry, _Short Hist. of - Astr._ p. 282; J. S. Putter, _Geschichte von der Universitat zu - Gottingen_, i. 68; J. Gehler, _Physik. Worterbuch neu bearbeitet_, vi. - 746, 1039; Allg. _Deutsche Biographie_ (S. Gunther). (A. M. C.) - - - - -MAYER, JULIUS ROBERT (1814-1878), German physicist, was born at -Heilbronn on the 25th of November 1814, studied medicine at Tubingen, -Munich and Paris, and after a journey to Java in 1840 as surgeon of a -Dutch vessel obtained a medical post in his native town. He claims -recognition as an independent a priori propounder of the "First Law of -Thermodynamics," but more especially as having early and ably applied -that law to the explanation of many remarkable phenomena, both cosmical -and terrestrial. His first little paper on the subject, "_Bemerkungen -uber die Krafte der unbelebten Natur_," appeared in 1842 in Liebig's -_Annalen_, five years after the republication, in the same journal, of -an extract from K. F. Mohr's paper on the nature of heat, and three -years later he published _Die organische Bewegung in ihren Zusammenhange -mit dem Stoffwechsel_. - - It has been repeatedly claimed for Mayer that he calculated the value - of the dynamical equivalent of heat, indirectly, no doubt, but in a - manner altogether free from error, and with a result according almost - exactly with that obtained by J. P. Joule after years of patient - labour in direct experimenting. This claim on Mayer's behalf was first - shown to be baseless by W. Thomson (Lord Kelvin) and P. G. Tait in an - article on "Energy," published in _Good Words_ in 1862, which gave - rise to a long but lively discussion. A calm and judicial annihilation - of the claim is to be found in a brief article by Sir G. G. Stokes, - _Proc. Roy. Soc._, 1871, p. 54. See also Maxwell's _Theory of Heat_, - chap. xiii. Mayer entirely ignored the grand fundamental principle - laid down by Sadi Carnot--that nothing can be concluded as to the - relation between heat and work from an experiment in which the working - substance is left at the end of an operation in a different physical - state from that in which it was at the commencement. Mayer has also - been styled the discoverer of the fact that heat consists in (the - energy of) motion, a matter settled at the very end of the 18th - century by Count Rumford and Sir H. Davy; but in the teeth of this - statement we have Mayer's own words, "We might much rather assume the - contrary--that in order to become heat motion must cease to be - motion." - - Mayer's real merit consists in the fact that, having for himself made - out, on inadequate and even questionable grounds, the conservation of - energy, and having obtained (though by inaccurate reasoning) a - numerical result correct so far as his data permitted, he applied the - principle with great power and insight to the explanation of numerous - physical phenomena. His papers, which were republished in a single - volume with the title _Die Mechanik der Warme_ (3rd ed., 1893), are of - unequal merit. But some, especially those on _Celestial Dynamics_ and - _Organic Motion_, are admirable examples of what really valuable work - may be effected by a man of high intellectual powers, in spite of - imperfect information and defective logic. - - Different, and it would appear exaggerated, estimates of Mayer are - given in John Tyndall's papers in the _Phil. Mag._, 1863-1864 (whose - avowed object was "to raise a noble and a suffering man to the - position which his labours entitled him to occupy"), and in E. - Duhring's _Robert Mayer, der Galilei des neunzehnten Jahrhunderts_, - Chemnitz, 1880. Some of the simpler facts of the case are summarized - by Tait in the _Phil. Mag._, 1864, ii. 289. - - - - -MAYFLOWER, the vessel which carried from Southampton, England, to -Plymouth, Massachusetts, the Pilgrims who established the first -permanent colony in New England. It was of about 180 tons burden, and in -company with the "Speedwell" sailed from Southampton on the 5th of -August 1620, the two having on board 120 Pilgrims. After two trials the -"Speedwell" was pronounced unseaworthy, and the "Mayflower" sailed alone -from Plymouth, England, on the 6th of September with the 100 (or 102) -passengers, some 41 of whom on the 11th of November (O.S.) signed the -famous "Mayflower Compact" in Provincetown Harbor, and a small party of -whom, including William Bradford, sent to choose a place for settlement, -landed at what is now Plymouth, Massachusetts, on the 11th of December -(21st N.S.), an event which is celebrated, as Forefathers' Day, on the -22nd of December. A "General Society of Mayflower Descendants" was -organized in 1894 by lineal descendants of passengers of the "Mayflower" -to "preserve their memory, their records, their history, and all facts -relating to them, their ancestors and their posterity." Every lineal -descendant, over eighteen years of age, of any passenger of the -"Mayflower" is eligible to membership. Branch societies have since been -organized in several of the states and in the District of Columbia, and -a triennial congress is held in Plymouth. - - See Azel Ames, _The May-Flower and Her Log_ (Boston, 1901); Blanche - McManus, _The Voyage of the Mayflower_ (New York, 1897); _The General - Society of Mayflower: Meetings, Officers and Members, arranged in - State Societies, Ancestors and their Descendants_ (New York, 1901). - Also the articles PLYMOUTH, MASS.; MASSACHUSETTS, S_History_; PILGRIM; - and PROVINCETOWN, MASS. - - - - -MAY-FLY. The Mayflies belong to the Ephemeridae, a remarkable family of -winged insects, included by Linnaeus in his order Neuroptera, which -derive their scientific name from [Greek: ephemeros], in allusion to -their very short lives. In some species it is possible that they have -scarcely more than one day's existence, but others are far longer lived, -though the extreme limit is probably rarely more than a week. The family -has very sharply defined characters, which separate its members at once -from all other neuropterous (or pseudo-neuropterous) groups. - -These insects are universally aquatic in their preparatory states. The -eggs are dropped into the water by the female in large masses, -resembling, in some species, bunches of grapes in miniature. Probably -several months elapse before the young larvae are excluded. The -sub-aquatic condition lasts a considerable time: in _Cloeon_, a genus of -small and delicate species, Sir J. Lubbock (Lord Avebury) proved it to -extend over more than six months; but in larger and more robust genera -(e.g. _Palingenia_) there appears reason to believe that the greater -part of three years is occupied in preparatory conditions. - - The larva is elongate and campodeiform. The head is rather large, and - is furnished at first with five simple eyes of nearly equal size; but - as it increases in size the homologues of the facetted eyes of the - imago become larger, whereas those equivalent to the ocelli remain - small. The antennae are long and thread-like, composed at first of few - joints, but the number of these latter apparently increases at each - moult. The mouth parts are well developed, consisting of an upper lip, - powerful mandibles, maxillae with three-jointed palpi, and a deeply - quadrifid labium or lower lip with three-jointed labial palpi. - Distinct and conspicuous maxillulae are associated with the tongue or - hypopharynx. There are three distinct and large thoracic segments, - whereof the prothorax is narrower than the others; the legs are much - shorter and stouter than in the winged insect, with monomerous tarsi - terminated by a single claw. The abdomen consists of ten segments, the - tenth furnished with long and slender multi-articulate tails, which - appear to be only two in number at first, but an intermediate one - gradually develops itself (though this latter is often lost in the - winged insect). Respiration is effected by means of external gills - placed along both sides of the dorsum of the abdomen and hinder - segments of the thorax. These vary in form: in some species they are - entire plates, in others they are cut up into numerous divisions, in - all cases traversed by numerous tracheal ramifications. According to - the researches of Lubbock and of E. Joly, the very young larvae have - no breathing organs, and respiration is effected through the skin. - Lubbock traced at least twenty moults in _Cloeon_; at about the tenth - rudiments of the wing-cases began to appear. These gradually become - larger, and when so the creature may be said to have entered its - "nymph" stage; but there is no condition analogous to the pupa-stage - of insects with complete metamorphoses. - - There may be said to be three or four different modes of life in these - larvae: some are fossorial, and form tubes in the mud or clay in which - they live; others are found on or beneath stones; while others again - swim and crawl freely among water plants. It is probable that some are - carnivorous, either attacking other larvae or subsisting on more - minute forms of animal life; but others perhaps feed more exclusively - on vegetable matters of a low type, such as diatoms. - - The most aberrant type of larva is that of the genus _Prosopistoma_, - which was originally described as an entomostracous crustacean on - account of the presence of a large carapace overlapping the greater - part of the body. The dorsal skeletal elements of the thorax and of - the anterior six abdominal segments unite with the wing-cases to form - a large respiratory chamber, containing five pairs of tracheal gills, - with lateral slits for the inflow and a posterior orifice for the - outflow of water. Species of this genus occur in Europe, Africa and - Madagascar. - -When the aquatic insect has reached its full growth it emerges from the -water or seeks its surface; the thorax splits down the back and the -winged form appears. But this is not yet perfect, although it has all -the form of a perfect insect and is capable of flight; it is what is -variously termed a "pseud-imago," "sub-imago" or "pro-imago." Contrary -to the habits of all other insects, there yet remains a pellicle that -has to be shed, covering every part of the body. This final moult is -effected soon after the insect's appearance in the winged form; the -creature seeks a temporary resting-place, the pellicle splits down the -back, and the now perfect insect comes forth, often differing very -greatly in colours and markings from the condition in which it was only -a few moments before. If the observer takes up a suitable position near -water, his coat is often seen to be covered with the cast sub-imaginal -skins of these insects, which had chosen him as a convenient object upon -which to undergo their final change. In some few genera of very low type -it appears probable that, at any rate in the female, this final change -is never effected and that the creature dies a sub-imago. - - The winged insect differs considerably in form from its sub-aquatic - condition. The head is smaller, often occupied almost entirely above - in the male by the very large eyes, which in some species are - curiously double in that sex, one portion being pillared, and forming - what is termed a "turban," the mouth parts are aborted, for the - creature is now incapable of taking nutriment either solid or fluid; - the antennae are mere short bristles, consisting of two rather large - basal joints and a multi-articulate thread. The prothorax is much - narrowed, whereas the other segments (especially the mesothorax) are - greatly enlarged; the legs long and slender, the anterior pair often - very much longer in the male than in the female; the tarsi four- or - five-jointed; but in some genera (e.g. _Oligoneuria_ and allies) the - legs are aborted, and the creatures are driven helplessly about by the - wind. The wings are carried erect: the anterior pair large, with - numerous longitudinal nervures, and usually abundant transverse - reticulation; the posterior pair very much smaller, often lanceolate, - and frequently wanting absolutely. The abdomen consists of ten - segments; at the end are either two or three long multi-articulate - tails; in the male the ninth joint bears forcipated appendages; in the - female the oviducts terminate at the junction of the seventh and - eighth ventral segments. The independent opening of the genital ducts - and the absence of an ectodermal vagina and ejaculatory duct are - remarkable archaic features of these insects, as has been pointed out - by J. A. Palmen. The sexual act takes place in the air, and is of very - short duration, but is apparently repeated several times, at any rate - in some cases. - -_Ephemeridae_ are found all over the world, even up to high northern -latitudes. F. J. Pictet, A. E. Eaton and others have given us valuable -works or monographs on the family; but the subject still remains little -understood, partly owing to the great difficulty of preserving such -delicate insects; and it appears probable they can only be -satisfactorily investigated as moist preparations. The number of -described species is less than 200, spread over many genera. - -From the earliest times attention has been drawn to the enormous -abundance of species of the family in certain localities. Johann Anton -Scopoli, writing in the 18th century, speaks of them as so abundant in -one place in Carniola that in June twenty cartloads were carried away -for manure! _Polymitarcys virgo_, which, though not found in England, -occurs in many parts of Europe (and is common at Paris), emerges from -the water soon after sunset, and continues for several hours in such -myriads as to resemble snow showers, putting out lights, and causing -inconvenience to man, and annoyance to horses by entering their -nostrils. In other parts of the world they have been recorded in -multitudes that obscured passers-by on the other side of the street. And -similar records might be multiplied almost to any extent. In Britain, -although they are often very abundant, we have scarcely anything -analogous. - -Fish, as is well known, devour them greedily, and enjoy a veritable -feast during the short period in which any particular species appears. -By anglers the common English species of _Ephemera_ (_vulgata_ and -_danica_, but more especially the latter, which is more abundant) is -known as the "may-fly," but the terms "green drake" and "bastard drake" -are applied to conditions of the same species. Useful information on -this point will be found in Ronalds's _Fly-Fisher's Entomology_, edited -by Westwood. - -Ephemeridae belong to a very ancient type of insects, and fossil -imprints of allied forms occur even in the Devonian and Carboniferous -formations. - -There is much to be said in favour of the view entertained by some -entomologists that the structural and developmental characteristics of -may-flies are sufficiently peculiar to warrant the formation for them of -a special order of insects, for which the names Agnatha, Plectoptera and -Ephemeroptera have been proposed. (See HEXAPODA, NEUROPTERA.) - - BIBLIOGRAPHY.--Of especial value to students of these insects are A. - E. Eaton's monograph (_Trans. Linn. Soc._ (2) iii. 1883-1885) and A. - Vayssiere's "Recherches sur l'organisation des larves" (_Ann. Sci. - Nat. Zool._ (6) xiii. 1882 (7) ix. 1890). J. A. Palmen's memoirs _Zur - Morphologie des Tracheensystems_ (Leipzig, 1877) and _Uber paarige - Ausfuhrungsgange der Geschlechtsorgane bei Insekten_ (Helsingfors, - 1884), contain important observations on may-flies. See also L. C. - Miall, _Nat. Hist. Aquatic Insects_ (London, 1895); J. G. Needham and - others (New York State Museum, Bull. 86, 1905). (R. M'L.; G. H. C.) - - - - -MAYHEM (for derivation see MAIMING), an old Anglo-French term of the law -signifying an assault whereby the injured person is deprived of a member -proper for his defence in fight, e.g. an arm, a leg, a fore tooth, &c. -The loss of an ear, jaw tooth, &c., was not mayhem. The most ancient -punishment in English law was retaliative--_membrum pro membro_, but -ultimately at common law fine and imprisonment. Various statutes were -passed aimed at the offence of maiming and disfiguring, which is now -dealt with by section 18 of the Offences against the Person Act 1861. -Mayhem may also be the ground of a civil action, which had this -peculiarity that the court on sight of the wound might increase the -damages awarded by the jury. - - - - -MAYHEW, HENRY (1812-1887), English author and journalist, son of a -London solicitor, was born in 1812. He was sent to Westminster school, -but ran away to sea. He sailed to India, and on his return studied law -for a short time under his father. He began his journalistic career by -founding, with Gilbert a Beckett, in 1831, a weekly paper, _Figaro in -London_. This was followed in 1832 by a short-lived paper called _The -Thief_; and he produced one or two successful farces. His brothers -Horace (1816-1872) and Augustus Septimus (1826-1875) were also -journalists, and with them Henry occasionally collaborated, notably with -the younger in _The Greatest Plague of Life_ (1847) and in _Acting -Charades_ (1850). In 1841 Henry Mayhew was one of the leading spirits -in the foundation of _Punch_, of which he was for the first two years -joint-editor with Mark Lemon. He afterwards wrote on all kinds of -subjects, and published a number of volumes of no permanent -reputation--humorous stories, travel and practical handbooks. He is -credited with being the first to "write up" the poverty side of London -life from a philanthropic point of view; with the collaboration of John -Binny and others he published _London Labour and London Poor_ (1851; -completed 1864) and other works on social and economic questions. He -died in London, on the 25th of July 1887. Horace Mayhew was for some -years sub-editor of _Punch_, and was the author of several humorous -publications and plays. The books of Horace and Augustus Mayhew owe -their survival chiefly to Cruikshank's illustrations. - - - - -MAYHEW, JONATHAN (1720-1766), American clergyman, was born at Martha's -Vineyard on the 8th of October 1720, being fifth in descent from Thomas -Mayhew (1592-1682), an early settler and the grantee (1641) of Martha's -Vineyard. Thomas Mayhew (c. 1616-1657), the younger, his son John (d. -1689) and John's son, Experience (1673-1758), were active missionaries -among the Indians of Martha's Vineyard and the vicinity. Jonathan, the -son of Experience, graduated at Harvard in 1744. So liberal were his -theological views that when he was to be ordained minister of the West -Church in Boston in 1747 only two ministers attended the first council -called for the ordination, and it was necessary to summon a second -council. Mayhew's preaching made his church practically the first -"Unitarian" Congregational church in New England, though it was never -officially Unitarian. In 1763 he published _Observations on the Charter -and Conduct of the Society for Propagating the Gospel in Foreign Parts_, -an attack on the policy of the society in sending missionaries to New -England contrary to its original purpose of "Maintaining Ministers of -the Gospel" in places "wholly destitute and unprovided with means for -the maintenance of ministers and for the public worship of God;" the -_Observations_ marked him as a leader among those in New England who -feared, as Mayhew said (1762), "that there is a scheme forming for -sending a bishop into this part of the country, and that our -Governor,[1] a true churchman, is deeply in the plot." To an American -reply to the _Observations_, entitled _A Candid Examination_ (1763), -Mayhew wrote a _Defense_; and after the publication of an _Answer_, -anonymously published in London in 1764 and written by Thomas Seeker, -archbishop of Canterbury, he wrote a _Second Defense_. He bitterly -opposed the Stamp Act, and urged the necessity of colonial union (or -"communion") to secure colonial liberties. He died on the 9th of July -1766. Mayhew was Dudleian lecturer at Harvard in 1765, and in 1749 had -received the degree of D.D. from the University of Aberdeen. - - See Alden Bradford, _Memoir of the Life and Writings of Rev. Jonathan - Mayhew_ (Boston, 1838), and "An Early Pulpit Champion of Colonial - Rights," chapter vi., in vol. i. of M. C. Tyler's _Literary History of - the American Revolution_ (2 vols., New York, 1897). - - -FOOTNOTE: - - [1] Francis Bernard, whose project for a college at Northampton - seemed to Mayhew and others a move to strengthen Anglicanism. - - - - -MAYHEW, THOMAS, English 18th century cabinet-maker. Mayhew was the less -distinguished partner of William Ince (q.v.). The chief source of -information as to his work is supplied by his own drawings in the volume -of designs, _The universal system of household furniture_, which he -published in collaboration with his partner. The name of the firm -appears to have been Mayhew and Ince, but on the title page of this book -the names are reversed, perhaps as an indication that Ince was the more -extensive contributor. In the main Mayhew's designs are heavy and -clumsy, and often downright extravagant, but he had a certain lightness -of accomplishment in his applications of the bizarre Chinese style. Of -original talent he possessed little, yet it is certain that much of his -Chinese work has been attributed to Chippendale. It is indeed often only -by reference to books of design that the respective work of the English -cabinet-makers of the second half of the 18th century can be correctly -attributed. - - - - -MAYMYO, a hill sanatorium in India, in the Mandalay district of Upper -Burma, 3500 ft. above the sea, with a station on the Mandalay-Lashio -railway 422 m. from Rangoon. Pop. (1901), 6223. It consists of an -undulating plateau, surrounded by hills, which are covered with thin oak -forest and bracken. Though not entirely free from malaria, it has been -chosen for the summer residence of the lieutenant-governor; and it is -also the permanent headquarters of the lieutenant-general commanding the -Burma division, and of other officials. - - - - -MAYNARD, FRANCOIS DE (1582-1646), French poet, was born at Toulouse in -1582. His father was _conseiller_ in the parlement of the town, and -Francois was also trained for the law, becoming eventually president of -Aurillac. He became secretary to Margaret of Valois, wife of Henry IV., -for whom his early poems are written. He was a disciple of Malherbe, who -said that in the workmanship of his lines he excelled Racan, but lacked -his rival's energy. In 1634 he accompanied the Cardinal de Noailles to -Rome and spent about two years in Italy. On his return to France he made -many unsuccessful efforts to obtain the favour of Richelieu, but was -obliged to retire to Toulouse. He never ceased to lament his exile from -Paris and his inability to be present at the meetings of the Academy, of -which he was one of the earliest members. The best of his poems is in -imitation of Horace, "Alcippe, reviens dans nos bois." He died at -Toulouse on the 23rd of December 1646. - - His works consist of odes, epigrams, songs and letters, and were - published in 1646 by Marin le Roy de Gomberville. - - - - -MAYNE, JASPER (1604-1672), English author, was baptized at Hatherleigh, -Devonshire, on the 23rd of November 1604. He was educated at Westminster -School and at Christ Church, Oxford, where he had a distinguished -career. He was presented to two college livings in Oxfordshire, and was -made D.D. in 1646. During the Commonwealth he was dispossessed, and -became chaplain to the duke of Devonshire. At the Restoration he was -made canon of Christ Church, archdeacon of Chichester and chaplain in -ordinary to the king. He wrote a farcical domestic comedy, _The City -Match_ (1639), which is reprinted in vol. xiii. of Hazlitt's edition of -Dodsley's _Old Plays_, and a fantastic tragi-comedy entitled _The -Amorous War_ (printed 1648). After receiving ecclesiastical preferment -he gave up poetry as unbefitting his profession. His other works -comprise some occasional gems, a translation of Lucian's _Dialogues_ -(printed 1664) and a number of sermons. He died on the 6th of December -1672 at Oxford. - - - - -MAYNOOTH, a small town of county Kildare, Ireland, on the Midland Great -Western railway and the Royal Canal, 15 m. W. by N. of Dublin. Pop. -(1901), 948. The Royal Catholic College of Maynooth, founded by an Act -of the Irish parliament in 1795, is the chief seminary for the education -of the Roman Catholic clergy of Ireland. The building is a fine Gothic -structure by A. W. Pugin, erected by a parliamentary grant obtained in -1846. The chapel, with fine oak choir-stalls, mosaic pavements, marble -altars and stained glass, and with adjoining cloisters, was dedicated in -1890. The average number of students is about 500--the number specified -under the act of 1845--and the full course of instruction is eight -years. Near the college stand the ruins of Maynooth Castle, probably -built in 1176, but subsequently extended, and formerly the residence of -the Fitzgerald family. It was besieged in the reigns of Henry VIII. and -Edward VI., and during the Cromwellian Wars, when it was demolished. The -beautiful mansion of Carton is about a mile from the town. - - - - -MAYO, RICHARD SOUTHWELL BOURKE, 6TH EARL OF (1822-1872), British -statesman, son of Robert Bourke, the 5th earl (1797-1867), was born in -Dublin on the 21st of February, 1822, and was educated at Trinity -College, Dublin. After travelling in Russia he entered parliament, and -sat successively for Kildare, Coleraine and Cockermouth. He was chief -secretary for Ireland in three administrations, in 1852, 1858 and 1866, -and was appointed viceroy of India in January 1869. He consolidated the -frontiers of India and met Shere Ali, amir of Afghanistan, in durbar at -Umballa in March 1869. His reorganization of the finances of the country -put India on a paying basis; and he did much to promote irrigation, -railways, forests and other useful public works. Visiting the convict -settlement at Port Blair in the Andaman Islands, for the purpose of -inspection, the viceroy was assassinated by a convict on the 8th of -February 1872. His successor was his son, Dermot Robert Wyndham Bourke -(b. 1851) who became 7th earl of Mayo. - - See Sir W. W. Hunter, _Life of the Earl of Mayo_, (1876), and _The - Earl of Mayo_ in the Rulers of India Series (1891). - - - - -MAYO, a western county of Ireland, in the province of Connaught, bounded -N. and W. by the Atlantic Ocean, N.E. by Sligo, E. by Roscommon, S.E. -and S. by Galway. The area is 1,380,390 acres, or about 2157 sq. m., the -county being the largest in Ireland after Cork and Galway. About -two-thirds of the boundary of Mayo is formed by sea, and the coast is -very much indented, and abounds in picturesque scenery. The principal -inlets are Killary Harbour between Mayo and Galway; Clew Bay, in which -are the harbours of Westport and Newport; Blacksod Bay and Broad Haven, -which form the peninsula of the Mullet; and Killala Bay between Mayo and -Sligo. The islands are very numerous, the principal being Inishturk, -near Killary Harbour; Clare Island, at the mouth of Clew Bay, where -there are many islets, all formed of drift; and Achill, the largest -island off Ireland. The coast scenery is not surpassed by that of -Donegal northward and Connemara southward, and there are several small -coast-towns, among which may be named Killala on the north coast, -Belmullet on the isthmus between Blacksod Bay and Broad Haven, Newport -and Westport on Clew Bay, with the watering-place of Mallaranny. The -majestic cliffs of the north coast, however, which reach an extreme -height in Benwee Head (892 ft.), are difficult of access and rarely -visited. In the eastern half of the county the surface is comparatively -level, with occasional hills; the western half is mountainous. Mweelrea -(2688 ft.) is included in a mountain range lying between Killary Harbour -and Lough Mask. The next highest summits are Nephin (2646 ft.), to the -west of Lough Conn, and Croagh Patrick (2510 ft.), to the south of Clew -Bay. The river Moy flows northwards, forming part of the boundary of the -county with Sligo, and falls into Killala Bay. The courses of the other -streams are short, and except when swollen by rains their volume is -small. The principal lakes are Lough Mask and Lough Corrib, on the -borders of the county with Galway, and Loughs Conn in the east, -Carrowmore in the north-west, Beltra in the west, and Carra adjoining -Lough Mask. These loughs and the smaller loughs, with the streams -generally, afford admirable sport with salmon, sea-trout and brown -trout, and Ballina is a favourite centre. - - _Geology._--The wild and barren west of this county, including the - great hills on Achill Island, is formed of "Dalradian" rocks, schists - and quartzites, highly folded and metamorphosed, with intrusions of - granite near Belmullet. At Blacksod Bay the granite has been quarried - as an ornamental stone. Nephin Beg, Nephin and Croagh Patrick are - typical quartzite summits, the last named belonging possibly to a - Silurian horizon but rising from a metamorphosed area on the south - side of Clew Bay. The schists and gneisses of the Ox Mountain axis - also enter the county north of Castlebar. The Muilrea and Ben Gorm - range, bounding the fine fjord of Killary Harbour, is formed of - terraced Silurian rocks, from Bala to Ludlow age. These beds, with - intercalated lavas, form the mountainous west shore of Lough Mask, the - east, like that of Lough Corrib, being formed of low Carboniferous - Limestone ground. Silurian rocks, with Old Red Sandstone over them, - come out at the west end of the Curlew range at Ballaghaderreen. Clew - Bay, with its islets capped by glacial drift, is a submerged part of a - synclinal of Carboniferous strata, and Old Red Sandstone comes out on - the north side of this, from near Achill to Lough Conn. The country - from Lough Conn northward to the sea is a lowland of Carboniferous - Limestone, with L. Carboniferous Sandstone against the Dalradian on - the west. - - _Industries._--There are some very fertile regions in the level - portions of the county, but in the mountainous districts the soil is - poor, the holdings are subdivided beyond the possibility of affording - proper sustenance to their occupiers, and, except where fishing is - combined with agricultural operations, the circumstances of the - peasantry are among the most wretched of any district of Ireland. The - proportion of tillage to pasturage is roughly as 1 to 3(1/2). Oats and - potatoes are the principal crops. Cattle, sheep, pigs and poultry are - reared. Coarse linen and woollen cloths are manufactured to a small - extent. At Foxford woollen-mills are established at a nunnery, in - connexion with a scheme of technical instruction. Keel, Belmullet and - Ballycastle are the headquarters of sea and coast fishing districts, - and Ballina of a salmon-fishing district, and these fisheries are of - some value to the poor inhabitants. A branch of the Midland Great - Western railway enters the county from Athlone, in the south-east, and - runs north to Ballina and Killala on the coast, branches diverging - from Claremorris to Ballinrobe, and from Manulla to Westport and - Achill on the west coast. The Limerick and Sligo line of the Great - Southern and Western passes from south to north-east by way of - Claremorris. - -_Population and Administration._--The population was 218,698 in 1891, -and 199,166 in 1901. The decrease of population and the number of -emigrants are slightly below the average of the Irish counties. Of the -total population about 97% are rural, and about the same percentage are -Roman Catholics. The chief towns are Ballina (pop. 4505), Westport -(3892) and Castlebar (3585), the county town. Ballaghaderreen, -Claremorris (Clare), Crossmolina and Swineford are lesser market towns; -and Newport and Westport are small seaports on Clew Bay. The county -includes nine baronies. Assizes are held at Castlebar, and quarter -sessions at Ballina, Ballinrobe, Belmullet, Castlebar, Claremorris, -Swineford and Westport. In the Irish parliament two members were -returned for the county, and two for the borough of Castlebar, but at -the union Castlebar was disfranchised. The division since 1885 is into -north, south, east and west parliamentary divisions, each returning one -member. The county is in the Protestant diocese of Tuam and the Roman -Catholic dioceses of Taum, Achonry, Galway and Kilmacduagh, and Killala. - -_History and Antiquities._--Erris in Mayo was the scene of the landing -of the chief colony of the Firbolgs, and the battle which is said to -have resulted in the overthrow and almost annihilation of this tribe -took place also in this county, at Moytura near Cong. At the close of -the 12th century what is now the county of Mayo was granted, with other -lands, by king John to William, brother of Hubert de Burgh. After the -murder of William de Burgh, 3rd earl of Ulster (1333), the Bourkes (de -Burghs) of the collateral male line, rejecting the claim of William's -heiress (the wife of Lionel, son of King Edward III.) to the succession, -succeeded in holding the bulk of the De Burgh possessions, what is now -Mayo falling to the branch known by the name of "MacWilliam Oughter," -who maintained their virtual independence till the time of Elizabeth. -Sir Henry Sydney, during his first viceroyalty, after making efforts to -improve communications between Dublin and Connaught in 1566, arranged -for the shiring of that province, and Mayo was made shire ground, taking -its name from the monastery of Maio or Mageo, which was the seat of a -bishop. Even after this period the MacWilliams continued to exercise -very great authority, which was regularized in 1603, when "the -MacWilliam Oughter," Theobald Bourke, surrendered his lands and received -them back, to hold them by English tenure, with the title of Viscount -Mayo (see BURGH, DE). Large confiscations of the estates in the county -were made in 1586, and on the termination of the wars of 1641; and in -1666 the restoration of his estates to the 4th Viscount Mayo involved -another confiscation, at the expense of Cromwell's settlers. Killala was -the scene of the landing of a French squadron in connexion with the -rebellion of 1798. In 1879 the village of Knock in the south-east -acquired notoriety from a story that the Virgin Mary had appeared in the -church, which became the resort of many pilgrims. - -There are round towers at Killala, Turlough, Meelick and Balla, and an -imperfect one at Aughagower. Killala was formerly a bishopric. The -monasteries were numerous, and many of them of considerable importance: -the principal being those at Mayo, Ballyhaunis, Cong, Ballinrobe, -Ballintober, Burrishoole, Cross or Holycross in the peninsula of Mullet, -Moyne, Roserk or Rosserick and Templemore or Strade. Of the old castles -the most notable are Carrigahooly near Newport, said to have been built -by the celebrated Grace O'Malley, and Deel Castle near Ballina, at one -time the residence of the earls of Arran. - - See Hubert Thomas Knox, _History of the County of Mayo_ (1908). - - - - -MAYOR, JOHN EYTON BICKERSTETH (1825- ), English classical scholar, was -born at Baddegama, Ceylon, on the 28th of January 1825, and educated in -England at Shrewsbury School and St John's College, Cambridge. From 1863 -to 1867 he was librarian of the university, and in 1872 succeeded H. A. -J. Munro in the professorship of Latin. His best-known work, an edition -of thirteen satires of Juvenal, is marked by an extraordinary wealth of -illustrative quotations. His _Bibliographical Clue to Latin Literature_ -(1873), based on E. Hubner's _Grundriss zu Vorlesungen uber die romische -Litteraturgeschichte_ is a valuable aid to the student, and his edition -of Cicero's _Second Philippic_ is widely used. He also edited the -English works of J. Fisher, bishop of Rochester, i. (1876); Thomas -Baker's _History of St John's College, Cambridge_ (1869); Richard of -Cirencester's _Speculum historiale de gestis regum Angliae 447-1066_ -(1863-1869); Roger Ascham's _Schoolmaster_ (new ed., 1883); the _Latin -Heptateuch_ (1889); and the _Journal of Philology_. - -His brother, JOSEPH BICKERSTETH MAYOR (1828- ), classical scholar and -theologian, was educated at Rugby and St John's College, Cambridge, and -from 1870 to 1879 was professor of classics at King's College, London. -His most important classical works are an edition of Cicero's _De natura -deorum_ (3 vols., 1880-1885) and _Guide to the Choice of Classical -Books_ (3rd ed., 1885, with supplement, 1896). He also devoted attention -to theological literature and edited the epistles of St James (2nd ed., -1892), St Jude and St Peter (1907), and the _Miscellanies_ of Clement of -Alexandria (with F. J. A. Hort, 1902). From 1887 to 1893 he was editor -of the _Classical Review_. His _Chapters on English Metre_ (1886) -reached a second edition in 1901. - - - - -MAYOR (Lat. _major_, greater), in modern times the title of a municipal -officer who discharges judicial and administrative functions. The French -form of the word is _maire_. In Germany the corresponding title is -_Burgermeister_, in Italy _sindico_, and in Spain _alcalde_. "Mayor" had -originally a much wider significance. Among the nations which arose on -the ruins of the Roman empire of the West, and which made use of the -Latin spoken by their "Roman" subjects as their official and legal -language, _major_ and the Low Latin feminine _majorissa_ were found to -be very convenient terms to describe important officials of both sexes -who had the superintendence of others. Any female servant or slave in -the household of a barbarian, whose business it was to overlook other -female servants or slaves, would be quite naturally called a -_majorissa_. So the male officer who governed the king's household would -be the _major domus_. In the households of the Frankish kings of the -Merovingian line, the _major domus_, who was also variously known as the -_gubernator_, _rector_, _moderator_ or _praefectus palatii_, was so -great an officer that he ended by evicting his master. He was the "mayor -of the palace" (q.v.). The fact that his office became hereditary in the -family of Pippin of Heristal made the fortune of the Carolingian line. -But besides the _major domus_ (the major-domo), there were other -officers who were _majores_, the _major cubiculi_, mayor of the -bedchamber, and _major equorum_, mayor of the horse. In fact a word -which could be applied so easily and with accuracy in so many -circumstances was certain to be widely used by itself, or in its -derivatives. The post-Augustine _majorinus_, "one of the larger kind," -was the origin of the medieval Spanish _merinus_, who in Castillian is -the _merino_, and sometimes the _merino mayor_, or chief merino. He was -a judicial and administrative officer of the king's. The _gregum -merinus_ was the superintendent of the flocks of the corporation of -sheep-owners called the _mesta_. From him the sheep, and then the wool, -have come to be known as _merinos_--a word identical in origin with the -municipal title of mayor. The latter came directly from the heads of -gilds, and other associations of freemen, who had their banner and -formed a group on the populations of the towns, the _majores baneriae_ -or _vexilli_. - -In England the major is the modern representative of the lord's bailiff -or reeve (see BOROUGH). We find the chief magistrate of London bearing -the title of portreeve for considerably more than a century after the -Conquest. This official was elected by popular choice, a privilege -secured from king John. By the beginning of the 11th century the title -of portreeve[1] gave way to that of mayor as the designation of the -chief officer of London,[2] and the adoption of the title by other -boroughs followed at various intervals. - - A mayor is now in England and America the official head of a municipal - government. In the United Kingdom the Municipal Corporations Act, - 1882, s. 15, regulates the election of mayors. He is to be a fit - person elected annually on the 9th of November by the council of the - borough from among the aldermen or councillors or persons qualified to - be such. His term of office is one year, but he is eligible for - re-election. He may appoint a deputy to act during illness or absence, - and such deputy must be either an alderman or councillor. A mayor who - is absent from the borough for more than two months becomes - disqualified and vacates his office. A mayor is _ex officio_ during - his year of office and the next year a justice of the peace for the - borough. He receives such remuneration as the council thinks - reasonable. The office of mayor in an English borough does not entail - any important administrative duties. It is generally regarded as an - honour conferred for past services. The mayor is expected to devote - much of his time to ornamental functions and to preside over meetings - which have for their object the advancement of the public welfare. His - administrative duties are merely to act as returning officer at - municipal elections, and as chairman of the meetings of the council. - - The position and power of an English mayor contrast very strongly with - those of the similar official in the United States. The latter is - elected directly by the voters within the city, usually for several - years; and he has extensive administrative powers. - - The English method of selecting a mayor by the council is followed for - the corresponding functionaries in France (except Paris), the more - important cities of Italy, and in Germany, where, however, the central - government must confirm the choice of the council. Direct appointment - by the central government exists in Belgium, Holland, Denmark, Norway, - Sweden and the smaller towns of Italy and Spain. As a rule, too, the - term of office is longer in other countries than in the United - Kingdom. In France election is for four years, in Holland for six, in - Belgium for an indefinite period, and in Germany usually for twelve - years, but in some cases for life. In Germany the post may be said to - be a professional one, the burgomaster being the head of the city - magistracy, and requiring, in order to be eligible, a training in - administration. German burgomasters are most frequently elected by - promotion from another city. In France the _maire_, and a number of - experienced members termed "adjuncts," who assist him as an executive - committee, are elected directly by the municipal council from among - their own number. Most of the administrative work is left in the hands - of the _maire_ and his adjuncts, the full council meeting - comparatively seldom. The _maire_ and the adjuncts receive no salary. - - Further information will be found in the sections on local government - in the articles on the various countries; see also A. Shaw, _Municipal - Government in Continental Europe_; J. A. Fairlie, _Municipal - Administration_; S. and B. Webb, _English Local Government_; Redlich - and Hirst, _Local Government in England_; A. L. Lowell, _The - Government of England_. - - -FOOTNOTES: - - [1] If a place was of mercantile importance it was called a port - (from _porta_, the city gate), and the reeve or bailiff, a - "portreeve." - - [2] The mayors of certain cities in the United Kingdom (London, York, - Dublin) have acquired by prescription the prefix of "lord." In the - case of London it seems to date from 1540. It has also been conferred - during the closing years of the 19th century by letters patent on - other cities--Birmingham, Liverpool, Manchester, Bristol, Sheffield, - Leeds, Cardiff, Bradford, Newcastle-on-Tyne, Belfast, Cork. In 1910 - it was granted to Norwich. Lord mayors are entitled to be addressed - as "right honourable." - - - - -MAYOR OF THE PALACE.--The office of mayor of the palace was an -institution peculiar to the Franks of the Merovingian period. A -landowner who did not manage his own estate placed it in the hands of a -steward (_major_), who superintended the working of the estate and -collected its revenues. If he had several estates, he appointed a chief -steward, who managed the whole of the estates and was called the _major -domus_. Each great personage had a _major domus_--the queen had hers, -the king his; and since the royal house was called the palace, this -officer took the name of "mayor of the palace." The mayor of the palace, -however, did not remain restricted to domestic functions; he had the -discipline of the palace and tried persons who resided there. Soon his -functions expanded. If the king were a minor, the mayor of the palace -supervised his education in the capacity of guardian (_nutricius_), and -often also occupied himself with affairs of state. When the king came of -age, the mayor exerted himself to keep this power, and succeeded. In the -7th century he became the head of the administration and a veritable -prime minister. He took part in the nomination of the counts and dukes; -in the king's absence he presided over the royal tribunal; and he often -commanded the armies. When the custom of commendation developed, the -king charged the mayor of the palace to protect those who had commended -themselves to him and to intervene at law on their behalf. The mayor of -the palace thus found himself at the head of the _commendati_, just as -he was at the head of the functionaries. - -It is difficult to trace the names of some of the mayors of the palace, -the post being of almost no significance in the time of Gregory of -Tours. When the office increased in importance the mayors of the palace -did not, as has been thought, pursue an identical policy. Some--for -instance, Otto, the mayor of the palace of Austrasia towards 640--were -devoted to the Crown. On the other hand, mayors like Flaochat (in -Burgundy) and Erkinoald (in Neustria) stirred up the great nobles, who -claimed the right to take part in their nomination, against the king. -Others again, sought to exercise the power in their own name both -against the king and against the great nobles--such as Ebroin (in -Neustria), and, later, the Carolingians Pippin II., Charles Martel, and -Pippin III., who, after making use of the great nobles, kept the -authority for themselves. In 751 Pippin III., fortified by his -consultation with Pope Zacharias, could quite naturally exchange the -title of mayor for that of king; and when he became king, he suppressed -the title of mayor of the palace. It must be observed that from 639 -there were generally separate mayors of Neustria, Austrasia and -Burgundy, even when Austrasia and Burgundy formed a single kingdom; the -mayor was a sign of the independence of the region. Each mayor, however, -sought to supplant the others; the Pippins and Charles Martel succeeded, -and their victory was at the same time the victory of Austrasia over -Neustria and Burgundy. - - See G. H. Pertz, _Geschichte der merowingischen Hausmeier_ (Hanover, - 1819); H. Bonnell, _De dignitate majoris domus_ (Berlin, 1858); E. - Hermann, _Das Hausmeieramt, ein echt germanisches Amt_, vol. ix. of - _Untersuchungen zur deutschen Staats- und Rechtsgeschichte_, ed. by O. - Gierke (Breslau, 1878, seq.); G. Waitz, _Deutsche - Verfassungsgeschichte_, 3rd ed., revised by K. Zeumer; and Fustel de - Coulanges, _Histoire des institutions politiques de l'ancienne France: - La monarchie franque_ (Paris, 1888). (C. Pf.) - - - - -MAYORUNA, a tribe of South American Indians of Panoan stock. Their -country is between the Ucayali and Javari rivers, north-eastern Peru. -They are a fine race, roaming the forests and living by hunting. They -cut their hair in a line across the forehead and let it hang down their -backs. Many have fair skins and beards, a peculiarity sometimes -explained by their alleged descent from Ursua's soldiers, but this -theory is improbable. They are famous for the potency of their blow-gun -poison. - - - - -MAYO-SMITH, RICHMOND (1854-1901), American economist, was born in Troy, -Ohio, on the 9th of February 1854. Educated at Amherst, and at Berlin -and Heidelberg, he became assistant professor of economics at Columbia -University in 1877. He was an adjunct professor from 1878 to 1883, when -he was appointed professor of political economy and social science, a -post which he held until his death on the 11th of November 1901. He -devoted himself especially to the study of statistics, and was -recognized as one of the foremost authorities on the subject. His works -include _Emigration and Immigration_ (1890); _Sociology and Statistics_ -(1895), and _Statistics and Economics_ (1899). - - - - -MAYOTTE, one of the Comoro Islands, in the Mozambique Channel between -Madagascar and the African mainland. It has belonged to France since -1843 (see COMORO ISLANDS). - - - - -MAYOW, JOHN (1643-1679), English chemist and physiologist, was born in -London in May 1643. At the age of fifteen he went up to Wadham College, -Oxford, of which he became a scholar a year later, and in 1660 he was -elected to a fellowship at All Souls. He graduated in law (bachelor, -1665, doctor, 1670), but made medicine his profession, and "became noted -for his practice therein, especially in the summer time, in the city of -Bath." In 1678, on the proposal of R. Hooke, he was chosen a fellow of -the Royal Society. The following year, after a marriage which was "not -altogether to his content," he died in London in September 1679. He -published at Oxford in 1668 two tracts, on respiration and rickets, and -in 1674 these were reprinted, the former in an enlarged and corrected -form, with three others "De sal-nitro et spiritu nitro-aereo," "De -respiratione foetus in utero et ovo," and "De motu musculari et -spiritibus animalibus" as _Tractatus quinque medico-physici_. The -contents of this work, which was several times republished and -translated into Dutch, German and French, show him to have been an -investigator much in advance of his time. - - Accepting as proved by Boyle's experiments that air is necessary for - combustion, he showed that fire is supported not by the air as a whole - but by a "more active and subtle part of it." This part he called - _spiritus igneo-aereus_, or sometimes _nitro-aereus_; for he - identified it with one of the constituents of the acid portion of - nitre which he regarded as formed by the union of fixed alkali with a - _spiritus acidus_. In combustion the _particulae nitro-aereae_--either - pre-existent in the thing consumed or supplied by the air--combined - with the material burnt; as he inferred from his observation that - antimony, strongly heated with a burning glass, undergoes an increase - of weight which can be attributed to nothing else but these particles. - In respiration he argued that the same particles are consumed, because - he found that when a small animal and a lighted candle were placed in - a closed vessel full of air the candle first went out and soon - afterwards the animal died, but if there was no candle present it - lived twice as long. He concluded that this constituent of the air is - absolutely necessary for life, and supposed that the lungs separate it - from the atmosphere and pass it into the blood. It is also necessary, - he inferred, for all muscular movements, and he thought there was - reason to believe that the sudden contraction of muscle is produced by - its combination with other combustible (salino-sulphureous) particles - in the body; hence the heart, being a muscle, ceases to beat when - respiration is stopped. Animal heat also is due to the union of - nitro-aerial particles, breathed in from the air, with the combustible - particles in the blood, and is further formed by the combination of - these two sets of particles in muscle during violent exertion. In - effect, therefore, Mayow--who also gives a remarkably correct - anatomical description of the mechanism of respiration--preceded - Priestley and Lavoisier by a century in recognizing the existence of - oxygen, under the guise of his _spiritus nitro-aereus_, as a separate - entity distinct from the general mass of the air; he perceived the - part it plays in combustion and in increasing the weight of the calces - of metals as compared with metals themselves; and, rejecting the - common notions of his time that the use of breathing is to cool the - heart, or assist the passage of the blood from the right to the left - side of the heart, or merely to agitate it, he saw in inspiration a - mechanism for introducing oxygen into the body, where it is consumed - for the production of heat and muscular activity, and even vaguely - conceived of expiration as an excretory process. - - - - -MAYSVILLE, a city and the county-seat of Mason county, Kentucky, U.S.A., -on the Ohio river, 60 m. by rail S.E. of Cincinnati. Pop. (1890) 5358; -(1900) 6423 (1155 negroes); (1910) 6141. It is served by the Louisville -& Nashville, and the Chesapeake & Ohio railways, and by steamboats on -the Ohio river. Among its principal buildings are the Mason county -public library (1878), the Federal building and Masonic and Odd Fellows' -temples. The city lies between the river and a range of hills; at the -back of the hills is a fine farming country, of which tobacco of -excellent quality is a leading product. There is a large plant of the -American Tobacco Company at Maysville, and among the city's manufactures -are pulleys, ploughs, whisky, flour, lumber, furniture, carriages, -cigars, foundry and machine-shop products, bricks and cotton goods. The -city is a distributing point for coal and other products brought to it -by Ohio river boats. Formerly it was one of the principal hemp markets -of the country. The place early became a landing point for immigrants to -Kentucky, and in 1784 a double log cabin and a blockhouse were erected -here. It was then called Limestone, from the creek which flows into the -Ohio here, but several years later the present name was adopted in -honour of John May, who with Simon Kenton laid out the town in 1787, and -who in 1790 was killed by the Indians. Maysville was incorporated as a -town in 1787, was chartered as a city in 1833, and became the -county-seat in 1848. - - In 1830, when the question of "internal improvements" by the National - government was an important political issue, Congress passed a bill - directing the government to aid in building a turnpike road from - Maysville to Lexington. President Andrew Jackson vetoed the bill on - the ground that the proposed improvement was a local rather than a - national one; but one-half the capital was then furnished privately, - the other half was furnished through several state appropriations, and - the road was completed in 1835 and marked the beginning of a system of - turnpike roads built with state aid. - - - - -MAZAGAN (_El Jadida_), a port on the Atlantic coast of Morocco in 33 -deg. 16' N. 8 deg. 26' W. Pop. (1908), about 12,000, of whom a fourth -are Jews and some 400 Europeans. It is the port for Marrakesh, from -which it is 110 m. nearly due north, and also for the fertile province -of Dukalla. Mazagan presents from the sea a very un-Moorish appearance; -it has massive Portuguese walls of hewn stone. The exports, which -include beans, almonds, maize, chick-peas, wool, hides, wax, eggs, &c., -were valued at L360,000 in 1900, L364,000 in 1904, and L248,000 in 1906. -The imports (cotton goods, sugar, tea, rice, &c.) were valued at -L280,000 in 1900, L286,000 in 1904, and L320,000 in 1906. About 46% of -the trade is with Great Britain and 34% with France. Mazagan was built -in 1506 by the Portuguese, who abandoned it to the Moors in 1769 and -established a colony, New Mazagan, on the shores of Para in Brazil. - - See A. H. Dye, "Les ports du Maroc" in _Bull. Soc. Geog. Comm. Paris_, - xxx. 325-332 (1908), and British consular reports. - - - - -MAZAMET, an industrial town of south-western France in the department of -Tarn, 41 m. S.S.E. of Albi by rail. Pop. (1906), town, 11,370; commune, -14,386. Mazamet is situated on the northern slope of the Montagnes -Noires and on the Arnette, a small sub-tributary of the Agout. Numerous -establishments are employed in wool-spinning and in the manufacture of -"swan-skins" and flannels, and clothing for troops, and hosiery, and -there are important tanneries and leather-dressing, glove and dye works. -Extensive commerce is carried on in wool and raw hides from Argentina, -Australia and Cape Colony. - - - - -MAZANDARAN, a province of northern Persia, lying between the Caspian Sea -and the Elburz range, and bounded E. and W. by the provinces of -Astarabad and Gilan respectively, 220 m. in length and 60 m. in (mean) -breadth, with an area of about 10,000 sq. m. and a population estimated -at from 150,000 to 200,000. Mazandaran comprises two distinct natural -regions presenting the sharpest contrasts in their relief, climate and -products. In the north the Caspian is encircled by the level and swampy -lowlands, varying in breadth from 10 to 30 m., partly under impenetrable -jungle, partly under rice, cotton, sugar and other crops. This section -is fringed northwards by the sandy beach of the Caspian, here almost -destitute of natural harbours, and rises somewhat abruptly inland to the -second section, comprising the northern slopes and spurs of the Elburz, -which approach at some points within 1 or 2 m. of the sea, and are -almost everywhere covered with dense forest. The lowlands, rising but a -few feet above the Caspian, and subject to frequent floodings, are -extremely malarious, while the highlands, culminating with the -magnificent Demavend (19,400 ft.), enjoy a tolerably healthy climate. -But the climate, generally hot and moist in summer, is everywhere -capricious and liable to sudden changes of temperature, whence the -prevalence of rheumatism, dropsy and especially ophthalmia, noticed by -all travellers. Snow falls heavily in the uplands, where it often lies -for weeks on the ground. The direction of the long sandbanks at the -river mouths, which project with remarkable uniformity from west to -east, shows that the prevailing winds blow from the west and north-west. -The rivers themselves, of which there are as many as fifty, are little -more than mountain torrents, all rising on the northern slopes of -Elburz, flowing mostly in independent channels to the Caspian, and -subject to sudden freshets and inundations along their lower course. The -chief are the Sardab-rud, Chalus, Herhaz (Lar in its upper course), -Babul, Tejen and Nika, and all are well stocked with trout, salmon -(_azad-mahi_), perch (_safid-mahi_), carp (_kupur_), bream (_subulu_), -sturgeon (_sag-mahi_) and other fish, which with rice form the staple -food of the inhabitants; the sturgeon supplies the caviare for the -Russian market. Near their mouths the rivers, running counter to the -prevailing winds and waves of the Caspian, form long sand-hills 20 to 30 -ft. high and about 200 yds. broad, behind which are developed the -so-called _murd-ab_, or "dead waters," stagnant pools and swamps -characteristic of this coast, and a main cause of its unhealthiness. - -The chief products are rice, cotton, sugar, a little silk, and fruits in -great variety, including several kinds of the orange, lemon and citron. -Some of the slopes are covered with extensive thickets of the -pomegranate, and the wild vine climbs to a great height round the trunks -of the forest trees. These woodlands are haunted by the tiger, panther, -bear, wolf and wild boar in considerable numbers. Of the domestic -animals, all remarkable for their small size, the chief are the black, -humped cattle somewhat resembling the Indian variety, and sheep and -goats. - - Kinneir, Fraser and other observers speak unfavourably of the - Mazandarani people, whom they describe as very ignorant and bigoted, - arrogant, rudely inquisitive and almost insolent towards strangers. - The peasantry, however, are far from dull, and betray much shrewdness - where their interests are concerned. In the healthy districts they are - stout and well made, and are considered a warlike race, furnishing - some cavalry (800 men) and eight battalions of infantry (5600 men) to - government. They speak a marked Persian dialect, but a Turki idiom - closely akin to the Turkoman is still current amongst the tribes, - although they have mostly already passed from the nomad to the settled - state. Of these tribes the most numerous are the Modaunlu, Khojehvand - and Abdul Maleki, originally of Lek or Kurd stock, besides branches of - the royal Afshar and Kajar tribes of Turki descent. All these are - exempt from taxes in consideration of their military service. - - The export trade is chiefly with Russia from Meshed-i-Sar, the - principal port of the province, to Baku, where European goods are - taken in exchange for the white and coloured calicoes, caviare, rice, - fruits and raw cotton of Mazandaran. Great quantities of rice are also - exported to the interior of Persia, principally to Teheran and Kazvin. - Owing to the almost impenetrable character of the country there are - scarcely any roads accessible to wheeled carriages, and the great - causeway of Shah Abbas along the coast has in many places even - disappeared under the jungle. Two routes, however, lead to Teheran, - one by Firuz Kuh, 180 m. long, the other by Larijan, 144 m. long, both - in tolerably good repair. Except where crossed by these routes the - Elburz forms an almost impassable barrier to the south. - - The administration is in the hands of a governor, who appoints the - sub-governors of the nine districts of Amol, Barfarush, Meshed-i-Sar, - Sari, Ashref, Farah-abad, Tunakabun, Kelarrustak and Kujur into which - the province is divided. There is fair security for life and property; - and, although otherwise indifferently administered, the country is - quite free from marauders; but local disturbances have latterly been - frequent in the two last-named districts. The revenue is about - L30,000, of which little goes to the state treasury, most being - required for the governors, troops and pensions. The capital is Sari, - the other chief towns being Barfarush, Meshed-i-Sar, Ashref and - Farah-abad. (A. H.-S.) - - - - -MAZARIN, JULES (1602-1661), French cardinal and statesman, elder son of -a Sicilian, Pietro Mazarini, the intendant of the household of Philip -Colonna, and of his wife Ortensia Buffalini, a connexion of the -Colonnas, was born at Piscina in the Abruzzi on the 14th of July 1602. -He was educated by the Jesuits at Rome till his seventeenth year, when -he accompanied Jerome Colonna as chamberlain to the university of Alcala -in Spain. There he distinguished himself more by his love of gambling -and his gallant adventures than by study, but made himself a thorough -master, not only of the Spanish language and character, but also of that -romantic fashion of Spanish love-making which was to help him greatly in -after life, when he became the servant of a Spanish queen. On his return -to Rome, about 1622, he took his degree as Doctor _utriusque juris_, and -then became captain of infantry in the regiment of Colonna, which took -part in the war in the Valtelline. During this war he gave proofs of -much diplomatic ability, and Pope Urban VIII. entrusted him, in 1629, -with the difficult task of putting an end to the war of the Mantuan -succession. His success marked him out for further distinction. He was -presented to two canonries in the churches of St John Lateran and Sta -Maria Maggiore, although he had only taken the minor orders, and had -never been consecrated priest; he negotiated the treaty of Turin between -France and Savoy in 1632, became vice-legate at Avignon in 1634, and -nuncio at the court of France from 1634 to 1636. But he began to wish -for a wider sphere than papal negotiations, and, seeing that he had no -chance of becoming a cardinal except by the aid of some great power, he -accepted Richelieu's offer of entering the service of the king of -France, and in 1639 became a naturalized Frenchman. - -In 1640 Richelieu sent him to Savoy, where the regency of Christine, the -duchess of Savoy, and sister of Louis XIII., was disputed by her -brothers-in-law, the princes Maurice and Thomas of Savoy, and he -succeeded not only in firmly establishing Christine but in winning over -the princes to France. This great service was rewarded by his promotion -to the rank of cardinal on the presentation of the king of France in -December 1641. On the 4th of December 1642 Cardinal Richelieu died, and -on the very next day the king sent a circular letter to all officials -ordering them to send in their reports to Cardinal Mazarin, as they had -formerly done to Cardinal Richelieu. Mazarin was thus acknowledged -supreme minister, but he still had a difficult part to play. The king -evidently could not live long, and to preserve power he must make -himself necessary to the queen, who would then be regent, and do this -without arousing the suspicions of the king or the distrust of the -queen. His measures were ably taken, and when the king died, on the 14th -of May 1643, to everyone's surprise her husband's minister remained the -queen's. The king had by a royal edict cumbered the queen-regent with a -council and other restrictions, and it was necessary to get the -parlement of Paris to overrule the edict and make the queen absolute -regent, which was done with the greatest complaisance. Now that the -queen was all-powerful, it was expected she would at once dismiss -Mazarin and summon her own friends to power. One of them, Potier, bishop -of Beauvais, already gave himself airs as prime minister, but Mazarin -had had the address to touch both the queen's heart by his Spanish -gallantry and her desire for her son's glory by his skilful policy -abroad, and he found himself able easily to overthrow the clique of -Importants, as they were called. That skilful policy was shown in every -arena on which the great Thirty Years' War was being fought out. Mazarin -had inherited the policy of France during the Thirty Years' War from -Richelieu. He had inherited his desire for the humiliation of the house -of Austria in both its branches, his desire to push the French frontier -to the Rhine and maintain a counterpoise of German states against -Austria, his alliances with the Netherlands and with Sweden, and his -four theatres of war--on the Rhine, in Flanders, in Italy and in -Catalonia. - -During the last five years of the great war it was Mazarin alone who -directed the French diplomacy of the period. He it was who made the -peace of Bromsebro between the Danes and the Swedes, and turned the -latter once again against the empire; he it was who sent Lionne to make -the peace of Castro, and combine the princes of North Italy against the -Spaniards, and who made the peace of Ulm between France and Bavaria, -thus detaching the emperor's best ally. He made one fatal mistake--he -dreamt of the French frontier being the Rhine and the Scheldt, and that -a Spanish princess might bring the Spanish Netherlands as dowry to Louis -XIV. This roused the jealousy of the United Provinces, and they made a -separate peace with Spain in January 1648; but the valour of the French -generals made the skill of the Spanish diplomatists of no avail, for -Turenne's victory at Zusmarshausen, and Conde's at Lens, caused the -peace of Westphalia to be definitely signed in October 1648. This -celebrated treaty belongs rather to the history of Germany than to a -life of Mazarin; but two questions have been often asked, whether -Mazarin did not delay the peace as long as possible in order to more -completely ruin Germany, and whether Richelieu would have made a similar -peace. To the first question Mazarin's letters, published by M. Cheruel, -prove a complete negative, for in them appears the zeal of Mazarin for -the peace. On the second point, Richelieu's letters in many places -indicate that his treatment of the great question of frontier would have -been more thorough, but then he would not have been hampered in France -itself. - -At home Mazarin's policy lacked the strength of Richelieu's. The Frondes -were largely due to his own fault. The arrest of Broussel threw the -people on the side of the parlement. His avarice and unscrupulous -plundering of the revenues of the realm, the enormous fortune which he -thus amassed, his supple ways, his nepotism, and the general lack of -public interest in the great foreign policy of Richelieu, made Mazarin -the especial object of hatred both by bourgeois and nobles. The -irritation of the latter was greatly Mazarin's own fault; he had tried -consistently to play off the king's brother Gaston of Orleans against -Conde, and their respective followers against each other, and had also, -as his _carnets_ prove, jealously kept any courtier from getting into -the good graces of the queen-regent except by his means, so that it was -not unnatural that the nobility should hate him, while the queen found -herself surrounded by his creatures alone. Events followed each other -quickly; the day of the barricades was followed by the peace of Ruel, -the peace of Ruel by the arrest of the princes, by the battle of Rethel, -and Mazarin's exile to Bruhl before the union of the two Frondes. It was -while in exile at Bruhl that Mazarin saw the mistake he had made in -isolating himself and the queen, and that his policy of balancing every -party in the state against each other had made every party distrust him. -So by his counsel the queen, while nominally in league with De Retz and -the parliamentary Fronde, laboured to form a purely royal party, wearied -by civil dissensions, who should act for her and her son's interest -alone, under the leadership of Mathieu Mole, the famous premier -president of the parlement of Paris. The new party grew in strength, and -in January 1652, after exactly a year's absence, Mazarin returned to the -court. Turenne had now become the royal general, and out-manoeuvred -Conde, while the royal party at last grew to such strength in Paris that -Conde had to leave the capital and France. In order to promote a -reconciliation with the parlement of Paris Mazarin had again retired -from court, this time to Sedan, in August 1652, but he returned finally -in February 1653. Long had been the trial, and greatly had Mazarin been -to blame in allowing the Frondes to come into existence, but he had -retrieved his position by founding that great royal party which steadily -grew until Louis XIV. could fairly have said "L'Etat, c'est moi." As the -war had progressed, Mazarin had steadily followed Richelieu's policy of -weakening the nobles on their country estates. Whenever he had an -opportunity he destroyed a feudal castle, and by destroying the towers -which commanded nearly every town in France, he freed such towns as -Bourges, for instance, from their long practical subjection to the -neighbouring great lord. - -The Fronde over, Mazarin had to build up afresh the power of France at -home and abroad. It is to his shame that he did so little at home. -Beyond destroying the brick-and-mortar remains of feudalism, he did -nothing for the people. But abroad his policy was everywhere successful, -and opened the way for the policy of Louis XIV. He at first, by means of -an alliance with Cromwell, recovered the north-western cities of France, -though at the price of yielding Dunkirk to the Protector. On the Baltic, -France guaranteed the Treaty of Oliva between her old allies Sweden, -Poland and Brandenburg, which preserved her influence in that quarter. -In Germany he, through Hugues de Lionne, formed the league of the Rhine, -by which the states along the Rhine bound themselves under the headship -of France to be on their guard against the house of Austria. By such -measures Spain was induced to sue for peace, which was finally signed in -the Isle of Pheasants on the Bidassoa, and is known as the Treaty of the -Pyrenees. By it Spain recovered Franche Comte, but ceded to France -Roussillon, and much of French Flanders; and, what was of greater -ultimate importance to Europe, Louis XIV. was to marry a Spanish -princess, who was to renounce her claims to the Spanish succession if -her dowry was paid, which Mazarin knew could not happen at present from -the emptiness of the Spanish exchequer. He returned to Paris in -declining health, and did not long survive the unhealthy sojourn on the -Bidassoa; after some political instruction to his young master he passed -away at Vincennes on the 9th of March 1661, leaving a fortune estimated -at from 18 to 40 million livres behind him, and his nieces married into -the greatest families of France and Italy. - - The man who could have had such success, who could have made the - Treaties of Westphalia and the Pyrenees, who could have weathered the - storm of the Fronde, and left France at peace with itself and with - Europe to Louis XIV., must have been a great man; and historians, - relying too much on the brilliant memoirs of his adversaries, like De - Retz, are apt to rank him too low. That he had many a petty fault - there can be no doubt; that he was avaricious and double-dealing was - also undoubted; and his _carnets_ show to what unworthy means he had - recourse to maintain his influence over the queen. What that influence - was will be always debated, but both his _carnets_ and the Bruhl - letters show that a real personal affection, amounting to passion on - the queen's part, existed. Whether they were ever married may be - doubted; but that hypothesis is made more possible by M. Cheruel's - having been able to prove from Mazarin's letters that the cardinal - himself had never taken more than the minor orders, which could always - be thrown off. With regard to France he played a more patriotic part - than Conde or Turenne, for he never treated with the Spaniards, and - his letters show that in the midst of his difficulties he followed - with intense eagerness every movement on the frontiers. It is that - immense mass of letters that prove the real greatness of the - statesman, and disprove De Retz's portrait, which is carefully - arranged to show off his enemy against the might of Richelieu. To - concede that the master was the greater man and the greater statesman - does not imply that Mazarin was but a foil to his predecessor. It is - true that we find none of those deep plans for the internal prosperity - of France which shine through Richelieu's policy. Mazarin was not a - Frenchman, but a citizen of the world, and always paid most attention - to foreign affairs; in his letters all that could teach a diplomatist - is to be found, broad general views of policy, minute details - carefully elaborated, keen insight into men's characters, cunning - directions when to dissimulate or when to be frank. Italian though he - was by birth, education and nature, France owed him a great debt for - his skilful management during the early years of Louis XIV., and the - king owed him yet more, for he had not only transmitted to him a - nation at peace, but had educated for him his great servants Le - Tellier, Lionne and Colbert. Literary men owed him also much; not only - did he throw his famous library open to them, but he pensioned all - their leaders, including Descartes, Vincent Voiture (1598-1648), Jean - Louis Guez de Balzac (1597-1654) and Pierre Corneille. The last-named - applied, with an adroit allusion to his birthplace, in the dedication - of his _Pompee_, the line of Virgil:-- - - "Tu regere imperio populos, Romane, memento." (H. M. S.) - - AUTHORITIES.--All the earlier works on Mazarin, and early accounts of - his administration, of which the best were Bazin's _Histoire de France - sous Louis XIII. et sous le Cardinal Mazarin_, 4 vols. (1846), and - Saint-Aulaire's _Histoire de la Fronde_, have been superseded by P. A. - Cheruel's admirable _Histoire de France pendant la minorite de Louis - XIV._, 4 vols. (1879-1880), which covers from 1643-1651, and its - sequel _Histoire de France sous le ministere de Cardinal Mazarin_, 2 - vols. (1881-1882), which is the first account of the period written by - one able to sift the statements of De Retz and the memoir writers, and - rest upon such documents as Mazarin's letters and _carnets_. Mazarin's - _Lettres_, which must be carefully studied by any student of the - history of France, have appeared in the _Collection des documents - inedits_, 9 vols. For his _carnets_ reference must be made to V. - Cousin's articles in the _Journal des Savants_, and Cheruel in _Revue - historique_ (1877), see also Cheruel's _Histoire de France pendant la - minorite_, &c., app. to vol. iii.; for his early life to Cousin's - _Jeunesse de Mazarin_ (1865) and for the careers of his nieces to - Renee's _Les Nieces de Mazarin_ (1856). For the Mazarinades or squibs - written against him in Paris during the Fronde, see C. Moreau's - _Bibliographie des mazarinades_ (1850), containing an account of 4082 - Mazarinades. See also A. Hassall, _Mazarin_ (1903). - - - - -MAZAR-I-SHARIF, a town of Afghanistan, the capital of the province of -Afghan Turkestan. Owing to the importance of the military cantonment of -Takhtapul, and its religious sanctity, it has long ago supplanted the -more ancient capital of Balkh. It is situated in a malarious, almost -desert plain, 9 m. E. of Balkh, and 30 m. S. of the Pata Kesar ferry on -the Oxus river. In this neighbourhood is concentrated most of the Afghan -army north of the Hindu Kush mountains, the fortified cantonment of -Dehdadi having been completed by Sirdar Ghulam Ali Khan and incorporated -with Mazar. Mazar-i-Sharif also contains a celebrated mosque, from which -the town takes its name. It is a huge ornate building with minarets and -a lofty cupola faced with shining blue tiles. It was built by Sultan Ali -Mirza about A.D. 1420, and is held in great veneration by all -Mussulmans, and especially by Shiites, because it is supposed to be the -tomb of Ali, the son-in-law of Mahomet. - - - - -MAZARRON, a town of eastern Spain, in the province of Murcia, 19 m. W. -of Cartagena. Pop. (1900), 23,284. There are soap and flour mills and -metallurgic factories in the town, and iron, copper and lead mines in -the neighbouring Sierra de Almenara. A railway 5 m. long unites Mazarron -to its port on the Mediterranean, where there is a suburb with 2500 -inhabitants (mostly engaged in fisheries and coasting trade), containing -barracks, a custom-house, and important leadworks. Outside of the suburb -there are saltpans, most of the proceeds of which are exported to -Galicia. - - - - -MAZATLAN, a city and port of the state of Sinaloa, Mexico, 120 m. -(direct) W.S.W. of the city of Durango, in lat. 23 deg. 12' N., long 106 -deg. 24' W. Pop. (1895), 15,852; (1900), 17,852. It is the Pacific -coast terminus of the International railway which crosses northern -Mexico from Ciudad Porfirio Diaz, and a port of call for the principal -steamship lines on this coast. The harbour is spacious, but the entrance -is obstructed by a bar. The city is built on a small peninsula. Its -public buildings include a fine town-hall, chamber of commerce, a -custom-house and two hospitals, besides which there is a nautical school -and a meteorological station, one of the first established in Mexico. -The harbour is provided with a sea-wall at Olas Altas. A government -wireless telegraph service is maintained between Mazatlan and La Paz, -Lower California. Among the manufactures are saw-mills, foundries, -cotton factories and ropeworks, and the exports are chiefly hides, -ixtle, dried and salted fish, gold, silver and copper (bars and ores), -fruit, rubber, tortoise-shell, and gums and resins. - - - - -MAZE, a network of winding paths, a labyrinth (q.v.). The word means -properly a state of confusion or wonder, and is probably of Scandinavian -origin; cf. Norw. _mas_, exhausting labour, also chatter, _masa_, to be -busy, also to worry, annoy; Swed. _masa_, to lounge, move slowly and -lazily, to dream, muse. Skeat (_Etym._ Dict.) takes the original sense -to be probably "to be lost in thought," "to dream," and connects with -the root _ma-man_-, to think, cf. "mind," "man," &c. The word "maze" -represents the addition of an intensive suffix. - - - - -MAZEPA-KOLEDINSKY, IVAN STEPANOVICH (1644?-1709), hetman of the -Cossacks, belonging to a noble Orthodox family, was born possibly at -Mazeptsina, either in 1629 or 1644, the latter being the more probable -date. He was educated at the court of the Polish king, John Casimir, and -completed his studies abroad. An intrigue with a Polish married lady -forced him to fly into the Ukraine. There is a trustworthy tradition -that the infuriated husband tied the naked youth to the back of a wild -horse and sent him forth into the steppe. He was rescued and cared for -by the Dnieperian Cossacks, and speedily became one of their ablest -leaders. In 1687, during a visit to Moscow, he won the favour of the -then all-powerful Vasily Golitsuin, from whom he virtually purchased the -hetmanship of the Cossacks (July 25). He took a very active part in the -Azov campaigns of Peter the Great and won the entire confidence of the -young tsar by his zeal and energy. He was also very serviceable to Peter -at the beginning of the Great Northern War, especially in 1705 and 1706, -when he took part in the Volhynian campaign and helped to construct the -fortress of Pechersk. The power and influence of Mazepa were fully -recognized by Peter the Great. No other Cossack hetman had ever been -treated with such deference at Moscow. He ranked with the highest -dignitaries in the state; he sat at the tsar's own table. He had been -made one of the first cavaliers of the newly established order of St -Andrew, and Augustus of Poland had bestowed upon him, at Peter's earnest -solicitation, the universally coveted order of the White Eagle. Mazepa -had no temptations to be anything but loyal, and loyal he would -doubtless have remained had not Charles XII. crossed the Russian -frontier. Then it was that Mazepa, who had had doubts of the issue of -the struggle all along, made up his mind that Charles, not Peter, was -going to win, and that it was high time he looked after his own -interests. Besides, he had his personal grievances against the tsar. He -did not like the new ways because they interfered with his old ones. He -was very jealous of the favourite (Menshikov), whom he suspected of a -design to supplant him. But he proceeded very cautiously. Indeed, he -would have preferred to remain neutral, but he was not strong enough to -stand alone. The crisis came when Peter ordered him to co-operate -actively with the Russian forces in the Ukraine. At this very time he -was in communication with Charles's first minister, Count Piper, and had -agreed to harbour the Swedes in the Ukraine and close it against the -Russians (Oct. 1708). The last doubt disappeared when Menshikov was sent -to supervise Mazepa. At the approach of his rival the old hetman -hastened to the Swedish outposts at Horki, in Severia. Mazepa's treason -took Peter completely by surprise. He instantly commanded Menshikov to -get a new hetman elected and raze Baturin, Mazepa's chief stronghold in -the Ukraine, to the ground. When Charles, a week later, passed Baturin -by, all that remained of the Cossack capital was a heap of smouldering -mills and ruined houses. The total destruction of Baturin, almost in -sight of the Swedes, overawed the bulk of the Cossacks into obedience, -and Mazepa's ancient prestige was ruined in a day when the metropolitan -of Kiev solemnly excommunicated him from the high altar, and his effigy, -after being dragged with contumely through the mud at Kiev, was publicly -burnt by the common hangman. Henceforth Mazepa, perforce, attached -himself to Charles. What part he took at the battle of Poltava is not -quite clear. After the catastrophe he accompanied Charles to Turkey with -some 1500 horsemen (the miserable remnant of his 80,000 warriors). The -sultan refused to surrender him to the tsar, though Peter offered -300,000 ducats for his head. He died at Bender on the 22nd of August -1709. - - See N. I. Kostomarov, _Mazepa and the Mazepanites_ (Russ.) (St - Petersburg), 1885; R. Nisbet Bain, _The First Romanovs_ (London, - 1905); S. M. Solovev, _History of Russia_ (Russ.), vol. xv. (St - Petersburg, 1895). (R. N. B.) - - - - -MAZER, the name of a special type of drinking vessel, properly made of -maple-wood, and so-called from the spotted or "birds-eye" marking on the -wood (Ger. _Maser_, spot, marking, especially on wood; cf. "measles"). -These drinking vessels are shallow bowls without handles, with a broad -flat foot and a knob or boss in the centre of the inside, known -technically as the "print." They were made from the 13th to the 16th -centuries, and were the most prized of the various wooden cups in use, -and so were ornamented with a rim of precious metal, generally of silver -or silver gilt; the foot and the "print" being also of metal. The depth -of the mazers seems to have decreased in course of time, those of the -16th century that survive being much shallower than the earlier -examples. There are examples with wooden covers with a metal handle, -such as the Flemish and German mazers in the Franks Bequest in the -British Museum. On the metal rim is usually an inscription, religious or -bacchanalian, and the "print" was also often decorated. The later mazers -sometimes had metal straps between the rim and the foot. - - A very fine mazer with silver gilt ornamentation 3 in. deep and 9(1/2) - in. in diameter was sold in the Braikenridge collection in 1908 for - L2300. It bears the London hall-mark of 1534. This example is - illustrated in the article PLATE: see also DRINKING VESSELS. - - - - -MAZURKA (Polish for a woman of the province of Mazovia), a lively dance, -originating in Poland, somewhat resembling the polka.It is danced in -couples, the music being in 3/8 or 3/4 time. - - - - -MAZZARA DEL VALLO, a town of Sicily, in the province of Trapani, on the -south-west coast of the island, 32 m. by rail S. of Trapani. Pop. -(1901), 20,130. It is the seat of a bishop; the cathedral, founded in -1093, was rebuilt in the 17th century. The castle, at the south-eastern -angle of the town walls, was erected in 1073. The mouth of the river, -which bears the same name, serves as a port for small ships only. -Mazzara was in origin a colony of Selinus: it was destroyed in 409, but -it is mentioned again as a Carthaginian fortress in the First Punic War -and as a post station on the Roman coast road, though whether it had -municipal rights is doubtful.[1] A few inscriptions of the imperial -period exist, but no other remains of importance. On the west bank of -the river are grottoes cut in the rock, of uncertain date: and there are -quarries in the neighbourhood resembling those of Syracuse, but on a -smaller scale. - - See A. Castiglione, _Sulle cose antiche della citta di Mazzara_ - (Alcamo, 1878). - - -FOOTNOTE: - - [1] Th. Mommsen in _Corpus inscr. lat._ (Berlin, 1883), x. 739. - - - - -MAZZINI, GIUSEPPE (1805-1872), Italian patriot, was born on the 22nd of -June 1805 at Genoa, where his father, Giacomo Mazzini, was a physician -in good practice, and a professor in the university. His mother is -described as having been a woman of great personal beauty, as well as of -active intellect and strong affections. During infancy and childhood his -health was extremely delicate, and it appears that he was nearly six -years of age before he was quite able to walk; but he had already begun -to devour books of all kinds and to show other signs of great -intellectual precocity. He studied Latin with his first tutor, an old -priest, but no one directed his extensive course of reading. He became a -student at the university of Genoa at an unusually early age, and -intended to follow his father's profession, but being unable to conquer -his horror of practical anatomy, he decided to graduate in law (1826). -His exceptional abilities, together with his remarkable generosity, -kindness and loftiness of character, endeared him to his fellow -students. As to his inner life during this period, we have only one -brief but significant sentence; "for a short time," he says, "my mind -was somewhat tainted by the doctrines of the foreign materialistic -school; but the study of history and the intuitions of conscience--the -only tests of truth--soon led me back to the spiritualism of our Italian -fathers." - -The natural bent of his genius was towards literature, and, in the -course of the four years of his nominal connexion with the legal -profession, he wrote a considerable number of essays and reviews, some -of which have been wholly or partially reproduced in the critical and -literary volumes of his _Life and Writings_. His first essay, -characteristically enough on "Dante's Love of Country," was sent to the -editor of the _Antologia fiorentina_ in 1826, but did not appear until -some years afterwards in the _Subalpino_. He was an ardent supporter of -romanticism as against what he called "literary servitude under the name -of classicism"; and in this interest all his critiques (as, for example, -that of Giannoni's "Exile" in the _Indicatore Livornese_, 1829) were -penned. But in the meantime the "republican instincts" which he tells us -he had inherited from his mother had been developing, and his sense of -the evils under which Italy was groaning had been intensified; and at -the same time he became possessed with the idea that Italians, and he -himself in particular, "_could_ and therefore _ought_ to struggle for -liberty of country." Therefore, he at once put aside his dearest -ambition, that of producing a complete history of religion, developing -his scheme of a new theology uniting the spiritual with the practical -life, and devoted himself to political thought. His literary articles -accordingly became more and more suggestive of advanced liberalism in -politics, and led to the suppression by government of the _Indicatore -Genovese_ and the _Indicatore Livornese_ successively. Having joined the -Carbonari, he soon rose to one of the higher grades in their hierarchy, -and was entrusted with a special secret mission into Tuscany; but, as -his acquaintance grew, his dissatisfaction with the organization of the -society increased, and he was already meditating the formation of a new -association stripped of foolish mysterious and theatrical formulae, -which instead of merely combating existing authorities should have a -definite and purely patriotic aim, when shortly after the French -revolution of 1830 he was betrayed, while initiating a new member, to -the Piedmontese authorities. He was imprisoned in the fortress of Savona -on the western Riviera for about six months, when, a conviction having -been found impracticable through deficiency of evidence, he was -released, but upon conditions involving so many restrictions of his -liberty that he preferred the alternative of leaving the country. He -withdrew accordingly into France, living chiefly in Marseilles. - -While in his lonely cell at Savona, in presence of "those symbols of the -infinite, the sky and the sea," with a greenfinch for his sole -companion, and having access to no books but "a Tacitus, a Byron, and a -Bible," he had finally become aware of the great mission or "apostolate" -(as he himself called it) of his life; and soon after his release his -prison meditations took shape in the programme of the organization which -was destined soon to become so famous throughout Europe, that of _La -Giovine Italia_, or Young Italy. Its publicly avowed aims were to be the -liberation of Italy both from foreign and domestic tyranny, and its -unification under a republican form of government; the means to be used -were education, and, where advisable, insurrection by guerrilla bands; -the motto was to be "God and the people," and the banner was to bear on -one side the words "Unity" and "Independence" and on the other -"Liberty," "Equality," and "Humanity," to describe respectively the -national and the international aims. In April 1831 Charles Albert, "the -ex-Carbonaro conspirator of 1821," succeeded Charles Felix on the -Sardinian throne, and towards the close of that year Mazzini, making -himself, as he afterwards confessed, "the interpreter of a hope which he -did not share," wrote the new king a letter, published at Marseilles, -urging him to take the lead in the impending struggle for Italian -independence. Clandestinely reprinted, and rapidly circulated all over -Italy, its bold and outspoken words produced a great sensation, but so -deep was the offence it gave to the Sardinian government that orders -were issued for the immediate arrest and imprisonment of the author -should he attempt to cross the frontier. Towards the end of the same -year appeared the important Young Italy "Manifesto," the substance of -which is given in the first volume of the _Life and Writings_ of -Mazzini; and this was followed soon afterwards by the society's -_Journal_, which, smuggled across the Italian frontier, had great -success in the objects for which it was written, numerous -"congregations" being formed at Genoa, Leghorn, and elsewhere. -Representations were consequently made by the Sardinian to the French -government, which issued in an order for Mazzini's withdrawal from -Marseilles (Aug. 1832); he lingered for a few months in concealment, but -ultimately found it necessary to retire into Switzerland. - -From this point it is somewhat difficult to follow the career of the -mysterious and terrible conspirator who for twenty years out of the next -thirty led a life of voluntary imprisonment (as he himself tells us) -"within the four walls of a room," and "kept no record of dates, made no -biographical notes, and preserved no copies of letters." In 1833, -however, he is known to have been concerned in an abortive revolutionary -movement which took place in the Sardinian army; several executions took -place, and he himself was laid under sentence of death. Before the close -of the same year a similar movement in Genoa had been planned, but -failed through the youth and inexperience of the leaders. At Geneva, -also in 1833, Mazzini set on foot _L'Europe Centrale_, a journal of -which one of the main objects was the emancipation of Savoy; but he did -not confine himself to a merely literary agitation for this end. Chiefly -through his agency a considerable body of German, Polish and Italian -exiles was organized, and an armed invasion of the duchy planned. The -frontier was actually crossed on the 1st of February 1834, but the -attack ignominiously broke down without a shot having been fired. -Mazzini, who personally accompanied the expedition, is no doubt correct -in attributing the failure to dissensions with the Carbonari leaders in -Paris, and to want of a cordial understanding between himself and the -Savoyard Ramorino, who had been chosen as military leader. - -In April 1834 the "Young Europe" association "of men believing in a -future of liberty, equality and fraternity for all mankind, and desirous -of consecrating their thoughts and actions to the realization of that -future" was formed also under the influence of Mazzini's enthusiasm; it -was followed soon afterwards by a "Young Switzerland" society, having -for its leading idea the formation of an Alpine confederation, to -include Switzerland, Tyrol, Savoy and the rest of the Alpine chain as -well. But _La Jeune Suisse_ newspaper was compelled to stop within a -year, and in other respects the affairs of the struggling patriot became -embarrassed. He was permitted to remain at Grenchen in Solothurn for a -while, but at last the Swiss diet, yielding to strong and persistent -pressure from abroad, exiled him about the end of 1836. In January 1837 -he arrived in London, where for many months he had to carry on a hard -fight with poverty and the sense of spiritual loneliness, so touchingly -described by himself in the first volume of the _Life and Writings_. -Ultimately, as he gained command of the English language, he began to -earn a livelihood by writing review articles, some of which have since -been reprinted, and are of a high order of literary merit; they include -papers on "Italian Literature since 1830" and "Paolo Sarpi" in the -_Westminster Review_, articles on "Lamennais," "George Sand," "Byron and -Goethe" in the _Monthly Chronicle_, and on "Lamartine," "Carlyle," and -"The Minor Works of Dante" in the _British and Foreign Review_. In 1839 -he entered into relations with the revolutionary committees sitting in -Malta and Paris, and in 1840 he originated a working men's association, -and the weekly journal entitled _Apostolato Popolare_, in which the -admirable popular treatise "On the Duties of Man" was commenced. Among -the patriotic and philanthropic labours undertaken by Mazzini during -this period of retirement in London may be mentioned a free evening -school conducted by himself and a few others for some years, at which -several hundreds of Italian children received at least the rudiments of -secular and religious education. He also exposed and combated the -infamous traffic carried on in southern Italy, where scoundrels bought -small boys from poverty-stricken parents and carried them off to England -and elsewhere to grind organs and suffer martyrdom at the hands of cruel -taskmasters. - -The most memorable episode in his life during the same period was -perhaps that which arose out of the conduct of Sir James Graham, the -home secretary, in systematically, for some months, opening Mazzini's -letters as they passed through the British post office, and -communicating their contents to the Neapolitan government--a proceeding -which was believed at the time to have led to the arrest and execution -of the brothers Bandiera, Austrian subjects, who had been planning an -expedition against Naples, although the recent publication of Sir James -Graham's life seems to exonerate him from the charge. The prolonged -discussions in parliament, and the report of the committee appointed to -inquire into the matter, did not, however, lead to any practical result, -unless indeed the incidental vindication of Mazzini's character, which -had been recklessly assailed in the course of debate. In this connexion -Thomas Carlyle wrote to _The Times_: "I have had the honour to know Mr -Mazzini for a series of years, and, whatever I may think of his -practical insight and skill in worldly affairs, I can with great freedom -testify that he, if I have ever seen one such, is a man of genius and -virtue, one of those rare men, numerable unfortunately but as units in -this world, who are worthy to be called martyr souls; who in silence, -piously in their daily life, practise what is meant by that." - -Mazzini did not share the enthusiastic hopes everywhere raised in the -ranks of the Liberal party throughout Europe by the first acts of Pius -IX., in 1846, but at the same time he availed himself, towards the end -of 1847, of the opportunity to publish a letter addressed to the new -pope, indicating the nature of the religious and national mission which -the Liberals expected him to undertake. The leaders of the revolutionary -outbreaks in Milan and Messina in the beginning of 1848 had long been in -secret correspondence with Mazzini; and their action, along with the -revolution in Paris, brought him early in the same year to Italy, where -he took a great and active interest in the events which dragged Charles -Albert into an unprofitable war with Austria; he actually for a short -time bore arms under Garibaldi immediately before the reoccupation of -Milan, but ultimately, after vain attempts to maintain the insurrection -in the mountain districts, found it necessary to retire to Lugano. In -the beginning of the following year he was nominated a member of the -short-lived provisional government of Tuscany formed after the flight of -the grand-duke, and almost simultaneously, when Rome had, in consequence -of the withdrawal of Pius IX., been proclaimed a republic, he was -declared a member of the constituent assembly there. A month afterwards, -the battle of Novara having again decided against Charles Albert in the -brief struggle with Austria, into which he had once more been drawn, -Mazzini was appointed a member of the Roman triumvirate, with supreme -executive power (March 23, 1849). The opportunity he now had for showing -the administrative and political ability which he was believed to -possess was more apparent than real, for the approach of the professedly -friendly French troops soon led to hostilities, and resulted in a siege -which terminated, towards the end of June, with the assembly's -resolution to discontinue the defence, and Mazzini's indignant -resignation. That he succeeded, however, for so long a time, and in -circumstances so adverse, in maintaining a high degree of order within -the turbulent city is a fact that speaks for itself. His diplomacy, -backed as it was by no adequate physical force, naturally showed at the -time to very great disadvantage, but his official correspondence and -proclamations can still be read with admiration and intellectual -pleasure, as well as his eloquent vindication of the revolution in his -published "Letter to MM. de Tocqueville and de Falloux." The surrender -of the city on the 30th of June was followed by Mazzini's not too -precipitate flight by way of Marseilles into Switzerland, whence he once -more found his way to London. Here in 1850 he became president of the -National Italian Committee, and at the same time entered into close -relations with Ledru-Rollin and Kossuth. He had a firm belief in the -value of revolutionary attempts, however hopeless they might seem; he -had a hand in the abortive rising at Mantua in 1852, and again, in -February 1853, a considerable share in the ill-planned insurrection at -Milan on the 6th of February 1853, the failure of which greatly weakened -his influence; once more, in 1854, he had gone far with preparations for -renewed action when his plans were completely disconcerted by the -withdrawal of professed supporters, and by the action of the French and -English governments in sending ships of war to Naples. - -The year 1857 found him yet once more in Italy, where, for complicity in -short-lived emeutes which took place at Genoa, Leghorn and Naples, he -was again laid under sentence of death. Undiscouraged in the pursuit of -the one great aim of his life by any such incidents as these, he -returned to London, where he edited his new journal _Pensiero ed -Azione_, in which the constant burden of his message to the overcautious -practical politicians of Italy was: "I am but a voice crying _Action_; -but the state of Italy cries for it also. So do the best men and people -of her cities. Do you wish to destroy my influence? _Act_." The same -tone was at a somewhat later date assumed in the letter he wrote to -Victor Emmanuel, urging him to put himself at the head of the movement -for Italian unity, and promising republican support. As regards the -events of 1859-1860, however, it may be questioned whether, through his -characteristic inability to distinguish between the ideally perfect and -the practically possible, he did not actually hinder more than he helped -the course of events by which the realization of so much of the great -dream of his life was at last brought about. If Mazzini was the prophet -of Italian unity, and Garibaldi its knight errant, to Cavour alone -belongs the honour of having been the statesman by whom it was finally -accomplished. After the irresistible pressure of the popular movement -had led to the establishment not of an Italian republic but of an -Italian kingdom, Mazzini could honestly enough write, "I too have -striven to realize unity under a monarchical flag," but candour -compelled him to add, "The Italian people are led astray by a delusion -at the present day, a delusion which has induced them to substitute -material for moral unity and their own reorganization. Not so I. I bow -my head sorrowfully to the sovereignty of the national will; but -monarchy will never number me amongst its servants or followers." In -1865, by way of protest against the still uncancelled sentence of death -under which he lay, Mazzini was elected by Messina as delegate to the -Italian parliament, but, feeling himself unable to take the oath of -allegiance to the monarchy, he never took his seat. In the following -year, when a general amnesty was granted after the cession of Venice to -Italy, the sentence of death was at last removed, but he declined to -accept such an "offer of oblivion and pardon for having loved Italy -above all earthly things." In May 1869 he was again expelled from -Switzerland at the instance of the Italian government for having -conspired with Garibaldi; after a few months spent in England he set out -(1870) for Sicily, but was promptly arrested at sea and carried to -Gaeta, where he was imprisoned for two months. Events soon made it -evident that there was little danger to fear from the contemplated -rising, and the occasion of the birth of a prince was seized for -restoring him to liberty. The remainder of his life, spent partly in -London and partly at Lugano, presents no noteworthy incidents. For some -time his health had been far from satisfactory, but the immediate cause -of his death was an attack of pleurisy with which he was seized at Pisa, -and which terminated fatally on the 10th of March 1872. The Italian -parliament by a unanimous vote expressed the national sorrow with which -the tidings of his death had been received, the president pronouncing an -eloquent eulogy on the departed patriot as a model of disinterestedness -and self-denial, and one who had dedicated his whole life ungrudgingly -to the cause of his country's freedom. A public funeral took place at -Pisa on the 14th of March, and the remains were afterwards conveyed to -Genoa. (J. S. Bl.) - - The published writings of Mazzini, mostly occasional, are very - voluminous. An edition was begun by himself and continued by A. Saffi, - _Scritti editi e inediti di Giuseppe Mazzini_, in 18 vols. (Milan and - Rome, 1861-1891); many of the most important are found in the - partially autobiographical _Life and Writings of Joseph Mazzini_ - (1864-1870) and the two most systematic--_Thoughts upon Democracy in - Europe_, a remarkable series of criticisms on Benthamism, St - Simonianism, Fourierism, and other economic and socialistic schools of - the day, and the treatise _On the Duties of Man_, an admirable primer - of ethics, dedicated to the Italian working class--will be found in - _Joseph Mazzini: a Memoir_, by Mrs E. A. Venturi (London, 1875). - Mazzini's "first great sacrifice," he tells us, was "the renunciation - of the career of literature for the more direct path of political - action," and as late as 1861 we find him still recurring to the - long-cherished hope of being able to leave the stormy arena of - politics and consecrate the last years of his life to the dream of his - youth. He had specially contemplated three considerable literary - undertakings--a volume of _Thoughts on Religion_, a popular _History - of Italy_, to enable the working classes to apprehend what he - conceived to be the "mission" of Italy in God's providential ordering - of the world, and a comprehensive collection of translations of - ancient and modern classics into Italian. None of these was actually - achieved. No one, however, can read even the briefest and most - occasional writing of Mazzini without gaining some impression of the - simple grandeur of the man, the lofty elevation of his moral tone, his - unwavering faith in the living God, who is ever revealing Himself in - the progressive development of humanity. His last public utterance is - to be found in a highly characteristic article on Renan's _Reforme - Morale et Intellectuelle_, finished on the 3rd of March 1872, and - published in the _Fortnightly Review_ for February 1874. Of the 40,000 - letters of Mazzini only a small part have been published. In 1887 two - hundred unpublished letters were printed at Turin (_Duecento lettere - inedite di Giuseppe Mazzini_), in 1895 the _Lettres intimes_ were - published in Paris, and in 1905 Francesco Rosso published _Lettre - inedite di Giuseppe Mazzini_ (Turin, 1905). A popular edition of - Mazzini's writings has been undertaken by order of the Italian - government. - - For Mazzini's biography see Jessie White Mario, _Della vita di - Giuseppe Mazzini_ (Milan, 1886), a useful if somewhat too enthusiastic - work; Bolton King, _Mazzini_ (London, 1903); Count von Schack, _Joseph - Mazzini und die italienische Einheit_ (Stuttgart, 1891). A. Luzio's - _Giuseppe Mazzini_ (Milan, 1905) contains a great deal of valuable - information, bibliographical and other, and Dora Melegari in _La - giovine Italia e Giuseppe Mazzini_ (Milan, 1906) publishes the - correspondence between Mazzini and Luigi A. Melegari during the early - days of "Young Italy." For the literary side of Mazzini's life see - Peretti, _Gli scritti letterarii di Giuseppe Mazzini_ (Turin, 1904). - (L. V.*) - - - - -MAZZONI, GIACOMO (1548-1598), Italian philosopher, was born at Cesena -and died at Ferrara. A member of a noble family and highly educated, he -was one of the most eminent savants of the period. He occupied chairs in -the universities of Pisa and Rome, was one of the founders of the Della -Crusca Academy, and had the distinction, it is said, of thrice -vanquishing the Admirable Crichton in dialectic. His chief work in -philosophy was an attempt to reconcile Plato and Aristotle, and in this -spirit he published in 1597 a treatise _In universam Platonis et -Aristotelis philosophiam praecludia_. He wrote also _De triplici hominum -vita_, wherein he outlined a theory of the infinite perfection and -development of nature. Apart from philosophy, he was prominent in -literature as the champion of Dante, and produced two works in the -poet's defence: _Discorso composto in difesa della comedia di Dante_ -(1572), and _Della difesa della comedia di Dante_ (1587, reprinted -1688). He was an authority on ancient languages and philology, and gave -a great impetus to the scientific study of the Italian language. - - - - -MAZZONI, GUIDO (1859- ), Italian poet, was born at Florence, and -educated at Pisa and Bologna. In 1887 he became professor of Italian at -Padua, and in 1894 at Florence. He was much influenced by Carducci, and -became prominent both as a prolific and well-read critic and as a poet -of individual distinction. His chief volumes of verse are _Versi_ -(1880), _Nuove poesie_ (1886), _Poesie_ (1891), _Voci della vita_ -(1893). - - - - -MEAD, LARKIN GOLDSMITH (1835- ), American sculptor, was born at -Chesterfield, New Hampshire, on the 3rd of January 1835. He was a pupil -(1853-1855) of Henry Kirke Brown. During the early part of the Civil -War he was at the front for six months, with the army of the Potomac, as -an artist for _Harper's Weekly_; and in 1862-1865 he was in Italy, being -for part of the time attached to the United States consulate at Venice, -while William D. Howells, his brother-in-law, was consul. He returned to -America in 1865, but subsequently went back to Italy and lived at -Florence. His first important work was a statue of Ethan Allen, now at -the State House, Montpelier, Vermont. His principal works are: the -monument to President Lincoln, Springfield, Illinois; "Ethan Allen" -(1876), National Hall of Statuary, Capitol, Washington; an heroic marble -statue, "The Father of Waters," New Orleans; and "Triumph of Ceres," -made for the Columbian Exposition, Chicago. - -His brother, WILLIAM RUTHERFORD MEAD (1846- ), graduated at Amherst -College in 1867, and studied architecture in New York under Russell -Sturgis, and also abroad. In 1879 he and J. F. McKim, with whom he had -been in partnership for two years as architects, were joined by Stanford -White, and formed the well-known firm of McKim, Mead & White. - - - - -MEAD, RICHARD (1673-1754), English physician, eleventh child of Matthew -Mead (1630-1699), Independent divine, was born on the 11th of August -1673 at Stepney, London. He studied at Utrecht for three years under J. -G. Graevius; having decided to follow the medical profession, he then -went to Leiden and attended the lectures of Paul Hermann and Archibald -Pitcairne. In 1695 he graduated in philosophy and physic at Padua, and -in 1696 he returned to London, entering at once on a successful -practice. His _Mechanical Account of Poisons_ appeared in 1702, and in -1703 he was admitted to the Royal Society, to whose _Transactions_ he -contributed in that year a paper on the parasitic nature of scabies. In -the same year he was elected physician to St Thomas's Hospital, and -appointed to read anatomical lectures at the Surgeons' Hall. On the -death of John Radcliffe in 1714 Mead became the recognized head of his -profession; he attended Queen Anne on her deathbed, and in 1727 was -appointed physician to George II., having previously served him in that -capacity when he was prince of Wales. He died in London on the 16th of -February 1754. - - Besides the _Mechanical Account of Poisons_ (2nd ed., 1708), Mead - published a treatise _De imperio solis et lunae in corpora humana et - morbis inde oriundis_ (1704), _A Short Discourse concerning - Pestilential Contagion, and the Method to be used to prevent it_ - (1720), _De variolis et morbillis dissertatio_ (1747), _Medica sacra, - sive de morbis insignioribus qui in bibliis memorantur commentarius_ - (1748), _On the Scurvy_ (1749), and _Monita et praecepta medica_ - (1751). A _Life_ of Mead by Dr Matthew Maty appeared in 1755. - - - - -MEAD. (1) A word now only used more or less poetically for the commoner -form "meadow," properly land laid down for grass and cut for hay, but -often extended in meaning to include pasture-land. "Meadow" represents -the oblique case, _maedwe_, of O. Eng. _maed_, which comes from the root -seen in "mow"; the word, therefore, means "mowed land." Cognate words -appear in other Teutonic languages, a familiar instance being Ger. -_matt_, seen in place-names such as Zermatt, Andermatt, &c. (See Grass.) -(2) The name of a drink made by the fermentation of honey mixed with -water. Alcoholic drinks made from honey were common in ancient times, -and during the middle ages throughout Europe. The Greeks and Romans knew -of such under the names of [Greek: hodromeli] and _hydromel_; _mulsum_ -was a form of mead with the addition of wine. The word is common to -Teutonic languages (cf. Du. _mede_, Ger. _Met_ or _Meth_), and is -cognate with Gr. [Greek: methu], wine, and Sansk. _madhu_, sweet drink. -"Metheglin," another word for mead, properly a medicated or spiced form -of the drink, is an adaptation of the Welsh _meddyglyn_, which is -derived from _meddyg_, healing (Lat. _medicus_) and _llyn_, liquor. It -therefore means "spiced or medicated drink," and is not etymologically -connected with "mead." - - - - -MEADE, GEORGE GORDON (1815-1872), American soldier, was born of American -parentage at Cadiz, Spain, on the 31st of December 1815. On graduation -at the United States Military Academy in 1835, he served in Florida with -the 3rd Artillery against the Seminoles. Resigning from the army in -1836, he became a civil engineer and constructor of railways, and was -engaged under the war department in survey work. In 1842 he was -appointed a second lieutenant in the corps of the topographical -engineers. In the war with Mexico he was on the staffs successively of -Generals Taylor, J. Worth and Robert Patterson, and was brevetted for -gallant conduct at Monterey. Until the Civil War he was engaged in -various engineering works, mainly in connexion with lighthouses, and -later as a captain of topographical engineers in the survey of the -northern lakes. In 1861 he was appointed brigadier-general of -volunteers, and had command of the 2nd brigade of the Pennsylvania -Reserves in the Army of the Potomac under General M'Call. He served in -the Seven Days, receiving a severe wound at the action of Frazier's -Farm. He was absent from his command until the second battle of Bull -Run, after which he obtained the command of his division. He -distinguished himself greatly at the battles of South Mountain and -Antietam. At Fredericksburg he and his division won great distinction by -their attack on the position held by Jackson's corps, and Meade was -promoted major-general of volunteers, to date from the 29th of November. -Soon afterwards he was placed in command of the V. corps. At -Chancellorsville he displayed great intrepidity and energy, and on the -eve of the battle of Gettysburg was appointed to succeed Hooker. The -choice was unexpected, but Meade justified it by his conduct of the -operations, and in the famous three days' battle he inflicted a complete -defeat on General Lee's army. His reward was the commission of -brigadier-general in the regular army. In the autumn of 1863 a war of -manoeuvre was fought between the two commanders, on the whole favourably -to the Union arms. Grant, commanding all the armies of the United -States, joined the Army of the Potomac in the spring of 1864, and -remained with it until the end of the war; but he continued Meade in his -command, and successfully urged his appointment as major-general in the -regular army (Aug. 18, 1864), eulogizing him as the commander who had -successfully met and defeated the best general and the strongest army on -the Confederate side. After the war Meade commanded successively the -military division of the Atlantic, the department of the east, the third -military district (Georgia and Alabama) and the department of the south. -He died at Philadelphia on the 6th of November, 1872. The degree of -LL.D. was conferred upon him by Harvard University, and his scientific -attainments were recognized by the American Philosophical Society and -the Philadelphia Academy of Natural Sciences. There are statues of -General Meade in Philadelphia and at Gettysburg. - - See I. R. Pennypacker, _General Meade_ ("Great Commanders" series, New - York, 1901). - - - - -MEADE, WILLIAM (1789-1862), American Protestant Episcopal bishop, the -son of Richard Kidder Meade (1746-1805), one of General Washington's -aides during the War of Independence, was born on the 11th of November -1789, near Millwood, in that part of Frederick county which is now -Clarke county, Virginia. He graduated as valedictorian in 1808 at the -college of New Jersey (Princeton); studied theology under the Rev. -Walter Addison of Maryland, and in Princeton; was ordained deacon in -1811 and priest in 1814; and preached both in the Stone Chapel, -Millwood, and in Christ Church, Alexandria, for some time. He became -assistant bishop of Virginia in 1829; was pastor of Christ Church, -Norfolk, in 1834-1836; in 1841 became bishop of Virginia; and in -1842-1862 was president of the Protestant Episcopal Theological Seminary -in Virginia, near Alexandria, delivering an annual course of lectures on -pastoral theology. In 1819 he had acted as the agent of the American -Colonization Society to purchase slaves, illegally brought into Georgia, -which had become the property of that state and were sold publicly at -Milledgeville. He had been prominent in the work of the Education -Society, which was organized in 1818 to advance funds to needy students -for the ministry of the American Episcopal Church, and in the -establishment of the Theological Seminary near Alexandria, as he was -afterwards in the work of the American Tract Society, and the Bible -Society. He was a founder and president of the Evangelical Knowledge -Society (1847), which, opposing what it considered the heterodoxy of -many of the books published by the Sunday School Union, attempted to -displace them by issuing works of a more evangelical type. A low -Churchman, he strongly opposed Tractarianism. He was active in the case -against Bishop Henry Ustick Onderdonk (1789-1858) of Pennsylvania, who -because of intemperance was forced to resign and was suspended from the -ministry in 1844; in that against Bishop Benjamin Tredwell Onderdonk -(1791-1861) of New York, who in 1845 was suspended from the ministry on -the charge of intoxication and improper conduct; and in that against -Bishop G. W. Doane of New Jersey. He fought against the threatening -secession of Virginia, but acquiesced in the decision of the state and -became presiding bishop of the Southern Church. He died in Richmond, -Virginia, on the 14th of March 1862. - - Among his publications, besides many sermons, were _A Brief Review of - the Episcopal Church in Virginia_ (1845); _Wilberforce, Cranmer, - Jewett and the Prayer Book on the Incarnation_ (1850); _Reasons for - Loving the Episcopal Church_ (1852); and _Old Churches, Ministers and - Families of Virginia_ (1857); a storehouse of material on the - ecclesiastical history of the state. - - See the _Life_ by John Johns (Baltimore, 1867). - - - - -MEADVILLE, a city and the county-seat of Crawford county, Pennsylvania, -U.S.A., on French Creek, 36 m. S. of Erie. Pop. (1900), 10,291, of whom -912 were foreign-born and 173 were negroes; (1910 census) 12,780. It is -served by the Erie, and the Bessemer & Lake Erie railways. Meadville has -three public parks, two general hospitals and a public library, and is -the seat of the Pennsylvania College of Music, of a commercial college, -of the Meadville Theological School (1844, Unitarian), and of Allegheny -College (co-educational), which was opened in 1815, came under the -general patronage of the Methodist Episcopal Church in 1833, and in 1909 -had 322 students (200 men and 122 women). Meadville is the commercial -centre of a good agricultural region, which also abounds in oil and -natural gas. The Erie Railroad has extensive shops here, which in 1905 -employed 46.7% of the total number of wage-earners, and there are -various manufactures. The factory product in 1905 was valued at -$2,074,600, being 24.4% more than that of 1900. Meadville, the oldest -settlement in N.W. Pennsylvania, was founded as a fortified post by -David Mead in 1793, laid out as a town in 1795, incorporated as a -borough in 1823 and chartered as a city in 1866. - - - - -MEAGHER, THOMAS FRANCIS (1823-1867), Irish nationalist and American -soldier, was born in Waterford, Ireland, on the 3rd of August 1823. He -graduated at Stonyhurst College, Lancashire, in 1843, and in 1844 began -the study of law at Dublin. He became a member of the Young Ireland -Party in 1845, and in 1847 was one of the founders of the Irish -Confederation. In March 1848 he made a speech before the Confederation -which led to his arrest for sedition, but at his trial the jury failed -to agree and he was discharged. In the following July the Confederation -created a "war directory" of five, of which Meagher was a member, and he -and William Smith O'Brien travelled through Ireland for the purpose of -starting a revolution. The attempt proved abortive; Meagher was arrested -in August, and in October was tried for high treason before a special -commission at Clonmel. He was found guilty and was condemned to death, -but his sentence was commuted to life imprisonment in Van Diemen's Land, -whither he was transported in the summer of 1849. Early in 1852 he -escaped, and in May reached New York City. He made a tour of the cities -of the United States as a popular lecturer, and then studied law and was -admitted to the New York bar in 1855. He made two unsuccessful ventures -in journalism, and in 1857 went to Central America, where he acquired -material for another series of lectures. In 1861 he was captain of a -company (which he had raised) in the 69th regiment of New York -volunteers and fought at the first battle of Bull Run; he then organized -an Irish brigade, of whose first regiment he was colonel until the 3rd -of February 1862, when he was appointed to the command of this -organization with the rank of brigadier-general. He took part in the -siege of Yorktown, the battle of Fair Oaks, the seven days' battle -before Richmond, and the battles of Antietam, Fredericksburg, where he -was wounded, and Chancellorsville, where his brigade was reduced in -numbers to less than a regiment, and General Meagher resigned his -commission. On the 23rd of December 1863 his resignation was cancelled, -and he was assigned to the command of the military district of Etowah, -with headquarters at Chattanooga. At the close of the war he was -appointed by President Johnson secretary of Montana Territory, and -there, in the absence of the territorial governor, he acted as governor -from September 1866 until his death from accidental drowning in the -Missouri River near Fort Benton, Montana, on the 1st of July 1867. He -published _Speeches on the Legislative Independence of Ireland_ (1852). - - W. F. Lyons, in _Brigadier-General Thomas Francis Meagher_ (New York, - 1870), gives a eulogistic account of his career. - - - - -MEAL. (1) (A word common to Teutonic languages, cf. Ger. _Mehl_, Du. -meel; the ultimate source is the root seen in various Teutonic words -meaning "to grind," and in Eng. "mill," Lat. _mola_, _molere_, Gr. -[Greek: myle]), a powder made from the edible part of any grain or -pulse, with the exception of wheat, which is known as "flour." In -America the word is specifically applied to the meal produced from -Indian corn or maize, as in Scotland and Ireland to that produced from -oats, while in South Africa the ears of the Indian corn itself are -called "mealies." (2) Properly, eating and drinking at regular stated -times of the day, as breakfast, dinner, &c., hence taking of food at any -time and also the food provided. The word was in O.E. _mael_, which also -had the meanings (now lost) of time, mark, measure, &c., which still -appear in many forms of the word in Teutonic languages; thus Ger. _mal_, -time, mark, cf. _Denkmal_, monument, _Mahl_, meal, repast, or Du. -_maal_, Swed. _mal_, also with both meanings. The ultimate source is the -pre-Teutonic root _me-_ _ma-_, to measure, and the word thus stood for a -marked-out point of time. - - - - -MEALIE, the South African name for Indian corn or maize. The word as -spelled represents the pronunciation of the Cape Dutch _milje_, an -adaptation of _milho_ (_da India_), the millet of India, the Portuguese -name for millet, used in South Africa for maize. - - - - -MEAN, an homonymous word, the chief uses of which may be divided thus. -(1) A verb with two principal applications, to intend, purpose or -design, and to signify. This word is in O.E. _maenan_, and cognate forms -appear in other Teutonic languages, cf. Du. _meenen_, Ger. _meinen_. The -ultimate origin is usually taken to be the root _men-_, to think, the -root of "mind." (2) An adjective and substantive meaning "that which is -in the middle." This is derived through the O. Fr. _men_, _meien_ or -_moien_, modern _moyen_, from the late Lat. adjective _medianus_, from -_medius_, middle. The law French form _mesne_ is still preserved in -certain legal phrases (see MESNE). The adjective "mean" is chiefly used -in the sense of "average," as in mean temperature, mean birth or death -rate, &c. - -"Mean" as a substantive has the following principal applications; it is -used of that quality, course of action, condition, state, &c., which is -equally distant from two extremes, as in such phrases as the "golden (or -happy) mean." For the philosophic application see ARISTOTLE and ETHICS. - -In mathematics, the term "mean," in its most general sense, is given to -some function of two or more quantities which (1) becomes equal to each -of the quantities when they themselves are made equal, and (2) is -unaffected in value when the quantities suffer any transpositions. The -three commonest means are the arithmetical, geometrical, and harmonic; -of less importance are the contraharmonical, arithmetico-geometrical, -and quadratic. - -From the sense of that which stands between two things, "mean," or the -plural "means," often with a singular construction, takes the further -significance of agency, instrument, &c., of which that produces some -result, hence resources capable of producing a result, particularly the -pecuniary or other resources by which a person is enabled to live, and -so used either of employment or of property, wealth, &c. There are many -adverbial phrases, such as "by all means," "by no means," &c., which are -extensions of "means" in the sense of agency. - -The word "mean" (like the French _moyen_) had also the sense of -middling, moderate, and this considerably influenced the uses of "mean" -(3). This, which is now chiefly used in the sense of inferior, low, -ignoble, or of avaricious, penurious, "stingy," meant originally that -which is common to more persons or things than one. The word in O. E. is -_gemaene_, and is represented in the modern Ger. _gemein_, common. It is -cognate with Lat. _communis_, from which "common" is derived. The -descent in meaning from that which is shared alike by several to that -which is inferior, vulgar or low, is paralleled by the uses of "common." - -In astronomy the "mean sun" is a fictitious sun which moves uniformly in -the celestial equator and has its right ascension always equal to the -sun's mean longitude. The time recorded by the mean sun is termed -mean-solar or clock time; it is regular as distinct from the non-uniform -solar or sun-dial time. The "mean moon" is a fictitious moon which moves -around the earth with a uniform velocity and in the same time as the -real moon. The "mean longitude" of a planet is the longitude of the -"mean" planet, i.e. a fictitious planet performing uniform revolutions -in the same time as the real planet. - - The arithmetical mean of n quantities is the sum of the quantities - divided by their number n. The geometrical mean of n quantities is the - nth root of their product. The harmonic mean of n quantities is the - arithmetical mean of their reciprocals. The significance of the word - "mean," i.e., middle, is seen by considering 3 instead of n - quantities; these will be denoted by a, b, c. The arithmetic mean b, - is seen to be such that the terms a, b, c are in arithmetical - progression, i.e. b = (1/2)(a + c); the geometrical mean b places a, - b, c in geometrical progression, i.e. in the proportion a : b :: b : c - or b^2 = ac; and the harmonic mean places the quantities in harmonic - proportion, i.e. a : c :: a - b : b - c, or b = 2ac/(a + c). The - contraharmonical mean is the quantity b given by the proportion a : c - :: b - c : a - b, i.e. b = (a^2 + c^2)/(a + c). The - arithmetico-geometrical mean of two quantities is obtained by first - forming the geometrical and arithmetical means, then forming the means - of these means, and repeating the process until the numbers become - equal. They were invented by Gauss to facilitate the computation of - elliptic integrals. The quadratic mean of n quantities is the square - root of the arithmetical mean of their squares. - - - - -MEASLES, (_Morbilli_, _Rubeola_; the M. E. word is _maseles_, properly a -diminutive of a word meaning "spot," O.H.G. _masa_, cf. "mazer"; the -equivalent is Ger. _Masern_; Fr. _Rougeole_), an acute infectious -disease occurring mostly in children. It is mentioned in the writings of -Rhazes and others of the Arabian physicians in the 10th century. For -long, however, it was held to be a variety of small-pox. After the -non-identity of these two diseases had been established, measles and -scarlet-fever continued to be confounded with each other; and in the -account given by Thomas Sydenham of epidemics of measles in London in -1670 and 1674 it is evident that even that accurate observer had not as -yet clearly perceived their pathological distinction, although it would -seem to have been made a century earlier by Giovanni Filippo Ingrassias -(1510-1580), a physician of Palermo. The specific micro-organism -responsible for measles has not been definitely isolated. - -Its progress is marked by several stages more or less sharply defined. -After the reception of the contagion into the system, there follows a -period of incubation or latency during which scarcely any disturbance of -the health is perceptible. This period generally lasts for from ten to -fourteen days, when it is followed by the invasion of the symptoms -specially characteristic of measles. These consist in the somewhat -sudden onset of acute catarrh of the mucous membranes. At this stage -minute white spots in the buccal mucous membrane frequently occur; when -they do, they are diagnostic of the disease. Sneezing, accompanied with -a watery discharge, sometimes bleeding, from the nose, redness and -watering of the eyes, cough of a short, frequent, and noisy character, -with little or no expectoration, hoarseness of the voice, and -occasionally sickness and diarrhoea, are the chief local phenomena of -this stage. With these there is well-marked febrile disturbance, the -temperature being elevated (102 deg.-104 deg. F.), and the pulse rapid, -while headache, thirst, and restlessness are usually present. In some -instances, these initial symptoms are slight, and the child is allowed -to associate with others at a time when, as will be afterwards seen, -the contagion of the disease is most active. In rare cases, especially -in young children, convulsions usher in, or occur in the course of, this -stage of invasion, which lasts as a rule for four or five days, the -febrile symptoms, however, showing some tendency to undergo abatement -after the second day. On the fourth or fifth day after the invasion, -sometimes later, rarely earlier, the characteristic eruption appears on -the skin, being first noticed on the brow, cheeks, chin, also behind the -ears, and on the neck. It consists of small spots of a dusky red or -crimson colour, just like flea-bites, slightly elevated above the -surface, at first isolated, but tending to become grouped into patches -of irregular, occasionally crescentic, outline, with portions of skin -free from the eruption intervening. The face acquires a swollen and -bloated appearance, which, taken with the catarrh of the nostrils and -eyes, is almost characteristic, and renders the diagnosis at this stage -a matter of no difficulty. The eruption spreads downwards over the body -and limbs, which are soon thickly studded with the red spots or patches. -Sometimes these become confluent over a considerable surface. The rash -continues to come out for two or three days, and then begins to fade in -the order in which it first showed itself, namely from above downwards. -By the end of about a week after its first appearance scarcely any trace -of the eruption remains beyond a faint staining of the skin. Usually -during convalescence slight peeling of the epidermis takes place, but -much less distinctly than is the case in scarlet fever. At the -commencement of the eruptive stage the fever, catarrh, and other -constitutional disturbance, which were present from the beginning, -become aggravated, the temperature often rising to 105 deg. or more, and -there is headache, thirst, furred tongue, and soreness of the throat, -upon which red patches similar to those on the surface of the body may -be observed. These symptoms usually decline as soon as the rash has -attained its maximum, and often there occurs a sudden and extensive fall -of temperature, indicating that the crisis of the disease has been -reached. In favourable cases convalescence proceeds rapidly, the patient -feeling perfectly well even before the rash has faded from the skin. - -Measles may, however, occur in a very malignant form, in which the -symptoms throughout are of urgent character, the rash but feebly -developed, and of dark purple hue, while there is great prostration, -accompanied with intense catarrh of the respiratory or gastro-intestinal -mucous membrane. Such cases are rare, occurring mostly in circumstances -of bad hygiene, both as regards the individual and his surroundings. On -the other hand, cases of measles are often of so mild a form throughout -that the patient can scarcely be persuaded to submit to treatment. - -Measles as a disease derives its chief importance from the risk, by no -means slight, of certain complications which are apt to arise during its -course, more especially inflammatory affections of the respiratory -organs. These are most liable to occur in the colder seasons of the year -and in very young and delicate children. It has been already stated that -irritation of the respiratory passages is one of the symptoms -characteristic of measles, but that this subsides with the decline of -the eruption. Not unfrequently, however, these symptoms, instead of -abating, become aggravated, and bronchitis of the capillary form (see -BRONCHITIS), or pneumonia, generally of the diffuse or lobular variety -(see PNEUMONIA), supervene. By far the greater proportion of the -mortality in measles is due to its complications, of which those just -mentioned are the most common, but which also include inflammatory -affections of the larynx, with attacks resembling croup, and also -diarrhoea assuming a dysenteric character. Or there may remain as direct -results of the disease chronic ophthalmia, or discharge from the ears -with deafness, and occasionally a form of gangrene affecting the tissues -of the mouth or cheeks and other parts of the body, leading to -disfigurement and gravely endangering life. - -Apart from those immediate risks there appears to be a tendency in many -cases for the disease to leave behind a weakened and vulnerable -condition of the general health, which may render children, previously -robust, delicate and liable to chest complaints, and is in not a few -instances the precursor of some of those tubercular affections to which -the period of childhood and youth is liable. These various effects or -sequelae of measles indicate that although in itself a comparatively -mild ailment, it should not be regarded with indifference. Indeed it is -doubtful whether any other disease of early life demands more careful -watching as to its influence on the health. Happily many of those -attending evils may by proper management be averted. - -Measles is a disease of the earlier years of childhood. Like other -infectious maladies, it is admittedly rare, though not unknown, in -nurslings or infants under six months old. It is comparatively seldom -met with in adults, but this is due to the fact that most persons have -undergone an attack in early life. Where this has not been the case, the -old suffer equally with the young. All races of men appear liable to -this disease, provided that which constitutes the essential factor in -its origin and spread exists, namely, contagion. Some countries enjoy -long immunity from outbreaks of measles, but it has frequently been -found in such cases that when the contagion has once been introduced the -disease extends with great rapidity and virulence. This was shown by the -epidemic in the Faroe Islands in 1846, where, within six months after -the arrival of a single case of measles, more than three-fourths of the -entire population were attacked and many perished; and the similarly -produced and still more destructive outbreak in Fiji in 1875, in which -it was estimated that about one-fourth of the inhabitants died from the -disease in about three months. In both these cases the great mortality -was due to the complications of the malady, specially induced by -overcrowding, insanitary surroundings, the absence of proper nourishment -and nursing for the sick, and the utter prostration and terror of the -people, and to the disease being specially malignant, occurring on what -might be termed virgin soil.[1] It may be regarded as an invariable rule -that the first epidemic of any disease in a community is specially -virulent, each successive attack conferring a certain immunity. - -In many lands, such as the United Kingdom, measles is rarely absent, -especially from large centres of population, where sporadic cases are -found at all seasons. Every now and then epidemics arise from the -extension of the disease among those members of a community who have not -been in some measure protected by a previous attack. There are few -diseases so contagious as measles, and its rapid spread in epidemic -outbreaks is no doubt due to the well-ascertained fact that contagion is -most potent in the earlier stages, even before its real nature has been -evinced by the characteristic appearances on the skin. Hence the -difficulty of timely isolation, and the readiness with which the disease -is spread in schools and families. The contagion is present in the skin -and the various secretions. While the contagion is generally direct, it -can also be conveyed by the particles from the nose and mouth which, -after being expelled, become dry and are conveyed as dust on clothes, -toys, &c. Fortunately the germs of measles do not retain their virulence -long under such conditions, comparing favourably with those of some -other diseases. - -_Treatment._--The treatment embraces the preventive measures to be -adopted by the isolation of the sick at as early a period as possible. -Epidemics have often, especially in limited localities, been curtailed -by such a precaution. In families with little house accommodation this -measure is frequently, for the reason given regarding the communicable -period of the disease, ineffectual; nevertheless where practicable it -ought to be tried. The unaffected children should be kept from school -for a time (probably about three weeks from the outbreak in the family -would suffice if no other case occur in the interval), and all clothing -in contact with the patient or nurses should be disinfected. In -extensive epidemics it is often desirable to close the schools for a -time. As regards special treatment, in an ordinary case of measles -little is required beyond what is necessary in febrile conditions -generally. Confinement to bed in a somewhat darkened room, into which, -however, air is freely admitted; light, nourishing, liquid diet (soups, -milk, &c.), water almost _ad lib._ to drink, and mild diaphoretic -remedies such as the acetate of ammonia or ipecacuanha, are all that is -necessary in the febrile stage. When the fever is very severe, sponging -the body generally or the chest and arms affords relief. The serious -chest complications of measles are to be dealt with by those measures -applicable for the relief of the particular symptoms (see BRONCHITIS; -PNEUMONIA). The preparations of ammonia are of special efficacy. During -convalescence the patient must be guarded from exposure to cold, and for -a time after recovery the state of the health ought to be watched with a -view of averting the evils, both local and constitutional, which too -often follow this disease. - - "German measles" (_Rotheln_, or _Epidemic Roseola_) is a term applied - to a contagious eruptive disorder having certain points of resemblance - to measles, and also to scarlet fever, but exhibiting its distinct - individuality in the fact that it protects from neither of these - diseases. It occurs most commonly in children, but frequently in - adults also, and is occasionally seen in extensive epidemics. Beyond - confinement to the house in the eruptive stage, which, from the slight - symptoms experienced, is often difficult of accomplishment, no special - treatment is called for. There is little doubt that the disease is - often mistaken for true measles, and many of the alleged second - attacks of the latter malady are probably cases of rotheln. The chief - points of difference are the following: (1) The absence of distinct - premonitory symptoms, the stage of invasion, which in measles is - usually of four days' duration, and accompanied with well-marked fever - and catarrh, being in rotheln either wholly absent or exceedingly - slight, enduring only for one day. (2) The eruption of rotheln, which, - although as regards its locality and manner of progress similar to - measles, differs somewhat in its appearance, the spots being of - smaller size, paler colour, and with less tendency to grouping in - crescentic patches. The rash attains its maximum in about one day, and - quickly disappears. There is not the same increase of temperature in - this stage as in measles. (3) The presence of white spots on the - buccal mucous membrane, in the case of measles. (4) The milder - character of the symptoms of rotheln throughout its whole course, and - the absence of complications and of liability to subsequent impairment - of health such as have been seen to appertain to measles. - - -FOOTNOTE: - - [1] _Transactions of the Epidemiological Society_ (London, 1877). - - - - -MEAT, a word originally applied to food in general, and so still used in -such phrases as "meat and drink"; but now, except as an archaism, -generally used of the flesh of certain domestic animals, slaughtered for -human food by butchers, "butcher's meat," as opposed to "game," that of -wild animals, "fish" or "poultry." Cognate forms of the O. Eng. _mete_ -are found in certain Teutonic languages, e.g. Swed. _mat_, Dan. _mad_ -and O. H. Ger. _Maz_. The ultimate origin has been disputed; the _New -English Dictionary_ considers probable a connexion with the root _med-_, -"to be fat," seen in Sansk. _meda_, Lat. _madere_, "to be wet," and Eng. -"mast," the fruit of the beech as food for pigs. - - See DIETETICS; FOOD PRESERVATION; PUBLIC HEALTH; AGRICULTURE; and the - sections dealing with agricultural statistics under the names of the - various countries. - - - - -MEATH (pronounced with _th_ soft, as in _the_), a county of Ireland in -the province of Leinster, bounded E. by the Irish Sea, S.E. by Dublin, -S. by Kildare and King's County, W. by Westmeath, N.W. by Cavan and -Monaghan, and N.E. by Louth. Area 579,320 acres, or about 905 sq. m. In -some districts the surface is varied by hills and swells, which to the -west reach a considerable elevation, although the general features of a -fine champain country are never lost. The coast, low and shelving, -extends about 10 m., but there is no harbour of importance. Laytown is a -small seaside resort, 5 m. S.E. of Drogheda. The Boyne enters the county -at its south-western extremity, and flowing north-east to Drogheda -divides it into two almost equal parts. At Navan it receives the -Blackwater, which flows south-west from Cavan. Both these rivers are -noted for their trout, and salmon are taken in the Boyne. The Boyne is -navigable for barges as far as Navan whence a canal is carried to Trim. -The Royal Canal passes along the southern boundary of the county from -Dublin. - - In the north is a broken country of Silurian rocks with much igneous - material, partly contemporaneous, partly intrusive, near Slane. - Carboniferous Limestone stretches from the Boyne valley to the Dublin - border, giving rise to a flat plain especially suitable for grazing. - Outliers of higher Carboniferous strata occur on the surface; but the - Coal Measures have all been removed by denudation. - - The climate is genial and favourable for all kinds of crops, there - being less rain than even in the neighbouring counties. Except a small - portion occupied by the Bog of Allen, the county is verdant and - fertile. The soil is principally a rich deep loam resting on limestone - gravel, but varies from a strong clayey loam to a light sandy gravel. - The proportion of tillage to pasturage is roughly as 1 to 3(1/2). Oats, - potatoes and turnips are the principal crops, but all decrease. The - numbers of cattle, sheep and poultry, however, are increasing or well - maintained. Agriculture is almost the sole industry, but coarse linen - is woven by hand-looms, and there are a few woollen manufactories. The - main line of the Midland Great Western railway skirts the southern - boundary, with a branch line north from Clonsilla to Navan and - Kingscourt (county Cavan). From Kilmessan on this line a branch serves - Trim and Athboy. From Drogheda (county Louth) a branch of the Great - Northern railway crosses the county from east to West by Navan and - Kells to Oldcastle. - - The population (76,111 in 1891; 67,497 in 1901) suffers a large - decrease, considerably above the average of Irish counties, and - emigration is heavy. Nearly 93% are Roman Catholics. The chief towns - are Navan (pop. 3839), Kells (2428) and Trim (1513), the county town. - Lesser market towns are Oldcastle and Athboy, an ancient town which - received a charter from Henry IV. The county includes eighteen - baronies. Assizes are held at Trim, and quarter sessions at Kells, - Navan and Trim. The county is in the Protestant dioceses of Armagh, - Kilmore and Meath, and in the Roman Catholic dioceses of Armagh and - Meath. Before the Union in 1800 it sent fourteen members to - parliament, but now only two members are returned, for the north and - south divisions of the county respectively. - -_History and Antiquities._--A district known as Meath (Midhe), and -including the present county of Meath as well as Westmeath and Longford, -with parts of Cavan, Kildare and King's County, was formed by Tuathal -(c. 130) into a kingdom to serve as mensal land or personal estate of -the Ard Ri or over-king of Ireland. Kings of Meath reigned until 1173, -and the title was claimed as late as the 15th century by their -descendants, but at the date mentioned Hugh de Lacy obtained the -lordship of the country and was confirmed in it by Henry II. Meath thus -came into the English "Pale." But though it was declared a county in the -reign of Edward I. (1296), and though it came by descent into the -possession of the Crown in the person of Edward IV., it was long before -it was fully subdued and its boundaries clearly defined. In 1543 -Westmeath was created a county apart from that of Meath, but as late as -1598 Meath was still regarded as a province by some, who included in it -the counties Westmeath, East Meath, Longford and Cavan. In the early -part of the 17th century it was at last established as a county, and no -longer considered as a fifth province of Ireland. - -There are two ancient round towers, the one at Kells and the other in -the churchyard of Donaghmore, near Navan. By the river Boyne near Slane -there is an extensive ancient burial-place called Brugh. Here are some -twenty burial mounds, the largest of which is that of New Grange, a -domed tumulus erected above a circular chamber, which is entered by a -narrow passage enclosed by great upright blocks of stone, covered with -carvings. The mound is surrounded by remains of a stone circle, and the -whole forms one of the most remarkable extant erections of its kind. -Tara (q.v.) is famous in history, especially as the seat of a royal -palace referred to in the well-known lines of Thomas Moore. Monastic -buildings were very numerous in Meath, among the more important ruins -being those of Duleek, which is said to have been the first -ecclesiastical building in Ireland of stone and mortar; the extensive -remains of Bective Abbey; and those of Clonard, where also were a -cathedral and a famous college. Of the old fortresses, the castle of -Trim still presents an imposing appearance. There are many fine old -mansions. - - - - -MEAUX, a town of northern France, capital of an arrondissement in the -department of Seine-et-Marne, and chief town of the agricultural region -of Brie, 28 m. E.N.E. of Paris by rail. Pop. (1906), 11,089. The town -proper stands on an eminence on the right bank of the Marne; on the left -bank lies the old suburb of Le Marche, with which it is united by a -bridge of the 16th century. Two rows of picturesque mills of the same -period are built across the river. The cathedral of St Stephen dates -from the 12th to the 16th centuries, and was restored in the 19th -century. Of the two western towers, the completed one is that to the -north of the facade, the other being disfigured by an unsightly slate -roof. The building, which is 275 ft. long and 105 ft. high, consists of -a short nave, with aisles, a fine transept, a choir and a sanctuary. The -choir contains the statue and the tomb of Bossuet, bishop from 1681 to -1704, and the pulpit of the cathedral has been reconstructed with the -panels of that from which the "eagle of Meaux" used to preach. The -transept terminates at each end in a fine portal surmounted by a -rose-window. The episcopal palace (17th century) has several curious old -rooms; the buildings of the choir school are likewise of some -archaeological interest. A statue of General Raoult (1870) stands in one -of the squares. - -Meaux is the centre of a considerable trade in cereals, wool, Brie -cheeses, and other farm-produce, while its mills provide much of the -flour with which Paris is supplied. Other industries are saw-milling, -metal-founding, distilling, the preparation of vermicelli and preserved -vegetables, and the manufacture of mustard, hosiery, plaster and -machinery. There are nursery-gardens in the vicinity. The Canal de -l'Ourcq, which surrounds the town, and the Marne furnish the means of -transport. Meaux is the seat of a bishopric dating from the 4th century, -and has among its public institutions a sub-prefecture, and tribunals of -first instance and of commerce. - -In the Roman period Meaux was the capital of the Meldi, a small Gallic -tribe, and in the middle ages of the Brie. It formed part of the kingdom -of Austrasia, and afterwards belonged to the counts of Vermandois and -Champagne, the latter of whom established important markets on the left -bank of the Marne. Its communal charter, received from them, is dated -1179. A treaty signed at Meaux in 1229 after the Albigensian War sealed -the submission of Raymond VII., count of Toulouse. The town suffered -much during the Jacquerie, the peasants receiving a severe check there -in 1358; during the Hundred Years' War; and also during the Religious -Wars, in which it was an important Protestant centre. It was the first -town which opened its gates to Henry IV. in 1594. On the high-road for -invaders marching on Paris from the east of France, Meaux saw its -environs ravaged by the army of Lorraine in 1652, and was laid under -heavy requisitions in 1814, 1815 and 1870. In September 1567 Meaux was -the scene of an attempt made by the Protestants to seize the French king -Charles IX., and his mother Catherine de' Medici. The plot, which is -sometimes called the "enterprise of Meaux," failed, the king and queen -with their courtiers escaping to Paris. This conduct, however, on the -part of the Huguenots had doubtless some share in influencing Charles to -assent to the massacre of St Bartholomew. - - - - -MECCA (Arab. _Makkah_),[1] the chief town of the Hejaz in Arabia, and -the great holy city of Islam. It is situated two camel marches (the -resting-place being Bahra or Hadda), or about 45 m. almost due E., from -Jidda on the Red Sea. Thus on a rough estimate Mecca lies in 21 deg. 25' -N., 39 deg. 50' E. It is said in the Koran (_Sur._ xiv. 40) that Mecca -lies in a sterile valley, and the old geographers observe that the whole -Haram or sacred territory round the city is almost without cultivation -or date palms, while fruit trees, springs, wells, gardens and green -valleys are found immediately beyond. Mecca in fact lies in the heart of -a mass of rough hills, intersected by a labyrinth of narrow valleys and -passes, and projecting into the Tehama or low country on the Red Sea, in -front of the great mountain wall that divides the coast-lands from the -central plateau, though in turn they are themselves separated from the -sea by a second curtain of hills forming the western wall of the great -Wadi Marr. The inner mountain wall is pierced by only two great passes, -and the valleys descending from these embrace on both sides the Mecca -hills. - -Holding this position commanding two great routes between the lowlands -and inner Arabia, and situated in a narrow and barren valley incapable -of supporting an urban population, Mecca must have been from the first a -commercial centre.[2] In the palmy days of South Arabia it was probably -a station on the great incense route, and thus Ptolemy may have learned -the name, which he writes Makoraba. At all events, long before Mahomet -we find Mecca established in the twofold quality of a commercial centre -and a privileged holy place, surrounded by an inviolable territory (the -Haram), which was not the sanctuary of a single tribe but a place of -pilgrimage, where religious observances were associated with a series of -annual fairs at different points in the vicinity. Indeed in the -unsettled state of the country commerce was possible only under the -sanctions of religion, and through the provisions of the sacred truce -which prohibited war for four months of the year, three of these being -the month of pilgrimage, with those immediately preceding and following. -The first of the series of fairs in which the Meccans had an interest -was at Okaz on the easier road between Mecca and Taif, where there was -also a sanctuary, and from it the visitors moved on to points still -nearer Mecca (Majanna, and finally Dhul-Majaz, on the flank of Jebel -Kabkab behind Arafa) where further fairs were held,[3] culminating in -the special religious ceremonies of the great feast at 'Arafa, Quzah -(Mozdalifa), and Mecca itself. The system of intercalation in the lunar -calendar of the heathen Arabs was designed to secure that the feast -should always fall at the time when the hides, fruits and other -merchandise were ready for market,[4] and the Meccans, who knew how to -attract the Bedouins by hospitality, bought up these wares in exchange -for imported goods, and so became the leaders of the international trade -of Arabia. Their caravans traversed the length and breadth of the -peninsula. Syria, and especially Gaza, was their chief goal. The Syrian -caravan intercepted, on its return, at Badr (see MAHOMET) represented -capital to the value of L20,000, an enormous sum for those days.[5] - -The victory of Mahommedanism made a vast change in the position of -Mecca. The merchant aristocracy became satraps or pensioners of a great -empire; but the seat of dominion was removed beyond the desert, and -though Mecca and the Hejaz strove for a time to maintain political as -well as religious predominance, the struggle was vain, and terminated on -the death of Ibn Zubair, the Meccan pretendant to the caliphate, when -the city was taken by Hajjaj (A.D. 692). The sanctuary and feast of -Mecca received, however, a new prestige from the victory of Islam. -Purged of elements obviously heathen, the Ka'ba became the holiest site, -and the pilgrimage the most sacred ritual observance of Mahommedanism, -drawing worshippers from so wide a circle that the confluence of the -petty traders of the desert was no longer the main feature of the holy -season. The pilgrimage retained its importance for the commercial -well-being of Mecca; to this day the Meccans live by the Hajj--letting -rooms, acting as guides and directors in the sacred ceremonies, as -contractors and touts for land and sea transport, as well as exploiting -the many benefactions that flow to the holy city; while the surrounding -Bedouins derive support from the camel-transport it demands and from the -subsidies by which they are engaged to protect or abstain from molesting -the pilgrim caravans. But the ancient "fairs of heathenism" were given -up, and the traffic of the pilgrim season, sanctioned by the Prophet in -_Sur._ ii. 194, was concentrated at Mina and Mecca, where most of the -pilgrims still have something to buy or sell, so that Mina, after the -sacrifice of the feast day, presents the aspect of a huge international -fancy fair.[6] In the middle ages this trade was much more important -than it is now. Ibn Jubair (ed. Wright, p. 118 seq.) in the 12th century -describes the mart of Mecca in the eight days following the feast as -full of gems, unguents, precious drugs, and all rare merchandise from -India, Irak, Khorasan, and every part of the Moslem world. - -The hills east and west of Mecca, which are partly built over and rise -several hundred feet above the valley, so enclose the city that the -ancient walls only barred the valley at three points, where three gates -led into the town. In the time of Ibn Jubair the gates still stood -though the walls were ruined, but now the gates have only left their -names to quarters of the town. At the northern or upper end was the Bab -el Ma'la, or gate of the upper quarter, whence the road continues up the -valley towards Mina and Arafa as well as towards Zeima and the Nejd. -Beyond the gate, in a place called the Hajun, is the chief cemetery, -commonly called el Ma'la, and said to be the resting-place of many of -the companions of Mahomet. Here a cross-road, running over the hill to -join the main Medina road from the western gate, turns off to the west -by the pass of Kada, the point from which the troops of the Prophet -stormed the city (A.H. 8).[7] Here too the body of Ibn Zubair was hung -on a cross by Hajjaj. The lower or southern gate, at the Masfala -quarter, opened on the Yemen road, where the rain-water from Mecca flows -off into an open valley. Beyond, there are mountains on both sides; on -that to the east, commanding the town, is the great castle, a fortress -of considerable strength. The third or western gate, Bab el-Omra -(formerly also Bab el-Zahir, from a village of that name), lay almost -opposite the great mosque, and opened on a road leading westwards round -the southern spurs of the Red Mountain. This is the way to Wadi Fatima -and Medina, the Jidda road branching off from it to the left. -Considerable suburbs now lie outside the quarter named after this gate; -in the middle ages a pleasant country road led for some miles through -partly cultivated land with good wells, as far as the boundary of the -sacred territory and gathering place of the pilgrims at Tanim, near the -mosque of Ayesha. This is the spot on the Medina road now called the -Omra, from a ceremonial connected with it which will be mentioned below. - -The length of the sinuous main axis of the city from the farthest -suburbs on the Medina road to the suburbs in the extreme north, now -frequented by Bedouins, is, according to Burckhardt, 3500 paces.[8] -About the middle of this line the longitudinal thoroughfares are pushed -aside by the vast courtyard and colonnades composing the great mosque, -which, with its spacious arcades surrounding the Ka'ba and other holy -places, and its seven minarets, forms the only prominent architectural -feature of the city. The mosque is enclosed by houses with windows -opening on the arcades and commanding a view of the Ka'ba. Immediately -beyond these, on the side facing Jebel Abu Kobais, a broad street runs -south-east and north-west across the valley. This is the Mas'a (sacred -course) between the eminences of Safa and Merwa, and has been from very -early times one of the most lively bazaars and the centre of Meccan -life. The other chief bazaars are also near the mosque in smaller -streets. The general aspect of the town is picturesque; the streets are -fairly spacious, though ill-kept and filthy; the houses are all of -stone, many of them well-built and four or five storeys high, with -terraced roofs and large projecting windows as in Jidda--a style of -building which has not varied materially since the 10th century -(Mukaddasi, p. 71), and gains in effect from the way in which the -dwellings run up the sides and spurs of the mountains. Of public -institutions there are baths, ribats, or hospices, for poor pilgrims -from India, Java, &c., a hospital and a public kitchen for the poor. - -The mosque is at the same time the university hall, where between two -pilgrim seasons lectures are delivered on Mahommedan law, doctrine and -connected branches of science. A poorly provided public library is open -to the use of students. The madrassehs or buildings around the mosque, -originally intended as lodgings for students and professors, have long -been let out to rich pilgrims. The minor places of visitation for -pilgrims, such as the birthplaces of the prophet and his chief -followers, are not notable.[9] Both these and the court of the great -mosque lie beneath the general level of the city, the site having been -gradually raised by accumulated rubbish. The town in fact has little air -of antiquity; genuine Arab buildings do not last long, especially in a -valley periodically ravaged by tremendous floods when the tropical rains -burst on the surrounding hills. The history of Mecca is full of the -record of these inundations, unsuccessfully combated by the great dam -drawn across the valley by the caliph Omar (_Kutbeddin_, p. 76), and -later works of Mahdi.[10] - -The fixed population of Mecca in 1878 was estimated by Assistant-Surgeon -'Abd el-Razzaq at 50,000 to 60,000; there is a large floating -population--and that not merely at the proper season of pilgrimage, the -pilgrims of one season often beginning to arrive before those of the -former season have all dispersed. At the height of the season the town -is much overcrowded, and the entire want of a drainage system is -severely felt. Fortunately good water is tolerably plentiful; for, -though the wells are mostly undrinkable, and even the famous Zamzam -water only available for medicinal or religious purposes, the -underground conduit from beyond Arafa, completed by Sultan Selim II. in -1571, supplies to the public fountains a sweet and light water, -containing, according to 'Abd el-Razzaq, a large amount of chlorides. -The water is said to be free to townsmen, but is sold to the pilgrims at -a rather high rate.[11] - -Medieval writers celebrate the copious supplies, especially of fine -fruits, brought to the city from Taif and other fertile parts of Arabia. -These fruits are still famous; rice and other foreign products are -brought by sea to Jidda; mutton, milk and butter are plentifully -supplied from the desert.[12] The industries all centre in the -pilgrimage; the chief object of every Meccan--from the notables and -sheikhs, who use their influence to gain custom for the Jidda -speculators in the pilgrim traffic, down to the cicerones, pilgrim -brokers, lodging-house keepers, and mendicants at the holy places--being -to pillage the visitor in every possible way. The fanaticism of the -Meccan is an affair of the purse; the mongrel population (for the town -is by no means purely Arab) has exchanged the virtues of the Bedouin for -the worst corruptions of Eastern town life, without casting off the -ferocity of the desert, and it is hardly possible to find a worse -certificate of character than the three parallel gashes on each cheek, -called Tashrit, which are the customary mark of birth in the holy city. -The unspeakable vices of Mecca are a scandal to all Islam, and a -constant source of wonder to pious pilgrims.[13] The slave trade has -connexions with the pilgrimage which are not thoroughly clear; but under -cover of the pilgrimage a great deal of importation and exportation of -slaves goes on. - -Since the fall of Ibn Zubair the political position of Mecca has always -been dependent on the movements of the greater Mahommedan world. In the -splendid times of the caliphs immense sums were lavished upon the -pilgrimage and the holy city; and conversely the decay of the central -authority of Islam brought with it a long period of faction, wars and -misery, in which the most notable episode was the sack of Mecca by the -Carmathians at the pilgrimage season of A.D. 930. The victors carried -off the "black stone," which was not restored for twenty-two years, and -then only for a great ransom, when it was plain that even the loss of -its palladium could not destroy the sacred character of the city. Under -the Fatimites Egyptian influence began to be strong in Mecca; it was -opposed by the sultans of Yemen, while native princes claiming descent -from the Prophet--the Hashimite amirs of Mecca, and after them the amirs -of the house of Qatada (since 1202)--attained to great authority and -aimed at independence; but soon after the final fall of the Abbasids the -Egyptian overlordship was definitely established by sultan Bibars (A.D. -1269). The Turkish conquest of Egypt transferred the supremacy to the -Ottoman sultans (1517), who treated Mecca with much favour, and during -the 16th century executed great works in the sanctuary and temple. The -Ottoman power, however, became gradually almost nominal, and that of the -amirs or sherifs increased in proportion, culminating under Ghalib, -whose accession dates from 1786. Then followed the wars of the Wahhabis -(see ARABIA and WAHHABIS) and the restoration of Turkish rule by the -troops of Mehemet 'Ali. By him the dignity of sherif was deprived of -much of its weight, and in 1827 a change of dynasty was effected by the -appointment of Ibn 'Aun. Afterwards Turkish authority again decayed. -Mecca is, however, officially the capital of a Turkish province, and has -a governor-general and a Turkish garrison, while Mahommedan law is -administered by a judge sent from Constantinople. But the real sovereign -of Mecca and the Hejaz is the sherif, who, as head of a princely family -claiming descent from the Prophet, holds a sort of feudal position. The -dignity of sherif (or grand sherif, as Europeans usually say for the -sake of distinction, since all the kin of the princely houses reckoning -descent from the Prophet are also named sherifs), although by no means a -religious pontificate, is highly respected owing to its traditional -descent in the line of Hasan, son of the fourth caliph 'Ali. From a -political point of view the sherif is the modern counterpart of the -ancient amirs of Mecca, who were named in the public prayers immediately -after the reigning caliph. When the great Mahommedan sultanates had -become too much occupied in internecine wars to maintain order in the -distant Hejaz, those branches of the Hassanids which from the beginning -of Islam had retained rural property in Arabia usurped power in the holy -cities and the adjacent Bedouin territories. About A.D. 960 they -established a sort of kingdom with Mecca as capital. The influence of -the princes of Mecca has varied from time to time, according to the -strength of the foreign protectorate in the Hejaz or in consequence of -feuds among the branches of the house; until about 1882 it was for most -purposes much greater than that of the Turks. The latter were strong -enough to hold the garrisoned towns, and thus the sultan was able within -certain limits--playing off one against the other the two rival branches -of the aristocracy, viz. the kin of Ghalib and the house of Ibn'Aun--to -assert the right of designating or removing the sherif, to whom in turn -he owed the possibility of maintaining, with the aid of considerable -pensions, the semblance of his much-prized lordship over the holy -cities. The grand sherif can muster a considerable force of freedmen and -clients, and his kin, holding wells and lands in various places through -the Hejaz, act as his deputies and administer the old Arabic customary -law to the Bedouin. To this influence the Hejaz owes what little of law -and order it enjoys. During the last quarter of the 19th century Turkish -influence became preponderant in western Arabia, and the railway from -Syria to the Hejaz tended to consolidate the sultan's supremacy. After -the sherifs, the principal family of Mecca is the house of Shaibah, -which holds the hereditary custodianship of the Ka'ba. - -_The Great Mosque and the Ka'ba._--Long before Mahomet the chief -sanctuary of Mecca was the Ka'ba, a rude stone building without windows, -and having a door 7 ft. from the ground; and so named from its -resemblance to a monstrous _astragalus_ (die) of about 40 ft. cube, -though the shapeless structure is not really an exact cube nor even -exactly rectangular.[14] The Ka'ba has been rebuilt more than once since -Mahomet purged it of idols and adopted it as the chief sanctuary of -Islam, but the old form has been preserved, except in secondary -details;[15] so that the "Ancient House," as it is titled, is still -essentially a heathen temple, adapted to the worship of Islam by the -clumsy fiction that it was built by Abraham and Ishmael by divine -revelation as a temple of pure monotheism, and that it was only -temporarily perverted to idol worship from the time when 'Amr ibn Lohai -introduced the statue of Hobal from Syria[16] till the victory of Islam. -This fiction has involved the superinduction of a new mythology over the -old heathen ritual, which remains practically unchanged. Thus the chief -object of veneration is the black stone, which is fixed in the external -angle facing Safa. The building is not exactly oriented, but it may be -called the south-east corner. Its technical name is the black corner, -the others being named the Yemen (south-west), Syrian (north-west), and -Irak (north-east) corners, from the lands to which they approximately -point. The black stone is a small dark mass a span long, with an aspect -suggesting volcanic or meteoric origin, fixed at such a height that it -can be conveniently kissed by a person of middle size. It was broken by -fire in the siege of A.D. 683 (not, as many authors relate, by the -Carmathians), and the pieces are kept together by a silver setting. The -history of this heavenly stone, given by Gabriel to Abraham, does not -conceal the fact that it was originally a fetish, the most venerated of -a multitude of idols and sacred stones which stood all round the -sanctuary in the time of Mahomet. The Prophet destroyed the idols, but -he left the characteristic form of worship--the _tawaf_, or sevenfold -circuit of the sanctuary, the worshipper kissing or touching the objects -of his veneration--and besides the black stone he recognized the -so-called "southern" stone, the same presumably as that which is still -touched in the tawaf at the Yemen corner (_Muh. in Med._ pp. 336, 425). -The ceremony of the tawaf and the worship of stone fetishes was common -to Mecca with other ancient Arabian sanctuaries.[17] It was, as it still -is, a frequent religious exercise of the Meccans, and the first duty of -one who returned to the city or arrived there under a vow of pilgrimage; -and thus the outside of the Ka'ba was and is more important than the -inside. Islam did away with the worship of idols; what was lost in -interest by their suppression has been supplied by the invention of -spots consecrated by recollections of Abraham, Ishmael and Hagar, or -held to be acceptable places of prayer. Thus the space of ten spans -between the black stone and the door, which is on the east side, between -the black and Irak corners, and a man's height from the ground, is -called the _Multazam_, and here prayer should be offered after the tawaf -with outstretched arms and breast pressed against the house. On the -other side of the door, against the same wall, is a shallow trough, -which is said to mark the original site of the stone on which Abraham -stood to build the Ka'ba. Here the growth of the legend can be traced, -for the place is now called the "kneading-place" (Ma'jan), where the -cement for the Ka'ba was prepared. This name and story do not appear in -the older accounts. Once more, on the north side of the Ka'ba, there -projects a low semicircular wall of marble, with an opening at each end -between it and the walls of the house. The space within is paved with -mosaic, and is called the Hijr. It is included in the tawaf, and two -slabs of _verde antico_ within it are called the graves of Ishmael and -Hagar, and are places of acceptable prayer. Even the golden or gilded -_mizab_ (water-spout) that projects into the Hijr marks a place where -prayer is heard, and another such place is the part of the west wall -close to the Yemen corner. - -The feeling of religious conservatism which has preserved the structural -rudeness of the Ka'ba did not prohibit costly surface decoration. In -Mahomet's time the outer walls were covered by a veil (or _kiswa_) of -striped Yemen cloth. The caliphs substituted a covering of figured -brocade, and the Egyptian government still sends with each pilgrim -caravan from Cairo a new kiswa of black brocade, adorned with a broad -band embroidered with golden inscriptions from the Koran, as well as a -richer curtain for the door.[18] The door of two leaves, with its posts -and lintel, is of silver gilt. - -The interior of the Ka'ba is now opened but a few times every year for -the general public, which ascends by the portable staircase brought -forward for the purpose. Foreigners can obtain admission at any time for -a special fee. The modern descriptions, from observations made under -difficulties, are not very complete. Little change, however, seems to -have been made since the time of Ibn Jubair, who describes the floor and -walls as overlaid with richly variegated marbles, and the upper half of -the walls as plated with silver thickly gilt, while the roof was veiled -with coloured silk. Modern writers describe the place as windowless, but -Ibn Jubair mentions five windows of rich stained glass from Irak. -Between the three pillars of teak hung thirteen silver lamps. A chest in -the corner to the left of one entering contained Korans, and at the Irak -corner a space was cut off enclosing the stair that leads to the roof. -The door to this stair (called the door of mercy--Bab el-Rahma) was -plated with silver by the caliph Motawakkil. Here, in the time of Ibn -Jubair, the _Maqam_ or standing stone of Abraham was usually placed for -better security, but brought out on great occasions.[19] - -The houses of ancient Mecca pressed close upon the Ka'ba, the noblest -families, who traced their descent from Kosai, the reputed founder of -the city, having their dwellings immediately round the sanctuary. To the -north of the Ka'ba was the Dar el-Nadwa, or place of assembly of the -Koreish. The multiplication of pilgrims after Islam soon made it -necessary to clear away the nearest dwellings and enlarge the place of -prayer around the Ancient House. Omar, Othman and Ibn Jubair had all a -share in this work, but the great founder of the mosque in its present -form, with its spacious area and deep colonnades, was the caliph Mahdi, -who spent enormous sums in bringing costly pillars from Egypt and Syria. -The work was still incomplete at his death in A.D. 785, and was finished -in less sumptuous style by his successor. Subsequent repairs and -additions, extending down to Turkish times, have left little of Mahdi's -work untouched, though a few of the pillars probably date from his days. -There are more than five hundred pillars in all, of very various style -and workmanship, and the enclosure--250 paces in length and 200 in -breadth, according to Burckhardt's measurement--is entered by nineteen -archways irregularly disposed. - -After the Ka'ba the principal points of interest in the mosque are the -well Zamzam and the Maqam Ibrahim. The former is a deep shaft enclosed -in a massive vaulted building paved with marble, and, according to -Mahommedan tradition, is the source (corresponding to the Beer-lahai-roi -of Gen. xvi. 14) from which Hagar drew water for her son Ishmael. The -legend tells that the well was long covered up and rediscovered by 'Abd -al-Mot[t.]alib, the grandfather of the Prophet. Sacred wells are -familiar features of Semitic sanctuaries, and Islam, retaining the well, -made a quasi-biblical story for it, and endowed its tepid waters with -miraculous curative virtues. They are eagerly drunk by the pilgrims, or -when poured over the body are held to give a miraculous refreshment -after the fatigues of religious exercise; and the manufacture of bottles -or jars for carrying the water to distant countries is quite a trade. -Ibn Jubair mentions a curious superstition of the Meccans, who believed -that the water rose in the shaft at the full moon of the month Shaban. -On this occasion a great crowd, especially of young people, thronged -round the well with shouts of religious enthusiasm, while the servants -of the well dashed buckets of water over their heads. The Maqam of -Abraham is also connected with a relic of heathenism, the ancient holy -stone which once stood on the Ma'jan, and is said to bear the prints of -the patriarch's feet. The whole legend of this stone, which is full of -miraculous incidents, seems to have arisen from a misconception, the -Maqam Ibrahim in the Koran meaning the sanctuary itself; but the stone, -which is a block about 3 spans in height and 2 in breadth, and in shape -"like a potter's furnace" (Ibn Jubair), is certainly very ancient. No -one is now allowed to see it, though the box in which it lies can be -seen or touched through a grating in the little chapel that surrounds -it. In the middle ages it was sometimes shown, and Ibn Jubair describes -the pious enthusiasm with which he drank Zamzam water poured on the -footprints. It was covered with inscriptions in an unknown character, -one of which was copied by Fakihi in his history of Mecca. To judge by -the facsimile in Dozy's _Israeliten te Mekka_, the character is probably -essentially one with that of the Syrian Safa inscriptions, which -extended through the Nejd and into the Hejaz.[20] - - _Safa and Merwa._--In religious importance these two points or - "hills," connected by the Mas'a, stand second only to the Ka'ba. Safa - is an elevated platform surmounted by a triple arch, and approached by - a flight of steps.[21] It lies south-east of the Ka'ba, facing the - black corner, and 76 paces from the "Gate of Safa," which is - architecturally the chief gate of the mosque. Merwa is a similar - platform, formerly covered with a single arch, on the opposite side of - the valley. It stands on a spur of the Red Mountain called Jebel - Kuaykian. The course between these two sacred points is 493 paces - long, and the religious ceremony called the "sa'y" consists in - traversing it seven times, beginning and ending at Safa. The lowest - part of the course, between the so-called green milestones, is done at - a run. This ceremony, which, as we shall presently see, is part of the - omra, is generally said to be performed in memory of Hagar, who ran to - and fro between the two eminences vainly seeking water for her son. - The observance, however, is certainly of pagan origin; and at one time - there were idols on both the so-called hills (see especially Azraqi, - pp. 74, 78). - - _The Ceremonies and the Pilgrimage._--Before Islam the Ka'ba was the - local sanctuary of the Meccans, where they prayed and did sacrifice, - where oaths were administered and hard cases submitted to divine - sentence according to the immemorial custom of Semitic shrines. But, - besides this, Mecca was already a place of pilgrimage. Pilgrimage with - the ancient Arabs was the fulfilment of a vow, which appears to have - generally terminated--at least on the part of the well-to-do--in a - sacrificial feast. A vow of pilgrimage might be directed to other - sanctuaries than Mecca--the technical word for it (_ihlal_) is - applied, for example, to the pilgrimage to Manat (_Bakri_, p. 519). He - who was under such a vow was bound by ceremonial observances of - abstinence from certain acts (e.g. hunting) and sensual pleasures, and - in particular was forbidden to shear or comb his hair till the - fulfilment of the vow. This old Semitic usage has its close parallel - in the vow of the Nazarite. It was not peculiarly connected with - Mecca; at Taif, for example, it was customary on return to the city - after an absence to present oneself at the sanctuary, and there shear - the hair (_Muh. in Med._, p. 381). Pilgrimages to Mecca were not tied - to a single time, but they were naturally associated with festive - occasions, and especially with the great annual feast and market. The - pilgrimage was so intimately connected with the well-being of Mecca, - and had already such a hold on the Arabs round about, that Mahomet - could not afford to sacrifice it to an abstract purity of religion, - and thus the old usages were transplanted into Islam in the double - form of the omra or vow of pilgrimage to Mecca, which can be - discharged at any time, and the hajj or pilgrimage at the great annual - feast. The latter closes with a visit to the Ka'ba, but its essential - ceremonies lie outside Mecca, at the neighbouring shrines where the - old Arabs gathered before the Meccan fair. - - The omra begins at some point outside the Haram (or holy territory), - generally at Tanim, both for convenience sake and because Ayesha began - the omra there in the year 10 of the Hegira. The pilgrim enters the - Haram in the antique and scanty pilgrimage dress (ihram), consisting - of two cloths wound round his person in a way prescribed by ritual. - His devotion is expressed in shouts of "Labbeyka" (a word of obscure - origin and meaning); he enters the great mosque, performs the tawaf - and the sa'y[22] and then has his head shaved and resumes his common - dress. This ceremony is now generally combined with the hajj, or is - performed by every stranger or traveller when he enters Mecca, and the - ihram (which involves the acts of abstinence already referred to) is - assumed at a considerable distance from the city. But it is also - proper during one's residence in the holy city to perform at least one - omra from Tanim in connexion with a visit to the mosque of Ayesha - there. The triviality of these rites is ill concealed by the legends - of the sa'y of Hagar and of the tawaf being first performed by Adam in - imitation of the circuit of the angels about the throne of God; the - meaning of their ceremonies seems to have been almost a blank to the - Arabs before Islam, whose religion had become a mere formal tradition. - We do not even know to what deity the worship expressed in the tawaf - was properly addressed. There is a tradition that the Ka'ba was a - temple of Saturn (Shahrastani, p. 431); perhaps the most distinctive - feature of the shrine may be sought in the sacred doves which still - enjoy the protection of the sanctuary. These recall the sacred doves - of Ascalon (Philo vi. 200 of Richter's ed.), and suggests - Venus-worship as at least one element (cf. Herod i. 131, iii. 8; Ephr. - Syr., _Op. Syr._ ii. 457). - - To the ordinary pilgrim the omra has become so much an episode of the - hajj that it is described by some European pilgrims as a mere visit to - the mosque of Ayesha; a better conception of its original significance - is got from the Meccan feast of the seventh month (Rajab), graphically - described by Ibn Jubair from his observations in A.D. 1184. Rajab was - one of the ancient sacred months, and the feast, which extended - through the whole month and was a joyful season of hospitality and - thanksgiving, no doubt represents the ancient feasts of Mecca more - exactly than the ceremonies of the hajj, in which old usage has been - overlaid by traditions and glosses of Islam. The omra was performed by - crowds from day to day, especially at new and full moon.[23] The new - moon celebration was nocturnal; the road to Tanim, the Mas'a, and the - mosque were brilliantly illuminated; and the appearing of the moon was - greeted with noisy music. A genuine old Arab market was held, for the - wild Bedouins of the Yemen mountains came in thousands to barter their - cattle and fruits for clothing, and deemed that to absent themselves - would bring drought and cattle plague in their homes. Though ignorant - of the legal ritual and prayers, they performed the tawaf with - enthusiasm, throwing themselves against the Ka'ba and clinging to its - curtains as a child clings to its mother. They also made a point of - entering the Ka'ba. The 29th of the month was the feast day of the - Meccan women, when they and their little ones had the Ka'ba to - themselves without the presence even of the Sheybas. - - The central and essential ceremonies of the hajj or greater pilgrimage - are those of the day of Arafa, the 9th of the "pilgrimage month" - (Dhu'l Hijja), the last of the Arab year; and every Moslem who is his - own master, and can command the necessary means, is bound to join in - these once in his life, or to have them fulfilled by a substitute on - his behalf and at his expense. By them the pilgrim becomes as pure - from sin as when he was born, and gains for the rest of his life the - honourable title of hajj. Neglect of many other parts of the pilgrim - ceremonial may be compensated by offerings, but to miss the "stand" - (_woquf_) at Arafa is to miss the pilgrimage. Arafa or Arafat is a - space, artificially limited, round a small isolated hill called the - Hill of Mercy, a little way outside the holy territory, on the road - from Mecca to Taif. One leaving Mecca after midday can easily reach - the place on foot the same evening. The road is first northwards along - the Mecca valley and then turns eastward. It leads through the - straggling village of Mina, occupying a long narrow valley (Wadi - Mina), two to three hours from Mecca, and thence by the mosque of - Mozdalifa over a narrow pass opening out into the plain of Arafa, - which is an expansion of the great Wadi Naman, through which the Taif - road descends from Mount Kara. The lofty and rugged mountains of the - Hodheyl tower over the plain on the north side and overshadow the - little Hill of Mercy, which is one of those bosses of weathered - granite so common in the Hejaz. Arafa lay quite near Dhul-Majaz, - where, according to Arabian tradition, a great fair was held from the - 1st to the 8th of the pilgrimage month; and the ceremonies from which - the hajj was derived were originally an appendix to this fair. Now, on - the contrary, the pilgrim is expected to follow as closely as may be - the movements of the prophet at his "farewell pilgrimage" in the year - 10 of the Hegira (A.D. 632). He therefore leaves Mecca in pilgrim garb - on the 8th of Dhu'l Hijja, called the day of _tarwiya_ (an obscure and - pre-Islamic name), and, strictly speaking, should spend the night at - Mina. It is now, however, customary to go right on and encamp at once - at Arafa. The night should be spent in devotion, but the coffee booths - do a lively trade, and songs are as common as prayers. Next forenoon - the pilgrim is free to move about, and towards midday he may if he - please hear a sermon. In the afternoon the essential ceremony begins; - it consists simply in "standing" on Arafa shouting "Labbeyka" and - reciting prayers and texts till sunset. After the sun is down the vast - assemblage breaks up, and a rush (technically _ifada_, _daf'_, _nafr_) - is made in the utmost confusion to Mozdalifa, where the night prayer - is said and the night spent. Before sunrise next morning (the 10th) a - second "stand" like that on Arafa is made for a short time by - torchlight round the mosque of Mozdalifa, but before the sun is fairly - up all must be in motion in the second _ifada_ towards Mina. The day - thus begun is the "day of sacrifice," and has four ceremonies--(1) to - pelt with seven stones a cairn (_jamrat al 'aqaba_) at the eastern end - of W. Mina, (2) to slay a victim at Mina and hold a sacrificial meal, - part of the flesh being also dried and so preserved, or given to the - poor,[24] (3) to be shaved and so terminate the _ihram_, (4) to make - the third _ifada_, i.e. go to Mecca and perform the tawaf and sa'y - (_'omrat al-ifada_), returning thereafter to Mina. The sacrifice and - visit to Mecca may, however, be delayed till the 11th, 12th or 13th. - These are the days of Mina, a fair and joyous feast, with no special - ceremony except that each day the pilgrim is expected to throw seven - stones at the _jamrat al 'aqaba_, and also at each of two similar - cairns in the valley. The stones are thrown in the name of Allah, and - are generally thought to be directed at the devil. This is, however, a - custom older than Islam, and a tradition in Azraqi, p. 412, represents - it as an act of worship to idols at Mina. As the stones are thrown on - the days of the fair, it is not unlikely that they have something to - do with the old Arab mode of closing a sale by the purchaser throwing - a stone (Biruni, p. 328).[25] The pilgrims leave Mina on the 12th or - 13th, and the hajj is then over. (See further MAHOMMEDAN RELIGION.) - - The colourless character of these ceremonies is plainly due to the - fact that they are nothing more than expurgated heathen rites. In - Islam proper they have no _raison d'etre_; the legends about Adam and - Eve on Arafa, about Abraham's sacrifice of the ram at Thabii by Mina, - imitated in the sacrifices of the pilgrimage, are clumsy - afterthoughts, as appears from their variations and only partial - acceptance. It is not so easy to get at the nature of the original - rites, which Islam was careful to suppress. But we find mention of - practices condemned by the orthodox, or forming no part of the Moslem - ritual, which may be regarded as traces of an older ceremonial. Such - are nocturnal illuminations at Mina (Ibn Batuta i. 396), Arafa and - Mozdalifa (Ibn Jubair, 179), and tawafs performed by the ignorant at - holy spots at Arafa not recognized by law (Snouck-Hurgronje p. 149 - sqq.). We know that the rites at Mozdalifa were originally connected - with a holy hill bearing the name of the god Quzah (the Edomite Koze) - whose bow is the rainbow, and there is reason to think that the - _ifadas_ from Arafa and Quzah, which were not made as now after sunset - and before sunrise, but when the sun rested on the tops of the - mountains, were ceremonies of farewell and salutation to the sun-god. - - The statistics of the pilgrimage cannot be given with certainty and - vary much from year to year. The quarantine office keeps a record of - arrivals by sea at Jidda (66,000 for 1904); but to these must be added - those travelling by land from Cairo, Damascus and Irak, the pilgrims - who reach Medina from Yanbu and go on to Mecca, and those from all - parts of the peninsula. Burckhardt in 1814 estimated the crowd at - Arafa at 70,000, Burton in 1853 at 50,000, 'Abd el-Razzak in 1858 at - 60,000. This great assemblage is always a dangerous centre of - infection, and the days of Mina especially, spent under circumstances - originally adapted only for a Bedouin fair, with no provisions for - proper cleanliness, and with the air full of the smell of putrefying - offal and flesh drying in the sun, produce much sickness. - - LITERATURE.--Besides the Arabic geographers and cosmographers, we have - Ibn 'Abd Rabbih's description of the mosque, early in the 10th century - (_'Ikd Farid_, Cairo ed., iii. 362 sqq.), but above all the admirable - record of Ibn Jubair (A.D. 1184), by far the best account extant of - Mecca and the pilgrimage. It has been much pillaged by Ibn Batuta. The - Arabic historians are largely occupied with fabulous matter as to - Mecca before Islam; for these legends the reader may refer to C. de - Perceval's _Essai_. How little confidence can be placed in the - pre-Islamic history appears very clearly from the distorted accounts - of Abraha's excursion against the Hejaz, which fell but a few years - before the birth of the Prophet, and is the first event in Meccan - history which has confirmation from other sources. See Noldeke's - version of Tabari, p. 204 sqq. For the period of the Prophet, Ibn - Hisham and Wakidi are valuable sources in topography as well as - history. Of the special histories and descriptions of Mecca published - by Wustenfeld (_Chroniken der Stadt Mekka_, 3 vols., 1857-1859, with - an abstract in German, 1861), the most valuable is that of Azraqi. It - has passed through the hands of several editors, but the oldest part - goes back to the beginning of the 9th Christian century. Kutbeddin's - history (vol. iii. of the _Chroniken_) goes down with the additions of - his nephew to A.D. 1592. - - Of European descriptions of Mecca from personal observation the best - is Burckhardt's _Travels in Arabia_ (cited above from the 8vo ed., - 1829). _The Travels of Aly Bey_ (Badia, London, 1816) describe a visit - in 1807; Burton's _Pilgrimage_ (3rd ed., 1879) often supplements - Burckhardt; Von Maltzan's _Wallfahrt nach Mekka_ (1865) is lively but - very slight. 'Abd el-Razzaq's report to the government of India on the - pilgrimage of 1858 is specially directed to sanitary questions; C. - Snouck-Hurgronje, _Mekka_ (2 vols., and a collection of photographs, - The Hague, 1888-1889), gives a description of the Meccan sanctuary and - of the public and private life of the Meccans as observed by the - author during a sojourn in the holy city in 1884-1885 and a political - history of Mecca from native sources from the Hegira till 1884. For - the pilgrimage see particularly Snouck-Hurgronje, _Het Mekkaansche - Feest_ (Leiden, 1880). (W. R. S.) - - -FOOTNOTES: - - [1] A variant of the name Makkah is Bakkah (_Sur._ iii. 90; Bakri, - 155 seq.). For other names and honorific epithets of the city see - Bakri, _ut supra_, Azraqi, p. 197, Yaqut iv. 617 seq. The lists are - in part corrupt, and some of the names (Kutha and 'Arsh or 'Ursh, - "the huts") are not properly names of the town as a whole. - - [2] Mecca, says one of its citizens, in Waqidi (Kremer's ed., p. 196, - or _Muh. in Med._ p. 100), is a settlement formed for trade with - Syria in summer and Abyssinia in winter, and cannot continue to exist - if the trade is interrupted. - - [3] The details are variously related. See Biruni, p. 328 (E. T., p. - 324); Asma'i in Yaqut, iii. 705, iv. 416, 421; Azraqi, p. 129 seq.; - Bakri, p. 661. Jebel Kabkab is a great mountain occupying the angle - between W. Naman and the plain of Arafa. The peak is due north of - Sheddad, the hamlet which Burckhardt (i. 115) calls Shedad. According - to Azraqi, p. 80, the last shrine visited was that of the three trees - of Uzza in W. Nakhla. - - [4] So we are told by Biruni, p. 62 (E. T., 73). - - [5] Waqidi, ed. Kremer, pp. 20, 21; _Muh. in Med._ p. 39. - - [6] The older fairs were not entirely deserted till the troubles of - the last days of the Omayyads (Azraqi, p. 131). - - [7] This is the cross-road traversed by Burckhardt (i. 109), and - described by him as cut through the rocks with much labour. - - [8] Istakhri gives the length of the city proper from north to south - as 2 m., and the greatest breadth from the Jiyad quarter east of the - great mosque across the valley and up the western slopes as - two-thirds of the length. - - [9] For details as to the ancient quarters of Mecca, where the - several families or septs lived apart, see Azraqi, 455 pp. seq., and - compare Ya'qubi, ed. Juynboll, p. 100. The minor sacred places are - described at length by Azraqi and Ibn Jubair. They are either - connected with genuine memories of the Prophet and his times, or have - spurious legends to conceal the fact that they were originally holy - stones, wells, or the like, of heathen sanctity. - - [10] Baladhuri, in his chapter on the floods of Mecca (pp. 53 seq.), - says that 'Omar built two dams. - - [11] The aqueduct is the successor of an older one associated with - the names of Zobaida, wife of Harun al-Rashid, and other benefactors. - But the old aqueduct was frequently out of repair, and seems to have - played but a secondary part in the medieval water supply. Even the - new aqueduct gave no adequate supply in Burckhardt's time. - - [12] In Ibn Jubair's time large supplies were brought from the Yemen - mountains. - - [13] The corruption of manners in Mecca is no new thing. See the - letter of the caliph Mahdi on the subject; Wustenfeld, _Chron. Mek._, - iv. 168. - - [14] The exact measurements (which, however, vary according to - different authorities) are stated to be: sides 37 ft. 2 in. and 38 - ft. 4 in.; ends 31 ft. 7 in. and 29 ft.; height 35 ft. - - [15] The Ka'ba of Mahomet's time was the successor of an older - building, said to have been destroyed by fire. It was constructed in - the still usual rude style of Arabic masonry, with string courses of - timber between the stones (like Solomon's Temple). The roof rested on - six pillars; the door was raised above the ground and approached by a - stair (probably on account of the floods which often swept the - valley); and worshippers left their shoes under the stair before - entering. During the first siege of Mecca (A.D. 683), the building - was burned down, the Ibn Zubair reconstructed it on an enlarged scale - and in better style of solid ashlar-work. After his death his most - glaring innovations (the introduction of two doors on a level with - the ground, and the extension of the building lengthwise to include - the Hijr) were corrected by Hajjaj, under orders from the caliph, but - the building retained its more solid structure. The roof now rested - on three pillars, and the height was raised one-half. The Ka'ba was - again entirely rebuilt after the flood of A.D. 1626, but since Hajjaj - there seem to have been no structural changes. - - [16] Hobal was set up within the Temple over the pit that contained - the sacred treasures. His chief function was connected with the - sacred lot to which the Meccans were accustomed to betake themselves - in all matters of difficulty. - - [17] See Ibn Hisham i. 54, Azraki p. 80 ('Uzza in Batn Marr); Yakut - iii. 705 (Otheyda); Bar Hebraeus on Psalm xii. 9. Stones worshipped - by circling round them bore the name _dawar_ or _duwar_ (Krehl, _Rel. - d. Araber_, p. 69). The later Arabs not unnaturally viewed such - cultus as imitated from that of Mecca (Yaqut iv. 622, cf. Dozy, - _Israeliten te Mekka_, p. 125, who draws very perverse inferences). - - [18] The old _kiswa_ is removed on the 25th day of the month before - the pilgrimage, and fragments of it are bought by the pilgrims as - charms. Till the 10th day of the pilgrimage month the Ka'ba is bare. - - [19] Before Islam the Ka'ba was opened every Monday and Thursday; in - the time of Ibn Jubair it was opened with considerable ceremony every - Monday and Friday, and daily in the month Rajab. But, though prayer - within the building is favoured by the example of the Prophet, it is - not compulsory on the Moslem, and even in the time of Ibn Batuta the - opportunities of entrance were reduced to Friday and the birthday of - the Prophet. - - [20] See De Vogue, _Syrie centrale: inscr. sem._; Lady Anne Blunt - _Pilgrimage of Nejd_, ii., and W. R. Smith, in the _Athenaeum_, March - 20, 1880. - - [21] Ibn Jubair speaks of fourteen steps, Ali Bey of four, Burckhardt - of three. The surrounding ground no doubt has risen so that the old - name "hill of Safa" is now inapplicable. - - [22] The latter perhaps was no part of the ancient omra; see - Snouck-Hurgronje, _Het Mekkaansche Feest_ (1880) p. 115 sqq. - - [23] The 27th was also a great day, but this day was in commemoration - of the rebuilding of the Ka'ba by Ibn Jubair. - - [24] The sacrifice is not indispensable except for those who can - afford it and are combining the hajj with the omra. - - [25] On the similar pelting of the supposed graves of Abu Lahab and - his wife (Ibn Jubair, p. 110) and of Abu Righal at Mughammas, see - Noldeke's translation of Tabari, 208. - - - - -MECHANICS. The subject of mechanics may be divided into two parts: (1) -theoretical or abstract mechanics, and (2) applied mechanics. - - -1. THEORETICAL MECHANICS - -Historically theoretical mechanics began with the study of practical -contrivances such as the lever, and the name _mechanics_ (Gr. [Greek: ta -mechanika]), which might more properly be restricted to the theory of -mechanisms, and which was indeed used in this narrower sense by Newton, -has clung to it, although the subject has long attained a far wider -scope. In recent times it has been proposed to adopt the term _dynamics_ -(from Gr. [Greek: dynamis] force,) as including the whole science of the -action of force on bodies, whether at rest or in motion. The subject is -usually expounded under the two divisions of _statics_ and _kinetics_, -the former dealing with the conditions of rest or equilibrium and the -latter with the phenomena of motion as affected by force. To this latter -division the old name of _dynamics_ (in a restricted sense) is still -often applied. The mere geometrical description and analysis of various -types of motion, apart from the consideration of the forces concerned, -belongs to _kinematics_. This is sometimes discussed as a separate -theory, but for our present purposes it is more convenient to introduce -kinematical motions as they are required. We follow also the traditional -practice of dealing first with statics and then with kinetics. This is, -in the main, the historical order of development, and for purposes of -exposition it has many advantages. The laws of equilibrium are, it is -true, necessarily included as a particular case under those of motion; -but there is no real inconvenience in formulating as the basis of -statics a few provisional postulates which are afterwards seen to be -comprehended in a more general scheme. - -The whole subject rests ultimately on the Newtonian laws of motion and -on some natural extensions of them. As these laws are discussed under a -separate heading (MOTION, LAWS OF), it is here only necessary to -indicate the standpoint from which the present article is written. It is -a purely empirical one. Guided by experience, we are able to frame -rules which enable us to say with more or less accuracy what will be the -consequences, or what were the antecedents, of a given state of things. -These rules are sometimes dignified by the name of "laws of nature," but -they have relation to our present state of knowledge and to the degree -of skill with which we have succeeded in giving more or less compact -expression to it. They are therefore liable to be modified from time to -time, or to be superseded by more convenient or more comprehensive modes -of statement. Again, we do not aim at anything so hopeless, or indeed so -useless, as a _complete_ description of any phenomenon. Some features -are naturally more important or more interesting to us than others; by -their relative simplicity and evident constancy they have the first hold -on our attention, whilst those which are apparently accidental and vary -from one occasion to another arc ignored, or postponed for later -examination. It follows that for the purposes of such description as is -possible some process of abstraction is inevitable if our statements are -to be simple and definite. Thus in studying the flight of a stone -through the air we replace the body in imagination by a mathematical -point endowed with a mass-coefficient. The size and shape, the -complicated spinning motion which it is seen to execute, the internal -strains and vibrations which doubtless take place, are all sacrificed in -the mental picture in order that attention may be concentrated on those -features of the phenomenon which are in the first place most interesting -to us. At a later stage in our subject the conception of the ideal rigid -body is introduced; this enables us to fill in some details which were -previously wanting, but others are still omitted. Again, the conception -of a force as concentrated in a mathematical line is as unreal as that -of a mass concentrated in a point, but it is a convenient fiction for -our purpose, owing to the simplicity which it lends to our statements. - -The laws which are to be imposed on these ideal representations are in -the first instance largely at our choice. Any scheme of abstract -dynamics constructed in this way, provided it be self-consistent, is -mathematically legitimate; but from the physical point of view we -require that it should help us to picture the sequence of phenomena as -they actually occur. Its success or failure in this respect can only be -judged a posteriori by comparison of the results to which it leads with -the facts. It is to be noticed, moreover, that all available tests apply -only to the scheme as a whole; owing to the complexity of phenomena we -cannot submit any one of its postulates to verification apart from the -rest. - -It is from this point of view that the question of relativity of motion, -which is often felt to be a stumbling-block on the very threshold of the -subject, is to be judged. By "motion" we mean of necessity motion -relative to some frame of reference which is conventionally spoken of as -"fixed." In the earlier stages of our subject this may be any rigid, or -apparently rigid, structure fixed relatively to the earth. If we meet -with phenomena which do not fit easily into this view, we have the -alternatives either to modify our assumed laws of motion, or to call to -our aid adventitious forces, or to examine whether the discrepancy can -be reconciled by the simpler expedient of a new basis of reference. It -is hardly necessary to say that the latter procedure has hitherto been -found to be adequate. As a first step we adopt a system of rectangular -axes whose origin is fixed in the earth, but whose directions are fixed -by relation to the stars; in the planetary theory the origin is -transferred to the sun, and afterwards to the mass-centre of the solar -system; and so on. At each step there is a gain in accuracy and -comprehensiveness; and the conviction is cherished that _some_ system of -rectangular axes exists with respect to which the Newtonian scheme holds -with all imaginable accuracy. - -A similar account might be given of the conception of time as a -measurable quantity, but the remarks which it is necessary to make under -this head will find a place later. - - The following synopsis shows the scheme on which the treatment is - based:-- - - _Part 1.--Statics._ - - 1. Statics of a particle. - 2. Statics of a system of particles. - 3. Plane kinematics of a rigid body. - 4. Plane statics. - 5. Graphical statics. - 6. Theory of frames. - 7. Three-dimensional kinematics of a rigid body. - 8. Three-dimensional statics. - 9. Work. - 10. Statics of inextensible chains. - 11. Theory of mass-systems. - - _Part 2.--Kinetics._ - - 12. Rectilinear motion. - 13. General motion of a particle. - 14. Central forces. Hodograph. - 15. Kinetics of a system of discrete particles. - 16. Kinetics of a rigid body. Fundamental principles. - 17. Two-dimensional problems. - 18. Equations of motion in three dimensions. - 19. Free motion of a solid. - 20. Motion of a solid of revolution. - 21. Moving axes of reference. - 22. Equations of motion in generalized co-ordinates. - 23. Stability of equilibrium. Theory of vibrations. - - -PART I.--STATICS - -S 1. _Statics of a Particle._--By a _particle_ is meant a body whose -position can for the purpose in hand be sufficiently specified by a -mathematical point. It need not be "infinitely small," or even small -compared with ordinary standards; thus in astronomy such vast bodies as -the sun, the earth, and the other planets can for many purposes be -treated merely as points endowed with mass. - -A _force_ is conceived as an effort having a certain direction and a -certain magnitude. It is therefore adequately represented, for -mathematical purposes, by a straight line AB drawn in the direction in -question, of length proportional (on any convenient scale) to the -magnitude of the force. In other words, a force is mathematically of the -nature of a "vector" (see VECTOR ANALYSIS, QUATERNIONS). In most -questions of pure statics we are concerned only with the _ratios_ of the -various forces which enter into the problem, so that it is indifferent -what _unit_ of force is adopted. For many purposes a gravitational -system of measurement is most natural; thus we speak of a force of so -many pounds or so many kilogrammes. The "absolute" system of measurement -will be referred to below in PART II., KINETICS. It is to be remembered -that all "force" is of the nature of a push or a pull, and that -according to the accepted terminology of modern mechanics such phrases -as "force of inertia," "accelerating force," "moving force," once -classical, are proscribed. This rigorous limitation of the meaning of -the word is of comparatively recent origin, and it is perhaps to be -regretted that some more technical term has not been devised, but the -convention must now be regarded as established. - -[Illustration: FIG. 1.] - -The fundamental postulate of this part of our subject is that the two -forces acting on a particle may be compounded by the "parallelogram -rule." Thus, if the two forces P,Q be represented by the lines OA, OB, -they can be replaced by a single force R represented by the diagonal OC -of the parallelogram determined by OA, OB. This is of course a physical -assumption whose propriety is justified solely by experience. We shall -see later that it is implied in Newton's statement of his Second Law of -motion. In modern language, forces are compounded by "vector-addition"; -thus, if we draw in succession vectors [->HK], [->KL] to represent P, Q, -the force R is represented by the vector [->HL] which is the "geometric -sum" of [->HK], [->KL]. - -By successive applications of the above rule any number of forces acting -on a particle may be replaced by a single force which is the vector-sum -of the given forces: this single force is called the _resultant_. Thus -if [->AB], [->BC], [->CD] ..., [->HK] be vectors representing the given -forces, the resultant will be given by [->AK]. It will be understood -that the figure ABCD ... K need not be confined to one plane. - -[Illustration: FIG. 2.] - -If, in particular, the point K coincides with A, so that the resultant -vanishes, the given system of forces is said to be in _equilibrium_--i.e. -the particle could remain permanently at rest under its action. This is -the proposition known as the _polygon of forces_. In the particular case -of three forces it reduces to the _triangle of forces_, viz. "If three -forces acting on a particle are represented as to magnitude and direction -by the sides of a triangle taken in order, they are in equilibrium." - -A sort of converse proposition is frequently useful, viz. if three -forces acting on a particle be in equilibrium, and any triangle be -constructed whose sides are respectively parallel to the forces, the -magnitudes of the forces will be to one another as the corresponding -sides of the triangle. This follows from the fact that all such -triangles are necessarily similar. - -[Illustration: FIG. 3.] - - As a simple example of the geometrical method of treating statical - problems we may consider the equilibrium of a particle on a "rough" - inclined plane. The usual empirical law of sliding friction is that - the mutual action between two plane surfaces in contact, or between a - particle and a curve or surface, cannot make with the normal an angle - exceeding a certain limit [lambda] called the _angle of friction_. If - the conditions of equilibrium require an obliquity greater than this, - sliding will take place. The precise value of [lambda] will vary with - the nature and condition of the surfaces in contact. In the case of a - body simply resting on an inclined plane, the reaction must of course - be vertical, for equilibrium, and the slope [alpha] of the plane must - therefore not exceed [lambda]. For this reason [lambda] is also known - as the _angle of repose_. If [alpha] > [lambda], a force P must be - applied in order to maintain equilibrium; let [theta] be the - inclination of P to the plane, as shown in the left-hand diagram. The - relations between this force P, the gravity W of the body, and the - reaction S of the plane are then determined by a triangle of forces - HKL. Since the inclination of S to the normal cannot exceed [lambda] - on either side, the value of P must lie between two limits which are - represented by L1H, L2H, in the right-hand diagram. Denoting these - limits by P1, P2, we have - - P1/W = L1H/HK = sin ([alpha] - [lambda])/cos ([theta] + [lambda]), - P2/W = L2H/HK = sin ([alpha] + [lambda])/cos ([theta] - [lambda]). - - It appears, moreover, that if [theta] be varied P will be least when - L1H is at right angles to KL1, in which case P1 = W sin ([alpha] - - [lambda]), corresponding to [theta] = -[lambda]. - -[Illustration: FIG. 4.] - -Just as two or more forces can be combined into a single resultant, so a -single force may be _resolved_ into _components_ acting in assigned -directions. Thus a force can be uniquely resolved into two components -acting in two assigned directions in the same plane with it by an -inversion of the parallelogram construction of fig. 1. If, as is usually -most convenient, the two assigned directions are at right angles, the -two components of a force P will be P cos [theta], P sin [theta], where -[theta] is the inclination of P to the direction of the former -component. This leads to formulae for the analytical reduction of a -system of coplanar forces acting on a particle. Adopting rectangular -axes Ox, Oy, in the plane of the forces, and distinguishing the various -forces of the system by suffixes, we can replace the system by two -forces X, Y, in the direction of co-ordinate axes; viz.-- - - X = P1 cos [theta]1 + P2 cos [theta]2 + ... = [Sigma](P cos [theta]), } - Y = P1 sin [theta]1 + P2 sin [theta]2 + ... = [Sigma](P sin [theta]). } (1) - -These two forces X, Y, may be combined into a single resultant R making -an angle [phi] with Ox, provided - - X = R cos [phi], Y = R sin [phi], (2) - -whence - - R^2 = X^2 + Y^2, tan [phi] = Y/X. (3) - -For equilibrium we must have R = 0, which requires X = 0, Y = 0; in -words, the sum of the components of the system must be zero for each of -two perpendicular directions in the plane. - -[Illustration: FIG. 5.] - -A similar procedure applies to a three-dimensional system. Thus if, O -being the origin, [->OH] represent any force P of the system, the planes -drawn through H parallel to the co-ordinate planes will enclose with the -latter a parallelepiped, and it is evident that [->OH] is the geometric -sum of [->OA], [->AN], [->NH], or [->OA], [->OB], [->OC], in the figure. -Hence P is equivalent to three forces Pl, Pm, Pn acting along Ox, Oy, -Oz, respectively, where l, m, n, are the "direction-ratios" of [->OH]. -The whole system can be reduced in this way to three forces - - X = [Sigma] (Pl), Y = [Sigma] (Pm), Z = [Sigma] (Pn), (4) - -acting along the co-ordinate axes. These can again be combined into a -single resultant R acting in the direction ([lambda], [mu], [nu]), -provided - - X = R[lambda], Y = R[mu], Z = R[nu]. (5) - -If the axes are rectangular, the direction-ratios become -direction-cosines, so that [lambda]^2 + [mu]^2 + [nu]^2 = 1, whence - - R^2 = X^2 + Y^2 + Z^2. (6) - -The conditions of equilibrium are X = 0, Y = 0, Z = 0. - -S 2. _Statics of a System of Particles._--We assume that the mutual -forces between the pairs of particles, whatever their nature, are -subject to the "Law of Action and Reaction" (Newton's Third Law); i.e. -the force exerted by a particle A on a particle B, and the force exerted -by B on A, are equal and opposite in the line AB. The problem of -determining the possible configurations of equilibrium of a system of -particles subject to extraneous forces which are known functions of the -positions of the particles, and to internal forces which are known -functions of the distances of the pairs of particles between which they -act, is in general determinate. For if n be the number of particles, the -3n conditions of equilibrium (three for each particle) are equal in -number to the 3n Cartesian (or other) co-ordinates of the particles, -which are to be found. If the system be subject to frictionless -constraints, e.g. if some of the particles be constrained to lie on -smooth surfaces, or if pairs of particles be connected by inextensible -strings, then for each geometrical relation thus introduced we have an -unknown reaction (e.g. the pressure of the smooth surface, or the -tension of the string), so that the problem is still determinate. - -[Illustration: FIG. 6.] - -[Illustration: FIG. 7.] - - The case of the _funicular polygon_ will be of use to us later. A - number of particles attached at various points of a string are acted - on by given extraneous forces P1, P2, P3 ... respectively. The - relation between the three forces acting on any particle, viz. the - extraneous force and the tensions in the two adjacent portions of the - string can be exhibited by means of a triangle of forces; and if the - successive triangles be drawn to the same scale they can be fitted - together so as to constitute a single _force-diagram_, as shown in - fig. 6. This diagram consists of a polygon whose successive sides - represent the given forces P1, P2, P3 ..., and of a series of lines - connecting the vertices with a point O. These latter lines measure the - tensions in the successive portions of string. As a special, but very - important case, the forces P1, P2, P3 ... may be parallel, e.g. they - may be the weights of the several particles. The polygon of forces is - then made up of segments of a vertical line. We note that the tensions - have now the same horizontal projection (represented by the dotted - line in fig. 7). It is further of interest to note that if the weights - be all equal, and at equal horizontal intervals, the vertices of the - funicular will lie on a parabola whose axis is vertical. To prove this - statement, let A, B, C, D ... be successive vertices, and let H, K ... - be the middle points of AC, BD ...; then BH, CK ... will be vertical - by the hypothesis, and since the geometric sum of [->BA], [->BC] is - represented by 2[->BH], the tension in BA: tension in BC: weight at B - - as BA: BC: 2BH. - - [Illustration: FIG. 8.] - - The tensions in the successive portions of the string are therefore - proportional to the respective lengths, and the lines BH, CK ... are - all equal. Hence AD, BC are parallel and are bisected by the same - vertical line; and a parabola with vertical axis can therefore be - described through A, B, C, D. The same holds for the four points B, C, - D, E and so on; but since a parabola is uniquely determined by the - direction of its axis and by three points on the curve, the successive - parabolas ABCD, BCDE, CDEF ... must be coincident. - -S 3. _Plane Kinematics of a Rigid Body._--The ideal _rigid body_ is one -in which the distance between any two points is invariable. For the -present we confine ourselves to the consideration of displacements in -two dimensions, so that the body is adequately represented by a thin -lamina or plate. - -[Illustration: FIG. 9.] - -The position of a lamina movable in its own plane is determinate when we -know the positions of any two points A, B of it. Since the four -co-ordinates (Cartesian or other) of these two points are connected by -the relation which expresses the invariability of the length AB, it is -plain that virtually three independent elements are required and suffice -to specify the position of the lamina. For instance, the lamina may in -general be fixed by connecting any three points of it by rigid links to -three fixed points in its plane. The three independent elements may be -chosen in a variety of ways (e.g. they may be the lengths of the three -links in the above example). They may be called (in a generalized sense) -the _co-ordinates_ of the lamina. The lamina when perfectly free to move -in its own plane is said to have _three degrees of freedom_. - -[Illustration: FIG. 10.] - -By a theorem due to M. Chasles any displacement whatever of the lamina -in its own plane is equivalent to a rotation about some finite or -infinitely distant point J. For suppose that in consequence of the -displacement a point of the lamina is brought from A to B, whilst the -point of the lamina which was originally at B is brought to C. Since AB, -BC, are two different positions of the same line in the lamina they are -equal, and it is evident that the rotation could have been effected by a -rotation about J, the centre of the circle ABC, through an angle AJB. As -a special case the three points A, B, C may be in a straight line; J is -then at infinity and the displacement is equivalent to a pure -_translation_, since every point of the lamina is now displaced parallel -to AB through a space equal to AB. - -[Illustration: FIG. 11.] - -Next, consider any continuous motion of the lamina. The latter may be -brought from any one of its positions to a neighbouring one by a -rotation about the proper centre. The limiting position J of this -centre, when the two positions are taken infinitely close to one -another, is called the _instantaneous centre_. If P, P' be consecutive -positions of the same point, and [delta][theta] the corresponding angle -of rotation, then ultimately PP' is at right angles to JP and equal to -JP.[delta][theta]. The instantaneous centre will have a certain locus in -space, and a certain locus in the lamina. These two loci are called -_pole-curves_ or _centrodes_, and are sometimes distinguished as the -_space-centrode_ and the _body-centrode_, respectively. In the -continuous motion in question the latter curve rolls without slipping on -the former (M. Chasles). Consider in fact any series of successive -positions 1, 2, 3... of the lamina (fig. 11); and let J12, J23, J34... -be the positions in space of the centres of the rotations by which the -lamina can be brought from the first position to the second, from the -second to the third, and so on. Further, in the position 1, let J12, -J'23, J'34 ... be the points of the lamina which have become the -successive centres of rotation. The given series of positions will be -assumed in succession if we imagine the lamina to rotate first about J12 -until J'23 comes into coincidence with J23, then about J23 until J'34 -comes into coincidence with J34, and so on. This is equivalent to -imagining the polygon J12 J'23 J'34 ..., supposed fixed in the lamina, -to roll on the polygon J12 J23 J34 ..., which is supposed fixed in -space. By imagining the successive positions to be taken infinitely -close to one another we derive the theorem stated. The particular case -where both centrodes are circles is specially important in mechanism. - -[Illustration: FIG. 12.] - - The theory may be illustrated by the case of "three-bar motion." Let - ABCD be any quadrilateral formed of jointed links. If, AB being held - fixed, the quadrilateral be slightly deformed, it is obvious that the - instantaneous centre J will be at the intersection of the straight - lines AD, BC, since the displacements of the points D, C are - necessarily at right angles to AD, BC, respectively. Hence these - displacements are proportional to JD, JC, and therefore to DD' CC', - where C'D' is any line drawn parallel to CD, meeting BC, AD in C', D', - respectively. The determination of the centrodes in three-bar motion - is in general complicated, but in one case, that of the "crossed - parallelogram" (fig. 13), they assume simple forms. We then have AB = - DC and AD = BC, and from the symmetries of the figure it is plain that - - AJ + JB = CJ + JD = AD. - - Hence the locus of J relative to AB, and the locus relative to CD are - equal ellipses of which A, B and C, D are respectively the foci. It - may be noticed that the lamina in fig. 9 is not, strictly speaking, - fixed, but admits of infinitesimal displacement, whenever the - directions of the three links are concurrent (or parallel). - -[Illustration: FIG. 13.] - -The matter may of course be treated analytically, but we shall only -require the formula for infinitely small displacements. If the origin of -rectangular axes fixed in the lamina be shifted through a space whose -projections on the original directions of the axes are [lambda], [mu], -and if the axes are simultaneously turned through an angle [epsilon], -the co-ordinates of a point of the lamina, relative to the original -axes, are changed from x, y to [lambda] + x cos [epsilon] - y sin -[epsilon], [mu] + x sin [epsilon] + y cos [epsilon], or [lambda] + x - -y[epsilon], [mu] + x[epsilon] + y, ultimately. Hence the component -displacements are ultimately - - [delta]x = [lambda] - y[epsilon], [delta]y = [mu] + x[epsilon] (1) - -If we equate these to zero we get the co-ordinates of the instantaneous -centre. - -S 4. _Plane Statics._--The statics of a rigid body rests on the -following two assumptions:-- - -(i) A force may be supposed to be applied indifferently at any point in -its line of action. In other words, a force is of the nature of a -"bound" or "localized" vector; it is regarded as resident in a certain -line, but has no special reference to any particular point of the line. - -(ii) Two forces in intersecting lines may be replaced by a force which -is their geometric sum, acting through the intersection. The theory of -parallel forces is included as a limiting case. For if O, A, B be any -three points, and m, n any scalar quantities, we have in vectors - - m . [->OA] + n.[->OB] = (m + n) [->OC], (1) - -provided - - m . [->CA] + n.[->CB] = 0. (2) - -Hence if forces P, Q act in OA, OB, the resultant R will pass through C, -provided - - m = P/OA, n = Q/OB; - -also - - R = P.OC/OA + Q.OC/OB, (3) - -and - - P.AC : Q.CB = OA : OB. (4) - -These formulae give a means of constructing the resultant by means of -any transversal AB cutting the lines of action. If we now imagine the -point O to recede to infinity, the forces P, Q and the resultant R are -parallel, and we have - - R = P + Q, P.AC = Q.CB. (5) - -[Illustration: FIG. 14.] - -When P, Q have opposite signs the point C divides AB externally on the -side of the greater force. The investigation fails when P + Q = 0, since -it leads to an infinitely small resultant acting in an infinitely -distant line. A combination of two equal, parallel, but oppositely -directed forces cannot in fact be replaced by anything simpler, and must -therefore be recognized as an independent entity in statics. It was -called by L. Poinsot, who first systematically investigated its -properties, a _couple_. - -We now restrict ourselves for the present to the systems of forces in -one plane. By successive applications of (ii) any such coplanar system -can in general be reduced to a _single resultant_ acting in a definite -line. As exceptional cases the system may reduce to a couple, or it may -be in equilibrium. - -[Illustration: FIG. 15.] - -The _moment_ of a force about a point O is the product of the force into -the perpendicular drawn to its line of action from O, this perpendicular -being reckoned positive or negative according as O lies on one side or -other of the line of action. If we mark off a segment AB along the line -of action so as to represent the force completely, the moment is -represented as to magnitude by twice the area of the triangle OAB, and -the usual convention as to sign is that the area is to be reckoned -positive or negative according as the letters O, A, B, occur in -"counter-clockwise" or "clockwise" order. - -[Illustration: FIG. 16.] - -The sum of the moments of two forces about any point O is equal to the -moment of their resultant (P. Varignon, 1687). Let AB, AC (fig. 16) -represent the two forces, AD their resultant; we have to prove that the -sum of the triangles OAB, OAC is equal to the triangle OAD, regard being -had to signs. Since the side OA is common, we have to prove that the sum -of the perpendiculars from B and C on OA is equal to the perpendicular -from D on OA, these perpendiculars being reckoned positive or negative -according as they lie to the right or left of AO. Regarded as a -statement concerning the orthogonal projections of the vectors [->AB] -and [->AC] (or BD), and of their sum [->AD], on a line perpendicular to -AO, this is obvious. - -It is now evident that in the process of reduction of a coplanar system -no change is made at any stage either in the sum of the projections of -the forces on any line or in the sum of their moments about any point. -It follows that the single resultant to which the system in general -reduces is uniquely determinate, i.e. it acts in a definite line and has -a definite magnitude and sense. Again it is necessary and sufficient for -equilibrium that the sum of the projections of the forces on each of two -perpendicular directions should vanish, and (moreover) that the sum of -the moments about some one point should be zero. The fact that three -independent conditions must hold for equilibrium is important. The -conditions may of course be expressed in different (but equivalent) -forms; e.g. the sum of the moments of the forces about each of the three -points which are not collinear must be zero. - -[Illustration: FIG. 17.] - -The particular case of three forces is of interest. If they are not all -parallel they must be concurrent, and their vector-sum must be zero. -Thus three forces acting perpendicular to the sides of a triangle at the -middle points will be in equilibrium provided they are proportional to -the respective sides, and act all inwards or all outwards. This result -is easily extended to the case of a polygon of any number of sides; it -has an important application in hydrostatics. - - Again, suppose we have a bar AB resting with its ends on two smooth - inclined planes which face each other. Let G be the centre of gravity - (S 11), and let AG = a, GB = b. Let [alpha], [beta] be the - inclinations of the planes, and [theta] the angle which the bar makes - with the vertical. The position of equilibrium is determined by the - consideration that the reactions at A and B, which are by hypothesis - normal to the planes, must meet at a point J on the vertical through - G. Hence - - JG/a = sin ([theta] - [alpha])/sin [alpha], JG/b = sin ([theta] + [beta])/sin [beta], - - whence - - a cot [alpha] - b cot [beta] - cot [theta] = ----------------------------. (6) - a + b - - If the bar is uniform we have a = b, and - - cot [theta] = (1/2) (cot [alpha] - cot [beta]). (7) - - The problem of a rod suspended by strings attached to two points of it - is virtually identical, the tensions of the strings taking the place - of the reactions of the planes. - -[Illustration: FIG. 18.] - -Just as a system of forces is in general equivalent to a single force, -so a given force can conversely be replaced by combinations of other -forces, in various ways. For instance, a given force (and consequently a -system of forces) can be replaced in one and only one way by three -forces acting in three assigned straight lines, provided these lines be -not concurrent or parallel. Thus if the three lines form a triangle ABC, -and if the given force F meet BC in H, then F can be resolved into two -components acting in HA, BC, respectively. And the force in HA can be -resolved into two components acting in BC, CA, respectively. A simple -graphical construction is indicated in fig. 19, where the dotted lines -are parallel. As an example, any system of forces acting on the lamina -in fig. 9 is balanced by three determinate tensions (or thrusts) in the -three links, provided the directions of the latter are not concurrent. - -[Illustration: FIG. 19.] - - If P, Q, R, be any three forces acting along BC, CA, AB, respectively, - the line of action of the resultant is determined by the consideration - that the sum of the moments about any point on it must vanish. Hence - in "trilinear" co-ordinates, with ABC as fundamental triangle, its - equation is P[alpha] + Q[beta] + R[gamma] = 0. If P : Q : R = a : b : - c, where a, b, c are the lengths of the sides, this becomes the "line - at infinity," and the forces reduce to a couple. - -[Illustration: FIG. 20.] - -The sum of the moments of the two forces of a couple is the same about -any point in the plane. Thus in the figure the sum of the moments about -O is P.OA - P.OB or P.AB, which is independent of the position of O. -This sum is called the _moment of the couple_; it must of course have -the proper sign attributed to it. It easily follows that any two couples -of the same moment are equivalent, and that any number of couples can be -replaced by a single couple whose moment is the sum of their moments. -Since a couple is for our purposes sufficiently represented by its -moment, it has been proposed to substitute the name _torque_ (or -twisting effort), as free from the suggestion of any special pair of -forces. - -A system of forces represented completely by the sides of a plane -polygon taken in order is equivalent to a couple whose moment is -represented by twice the area of the polygon; this is proved by taking -moments about any point. If the polygon intersects itself, care must be -taken to attribute to the different parts of the area their proper -signs. - -[Illustration: FIG. 21.] - -Again, any coplanar system of forces can be replaced by a single force R -acting at any assigned point O, together with a couple G. The force R is -the geometric sum of the given forces, and the moment (G) of the couple -is equal to the sum of the moments of the given forces about O. The -value of G will in general vary with the position of O, and will vanish -when O lies on the line of action of the single resultant. - -[Illustration: FIG. 22.] - -The formal analytical reduction of a system of coplanar forces is as -follows. Let (x1, y1), (x2, y2), ... be the rectangular co-ordinates of -any points A1, A2, ... on the lines of action of the respective forces. -The force at A1 may be replaced by its components X1, Y1, parallel to -the co-ordinate axes; that at A2 by its components X2, Y2, and so on. -Introducing at O two equal and opposite forces [+-]X1 in Ox, we see that X1 -at A1 may be replaced by an equal and parallel force at O together with -a couple -y1X1. Similarly the force Y1 at A1 may be replaced by a force -Y1 at O together with a couple x1Y1. The forces X1, Y1, at O can thus be -transferred to O provided we introduce a couple x1Y1 - y1X1. Treating -the remaining forces in the same way we get a force X1 + X2 + ... or -[Sigma](X) along Ox, a force Y1 + Y2 + ... or [Sigma](Y) along Oy, and a -couple (x1Y1 - y1X1) + (x2Y2 - y2X2) + ... or [Sigma](xY - yX). The -three conditions of equilibrium are therefore - - [Sigma](X) = 0, [Sigma](Y) = 0, [Sigma](xY - yX) = 0. (8) - -If O' be a point whose co-ordinates are ([xi], [eta]), the moment of the -couple when the forces are transferred to O' as a new origin will be -[Sigma]{(x - [xi]) Y - (y - [eta]) X}. This vanishes, i.e. the system -reduces to a single resultant through O', provided - - -[xi].[Sigma](Y) + [eta].[Sigma](X) + [Sigma](xY - yX) = 0. (9) - -If [xi], [eta] be regarded as current co-ordinates, this is the equation -of the line of action of the single resultant to which the system is in -general reducible. - -If the forces are all parallel, making say an angle [theta] with Ox, we -may write X1 = P1 cos [theta], Y1 = P1 sin [theta], X2 = P2 cos [theta], -Y2 = P2 sin [theta], .... The equation (9) then becomes - - {[Sigma](xP) - [xi].[Sigma](P)} sin [theta] - {[Sigma](yP) - [eta].[Sigma](P)} cos [theta] = 0. (10) - -If the forces P1, P2, ... be turned in the same sense through the same -angle about the respective points A1, A2, ... so as to remain parallel, -the value of [theta] is alone altered, and the resultant [Sigma](P) -passes always through the point - - [Sigma](xP) [Sigma](yP) - [|x] = -----------, [|y] = -----------, (11) - [Sigma](P) [Sigma](P) - -which is determined solely by the configuration of the points A1, A2, -... and by the ratios P1: P2: ... of the forces acting at them -respectively. This point is called the _centre_ of the given system of -parallel forces; it is finite and determinate unless [Sigma](P) = 0. A -geometrical proof of this theorem, which is not restricted to a -two-dimensional system, is given later (S 11). It contains the theory of -the _centre of gravity_ as ordinarily understood. For if we have an -assemblage of particles whose mutual distances are small compared with -the dimensions of the earth, the forces of gravity on them constitute a -system of sensibly parallel forces, sensibly proportional to the -respective masses. If now the assemblage be brought into any other -position relative to the earth, without alteration of the mutual -distances, this is equivalent to a rotation of the directions of the -forces relatively to the assemblage, the ratios of the forces remaining -unaltered. Hence there is a certain point, fixed relatively to the -assemblage, through which the resultant of gravitational action always -passes; this resultant is moreover equal to the sum of the forces on the -several particles. - -[Illustration: FIG. 23.] - - The theorem that any coplanar system of forces can be reduced to a - force acting through any assigned point, together with a couple, has - an important illustration in the theory of the distribution of - shearing stress and bending moment in a horizontal beam, or other - structure, subject to vertical extraneous forces. If we consider any - vertical section P, the forces exerted across the section by the - portion of the structure on one side on the portion on the other may - be reduced to a vertical force F at P and a couple M. The force - measures the _shearing stress_, and the couple the _bending moment_ at - P; we will reckon these quantities positive when the senses are as - indicated in the figure. - - If the remaining forces acting on the portion of the structure on - either side of P are known, then resolving vertically we find F, and - taking moments about P we find M. Again if PQ be any segment of the - beam which is free from load, Q lying to the right of P, we find - - F_P = F_Q, M_P - M_Q = -F.PQ; (12) - - hence F is constant between the loads, whilst M decreases as we travel - to the right, with a constant gradient -F. If PQ be a short segment - containing an isolated load W, we have - - F_Q - F_P = -W, M_Q = M_P; (13) - - hence F is discontinuous at a concentrated load, diminishing by an - amount equal to the load as we pass the loaded point to the right, - whilst M is continuous. Accordingly the graph of F for any system of - isolated loads will consist of a series of horizontal lines, whilst - that of M will be a continuous polygon. - - [Illustration: FIG. 24.] - - To pass to the case of continuous loads, let x be measured - horizontally along the beam to the right. The load on an element - [delta]x of the beam may be represented by w[delta]x, where w is in - general a function of x. The equations (12) are now replaced by - - [delta]F = -w[delta]x, [delta]M = -F[delta]x, - - whence - _ _ - / Q / Q - F_Q - F_P = - | w dx, M_Q - M_P = - | F dx. (14) - _/P _/P - - The latter relation shows that the bending moment varies as the area - cut off by the ordinate in the graph of F. In the case of uniform load - we have - - F = -wx + A, M = (1/2)wx^2 - Ax + B, (15) - - where the arbitrary constants A,B are to be determined by the - conditions of the special problem, e.g. the conditions at the ends of - the beam. The graph of F is a straight line; that of M is a parabola - with vertical axis. In all cases the graphs due to different - distributions of load may be superposed. The figure shows the case of - a uniform heavy beam supported at its ends. - -[Illustration: FIG. 25.] - -[Illustration: FIG. 26.] - -S 5. _Graphical Statics._--A graphical method of reducing a plane system -of forces was introduced by C. Culmann (1864). It involves the -construction of two figures, a _force-diagram_ and a _funicular -polygon_. The force-diagram is constructed by placing end to end a -series of vectors representing the given forces in magnitude and -direction, and joining the vertices of the polygon thus formed to an -arbitrary _pole_ O. The funicular or link polygon has its vertices on -the lines of action of the given forces, and its sides respectively -parallel to the lines drawn from O in the force-diagram; in particular, -the two sides meeting in any vertex are respectively parallel to the -lines drawn from O to the ends of that side of the force-polygon which -represents the corresponding force. The relations will be understood -from the annexed diagram, where corresponding lines in the force-diagram -(to the right) and the funicular (to the left) are numbered similarly. -The sides of the force-polygon may in the first instance be arranged in -any order; the force-diagram can then be completed in a doubly infinite -number of ways, owing to the arbitrary position of O; and for each -force-diagram a simply infinite number of funiculars can be drawn. The -two diagrams being supposed constructed, it is seen that each of the -given systems of forces can be replaced by two components acting in the -sides of the funicular which meet at the corresponding vertex, and that -the magnitudes of these components will be given by the corresponding -triangle of forces in the force-diagram; thus the force 1 in the figure -is equivalent to two forces represented by 01 and 12. When this process -of replacement is complete, each terminated side of the funicular is the -seat of two forces which neutralize one another, and there remain only -two uncompensated forces, viz., those resident in the first and last -sides of the funicular. If these sides intersect, the resultant acts -through the intersection, and its magnitude and direction are given by -the line joining the first and last sides of the force-polygon (see fig. -26, where the resultant of the four given forces is denoted by R). As a -special case it may happen that the force-polygon is closed, i.e. its -first and last points coincide; the first and last sides of the -funicular will then be parallel (unless they coincide), and the two -uncompensated forces form a couple. If, however, the first and last -sides of the funicular coincide, the two outstanding forces neutralize -one another, and we have equilibrium. Hence the necessary and sufficient -conditions of equilibrium are that the force-polygon and the funicular -should both be closed. This is illustrated by fig. 26 if we imagine the -force R, reversed, to be included in the system of given forces. - -It is evident that a system of jointed bars having the shape of the -funicular polygon would be in equilibrium under the action of the given -forces, supposed applied to the joints; moreover any bar in which the -stress is of the nature of a tension (as distinguished from a thrust) -might be replaced by a string. This is the origin of the names -"link-polygon" and "funicular" (cf. S 2). - - If funiculars be drawn for two positions O, O' of the pole in the - force-diagram, their corresponding sides will intersect on a straight - line parallel to OO'. This is essentially a theorem of projective - geometry, but the following statical proof is interesting. Let AB - (fig. 27) be any side of the force-polygon, and construct the - corresponding portions of the two diagrams, first with O and then with - O' as pole. The force corresponding to AB may be replaced by the two - components marked x, y; and a force corresponding to BA may be - represented by the two components marked x', y'. Hence the forces x, - y, x', y' are in equilibrium. Now x, x' have a resultant through H, - represented in magnitude and direction by OO', whilst y, y' have a - resultant through K represented in magnitude and direction by O'O. - Hence HK must be parallel to OO'. This theorem enables us, when one - funicular has been drawn, to construct any other without further - reference to the force-diagram. - - [Illustration: FIG. 27.] - - The complete figures obtained by drawing first the force-diagrams of a - system of forces in equilibrium with two distinct poles O, O', and - secondly the corresponding funiculars, have various interesting - relations. In the first place, each of these figures may be conceived - as an orthogonal projection of a closed plane-faced polyhedron. As - regards the former figure this is evident at once; viz. the polyhedron - consists of two pyramids with vertices represented by O, O', and a - common base whose perimeter is represented by the force-polygon (only - one of these is shown in fig. 28). As regards the funicular diagram, - let LM be the line on which the pairs of corresponding sides of the - two polygons meet, and through it draw any two planes [omega], - [omega]'. Through the vertices A, B, C, ... and A', B', C', ... of the - two funiculars draw normals to the plane of the diagram, to meet - [omega] and [omega]' respectively. The points thus obtained are - evidently the vertices of a polyhedron with plane faces. - - [Illustration: FIG. 28.] - - [Illustration: FIG. 29.] - - To every line in either of the original figures corresponds of course - a parallel line in the other; moreover, it is seen that concurrent - lines in either figure correspond to lines forming a closed polygon in - the other. Two plane figures so related are called _reciprocal_, since - the properties of the first figure in relation to the second are the - same as those of the second with respect to the first. A still simpler - instance of reciprocal figures is supplied by the case of concurrent - forces in equilibrium (fig. 29). The theory of these reciprocal - figures was first studied by J. Clerk Maxwell, who showed amongst - other things that a reciprocal can always be drawn to any figure which - is the orthogonal projection of a plane-faced polyhedron. If in fact - we take the pole of each face of such a polyhedron with respect to a - paraboloid of revolution, these poles will be the vertices of a second - polyhedron whose edges are the "conjugate lines" of those of the - former. If we project both polyhedra orthogonally on a plane - perpendicular to the axis of the paraboloid, we obtain two figures - which are reciprocal, except that corresponding lines are orthogonal - instead of parallel. Another proof will be indicated later (S 8) in - connexion with the properties of the linear complex. It is convenient - to have a notation which shall put in evidence the reciprocal - character. For this purpose we may designate the points in one figure - by letters A, B, C, ... and the corresponding polygons in the other - figure by the same letters; a line joining two points A, B in one - figure will then correspond to the side common to the two polygons A, - B in the other. This notation was employed by R. H. Bow in connexion - with the theory of frames (S 6, and see also APPLIED MECHANICS below) - where reciprocal diagrams are frequently of use (cf. DIAGRAM). - - When the given forces are all parallel, the force-polygon consists of - a series of segments of a straight line. This case has important - practical applications; for instance we may use the method to find the - pressures on the supports of a beam loaded in any given manner. Thus - if AB, BC, CD represent the given loads, in the force-diagram, we - construct the sides corresponding to OA, OB, OC, OD in the funicular; - we then draw the _closing line_ of the funicular polygon, and a - parallel OE to it in the force diagram. The segments DE, EA then - represent the upward pressures of the two supports on the beam, which - pressures together with the given loads constitute a system of forces - in equilibrium. The pressures of the beam on the supports are of - course represented by ED, AE. The two diagrams are portions of - reciprocal figures, so that Bow's notation is applicable. - - [Illustration: FIG. 30.] - - [Illustration: FIG. 31.] - - A graphical method can also be applied to find the moment of a force, - or of a system of forces, about any assigned point P. Let F be a force - represented by AB in the force-diagram. Draw a parallel through P to - meet the sides of the funicular which correspond to OA, OB in the - points H, K. If R be the intersection of these sides, the triangles - OAB, RHK are similar, and if the perpendiculars OM, RN be drawn we - have - - HK.OM = AB.RN = F.RN, - - which is the moment of F about P. If the given forces are all parallel - (say vertical) OM is the same for all, and the moments of the several - forces about P are represented on a certain scale by the lengths - intercepted by the successive pairs of sides on the vertical through - P. Moreover, the moments are compounded by adding (geometrically) the - corresponding lengths HK. Hence if a system of vertical forces be in - equilibrium, so that the funicular polygon is closed, the length which - this polygon intercepts on the vertical through any point P gives the - sum of the moments about P of all the forces on one side of this - vertical. For instance, in the case of a beam in equilibrium under any - given loads and the reactions at the supports, we get a graphical - representation of the distribution of bending moment over the beam. - The construction in fig. 30 can easily be adjusted so that the closing - line shall be horizontal; and the figure then becomes identical with - the bending-moment diagram of S 4. If we wish to study the effects of - a movable load, or system of loads, in different positions on the - beam, it is only necessary to shift the lines of action of the - pressures of the supports relatively to the funicular, keeping them at - the same, distance apart; the only change is then in the position of - the closing line of the funicular. It may be remarked that since this - line joins homologous points of two "similar" rows it will envelope a - parabola. - -The "centre" (S 4) of a system of parallel forces of given magnitudes, -acting at given points, is easily determined graphically. We have only -to construct the line of action of the resultant for each of two -arbitrary directions of the forces; the intersection of the two lines -gives the point required. The construction is neatest if the two -arbitrary directions are taken at right angles to one another. - -S 6. _Theory of Frames._--A _frame_ is a structure made up of pieces, or -_members_, each of which has two _joints_ connecting it with other -members. In a two-dimensional frame, each joint may be conceived as -consisting of a small cylindrical pin fitting accurately and smoothly -into holes drilled through the members which it connects. This -supposition is a somewhat ideal one, and is often only roughly -approximated to in practice. We shall suppose, in the first instance, -that extraneous forces act on the frame at the joints only, i.e. on the -pins. - -On this assumption, the reactions on any member at its two joints must -be equal and opposite. This combination of equal and opposite forces is -called the _stress_ in the member; it may be a _tension_ or a _thrust_. -For diagrammatic purposes each member is sufficiently represented by a -straight line terminating at the two joints; these lines will be -referred to as the _bars_ of the frame. - -[Illustration: FIG. 32.] - -In structural applications a frame must be _stiff_, or _rigid_, i.e. it -must be incapable of deformation without alteration of length in at -least one of its bars. It is said to be _just rigid_ if it ceases to be -rigid when any one of its bars is removed. A frame which has more bars -than are essential for rigidity may be called _over-rigid_; such a frame -is in general self-stressed, i.e. it is in a state of stress -independently of the action of extraneous forces. A plane frame of n -joints which is just rigid (as regards deformation in its own plane) has -2n - 3 bars, for if one bar be held fixed the 2(n - 2) co-ordinates of -the remaining n - 2 joints must just be determined by the lengths of the -remaining bars. The total number of bars is therefore 2(n - 2) + 1. When -a plane frame which is just rigid is subject to a given system of -equilibrating extraneous forces (in its own plane) acting on the joints, -the stresses in the bars are in general uniquely determinate. For the -conditions of equilibrium of the forces on each pin furnish 2n -equations, viz. two for each point, which are linear in respect of the -stresses and the extraneous forces. This system of equations must -involve the three conditions of equilibrium of the extraneous forces -which are already identically satisfied, by hypothesis; there remain -therefore 2n - 3 independent relations to determine the 2n - 3 unknown -stresses. A frame of n joints and 2n - 3 bars may of course fail to be -rigid owing to some parts being over-stiff whilst others are deformable; -in such a case it will be found that the statical equations, apart from -the three identical relations imposed by the equilibrium of the -extraneous forces, are not all independent but are equivalent to less -than 2n - 3 relations. Another exceptional case, known as the _critical -case_, will be noticed later (S 9). - -A plane frame which can be built up from a single bar by successive -steps, at each of which a new joint is introduced by two new bars -meeting there, is called a _simple_ frame; it is obviously just rigid. -The stresses produced by extraneous forces in a simple frame can be -found by considering the equilibrium of the various joints in a proper -succession; and if the graphical method be employed the various polygons -of force can be combined into a single force-diagram. This procedure was -introduced by W. J. M. Rankine and J. Clerk Maxwell (1864). It may be -noticed that if we take an arbitrary pole in the force-diagram, and draw -a corresponding funicular in the skeleton diagram which represents the -frame together with the lines of action of the extraneous forces, we -obtain two complete reciprocal figures, in Maxwell's sense. It is -accordingly convenient to use Bow's notation (S 5), and to distinguish -the several compartments of the frame-diagram by letters. See fig. 33, -where the successive triangles in the diagram of forces may be -constructed in the order XYZ, ZXA, AZB. The class of "simple" frames -includes many of the frameworks used in the construction of roofs, -lattice girders and suspension bridges; a number of examples will be -found in the article BRIDGES. By examining the senses in which the -respective forces act at each joint we can ascertain which members are -in tension and which are in thrust; in fig. 33 this is indicated by the -directions of the arrowheads. - -[Illustration: FIG. 33.] - -[Illustration: FIG. 34.] - -When a frame, though just rigid, is not "simple" in the above sense, the -preceding method must be replaced, or supplemented, by one or other of -various artifices. In some cases the _method of sections_ is sufficient -for the purpose. If an ideal section be drawn across the frame, the -extraneous forces on either side must be in equilibrium with the forces -in the bars cut across; and if the section can be drawn so as to cut -only three bars, the forces in these can be found, since the problem -reduces to that of resolving a given force into three components acting -in three given lines (S 4). The "critical case" where the directions of -the three bars are concurrent is of course excluded. Another method, -always available, will be explained under "Work" (S 9). - - When extraneous forces act on the bars themselves the stress in each - bar no longer consists of a simple longitudinal tension or thrust. To - find the reactions at the joints we may proceed as follows. Each - extraneous force W acting on a bar may be replaced (in an infinite - number of ways) by two components P, Q in lines through the centres of - the pins at the extremities. In practice the forces W are usually - vertical, and the components P, Q are then conveniently taken to be - vertical also. We first alter the problem by transferring the forces - P, Q to the pins. The stresses in the bars, in the problem as thus - modified, may be supposed found by the preceding methods; it remains - to infer from the results thus obtained the reactions in the original - form of the problem. To find the pressure exerted by a bar AB on the - pin A we compound with the force in AB given by the diagram a force - equal to P. Conversely, to find the pressure of the pin A on the bar - AB we must compound with the force given by the diagram a force equal - and opposite to P. This question arises in practice in the theory of - "three-jointed" structures; for the purpose in hand such a structure - is sufficiently represented by two bars AB, BC. The right-hand figure - represents a portion of the force-diagram; in particular [->ZX] - represents the pressure of AB on B in the modified problem where the - loads W1 and W2 on the two bars are replaced by loads P1, Q1, and P2, - Q2 respectively, acting on the pins. Compounding with this [->XV], - which represents Q1, we get the actual pressure [->ZV] exerted by AB - on B. The directions and magnitudes of the reactions at A and C are - then easily ascertained. On account of its practical importance - several other graphical solutions of this problem have been devised. - -[Illustration: FIG. 35.] - -S 7. _Three-dimensional Kinematics of a Rigid Body._--The position of a -rigid body is determined when we know the positions of three points A, -B, C of it which are not collinear, for the position of any other point -P is then determined by the three distances PA, PB, PC. The nine -co-ordinates (Cartesian or other) of A, B, C are subject to the three -relations which express the invariability of the distances BC, CA, AB, -and are therefore equivalent to six independent quantities. Hence a -rigid body not constrained in any way is said to have six degrees of -freedom. Conversely, any six geometrical relations restrict the body in -general to one or other of a series of definite positions, none of which -can be departed from without violating the conditions in question. For -instance, the position of a theodolite is fixed by the fact that its -rounded feet rest in contact with six given plane surfaces. Again, a -rigid three-dimensional frame can be rigidly fixed relatively to the -earth by means of six links. - -[Illustration: FIG. 36.] - -[Illustration: FIG. 37.] - - The six independent quantities, or "co-ordinates," which serve to - specify the position of a rigid body in space may of course be chosen - in an endless variety of ways. We may, for instance, employ the three - Cartesian co-ordinates of a particular point O of the body, and three - angular co-ordinates which express the orientation of the body with - respect to O. Thus in fig. 36, if OA, OB, OC be three mutually - perpendicular lines in the solid, we may denote by [theta] the angle - which OC makes with a fixed direction OZ, by [psi] the azimuth of the - plane ZOC measured from some fixed plane through OZ, and by [phi] the - inclination of the plane COA to the plane ZOC. In fig. 36 these - various lines and planes are represented by their intersections with a - unit sphere having O as centre. This very useful, although - unsymmetrical, system of angular co-ordinates was introduced by L. - Euler. It is exemplified in "Cardan's suspension," as used in - connexion with a compass-bowl or a gyroscope. Thus in the gyroscope - the "flywheel" (represented by the globe in fig. 37) can turn about a - diameter OC of a ring which is itself free to turn about a diametral - axis OX at right angles to the former; this axis is carried by a - second ring which is free to turn about a fixed diameter OZ, which is - at right angles to OX. - -[Illustration: FIG. 10.] - -We proceed to sketch the theory of the finite displacements of a rigid -body. It was shown by Euler (1776) that any displacement in which one -point O of the body is fixed is equivalent to a pure _rotation_ about -some axis through O. Imagine two spheres of equal radius with O as their -common centre, one fixed in the body and moving with it, the other fixed -in space. In any displacement about O as a fixed point, the former -sphere slides over the latter, as in a "ball-and-socket" joint. Suppose -that as the result of the displacement a point of the moving sphere is -brought from A to B, whilst the point which was at B is brought to C -(cf. fig. 10). Let J be the pole of the circle ABC (usually a "small -circle" of the fixed sphere), and join JA, JB, JC, AB, BC by -great-circle arcs. The spherical isosceles triangles AJB, BJC are -congruent, and we see that AB can be brought into the position BC by a -rotation about the axis OJ through an angle AJB. - -[Illustration: FIG. 38.] - -[Illustration: FIG. 39.] - -It is convenient to distinguish the two senses in which rotation may -take place about an axis OA by opposite signs. We shall reckon a -rotation as positive when it is related to the direction from O to A as -the direction of rotation is related to that of translation in a -right-handed screw. Thus a negative rotation about OA may be regarded as -a positive rotation about OA', the prolongation of AO. Now suppose that -a body receives first a positive rotation [alpha] about OA, and secondly -a positive rotation [beta] about OB; and let A, B be the intersections -of these axes with a sphere described about O as centre. If we construct -the spherical triangles ABC, ABC' (fig. 38), having in each case the -angles at A and B equal to (1/2)[alpha] and (1/2)[beta] respectively, it -is evident that the first rotation will bring a point from C to C' and -that the second will bring it back to C; the result is therefore -equivalent to a rotation about OC. We note also that if the given -rotations had been effected in the inverse order, the axis of the -resultant rotation would have been OC', so that finite rotations do not -obey the "commutative law." To find the angle of the equivalent -rotation, in the actual case, suppose that the second rotation (about -OB) brings a point from A to A'. The spherical triangles ABC, A'BC (fig. -39) are "symmetrically equal," and the angle of the resultant rotation, -viz. ACA', is 2[pi] - 2C. This is equivalent to a negative rotation 2C -about OC, whence the theorem that the effect of three successive -positive rotations 2A, 2B, 2C about OA, OB, OC, respectively, is to -leave the body in its original position, provided the circuit ABC is -left-handed as seen from O. This theorem is due to O. Rodrigues (1840). -The composition of finite rotations about parallel axes is a particular -case of the preceding; the radius of the sphere is now infinite, and the -triangles are plane. - -In any continuous motion of a solid about a fixed point O, the limiting -position of the axis of the rotation by which the body can be brought -from any one of its positions to a consecutive one is called the -_instantaneous axis_. This axis traces out a certain cone in the body, -and a certain cone in space, and the continuous motion in question may -be represented as consisting in a rolling of the former cone on the -latter. The proof is similar to that of the corresponding theorem of -plane kinematics (S 3). - -It follows from Euler's theorem that the most general displacement of a -rigid body may be effected by a pure translation which brings any one -point of it to its final position O, followed by a pure rotation about -some axis through O. Those planes in the body which are perpendicular to -this axis obviously remain parallel to their original positions. Hence, -if [sigma], [sigma]' denote the initial and final positions of any -figure in one of these planes, the displacement could evidently have -been effected by (1) a translation perpendicular to the planes in -question, bringing [sigma] into some position [sigma]" in the plane of -[sigma]', and (2) a rotation about a normal to the planes, bringing -[sigma]" into coincidence with [sigma] (S 3). In other words, the most -general displacement is equivalent to a translation parallel to a -certain axis combined with a rotation about that axis; i.e. it may be -described as a _twist_ about a certain _screw_. In particular cases, of -course, the translation, or the rotation, may vanish. - - The preceding theorem, which is due to Michel Chasles (1830), may be - proved in various other interesting ways. Thus if a point of the body - be displaced from A to B, whilst the point which was at B is displaced - to C, and that which was at C to D, the four points A, B, C, D lie on - a helix whose axis is the common perpendicular to the bisectors of the - angles ABC, BCD. This is the axis of the required screw; the amount of - the translation is measured by the projection of AB or BC or CD on the - axis; and the angle of rotation is given by the inclination of the - aforesaid bisectors. This construction was given by M. W. Crofton. - Again, H. Wiener and W. Burnside have employed the _half-turn_ (i.e. a - rotation through two right angles) as the fundamental operation. This - has the advantage that it is completely specified by the axis of the - rotation, the sense being immaterial. Successive half-turns about - parallel axes a, b are equivalent to a translation measured by double - the distance between these axes in the direction from a to b. - Successive half-turns about intersecting axes a, b are equivalent to a - rotation about the common perpendicular to a, b at their intersection, - of amount equal to twice the acute angle between them, in the - direction from a to b. Successive half-turns about two skew axes a, b - are equivalent to a twist about a screw whose axis is the common - perpendicular to a, b, the translation being double the shortest - distance, and the angle of rotation being twice the acute angle - between a, b, in the direction from a to b. It is easily shown that - any displacement whatever is equivalent to two half-turns and - therefore to a screw. - -[Illustration: FIG. 16.] - -In mechanics we are specially concerned with the theory of infinitesimal -displacements. This is included in the preceding, but it is simpler in -that the various operations are commutative. An infinitely small -rotation about any axis is conveniently represented geometrically by a -length AB measures along the axis and proportional to the angle of -rotation, with the convention that the direction from A to B shall be -related to the rotation as is the direction of translation to that of -rotation in a right-handed screw. The consequent displacement of any -point P will then be at right angles to the plane PAB, its amount will -be represented by double the area of the triangle PAB, and its sense -will depend on the cyclical order of the letters P, A, B. If AB, AC -represent infinitesimal rotations about intersecting axes, the -consequent displacement of any point O in the plane BAC will be at right -angles to this plane, and will be represented by twice the sum of the -areas OAB, OAC, taken with proper signs. It follows by analogy with the -theory of moments (S 4) that the resultant rotation will be represented -by AD, the vector-sum of AB, AC (see fig. 16). It is easily inferred as -a limiting case, or proved directly, that two infinitesimal rotations -[alpha], [beta] about parallel axes are equivalent to a rotation [alpha] -+ [beta] about a parallel axis in the same plane with the two former, -and dividing a common perpendicular AB in a point C so that AC/CB = -[beta]/[alpha]. If the rotations are equal and opposite, so that [alpha] -+ [beta] = 0, the point C is at infinity, and the effect is a -translation perpendicular to the plane of the two given axes, of amount -[alpha].AB. It thus appears that an infinitesimal rotation is of the -nature of a "localized vector," and is subject in all respects to the -same mathematical laws as a force, conceived as acting on a rigid body. -Moreover, that an infinitesimal translation is analogous to a couple and -follows the same laws. These results are due to Poinsot. - -The analytical treatment of small displacements is as follows. We first -suppose that one point O of the body is fixed, and take this as the -origin of a "right-handed" system of rectangular co-ordinates; i.e. the -positive directions of the axes are assumed to be so arranged that a -positive rotation of 90 deg. about Ox would bring Oy into the position of -Oz, and so on. The displacement will consist of an infinitesimal -rotation [epsilon] about some axis through O, whose direction-cosines -are, say, l, m, n. From the equivalence of a small rotation to a -localized vector it follows that the rotation [epsilon] will be -equivalent to rotations [xi], [eta], [zeta] about Ox, Oy, Oz, -respectively, provided - - [xi] = l[epsilon], [eta] = m[epsilon], [zeta] = n[epsilon], (1) - -and we note that - - [xi]^2 + [eta]^2 + [zeta]^2 = [epsilon]^2. (2) - - Thus in the case of fig. 36 it may be required to connect the - infinitesimal rotations [xi], [eta], [zeta] about OA, OB, OC with the - variations of the angular co-ordinates [theta], [psi], [phi]. The - displacement of the point C of the body is made up of [delta][theta] - tangential to the meridian ZC and sin [theta] [delta][psi] - perpendicular to the plane of this meridian. Hence, resolving along - the tangents to the arcs BC, CA, respectively, we have - - [xi] = [delta][theta] sin [phi] - sin [theta] [delta][psi] cos [phi], - [eta] = [delta][theta] cos [phi] + sin [theta] [delta][psi] sin [phi]. (3) - - Again, consider the point of the solid which was initially at A' in - the figure. This is displaced relatively to A' through a space - [delta][psi] perpendicular to the plane of the meridian, whilst A' - itself is displaced through a space cos [theta] [delta][psi] in the - same direction. Hence - - [zeta] = [delta][phi] + cos [theta] [delta][psi]. (4) - -[Illustration: FIG. 40.] - -To find the component displacements of a point P of the body, whose -co-ordinates are x, y, z, we draw PL normal to the plane yOz, and LH, LK -perpendicular to Oy, Oz, respectively. The displacement of P parallel to -Ox is the same as that of L, which is made up of [eta]z and -[zeta]y. In -this way we obtain the formulae - - [delta]x = [eta]z - [zeta]y, [delta]y = [zeta]x - [xi]z, [delta]z = [xi]y - [eta]x. (5) - -The most general case is derived from this by adding the component -displacements [lambda], [mu], [nu] (say) of the point which was at O; -thus - - [delta]x = [lambda] + [eta]z - [zeta]y, \ - [delta]y = [mu] + [zeta]x - [xi]z, > (6) - [delta]z = [nu] + [xi]y - [eta]x. / - -The displacement is thus expressed in terms of the six independent -quantities [xi], [eta], [zeta], [lambda], [mu], [nu]. The points whose -displacements are in the direction of the resultant axis of rotation are -determined by [delta]x:[delta]y:[delta]z = [xi]:[eta]:[zeta], or - - ([lambda] + [eta]z - [zeta]y)/([xi] = [mu] + [zeta]x - [xi]z)/[eta] = ([nu] + [xi]y - [eta]x)/[zeta]. (7) - -These are the equations of a straight line, and the displacement is in -fact equivalent to a twist about a screw having this line as axis. The -translation parallel to this axis is - - l[delta]x + m[delta]y + n[delta]z = ([lambda][xi] + [mu][eta] + [nu][zeta])/[epsilon]. (8) - -The linear magnitude which measures the ratio of translation to rotation -in a screw is called the _pitch_. In the present case the pitch is - - ([lambda][xi] + [mu][eta] + [nu][zeta])/([xi]^2 + [eta]^2 + [zeta]^2). (9) - -Since [xi]^2 + [eta]^2 + [zeta]^2, or [epsilon]^2, is necessarily an -absolute invariant for all transformations of the (rectangular) -co-ordinate axes, we infer that [lambda][xi] + [mu][eta] + [nu][zeta] is -also an absolute invariant. When the latter invariant, but not the -former, vanishes, the displacement is equivalent to a pure rotation. - - If the small displacements of a rigid body be subject to one - constraint, e.g. if a point of the body be restricted to lie on a - given surface, the mathematical expression of this fact leads to a - homogeneous linear equation between the infinitesimals [xi], [eta], - [zeta], [lambda], [mu], [nu], say - - A[xi] + B[eta] + C[zeta] + F[lambda] + G[mu] + H[nu] = 0. (10) - - The quantities [xi], [eta], [zeta], [lambda], [mu], [nu] are no longer - independent, and the body has now only five degrees of freedom. Every - additional constraint introduces an additional equation of the type - (10) and reduces the number of degrees of freedom by one. In Sir R. S. - Ball's _Theory of Screws_ an analysis is made of the possible - displacements of a body which has respectively two, three, four, five - degrees of freedom. We will briefly notice the case of two degrees, - which involves an interesting generalization of the method (already - explained) of compounding rotations about intersecting axes. We assume - that the body receives arbitrary twists about two given screws, and - it is required to determine the character of the resultant - displacement. We examine first the case where the axes of the two - screws are at right angles and intersect. We take these as axes of x - and y; then if [xi], [eta] be the component rotations about them, we - have - - [lambda] = h[xi], [mu] = k[eta], [nu] = 0, (11) - - where h, k, are the pitches of the two given screws. The equations (7) - of the axis of the resultant screw then reduce to - - x/[xi] = y/[eta], z([xi]^2 + [eta]^2) = (k - h)[xi][eta]. (12) - - Hence, whatever the ratio [xi] : [eta], the axis of the resultant - screw lies on the conoidal surface - - z(x^2 + y^2) = cxy, (13) - - where c = (1/2)(k - h). The co-ordinates of any point on (13) may be - written - - x = r cos [theta], y = r sin [theta], z = c sin 2[theta]; (14) - - hence if we imagine a curve of sines to be traced on a circular - cylinder so that the circumference just includes two complete - undulations, a straight line cutting the axis of the cylinder at right - angles and meeting this curve will generate the surface. This is - called a _cylindroid_. Again, the pitch of the resultant screw is - - p = ([lambda][xi] + [mu][eta])/([xi]^2 + [eta]^2) = h cos^2 [theta] + k sin^2 [theta]. (15) - - [Illustration: From Sir Robert S. Ball's _Theory of Screws_. - - FIG. 41.] - - The distribution of pitch among the various screws has therefore a - simple relation to the _pitch-conic_ - - hx^2 + ky^2 = const; (16) - - viz. the pitch of any screw varies inversely as the square of that - diameter of the conic which is parallel to its axis. It is to be - noticed that the parameter c of the cylindroid is unaltered if the two - pitches h, k be increased by equal amounts; the only change is that - all the pitches are increased by the same amount. It remains to show - that a system of screws of the above type can be constructed so as to - contain any two given screws whatever. In the first place, a - cylindroid can be constructed so as to have its axis coincident with - the common perpendicular to the axes of the two given screws and to - satisfy three other conditions, for the position of the centre, the - parameter, and the orientation about the axis are still at our - disposal. Hence we can adjust these so that the surface shall contain - the axes of the two given screws as generators, and that the - difference of the corresponding pitches shall have the proper value. - It follows that when a body has two degrees of freedom it can twist - about any one of a singly infinite system of screws whose axes lie on - a certain cylindroid. In particular cases the cylindroid may - degenerate into a plane, the pitches being then all equal. - -S 8. _Three-dimensional Statics._--A system of parallel forces can be -combined two and two until they are replaced by a single resultant equal -to their sum, acting in a certain line. As special cases, the system may -reduce to a couple, or it may be in equilibrium. - -In general, however, a three-dimensional system of forces cannot be -replaced by a single resultant force. But it may be reduced to simpler -elements in a variety of ways. For example, it may be reduced to two -forces in perpendicular skew lines. For consider any plane, and let each -force, at its intersection with the plane, be resolved into two -components, one (P) normal to the plane, the other (Q) in the plane. The -assemblage of parallel forces P can be replaced in general by a single -force, and the coplanar system of forces Q by another single force. - -If the plane in question be chosen perpendicular to the direction of the -vector-sum of the given forces, the vector-sum of the components Q is -zero, and these components are therefore equivalent to a couple (S 4). -Hence any three-dimensional system can be reduced to a single force R -acting in a certain line, together with a couple G in a plane -perpendicular to the line. This theorem was first given by L. Poinsot, -and the line of action of R was called by him the _central axis_ of the -system. The combination of a force and a couple in a perpendicular plane -is termed by Sir R. S. Ball a _wrench_. Its type, as distinguished from -its absolute magnitude, may be specified by a screw whose axis is the -line of action of R, and whose pitch is the ratio G/R. - -[Illustration: FIG. 42.] - - The case of two forces may be specially noticed. Let AB be the - shortest distance between the lines of action, and let AA', BB' (fig. - 42) represent the forces. Let [alpha], [beta] be the angles which AA', - BB' make with the direction of the vector-sum, on opposite sides. - Divide AB in O, so that - - AA'.cos [alpha].AO = BB'.cos [beta].OB, (1) - - and draw OC parallel to the vector-sum. Resolving AA', BB' each into - two components parallel and perpendicular to OC, we see that the - former components have a single resultant in OC, of amount - - R = AA' cos [alpha] + BB' cos [beta], (2) - - whilst the latter components form a couple of moment - - G = AA'.AB.sin [alpha] = BB'.AB.sin [beta]. (3) - - Conversely it is seen that any wrench can be replaced in an infinite - number of ways by two forces, and that the line of action of one of - these may be chosen quite arbitrarily. Also, we find from (2) and (3) - that - - G.R = AA'.BB'.AB.sin ([alpha] + [beta]). (4) - - The right-hand expression is six times the volume of the tetrahedron - of which the lines AA', BB' representing the forces are opposite - edges; and we infer that, in whatever way the wrench be resolved into - two forces, the volume of this tetrahedron is invariable. - -To define the _moment_ of a force _about an axis_ HK, we project the -force orthogonally on a plane perpendicular to HK and take the moment of -the projection about the intersection of HK with the plane (see S 4). -Some convention as to sign is necessary; we shall reckon the moment to -be positive when the tendency of the force is right-handed as regards -the direction from H to K. Since two concurrent forces and their -resultant obviously project into two concurrent forces and their -resultant, we see that the sum of the moments of two concurrent forces -about any axis HK is equal to the moment of their resultant. Parallel -forces may be included in this statement as a limiting case. Hence, in -whatever way one system of forces is by successive steps replaced by -another, no change is made in the sum of the moments about any assigned -axis. By means of this theorem we can show that the previous reduction -of any system to a wrench is unique. - -From the analogy of couples to translations which was pointed out in S -7, we may infer that a couple is sufficiently represented by a "free" -(or non-localized) vector perpendicular to its plane. The length of the -vector must be proportional to the moment of the couple, and its sense -must be such that the sum of the moments of the two forces of the couple -about it is positive. In particular, we infer that couples of the same -moment in parallel planes are equivalent; and that couples in any two -planes may be compounded by geometrical addition of the corresponding -vectors. Independent statical proofs are of course easily given. Thus, -let the plane of the paper be perpendicular to the planes of two -couples, and therefore perpendicular to the line of intersection of -these planes. By S 4, each couple can be replaced by two forces [+-] P -(fig. 43) perpendicular to the plane of the paper, and so that one force -of each couple is in the line of intersection (B); the arms (AB, BC) -will then be proportional to the respective moments. The two forces at B -will cancel, and we are left with a couple of moment P . AC in the plane -AC. If we draw three vectors to represent these three couples, they will -be perpendicular and proportional to the respective sides of the -triangle ABC; hence the third vector is the geometric sum of the other -two. Since, in this proof the magnitude of P is arbitrary, It follows -incidentally that couples of the same moment in parallel planes, e.g. -planes parallel to AC, are equivalent. - -[Illustration: FIG. 43.] - -[Illustration: FIG. 44.] - -Hence a couple of moment G, whose axis has the direction (l, m, n) -relative to a right-handed system of rectangular axes, is equivalent to -three couples lG, mG, nG in the co-ordinate planes. The analytical -reduction of a three-dimensional system can now be conducted as follows. -Let (x1, y1, z1) be the co-ordinates of a point P1 on the line of action -of one of the forces, whose components are (say) X1, Y1, Z1. Draw P1H -normal to the plane zOx, and HK perpendicular to Oz. In KH introduce two -equal and opposite forces [+-] X1. The force X1 at P1 with -X1 in KH forms -a couple about Oz, of moment -y1X1. Next, introduce along Ox two equal -and opposite forces [+-]X1. The force X1 in KH with -X1 in Ox forms a -couple about Oy, of moment z1X1. Hence the force X1 can be transferred -from P1 to O, provided we introduce couples of moments z1X1 about Oy and --y1X1, about Oz. Dealing in the same way with the forces Y1, Z1 at P1, -we find that all three components of the force at P1 can be transferred -to O, provided we introduce three couples L1, M1, N1 about Ox, Oy, Oz -respectively, viz. - - L1 = y1Z1 - z1Y1, M1 = z1X1 - x1Z1, N1 = x1Y1 - y1X1. (5) - -It is seen that L1, M1, N1 are the moments of the original force at P1 -about the co-ordinate axes. Summing up for all the forces of the given -system, we obtain a force R at O, whose components are - - X = [Sigma](X_r), Y = [Sigma](Y_r), Z = [Sigma](Z_r), (6) - -and a couple G whose components are - - L = [Sigma](L_r), M = [Sigma](M_r), N = [Sigma](N_r), (7) - -where r= 1, 2, 3 ... Since R^2 = X^2 + Y^2 + Z^2, G^2 = L^2 + M^2 + N^2, -it is necessary and sufficient for equilibrium that the six quantities -X, Y, Z, L, M, N, should all vanish. In words: the sum of the -projections of the forces on each of the co-ordinate axes must vanish; -and, the sum of the moments of the forces about each of these axes must -vanish. - -If any other point O', whose co-ordinates are x, y, z, be chosen in -place of O, as the point to which the forces are transferred, we have to -write x1 - x, y1 - y, z1 - z for x1, y1, z1, and so on, in the preceding -process. The components of the resultant force R are unaltered, but the -new components of couple are found to be - - L' = L - yZ + zY, \ - M' = M - zX + xZ, > (8) - N' = N - xY + yX. / - -By properly choosing O' we can make the plane of the couple -perpendicular to the resultant force. The conditions for this are L' : -M' : N' = X : Y : Z, or - - L - yZ + zY M - zX + xZ N - xY + yX - ----------- = ----------- = ----------- (9) - X Y Z - -These are the equations of the central axis. Since the moment of the -resultant couple is now - - X Y Z LX + MY + NZ - G' = --- L' + --- M' + --- N' = ------------, (10) - R R R R - -the pitch of the equivalent wrench is - - (LX + MY + NZ)/(X^2 + Y^2 + Z^2). - -It appears that X^2 + Y^2 + Z^2 and LX + MY + NZ are absolute invariants -(cf. S 7). When the latter invariant, but not the former, vanishes, the -system reduces to a single force. - -The analogy between the mathematical relations of infinitely small -displacements on the one hand and those of force-systems on the other -enables us immediately to convert any theorem in the one subject into a -theorem in the other. For example, we can assert without further proof -that any infinitely small displacement may be resolved into two -rotations, and that the axis of one of these can be chosen arbitrarily. -Again, that wrenches of arbitrary amounts about two given screws -compound into a wrench the locus of whose axis is a cylindroid. - - The mathematical properties of a twist or of a wrench have been the - subject of many remarkable investigations, which are, however, of - secondary importance from a physical point of view. In the - "Null-System" of A. F. Mobius (1790-1868), a line such that the moment - of a given wrench about it is zero is called a _null-line_. The triply - infinite system of null-lines form what is called in line-geometry a - "complex." As regards the configuration of this complex, consider a - line whose shortest distance from the central axis is r, and whose - inclination to the central axis is [theta]. The moment of the - resultant force R of the wrench about this line is - Rr sin [theta], - and that of the couple G is G cos [theta]. Hence the line will be a - null-line provided - - tan [theta] = k/r, (11) - - where k is the pitch of the wrench. The null-lines which are at a - given distance r from a point O of the central axis will therefore - form one system of generators of a hyperboloid of revolution; and by - varying r we get a series of such hyperboloids with a common centre - and axis. By moving O along the central axis we obtain the whole - complex of null-lines. It appears also from (11) that the null-lines - whose distance from the central axis is r are tangent lines to a - system of helices of slope tan^-1 (r/k); and it is to be noticed that - these helices are left-handed if the given wrench is right-handed, and - vice versa. - - Since the given wrench can be replaced by a force acting through any - assigned point P, and a couple, the locus of the null-lines through P - is a plane, viz. a plane perpendicular to the vector which represents - the couple. The complex is therefore of the type called "linear" (in - relation to the degree of this locus). The plane in question is called - the _null-plane_ of P. If the null-plane of P pass through Q, the - null-plane of Q will pass through P, since PQ is a null-line. Again, - any plane [omega] is the locus of a system of null-lines meeting in a - point, called the _null-point_ of [omega]. If a plane revolve about a - fixed straight line p in it, its null-point describes another straight - line p', which is called the _conjugate line_ of p. We have seen that - the wrench may be replaced by two forces, one of which may act in any - arbitrary line p. It is now evident that the second force must act in - the conjugate line p', since every line meeting p, p' is a null-line. - Again, since the shortest distance between any two conjugate lines - cuts the central axis at right angles, the orthogonal projections of - two conjugate lines on a plane perpendicular to the central axis will - be parallel (fig. 42). This property was employed by L. Cremona to - prove the existence under certain conditions of "reciprocal figures" - in a plane (S 5). If we take any polyhedron with plane faces, the - null-planes of its vertices with respect to a given wrench will form - another polyhedron, and the edges of the latter will be conjugate (in - the above sense) to those of the former. Projecting orthogonally on a - plane perpendicular to the central axis we obtain two reciprocal - figures. - - In the analogous theory of infinitely small displacements of a solid, - a "null-line" is a line such that the lengthwise displacement of any - point on it is zero. - - Since a wrench is defined by six independent quantities, it can in - general be replaced by any system of forces which involves six - adjustable elements. For instance, it can in general be replaced by - six forces acting in six given lines, e.g. in the six edges of a given - tetrahedron. An exception to the general statement occurs when the six - lines are such that they are possible lines of action of a system of - six forces in equilibrium; they are then said to be _in involution_. - The theory of forces in involution has been studied by A. Cayley, J. - J. Sylvester and others. We have seen that a rigid structure may in - general be rigidly connected with the earth by six links, and it now - appears that any system of forces acting on the structure can in - general be balanced by six determinate forces exerted by the links. - If, however, the links are in involution, these forces become infinite - or indeterminate. There is a corresponding kinematic peculiarity, in - that the connexion is now not strictly rigid, an infinitely small - relative displacement being possible. See S 9. - -When parallel forces of given magnitudes act at given points, the -resultant acts through a definite point, or _centre of parallel forces_, -which is independent of the special direction of the forces. If P_r be -the force at (x_r, y_r, z_r), acting in the direction (l, m, n), the -formulae (6) and (7) reduce to - - X = [Sigma](P).l, Y = [Sigma](P).m, Z = [Sigma](P).n, (12) - -and - - L = [Sigma](P).(n[|y] - m[|z]), M = [Sigma](P).(l[|z] - n[|x]), N = [Sigma](P).(m[|x] - l[|y]), (13) - -provided - - [Sigma](Px) [Sigma](Py) [Sigma](Pz) - [|x] = -----------, [|y] = -----------, [|z] = -----------. (14) - [Sigma](P) [Sigma](P) [Sigma](P) - -These are the same as if we had a single force [Sigma](P) acting at the -point ([|x], [|y], [|z]), which is the same for all directions (l, m, -n). We can hence derive the theory of the centre of gravity, as in S 4. -An exceptional case occurs when [Sigma](P) = 0. - - If we imagine a rigid body to be acted on at given points by forces of - given magnitudes in directions (not all parallel) which are fixed in - space, then as the body is turned about the resultant wrench will - assume different configurations in the body, and will in certain - positions reduce to a single force. The investigation of such - questions forms the subject of "Astatics," which has been cultivated - by Mobius, Minding, G. Darboux and others. As it has no physical - bearing it is passed over here. - -[Illustration: FIG. 45.] - -S 9. _Work._--The _work_ done by a force acting on a particle, in any -infinitely small displacement, is defined as the product of the force -into the orthogonal projection of the displacement on the direction of -the force; i.e. it is equal to F.[delta]s cos [theta], where F is the -force, [delta]s the displacement, and [theta] is the angle between the -directions of F and [delta]s. In the language of vector analysis (q.v.) -it is the "scalar product" of the vector representing the force and the -displacement. In the same way, the work done by a force acting on a -rigid body in any infinitely small displacement of the body is the -scalar product of the force into the displacement of any point on the -line of action. This product is the same whatever point on the line of -action be taken, since the lengthwise components of the displacements of -any two points A, B on a line AB are equal, to the first order of small -quantities. To see this, let A', B' be the displaced positions of A, B, -and let [phi] be the infinitely small angle between AB and A'B'. Then if -[alpha], [beta] be the orthogonal projections of A', B' on AB, we have - - A[alpha] - B[beta] = AB - [alpha][beta] = AB(1 - cos [phi]) = (1/2)AB.[phi]^2, - -ultimately. Since this is of the second order, the products F.A[alpha] -and F.B[beta] are ultimately equal. - -[Illustration: FIG. 46.] - -[Illustration: FIG. 47.] - -The total work done by two concurrent forces acting on a particle, or on -a rigid body, in any infinitely small displacement, is equal to the work -of their resultant. Let AB, AC (fig. 46) represent the forces, AD their -resultant, and let AH be the direction of the displacement [delta]s of -the point A. The proposition follows at once from the fact that the sum -of orthogonal projections of [->AB], [->AC] on AH is equal to the -projection of [->AD]. It is to be noticed that AH need not be in the -same plane with AB, AC. - -It follows from the preceding statements that any two systems of forces -which are statically equivalent, according to the principles of SS 4, 8, -will (to the first order of small quantities) do the same amount of work -in any infinitely small displacement of a rigid body to which they may -be applied. It is also evident that the total work done in two or more -successive infinitely small displacements is equal to the work done in -the resultant displacement. - -The work of a couple in any infinitely small rotation of a rigid body -about an axis perpendicular to the plane of the couple is equal to the -product of the moment of the couple into the angle of rotation, proper -conventions as to sign being observed. Let the couple consist of two -forces P, P (fig. 47) in the plane of the paper, and let J be the point -where this plane is met by the axis of rotation. Draw JBA perpendicular -to the lines of action, and let [epsilon] be the angle of rotation. The -work of the couple is - - P.JA.[epsilon] - P.JB.[epsilon] = P.AB.[epsilon] = G[epsilon], - -if G be the moment of the couple. - -The analytical calculation of the work done by a system of forces in any -infinitesimal displacement is as follows. For a two-dimensional system -we have, in the notation of SS 3, 4, - - [Sigma](X[delta]x + Y[delta]y) = [Sigma]{X([lambda] - y[epsilon]) + Y([mu] + x[epsilon])} - = [Sigma](X).[lambda] + [Sigma](Y).[mu] + [Sigma](xY - yX)[epsilon] - = X[lambda] + Y[mu] + N[epsilon]. (1) - -Again, for a three-dimensional system, in the notation of SS 7, 8, - - [Sigma](X[delta]x + Y[delta]y + Z[delta]z) - = [Sigma]{(X([lambda] + [eta]z - [zeta]y) + Y([mu] + [zeta]x - [xi]x) + Z([nu] + [xi]y - [eta]x)} - = [Sigma](X).[lambda] + [Sigma](Y).[mu] + [Sigma](Z).[nu] + [Sigma](yZ - zY).[xi] - + [Sigma](zX - xZ).[eta] + [Sigma](xY - yX).[zeta] - = X[lambda] + Y[mu] + Z[nu] + L[xi] + M[eta] + N[zeta]. (2) - -This expression gives the work done by a given wrench when the body -receives a given infinitely small twist; it must of course be an -absolute invariant for all transformations of rectangular axes. The -first three terms express the work done by the components of a force (X, -Y, Z) acting at O, and the remaining three terms express the work of a -couple (L, M, N). - -[Illustration: FIG. 48.] - - The work done by a wrench about a given screw, when the body twists - about a second given screw, may be calculated directly as follows. In - fig. 48 let R, G be the force and couple of the wrench, - [epsilon],[tau] the rotation and translation in the twist. Let the - axes of the wrench and the twist be inclined at an angle [theta], and - let h be the shortest distance between them. The displacement of the - point H in the figure, resolved in the direction of R, is [tau] cos - [theta] - [epsilon]h sin [theta]. The work is therefore - - R([tau] cos [theta] - [epsilon]h sin [theta]) + G cos [theta] - = R[epsilon]{(p + p') cos [theta] - h sin [theta]}, (3) - - if G = pR, [tau] = p'[epsilon], i.e. p, p' are the pitches of the two - screws. The factor (p + p') cos[theta] - h sin[theta] is called the - _virtual coefficient_ of the two screws which define the types of the - wrench and twist, respectively. - - A screw is determined by its axis and its pitch, and therefore - involves five Independent elements. These may be, for instance, the - five ratios [xi]:[eta]:[zeta]:[lambda]:[mu]:[nu] of the six quantities - which specify an infinitesimal twist about the screw. If the twist is - a pure rotation, these quantities are subject to the relation - - [lambda][xi] + [mu][eta] + [nu][zeta] = 0. (4) - - In the analytical investigations of line geometry, these six - quantities, supposed subject to the relation (4), are used to specify - a line, and are called the six "co-ordinates" of the line; they are of - course equivalent to only four independent quantities. If a line is a - null-line with respect to the wrench (X, Y, Z, L, M, N), the work done - in an infinitely small rotation about it is zero, and its co-ordinates - are accordingly subject to the further relation - - L[xi] + M[eta] + N[zeta] + X[lambda] + Y[mu] + Z[nu] = 0, (5) - - where the coefficients are constant. This is the equation of a "linear - complex" (cf. S 8). - - Two screws are _reciprocal_ when a wrench about one does no work on a - body which twists about the other. The condition for this is - - [lambda][xi]' + [mu][eta]' + [nu][zeta]' + [lambda]'[xi] + [mu]'[eta] + [nu]'[zeta] = 0, (6) - - if the screws be defined by the ratios [xi] : [eta] : [zeta] : - [lambda] : [mu] : [nu] and [xi]' : [eta]' : [zeta]' : [lambda]' : - [mu]' : [nu]', respectively. The theory of the screw-systems which are - reciprocal to one, two, three, four given screws respectively has been - investigated by Sir R. S. Ball. - -Considering a rigid body in any given position, we may contemplate the -whole group of infinitesimal displacements which might be given to it. -If the extraneous forces are in equilibrium the total work which they -would perform in any such displacement would be zero, since they reduce -to a zero force and a zero couple. This is (in part) the celebrated -principle of _virtual velocities_, now often described as the principle -of _virtual work_, enunciated by John Bernoulli (1667-1748). The word -"virtual" is used because the displacements in question are not regarded -as actually taking place, the body being in fact at rest. The -"velocities" referred to are the velocities of the various points of the -body in any imagined motion of the body through the position in -question; they obviously bear to one another the same ratios as the -corresponding infinitesimal displacements. Conversely, we can show that -if the virtual work of the extraneous forces be zero for every -infinitesimal displacement of the body as rigid, these forces must be in -equilibrium. For by giving the body (in imagination) a displacement of -translation we learn that the sum of the resolved parts of the forces in -any assigned direction is zero, and by giving it a displacement of pure -rotation we learn that the sum of the moments about any assigned axis is -zero. The same thing follows of course from the analytical expression -(2) for the virtual work. If this vanishes for all values of [lambda], -[mu], [nu], [xi], [eta], [zeta] we must have X, Y, Z, L, M, N = 0, which -are the conditions of equilibrium. - -The principle can of course be extended to any system of particles or -rigid bodies, connected together in any way, provided we take into -account the internal stresses, or reactions, between the various parts. -Each such reaction consists of two equal and opposite forces, both of -which may contribute to the equation of virtual work. - -The proper significance of the principle of virtual work, and of its -converse, will appear more clearly when we come to kinetics (S 16); for -the present it may be regarded merely as a compact and (for many -purposes) highly convenient summary of the laws of equilibrium. Its -special value lies in this, that by a suitable adjustment of the -hypothetical displacements we are often enabled to eliminate unknown -reactions. For example, in the case of a particle lying on a smooth -curve, or on a smooth surface, if it be displaced along the curve, or on -the surface, the virtual work of the normal component of the pressure -may be ignored, since it is of the second order. Again, if two bodies -are connected by a string or rod, and if the hypothetical displacements -be adjusted so that the distance between the points of attachment is -unaltered, the corresponding stress may be ignored. This is evident from -fig. 45; if AB, A'B' represent the two positions of a string, and T be -the tension, the virtual work of the two forces [+-]T at A, B is T(A[alpha] -- B[beta]), which was shown to be of the second order. Again, the normal -pressure between two surfaces disappears from the equation, provided the -displacements be such that one of these surfaces merely slides -relatively to the other. It is evident, in the first place, that in any -displacement common to the two surfaces, the work of the two equal and -opposite normal pressures will cancel; moreover if, one of the surfaces -being fixed, an infinitely small displacement shifts the point of -contact from A to B, and if A' be the new position of that point of the -sliding body which was at A, the projection of AA' on the normal at A is -of the second order. It is to be noticed, in this case, that the -tangential reaction (if any) between the two surfaces is not eliminated. -Again, if the displacements be such that one curved surface rolls -without sliding on another, the reaction, whether normal or tangential, -at the point of contact may be ignored. For the virtual work of two -equal and opposite forces will cancel in any displacement which is -common to the two surfaces; whilst, if one surface be fixed, the -displacement of that point of the rolling surface which was in contact -with the other is of the second order. We are thus able to imagine a -great variety of mechanical systems to which the principle of virtual -work can be applied without any regard to the internal stresses, -provided the hypothetical displacements be such that none of the -connexions of the system are violated. - -If the system be subject to gravity, the corresponding part of the -virtual work can be calculated from the displacement of the centre of -gravity. If W1, W2, ... be the weights of a system of particles, whose -depths below a fixed horizontal plane of reference are z1, z2, ..., -respectively, the virtual work of gravity is - - W1[delta].z1 + W2[delta]z2 + ... = [delta](W1z1 + W2z2 + ...) (7) - = (W1 + W2 + ...) [delta][|z], - -where [|z] is the depth of the centre of gravity (see S 8 (14) and S 11 -(6)). This expression is the same as if the whole mass were concentrated -at the centre of gravity, and displaced with this point. An important -conclusion is that in any displacement of a system of bodies in -equilibrium, such that the virtual work of all forces except gravity may -be ignored, the depth of the centre of gravity is "stationary." - -The question as to stability of equilibrium belongs essentially to -kinetics; but we may state by anticipation that in cases where gravity -is the only force which does work, the equilibrium of a body or system -of bodies is stable only if the depth of the centre of gravity be a -maximum. - -[Illustration: FIG. 49.] - - Consider, for instance, the case of a bar resting with its ends on two - smooth inclines (fig. 18). If the bar be displaced in a vertical plane - so that its ends slide on the two inclines, the instantaneous centre - is at the point J. The displacement of G is at right angles to JG; - this shows that for equilibrium JG must be vertical. Again, the locus - of G is an arc of an ellipse whose centre is in the intersection of - the planes; since this arc is convex upwards the equilibrium is - unstable. A general criterion for the case of a rigid body movable in - two dimensions, with one degree of freedom, can be obtained as - follows. We have seen (S 3) that the sequence of possible positions is - obtained if we imagine the "body-centrode" to roll on the - "space-centrode." For equilibrium, the altitude of the centre of - gravity G must be stationary; hence G must lie in the same vertical - line with the point of contact J of the two curves. Further, it is - known from the theory of "roulettes" that the locus of G will be - concave or convex upwards according as - - cos[phi] 1 1 - ------- = ----- + ------, (8) - h [rho] [rho]' - - where [rho], [rho]' are the radii of curvature of the two curves at J, - [phi] is the inclination of the common tangent at J to the horizontal, - and h is the height of G above J. The signs of [rho], [rho]' are to be - taken positive when the curvatures are as in the standard case shown - in fig. 49. Hence for stability the upper sign must obtain in (8). The - same criterion may be arrived at in a more intuitive manner as - follows. If the body be supposed to roll (say to the right) until the - curves touch at J', and if JJ' = [delta]s, the angle through which the - upper figure rotates is [delta]s/[rho] + [delta]s/[rho]', and the - horizontal displacement of G is equal to the product of this - expression into h. If this displacement be less than the horizontal - projection of JJ', viz. [delta]s cos[phi], the vertical through the - new position of G will fall to the left of J' and gravity will tend to - restore the body to its former position. It is here assumed that the - remaining forces acting on the body in its displaced position have - zero moment about J'; this is evidently the case, for instance, in the - problem of "rocking stones." - -The principle of virtual work is specially convenient in the theory of -frames (S 6), since the reactions at smooth joints and the stresses in -inextensible bars may be left out of account. In particular, in the case -of a frame which is just rigid, the principle enables us to find the -stress in any one bar independently of the rest. If we imagine the bar -in question to be removed, equilibrium will still persist if we -introduce two equal and opposite forces S, of suitable magnitude, at the -joints which it connected. In any infinitely small deformation of the -frame as thus modified, the virtual work of the forces S, together with -that of the original extraneous forces, must vanish; this determines S. - - As a simple example, take the case of a light frame, whose bars form - the slides of a rhombus ABCD with the diagonal BD, suspended from A - and carrying a weight W at C; and let it be required to find the - stress in BD. If we remove the bar BD, and apply two equal and - opposite forces S at B and D, the equation is - - W.[delta](2l cos[theta]) + 2S.[delta](l sin [theta]) = 0, - - where l is the length of a side of the rhombus, and [theta] its - inclination to the vertical. Hence - - S = W tan [theta] = W.BD/AC. (8) - - [Illustration: FIG. 50.] - - The method is specially appropriate when the frame, although just - rigid, is not "simple" in the sense of S 6, and when accordingly the - method of reciprocal figures is not immediately available. To avoid - the intricate trigonometrical calculations which would often be - necessary, graphical devices have been introduced by H. Muller-Breslau - and others. For this purpose the infinitesimal displacements of the - various joints are replaced by finite lengths proportional to them, - and therefore proportional to the velocities of the joints in some - imagined motion of the deformable frame through its actual - configuration; this is really (it may be remarked) a reversion to the - original notion of "virtual velocities." Let J be the instantaneous - centre for any bar CD (fig. 12), and let s1, s2 represent the virtual - velocities of C, D. If these lines be turned through a right angle in - the same sense, they take up positions such as CC', DD', where C', D' - are on JC, JD, respectively, and C'D' is parallel to CD. Further, if - F1 (fig. 51) be any force acting on the joint C, its virtual work will - be equal to the moment of F1 about C'; the equation of virtual work is - thus transformed into an equation of moments. - - [Illustration: FIG. 12.] - - [Illustration: FIG. 51.] - - [Illustration: FIG. 52.] - - Consider, for example, a frame whose sides form the six sides of a - hexagon ABCDEF and the three diagonals AD, BE, CF; and suppose that it - is required to find the stress in CF due to a given system of - extraneous forces in equilibrium, acting on the joints. Imagine the - bar CF to be removed, and consider a deformation in which AB is fixed. - The instantaneous centre of CD will be at the intersection of AD, BC, - and if C'D' be drawn parallel to CD, the lines CC', DD' may be taken - to represent the virtual velocities of C, D turned each through a - right angle. Moreover, if we draw D'E' parallel to DE, and E'F' - parallel to EF, the lines CC', DD', EE', FF' will represent on the - same scale the virtual velocities of the points C, D, E, F, - respectively, turned each through a right angle. The equation of - virtual work is then formed by taking moments about C', D', E', F' of - the extraneous forces which act at C, D, E, F, respectively. Amongst - these forces we must include the two equal and opposite forces S which - take the place of the stress in the removed bar FC. - - The above method lends itself naturally to the investigation of the - _critical forms_ of a frame whose general structure is given. We have - seen that the stresses produced by an equilibrating system of - extraneous forces in a frame which is just rigid, according to the - criterion of S 6, are in general uniquely determinate; in particular, - when there are no extraneous forces the bars are in general free from - stress. It may however happen that owing to some special relation - between the lengths of the bars the frame admits of an infinitesimal - deformation. The simplest case is that of a frame of three bars, when - the three joints A, B, C fall into a straight line; a small - displacement of the joint B at right angles to AC would involve - changes in the lengths of AB, BC which are only of the second order of - small quantities. Another example is shown in fig. 53. The graphical - method leads at once to the detection of such cases. Thus in the - hexagonal frame of fig. 52, if an infinitesimal deformation is - possible without removing the bar CF, the instantaneous centre of CF - (when AB is fixed) will be at the intersection of AF and BC, and since - CC', FF' represent the virtual velocities of the points C, F, turned - each through a right angle, C'F' must be parallel to CF. Conversely, - if this condition be satisfied, an infinitesimal deformation is - possible. The result may be generalized into the statement that a - frame has a critical form whenever a frame of the same structure can - be designed with corresponding bars parallel, but without complete - geometric similarity. In the case of fig. 52 it may be shown that an - equivalent condition is that the six points A, B, C, D, E, F should - lie on a conic (M. W. Crofton). This is fulfilled when the opposite - sides of the hexagon are parallel, and (as a still more special case) - when the hexagon is regular. - - [Illustration: FIG. 53.] - - When a frame has a critical form it may be in a state of stress - independently of the action of extraneous forces; moreover, the - stresses due to extraneous forces are indeterminate, and may be - infinite. For suppose as before that one of the bars is removed. If - there are no extraneous forces the equation of virtual work reduces to - S.[delta]s = 0, where S is the stress in the removed bar, and [delta]s - is the change in the distance between the joints which it connected. - In a critical form we have [delta]s = 0, and the equation is satisfied - by an arbitrary value of S; a consistent system of stresses in the - remaining bars can then be found by preceding rules. Again, when - extraneous forces P act on the joints, the equation is - - [Sigma](P.[delta]p) + S.[delta]s = 0, - - where [delta]p is the displacement of any joint in the direction of - the corresponding force P. If [Sigma](P.[delta]p) = 0, the stresses - are merely indeterminate as before; but if [Sigma] (P.[delta]p) does - not vanish, the equation cannot be satisfied by any finite value of S, - since [delta]s = 0. This means that, if the material of the frame were - absolutely unyielding, no finite stresses in the bars would enable it - to withstand the extraneous forces. With actual materials, the frame - would yield elastically, until its configuration is no longer - "critical." The stresses in the bars would then be comparatively very - great, although finite. The use of frames which approximate to a - critical form is of course to be avoided in practice. - - A brief reference must suffice to the theory of three dimensional - frames. This is important from a technical point of view, since all - structures are practically three-dimensional. We may note that a frame - of n joints which is just rigid must have 3n - 6 bars; and that the - stresses produced in such a frame by a given system of extraneous - forces in equilibrium are statically determinate, subject to the - exception of "critical forms." - -S 10. _Statics of Inextensible Chains._--The theory of bodies or -structures which are deformable in their smallest parts belongs properly -to elasticity (q.v.). The case of inextensible strings or chains is, -however, so simple that it is generally included in expositions of pure -statics. - -It is assumed that the form can be sufficiently represented by a plane -curve, that the stress (tension) at any point P of the curve, between -the two portions which meet there, is in the direction of the tangent at -P, and that the forces on any linear element [delta]s must satisfy the -conditions of equilibrium laid down in S 1. It follows that the forces -on any finite portion will satisfy the conditions of equilibrium which -apply to the case of a rigid body (S 4). - -[Illustration: FIG. 54.] - -We will suppose in the first instance that the curve is plane. It is -often convenient to resolve the forces on an element PQ (= [delta]s) in -the directions of the tangent and normal respectively. If T, T + -[delta]T be the tensions at P, Q, and [delta][psi] be the angle between -the directions of the curve at these points, the components of the -tensions along the tangent at P give (T + [delta]T) cos [psi] - T, or -[delta]T, ultimately; whilst for the component along the normal at P we -have (T + [delta]T) sin [delta][psi], or T[delta][psi], or -T[delta]s/[rho], where [rho] is the radius of curvature. - -Suppose, for example, that we have a light string stretched over a -smooth curve; and let R[delta]s denote the normal pressure (outwards -from the centre of curvature) on [delta]s. The two resolutions give -[delta]T = 0, T[delta][psi] = R[delta]s, or - - T = const., R = T/[rho]. (1) - -The tension is constant, and the pressure per unit length varies as the -curvature. - -Next suppose that the curve is "rough"; and let F[delta]s be the -tangential force of friction on [delta]s. We have [delta]T [+-] F[delta]s = -0, T[delta][psi] = R[delta]s, where the upper or lower sign is to be -taken according to the sense in which F acts. We assume that in -limiting equilibrium we have F = [mu]R, everywhere, where [mu] is the -coefficient of friction. If the string be on the point of slipping in -the direction in which [psi] increases, the lower sign is to be taken; -hence [delta]T = F[delta]s = [mu]T[delta][psi], whence - - T = T0 e^([mu][psi]), (2) - -if T0 be the tension corresponding to [psi] = 0. This illustrates the -resistance to dragging of a rope coiled round a post; e.g. if we put -[mu] = .3, [psi] = 2[pi], we find for the change of tension in one turn -T/T0 = 6.5. In two turns this ratio is squared, and so on. - -Again, take the case of a string under gravity, in contact with a smooth -curve in a vertical plane. Let [psi] denote the inclination to the -horizontal, and w [delta]s the weight of an element [delta]s. The -tangential and normal components of w[delta]s are -s sin [psi] and --w [delta]s cos [psi]. Hence - - [delta]T = w [delta]s sin [psi], T [delta][psi] = w [delta]s cos [psi] + R[delta]s. (3) - -If we take rectangular axes Ox, Oy, of which Oy is drawn vertically -upwards, we have [delta]y = sin[psi] [delta]s, whence [delta]T = -w[delta]y. If the string be uniform, w is constant, and - - T = wy + const. = w(y - y0), (4) - -say; hence the tension varies as the height above some fixed level (y0). -The pressure is then given by the formula - - d[psi] - R = T ------ - w cos [psi]. (5) - ds - -In the case of a chain hanging freely under gravity it is usually -convenient to formulate the conditions of equilibrium of a finite -portion PQ. The forces on this reduce to three, viz. the weight of PQ -and the tensions at P, Q. Hence these three forces will be concurrent, -and their ratios will be given by a triangle of forces. In particular, -if we consider a length AP beginning at the lowest point A, then -resolving horizontally and vertically we have - - T cos [psi] = T0, T sin [psi] = W, (6) - -where T0 is the tension at A, and W is the weight of PA. The former -equation expresses that the horizontal tension is constant. - -[Illustration: FIG. 55.] - -If the chain be uniform we have W = ws, where s is the arc AP: hence ws -= T0 tan[psi]. If we write T0 = wa, so that a is the length of a portion -of the chain whose weight would equal the horizontal tension, this -becomes - - s = a tan [psi]. (7) - -This is the "intrinsic" equation of the curve. If the axes of x and y be -taken horizontal and vertical (upwards), we derive - - x = a log (sec [psi] + tan [psi]), y = a sec [psi]. (8) - -Eliminating [psi] we obtain the Cartesian equation - - x - y = a cosh --- (9) - a - -of the _common catenary_, as it is called (fig. 56). The omission of the -additive arbitrary constants of integration in (8) is equivalent to a -special choice of the origin O of co-ordinates; viz. O is at a distance -a vertically below the lowest point ([psi] = 0) of the curve. The -horizontal line through O is called the _directrix_. The relations - - s = a sinh x/a, y^2 = a^2 + s^2, T = T0 sec [psi] = wy, (10) - -[Illustration: FIG. 56.] - -which are involved in the preceding formulae are also noteworthy. It is -a classical problem in the calculus of variations to deduce the equation -(9) from the condition that the depth of the centre of gravity of a -chain of given length hanging between fixed points must be stationary (S -9). The length a is called the _parameter_ of the catenary; it -determines the scale of the curve, all catenaries being geometrically -similar. If weights be suspended from various points of a hanging chain, -the intervening portions will form arcs of equal catenaries, since the -horizontal tension (wa) is the same for all. Again, if a chain pass over -a perfectly smooth peg, the catenaries in which it hangs on the two -sides, though usually of different parameters, will have the same -directrix, since by (10) y is the same for both at the peg. - - As an example of the use of the formulae we may determine the maximum - span for a wire of given material. The condition is that the tension - must not exceed the weight of a certain length [lambda] of the wire. - At the ends we shall have y = [lambda], or - - x - [lambda] = a cosh ---, (11) - a - - and the problem is to make x a maximum for variations of a. - Differentiating (11) we find that, if dx/da = 0, - - x x - --- tanh --- = 1. (12) - a a - - It is easily seen graphically, or from a table of hyperbolic tangents, - that the equation u tanh u = 1 has only one positive root (u = 1.200); - the span is therefore - - 2x = 2au = 2[lambda]/sinh u = 1.326[lambda], - - and the length of wire is - - 2s = 2[lambda]/u = 1.667 [lambda]. - - The tangents at the ends meet on the directrix, and their inclination - to the horizontal is 56 deg. 30'. - - [Illustration: FIG. 57.] - - The relation between the sag, the tension, and the span of a wire - (e.g. a telegraph wire) stretched nearly straight between two points - A, B at the same level is determined most simply from first - principles. If T be the tension, W the total weight, k the sag in the - middle, and [psi] the inclination to the horizontal at A or B, we have - 2T[psi] = W, AB = 2[rho][psi], approximately, where [rho] is the - radius of curvature. Since 2k[rho] = ((1/2)AB)^2, ultimately, we have - - k = (1/8)W.AB/T. (13) - - The same formula applies if A, B be at different levels, provided k be - the sag, measured vertically, half way between A and B. - -In relation to the theory of suspension bridges the case where the -weight of any portion of the chain varies as its horizontal projection -is of interest. The vertical through the centre of gravity of the arc AP -(see fig. 55) will then bisect its horizontal projection AN; hence if PS -be the tangent at P we shall have AS = SN. This property is -characteristic of a parabola whose axis is vertical. If we take A as -origin and AN as axis of x, the weight of AP may be denoted by wx, where -w is the weight per unit length at A. Since PNS is a triangle of forces -for the portion AP of the chain, we have wx/T0 = PN/NS, or - - y = w.x^2/2T0, (14) - -which is the equation of the parabola in question. The result might of -course have been inferred from the theory of the parabolic funicular in -S 2. - - Finally, we may refer to the _catenary of uniform strength_, where the - cross-section of the wire (or cable) is supposed to vary as the - tension. Hence w, the weight per foot, varies as T, and we may write - T = w[lambda], where [lambda] is a constant length. Resolving along - the normal the forces on an element [delta]s, we find T[delta][psi] = - w[delta]s cos[psi], whence - - ds - p = ------ = [lambda] sec [psi]. (15) - d[psi] - - From this we derive - - x - x = [lambda][psi], y = [lambda] log sec --------, (16) - [lambda] - - where the directions of x and y are horizontal and vertical, and the - origin is taken at the lowest point. The curve (fig. 58) has two - vertical asymptotes x = [+-] (1/2)[pi][lambda]; this shows that - however the thickness of a cable be adjusted there is a limit - [pi][lambda] to the horizontal span, where [lambda] depends on the - tensile strength of the material. For a uniform catenary the limit was - found above to be 1.326[lambda]. - -[Illustration: FIG. 58.] - -For investigations relating to the equilibrium of a string in three -dimensions we must refer to the textbooks. In the case of a string -stretched over a smooth surface, but in other respects free from -extraneous force, the tensions at the ends of a small element [delta]s -must be balanced by the normal reaction of the surface. It follows that -the osculating plane of the curve formed by the string must contain the -normal to the surface, i.e. the curve must be a "geodesic," and that the -normal pressure per unit length must vary as the principal curvature of -the curve. - -S 11. _Theory of Mass-Systems._--This is a purely geometrical subject. -We consider a system of points P1, P2 ..., P_n, with which are -associated certain coefficients m1, m2, ... m_n, respectively. In the -application to mechanics these coefficients are the masses of particles -situate at the respective points, and are therefore all positive. We -shall make this supposition in what follows, but it should be remarked -that hardly any difference is made in the theory if some of the -coefficients have a different sign from the rest, except in the special -case where [Sigma](m) = 0. This has a certain interest in magnetism. - -In a given mass-system there exists one and only one point G such that - - [Sigma](m.[->GP]) = 0. (1) - -For, take any point O, and construct the vector - - [Sigma](m.[->OP]) - [->OG] = -----------------. (2) - [Sigma](m) - -Then - - [Sigma](m.[->GP]) = [Sigma]{m([->GO] + [->OP])} = [Sigma](m).[->GO] + [Sigma](m).[->OP] = 0. (3) - -Also there cannot be a distinct point G' such that [Sigma](m.G'P) = 0, -for we should have, by subtraction, - - [Sigma]{m([->GP] + [->PG'])} = 0, or [Sigma](m).GG' = 0; (4) - -i.e. G' must coincide with G. The point G determined by (1) is called -the _mass-centre_ or _centre of inertia_ of the given system. It is -easily seen that, in the process of determining the mass-centre, any -group of particles may be replaced by a single particle whose mass is -equal to that of the group, situate at the mass-centre of the group. - -If through P1, P2, ... P_n we draw any system of parallel planes meeting -a straight line OX in the points M1, M2 ... M_n, the collinear vectors -[->OM1], [->OM2] ... [->OM_n] may be called the "projections" of -[->OP1], [->OP2], ... [->OP_n] on OX. Let these projections be denoted -algebraically by x1, x2, ... x_n, the sign being positive or negative -according as the direction is that of OX or the reverse. Since the -projection of a vector-sum is the sum of the projections of the several -vectors, the equation (2) gives - - [Sigma](mx) - [|x] = -----------, (5) - [Sigma](m) - -if [|x] be the projection of [->OG]. Hence if the Cartesian co-ordinates -of P1, P2, ... P_n relative to any axes, rectangular or oblique be (x1, -y1, z1), (x2, y2, z2), ..., (x_n, y_n, z_n), the mass-centre ([|x], -[|y], [|z]) is determined by the formulae - - [Sigma](mx) [Sigma](my) [Sigma](mz) - [|x] = -----------, [|y] = -----------, [|z] = -----------. (6) - [Sigma](m) [Sigma](m) [Sigma](m) - -If we write x = [|x] + [xi], y = [|y] + [eta], z = [|z] + [zeta], so -that [xi], [eta], [zeta] denote co-ordinates relative to the mass-centre -G, we have from (6) - - [Sigma](m[xi]) = 0, [Sigma](m[eta]) = 0, [Sigma](m[zeta]) = 0. (7) - - One or two special cases may be noticed. If three masses [alpha], - [beta], [gamma] be situate at the vertices of a triangle ABC, the - mass-centre of [beta] and [gamma] is at a point A' in BC, such that - [beta].BA' = [gamma].A'C. The mass-centre (G) of [alpha], [beta], - [gamma] will then divide AA' so that [alpha].AG = ([beta] + [gamma]) - GA'. It is easily proved that - - [alpha] : [beta] : [gamma] = [Delta]BGA : [Delta]GCA : [Delta]GAB; - - also, by giving suitable values (positive or negative) to the ratios - [alpha] : [beta] : [gamma] we can make G assume any assigned position - in the plane ABC. We have here the origin of the "barycentric - co-ordinates" of Mobius, now usually known as "areal" co-ordinates. If - [alpha] + [beta] + [gamma] = 0, G is at infinity; if [alpha] = [beta] - = [gamma], G is at the intersection of the median lines of the - triangle; if [alpha] : [beta] : [gamma] = a : b : c, G is at the - centre of the inscribed circle. Again, if G be the mass-centre of four - particles [alpha], [beta], [gamma], [delta] situate at the vertices of - a tetrahedron ABCD, we find - - [alpha] : [beta] : [gamma] : [delta] = tet^n GBCD : tet^n GCDA : tet^n GDAB : tet^n GABC, - - and by suitable determination of the ratios on the left hand we can - make G assume any assigned position in space. If [alpha] + [beta] + - [gamma] + [delta] = O, G is at infinity; if [alpha] = [beta] = [gamma] - = [delta], G bisects the lines joining the middle points of opposite - edges of the tetrahedron ABCD; if [alpha] : [beta] : [gamma] : [delta] - = [Delta]BCD : [Delta]CDA : [Delta]DAB : [Delta]ABC, G is at the - centre of the inscribed sphere. - - If we have a continuous distribution of matter, instead of a system of - discrete particles, the summations in (6) are to be replaced by - integrations. Examples will be found in textbooks of the calculus and - of analytical statics. As particular cases: the mass-centre of a - uniform thin triangular plate coincides with that of three equal - particles at the corners; and that of a uniform solid tetrahedron - coincides with that of four equal particles at the vertices. Again, - the mass-centre of a uniform solid right circular cone divides the - axis in the ratio 3 : 1; that of a uniform solid hemisphere divides - the axial radius in the ratio 3 : 5. - - It is easily seen from (6) that if the configuration of a system of - particles be altered by "homogeneous strain" (see ELASTICITY) the new - position of the mass-centre will be at that point of the strained - figure which corresponds to the original mass-centre. - -The formula (2) shows that a system of concurrent forces represented by -m1.[->OP1], m2.[->OP2], ... m_n.[->OP_n] will have a resultant -represented hy [Sigma](m).[->OG]. If we imagine O to recede to infinity -in any direction we learn that a system of parallel forces proportional -to m1, m2,... m_n, acting at P1, P2 ... P_n have a resultant -proportional to [Sigma](m) which acts always through a point G fixed -relatively to the given mass-system. This contains the theory of the -"centre of gravity" (SS 4, 9). We may note also that if P1, P2, ... P_n, -and P1', P2', ... P_n' represent two configurations of the series of -particles, then - - [Sigma](m.[->PP']) = Sigma(m).[->GG'], (8) - -where G, G' are the two positions of the mass-centre. The forces -m1.[->P1P1'], m2.[->P2P2'], ... m_n.[->P_nP_n'], considered as localized -vectors, do not, however, as a rule reduce to a single resultant. - -We proceed to the theory of the _plane_, _axial_ and _polar quadratic -moments_ of the system. The axial moments have alone a dynamical -significance, but the others are useful as subsidiary conceptions. If -h1, h2, ... h_n be the perpendicular distances of the particles from any -fixed plane, the sum [Sigma](mh^2) is the quadratic moment with respect -to the plane. If p1, p2, ... p_n be the perpendicular distances from any -given axis, the sum [Sigma](mp^2) is the quadratic moment with respect to -the axis; it is also called the _moment of inertia_ about the axis. If -r1, r2, ... r_n be the distances from a fixed point, the sum -[Sigma](mr^2) is the quadratic moment with respect to that point (or -pole). If we divide any of the above quadratic moments by the total -mass [Sigma](m), the result is called the _mean square_ of the distances -of the particles from the respective plane, axis or pole. In the case of -an axial moment, the square root of the resulting mean square is called -the _radius of gyration_ of the system about the axis in question. If we -take rectangular axes through any point O, the quadratic moments with -respect to the co-ordinate planes are - - I_x = [Sigma](mx^2), I_y = [Sigma](my^2), I_z = [Sigma](mz^2); (9) - -those with respect to the co-ordinate axes are - - I_yz = [Sigma]{m(y^2 + z^2)}, I_zx = [Sigma]{m(z^2 + x^2)}, - I_xy = [Sigma]{m(x^2 + y^2)}; (10) - -whilst the polar quadratic moment with respect to O is - - I0 = [Sigma]{m(x^2 + y^2 + z^2)}. (11) - -We note that - - I_yz = I_y + I_z, I_zx = I_z + I_x, I_xy = I_x + I_y, (12) - -and - - I0 = I_x + I_y + I_z = (1/2)(I_yz + I_zx + I_xy). (13) - - In the case of continuous distributions of matter the summations in - (9), (10), (11) are of course to be replaced by integrations. For a - uniform thin circular plate, we find, taking the origin at its centre, - and the axis of z normal to its plane, I0 = (1/2)Ma^2, where M is the - mass and a the radius. Since I_x = I_y, I_z = 0, we deduce I_zx = - (1/2)Ma^2, I_xy = (1/2)Ma^2; hence the value of the squared radius of - gyration is for a diameter (1/4)a^2, and for the axis of symmetry - (1/2)a^2. Again, for a uniform solid sphere having its centre at the - origin we find I0 = (3/5)Ma^2, I_x = I_y = I_z = (1/5)Ma^2, I_yz = - I_zx = l_xy = (3/5)Ma^2; i.e. the square of the radius of gyration - with respect to a diameter is (2/5)a^2. The method of homogeneous - strain can be applied to deduce the corresponding results for an - ellipsoid of semi-axes a, b, c. If the co-ordinate axes coincide with - the principal axes, we find I_x = (1/5)Ma^2, I_y = (1/5)Mb^2, I_z = - (1/5)Mc^2, whence I_yz = (1/5)M (b^2 + c^2), &c. - -If [phi](x, y, z) be any homogeneous quadratic function of x, y, z, we -have - - [Sigma]{m[phi](x, y, z)} = [Sigma] {m[phi]([|x] + [xi], [|y] + [eta], [|z] + [zeta])} - = [Sigma] {m[phi](x, y, z)} + [Sigma]{m[phi]([xi], [eta], [zeta])}, (14) - -since the terms which are bilinear in respect to [|x], [|y], [|z], and -[xi], [eta], [zeta] vanish, in virtue of the relations (7). Thus - - I_x = I[xi] + [Sigma](m)x^2, (15) - - I_yz = I[eta][zeta] + [Sigma](m).(y^2 + z^2), (16) - -with similar relations, and - - I_O = I_G + [Sigma](m).OG^2. (17) - -The formula (16) expresses that the squared radius of gyration about any -axis (Ox) exceeds the squared radius of gyration about a parallel axis -through G by the square of the distance between the two axes. The -formula (17) is due to J. L. Lagrange; it may be written - - [Sigma](m.OP^2) [Sigma](m.GP^2) - -------------- = -------------- + OG^2, (18) - [Sigma](m) [Sigma](m) - -and expresses that the mean square of the distances of the particles -from O exceeds the mean square of the distances from G by OG^2. The -mass-centre is accordingly that point the mean square of whose distances -from the several particles is least. If in (18) we make O coincide with -P1, P2, ... P_n in succession, we obtain - - 0 + m2.P1P2^2 + ... + mn.P1P_n^2 = [Sigma](m.GP^2) + [Sigma](m).GP1^2, \ - m1.P2P1^2 + 0 + ... + mn.P2P_n^2 = [Sigma](m.GP^2) + [Sigma](m).GP2^2, > (19) - ... ... ... ... ... | - m1.P_nP1^2 + m2.P_nP2^2 + ... + 0 = [Sigma](m.GP^2) + [Sigma](m).GP_n^2. / - -If we multiply these equations by m1, m2 ... m_n, respectively, and add, -we find - - [Sigma][Sigma](m_r m_s.P_r P_s^2) = [Sigma](m).[Sigma](m.GP^2), (20) - -provided the summation [Sigma][Sigma] on the left hand be understood to -include each pair of particles once only. This theorem, also due to -Lagrange, enables us to express the mean square of the distances of the -particles from the centre of mass in terms of the masses and mutual -distances. For instance, considering four equal particles at the -vertices of a regular tetrahedron, we can infer that the radius R of the -circumscribing sphere is given by R^2 = (3/8)a^2, if a be the length of -an edge. - -Another type of quadratic moment is supplied by the _deviation-moments_, -or _products of inertia_ of a distribution of matter. Thus the sum -[Sigma](m.yz) is called the "product of inertia" with respect to the -planes y = 0, z = 0. This may be expressed In terms of the product of -inertia with respect to parallel planes through G by means of the -formula (14); viz.:-- - - [Sigma](m.yz) = [Sigma](m.[eta][zeta]) + [Sigma](m).yz (21) - -The quadratic moments with respect to different planes through a fixed -point O are related to one another as follows. The moment with respect -to the plane - - [lambda]x + [mu]y + [nu]z = 0, (22) - -where [lambda], [mu], [nu] are direction-cosines, is - - [Sigma]{(m([lambda]x + [mu]y + [nu]z)^2} = [Sigma](mx^2).[lambda]^2 + [Sigma](my^2).[mu]^2 + [Sigma](mz^2).[nu]^2 - + 2[Sigma](myz).[mu][nu] + 2[Sigma](mzx).[nu][lambda] + 2[Sigma](mxy).[lambda][mu], (23) - -and therefore varies as the square of the perpendicular drawn from O to -a tangent plane of a certain quadric surface, the tangent plane in -question being parallel to (22). If the co-ordinate axes coincide with -the principal axes of this quadric, we shall have - - [Sigma](myz) = 0, [Sigma](mzx) = 0, [Sigma](mxy) = 0; (24) - -and if we write - - [Sigma](mx^2) = Ma^2, [Sigma](my^2) = Mb^2, [Sigma](mz^2) = Mc^2, (25) - -where M = [Sigma](m), the quadratic moment becomes M(a^2[lambda]^2 + -b^2[mu]^2 + c^2[nu]^2), or Mp^2, where p is the distance of the origin -from that tangent plane of the ellipsoid - - x^2 y^2 z^2 - --- + --- + --- = 1, (26) - a^2 b^2 c^2 - -which is parallel to (22). It appears from (24) that through any -assigned point O three rectangular axes can be drawn such that the -product of inertia with respect to each pair of co-ordinate planes -vanishes; these are called the _principal axes of inertia_ at O. The -ellipsoid (26) was first employed by J. Binet (1811), and may be called -"Binet's Ellipsoid" for the point O. Evidently the quadratic moment for -a variable plane through O will have a "stationary" value when, and only -when, the plane coincides with a principal plane of (26). It may further -be shown that if Binet's ellipsoid be referred to any system of -conjugate diameters as co-ordinate axes, its equation will be - - x'^2 y'^2 z'^2 - ---- + ---- + ---- = 1, (27) - a'^2 b'^2 c'^2 - -provided - - [Sigma](mx'^2) = Ma'^2, [Sigma](my'^2) Mb'^2, [Sigma](mz'^2) = Mc'^2; - -also that - - [Sigma](my'z') = 0, [Sigma](mz'x') = 0, [Sigma](mx'y') = 0. (28) - -Let us now take as co-ordinate axes the principal axes of inertia at the -mass-centre G. If a, b, c be the semi-axes of the Binet's ellipsoid of -G, the quadratic moment with respect to the plane [lambda]x + [mu]y + -[nu]z = 0 will be M(a^2[lambda]^2 + b^2[mu]^2 + c^2[nu]^2), and that with -respect to a parallel plane - - [lambda]x + [mu]y + [nu]z = p (29) - -will be M(a^2[lambda]^2 + b^2[mu]^2 + c^2[nu]^2 + p^2), by (15). This -will have a given value Mk^2, provided - - p^2 = (k^2 - a^2)[lambda]^2 + (k^2 - b^2)[mu]^2 + (k^2 - c^2)[nu]^2. (30) - -Hence the planes of constant quadratic moment Mk^2 will envelop the -quadric - - x^2 y^2 z^2 - --------- + --------- + --------- = 1, (31) - k^2 - a^2 k^2 - b^2 k^2 - c^2 - -and the quadrics corresponding to different values of k^2 will be -confocal. If we write - - k^2 = a^2 + b^2 + c^2 + [theta], - b^2 + c^2 = [alpha]^2, c^2 + a^2 = [beta]^2, a^2 + b^2 = [gamma]^2 (32) - -the equation (31) becomes - - x^2 y^2 z^2 - ------------------- + ------------------ + ------------------- = 1 (33) - [alpha]^2 + [theta] [beta]^2 + [theta] [gamma]^2 + [theta] - -for different values of [theta] this represents a system of quadrics -confocal with the ellipsoid - - x^2 y^2 z^2 - --------- + -------- + --------- = 1, (34) - [alpha]^2 [beta]^2 [gamma]^2 - -which we shall meet with presently as the "ellipsoid of gyration" at G. -Now consider the tangent plane [omega] at any point P of a confocal, the -tangent plane [omega]' at an adjacent point N', and a plane [omega]" -through P parallel to [omega]'. The distance between the planes [omega]' -and [omega]" will be of the second order of small quantities, and the -quadratic moments with respect to [omega]' and [omega]" will therefore -be equal, to the first order. Since the quadratic moments with respect -to [omega] and [omega]' are equal, it follows that [omega] is a plane of -stationary quadratic moment at P, and therefore a principal plane of -inertia at P. In other words, the principal axes of inertia at P arc the -normals to the three confocals of the system (33) which pass through P. -Moreover if x, y, z be the co-ordinates of P, (33) is an equation to -find the corresponding values of [theta]; and if [theta]1, [theta]2, -[theta]3 be the roots we find - - [theta]1 + [theta]2 + [theta]3 = r^2 - [alpha]^2 - [beta]^2 -[gamma]^2, (35) - -where r^2 = x^2 + y^2 + z^2. The squares of the radii of gyration about -the principal axes at P may be denoted by k2^2 + k3^2, k3^2 + k1^2, k1^2 -+ k2^2; hence by (32) and (35) they are r^2 - [theta]1, r^2 - [theta]2, -r^2 - [theta]3, respectively. - -To find the relations between the moments of inertia about different -axes through any assigned point O, we take O as origin. Since the square -of the distance of a point (x, y, z) from the axis - - x y z - -------- = ---- = ---- (36) - [lambda] [mu] [nu] - -is x^2 + y^2 + z^2 - ([lambda]x + [mu]y + [nu]z)^2, the moment of inertia -about this axis is - - I = [Sigma][m{([lambda]^2 + [mu]^2 + [nu]^2)(x^2 + y^2 + z^2) - ([lambda]x + [mu]y + [nu]z)^2}] - = A[lambda]^2 + B[mu]^2 + C[nu]^2 - 2F[mu][nu] - 2G[nu][lambda] - 2H[lambda][mu], (37) - -provided - - A = [Sigma]{m(y^2 + z^2)}, B = [Sigma]{m(z^2 + x^2)}, C = [Sigma]{m(x^2 + y^2)}, - F = [Sigma](myz), G = [Sigma](mzx), H = [Sigma](mxy); (38) - -i.e. A, B, C are the moments of inertia about the co-ordinate axes, and -F, G, H are the products of inertia with respect to the pairs of -co-ordinate planes. If we construct the quadric - - Ax^2 + By^2 + Cz^2 - 2Fyz - 2Gzx - 2Hxy = M[epsilon]^4 (39) - -where [epsilon] is an arbitrary linear magnitude, the intercept r which -it makes on a radius drawn in the direction [lambda], [mu], [nu] is -found by putting x, y, z = [lambda]r, [mu]r, [nu]r. Hence, by comparison -with (37), - - I = M[epsilon]^4/r^2. (40) - -The moment of inertia about any radius of the quadric (39) therefore -varies inversely as the square of the length of this radius. When -referred to its principal axes, the equation of the quadric takes the -form - - Ax^2 + By^2 + Cz^2 = M[epsilon]^4. (41) - -The directions of these axes are determined by the property (24), and -therefore coincide with those of the principal axes of inertia at O, as -already defined in connexion with the theory of plane quadratic moments. -The new A, B, C are called the _principal moments of inertia_ at O. -Since they are essentially positive the quadric is an ellipsoid; it is -called the _momental ellipsoid_ at O. Since, by (12), B + C > A, &c., -the sum of the two lesser principal moments must exceed the greatest -principal moment. A limitation is thus imposed on the possible forms of -the momental ellipsoid; e.g. in the case of symmetry about an axis it -appears that the ratio of the polar to the equatorial diameter of the -ellipsoid cannot be less than 1/[root]2. - -If we write A = M[alpha]^2, B = M[beta]^2, C = M[gamma]^2, the formula -(37), when referred to the principal axes at O, becomes - - I = M([alpha]^2[lambda]^2 + [beta]^2[mu]^2 + [gamma]^2[nu]^2) = Mp^2, (42) - -if p denotes the perpendicular drawn from O in the direction ([lambda], -[mu], [nu]) to a tangent plane of the ellipsoid - - x^2 y^2 z^2 - --------- + -------- + --------- = 1 (43) - [alpha]^2 [beta]^2 [gamma]^2 - -This is called the _ellipsoid of gyration_ at O; it was introduced into -the theory by J. MacCullagh. The ellipsoids (41) and (43) are reciprocal -polars with respect to a sphere having O as centre. - -If A = B = C, the momental ellipsoid becomes a sphere; all axes through -O are then principal axes, and the moment of inertia is the same for -each. The mass-system is then said to possess kinetic symmetry about O. - - If all the masses lie in a plane (z = 0) we have, in the notation of - (25), c^2 = 0, and therefore A = Mb^2, B = Ma^2, C = M(a^2 + b^2), so - that the equation of the momental ellipsoid takes the form - - b^2x^2 + a^2y^2 + (a^2 + b^2)z^2 = [epsilon]^4. (44) - - The section of this by the plane z = 0 is similar to - - x^2 y^2 - ---- + --- = 1, (45) - a^2 b^2 - - which may be called the _momental ellipse_ at O. It possesses the - property that the radius of gyration about any diameter is half the - distance between the two tangents which are parallel to that diameter. - In the case of a uniform triangular plate it may be shown that the - momental ellipse at G is concentric, similar and similarly situated - - to the ellipse which touches the sides of the triangle at their middle - points. - - [Illustration: FIG. 59.] - - [Illustration: FIG. 60.] - - The graphical methods of determining the moment of inertia of a plane - system of particles with respect to any line in its plane may be - briefly noticed. It appears from S 5 (fig. 31) that the linear moment - of each particle about the line may be found by means of a funicular - polygon. If we replace the mass of each particle by its moment, as - thus found, we can in like manner obtain the quadratic moment of the - system with respect to the line. For if the line in question be the - axis of y, the first process gives us the values of mx, and the second - the value of [Sigma](mx.x) or [Sigma](mx^2). The construction of a - second funicular may be dispensed with by the employment of a - planimeter, as follows. In fig. 59 p is the line with respect to which - moments are to be taken, and the masses of the respective particles - are indicated by the corresponding segments of a line in the - force-diagram, drawn parallel to p. The funicular ZABCD ... - corresponding to any pole O is constructed for a system of forces - acting parallel to p through the positions of the particles and - proportional to the respective masses; and its successive sides are - produced to meet p in the points H, K, L, M, ... As explained in S 5, - the moment of the first particle is represented on a certain scale by - HK, that of the second by KL, and so on. The quadratic moment of the - first particle will then be represented by twice the area AHK, that of - the second by twice the area BKL, and so on. The quadratic moment of - the whole system is therefore represented by twice the area AHEDCBA. - Since a quadratic moment is essentially positive, the various areas - are to taken positive in all cases. If k be the radius of gyration - about p we find - - k^2 = 2 X area AHEDCBA X ON / [alpha][beta], - - where [alpha][beta] is the line in the force-diagram which represents - the sum of the masses, and ON is the distance of the pole O from this - line. If some of the particles lie on one side of p and some on the - other, the quadratic moment of each set may be found, and the results - added. This is illustrated in fig. 60, where the total quadratic - moment is represented by the sum of the shaded areas. It is seen that - for a given direction of p this moment is least when p passes through - the intersection X of the first and last sides of the funicular; i.e. - when p goes through the mass-centre of the given system; cf. equation - (15). - - -PART II.--KINETICS - -S 12. _Rectilinear Motion._--Let x denote the distance OP of a moving -point P at time t from a fixed origin O on the line of motion, this -distance being reckoned positive or negative according as it lies to one -side or the other of O. At time t + [delta]t let the point be at Q, and -let OQ = x + [delta]x. The _mean velocity_ of the point in the interval -[delta]t is [delta]x/[delta]t. The limiting value of this when [delta]t -is infinitely small, viz. dx/dt, is adopted as the definition of the -_velocity_ at the instant t. Again, let u be the velocity at time t, u + -[delta]u that at time t + [delta]t. The mean rate of increase of -velocity, or the _mean acceleration_, in the interval [delta]t is then -[delta]u/[delta]t. The limiting value of this when [delta]t is -infinitely small, viz., du/dt, is adopted as the definition of the -_acceleration_ at the instant t. Since u = dx/dt, the acceleration is -also denoted by d^2x/dt^2. It is often convenient to use the "fluxional" -notation for differential coefficients with respect to time; thus the -velocity may be represented by [.x] and the acceleration by [.u] or -[:x]. There is another formula for the acceleration, in which u is -regarded as a function of the position; thus du/dt = (du/dx)(dx/dt) = -u(du/dx). The relation between x and t in any particular case may be -illustrated by means of a curve constructed with t as abscissa and x as -ordinate. This is called the _curve of positions_ or _space-time curve_; -its gradient represents the velocity. Such curves are often traced -mechanically in acoustical and other experiments. A, curve with t as -abscissa and u as ordinate is called the _curve of velocities_ or -_velocity-time curve_. Its gradient represents the acceleration, and the -area ([int]udt) included between any two ordinates represents the space -described in the interval between the corresponding instants (see fig. -62). - -So far nothing has been said about the measurement of time. From the -purely kinematic point of view, the t of our formulae may be any -continuous independent variable, suggested (it may be) by some physical -process. But from the dynamical standpoint it is obvious that equations -which represent the facts correctly on one system of time-measurement -might become seriously defective on another. It is found that for almost -all purposes a system of measurement based ultimately on the earth's -rotation is perfectly adequate. It is only when we come to consider such -delicate questions as the influence of tidal friction that other -standards become necessary. - -The most important conception in kinetics is that of "inertia." It is a -matter of ordinary observation that different bodies acted on by the -same force, or what is judged to be the same force, undergo different -changes of velocity in equal times. In our ideal representation of -natural phenomena this is allowed for by endowing each material particle -with a suitable _mass_ or _inertia-coefficient_ m. The product _mu_ of -the mass into the velocity is called the _momentum_ or (in Newton's -phrase) the _quantity of motion_. On the Newtonian system the motion of -a particle entirely uninfluenced by other bodies, when referred to a -suitable base, would be rectilinear, with constant velocity. If the -velocity changes, this is attributed to the action of force; and if we -agree to measure the force (X) by the rate of change of momentum which -it produces, we have the equation - - d - --- (mu) = X. (1) - dt - -From this point of view the equation is a mere truism, its real -importance resting on the fact that by attributing suitable values to -the masses m, and by making simple assumptions as to the value of X in -each case, we are able to frame adequate representations of whole -classes of phenomena as they actually occur. The question remains, of -course, as to how far the measurement of force here implied is -practically consistent with the gravitational method usually adopted in -statics; this will be referred to presently. - -The practical unit or standard of mass must, from the nature of the -case, be the mass of some particular body, e.g. the imperial pound, or -the kilogramme. In the "C.G.S." system a subdivision of the latter, viz. -the gramme, is adopted, and is associated with the centimetre as the -unit of length, and the mean solar second as the unit of time. The unit -of force implied in (1) is that which produces unit momentum in unit -time. On the C.G.S. system it is that force which acting on one gramme -for one second produces a velocity of one centimetre per second; this -unit is known as the _dyne_. Units of this kind are called _absolute_ on -account of their fundamental and invariable character as contrasted with -gravitational units, which (as we shall see presently) vary somewhat -with the locality at which the measurements are supposed to be made. - -If we integrate the equation (1) with respect to t between the limits t, -t' we obtain - _ - / t' - mu'- mu = | X dt. (2) - _/ t - -The time-integral on the right hand is called the _impulse_ of the force -on the interval t' - t. The statement that the increase of momentum is -equal to the impulse is (it maybe remarked) equivalent to Newton's own -formulation of his Second Law. The form (1) is deduced from it by -putting t'- t = [delta]t, and taking [delta]t to be infinitely small. In -problems of impact we have to deal with cases of practically -instantaneous impulse, where a very great and rapidly varying force -produces an appreciable change of momentum in an exceedingly minute -interval of time. - -In the case of a constant force, the acceleration [.u] or [:x] is, -according to (1), constant, and we have - - d^2x - ---- = [alpha], (3) - dt^2 - -say, the general solution of which is - - x = (1/2)[alpha]t^2 + At + B. (4) - -The "arbitrary constants" A, B enable us to represent the circumstances -of any particular case; thus if the velocity [.x] and the position x be -given for any one value of t, we have two conditions to determine A, B. -The curve of positions corresponding to (4) is a parabola, and that of -velocities is a straight line. We may take it as an experimental result, -although the best evidence is indirect, that a particle falling freely -under gravity experiences a constant acceleration which at the same -place is the same for all bodies. This acceleration is denoted by g; its -value at Greenwich is about 981 centimetre-second units, or 32.2 feet -per second. It increases somewhat with the latitude, the extreme -variation from the equator to the pole being about (1/2)%. We infer that -on our reckoning the force of gravity on a mass m is to be measured by -mg, the momentum produced per second when this force acts alone. Since -this is proportional to the mass, the relative masses to be attributed -to various bodies can be determined practically by means of the balance. -We learn also that on account of the variation of g with the locality a -gravitational system of force-measurement is inapplicable when more than -a moderate degree of accuracy is desired. - -[Illustration: FIG. 61.] - -We take next the case of a particle attracted towards a fixed point O in -the line of motion with a force varying as the distance from that point. -If [mu] be the acceleration at unit distance, the equation of motion -becomes - - d^2x - ---- = -[mu]x, (5) - dt^2 - -the solution of which may be written in either of the forms - - x = A cos [sigma]t + B sin [sigma]t, x = a cos ([sigma]t + [epsilon]), (6) - -where [sigma]= [root][mu], and the two constants A, B or a, [epsilon] -are arbitrary. The particle oscillates between the two positions x = [+-]a, -and the same point is passed through in the same direction with the same -velocity at equal intervals of time 2[pi]/[sigma]. The type of motion -represented by (6) is of fundamental importance in the theory of -vibrations (S 23); it is called a _simple-harmonic_ or (shortly) a -_simple_ vibration. If we imagine a point Q to describe a circle of -radius a with the angular velocity [sigma], its orthogonal projection P -on a fixed diameter AA' will execute a vibration of this character. The -angle [sigma]t + [epsilon] (or AOQ) is called the _phase_; the arbitrary -elements a, [epsilon] are called the _amplitude_ and _epoch_ (or initial -phase), respectively. In the case of very rapid vibrations it is usual -to specify, not the _period_ (2[pi]/[sigma]), but its reciprocal the -_frequency_, i.e. the number of complete vibrations per unit time. Fig. -62 shows the curves of position and velocity; they both have the form of -the "curve of sines." The numbers correspond to an amplitude of 10 -centimetres and a period of two seconds. - -The vertical oscillations of a weight which hangs from a fixed point by -a spiral spring come under this case. If M be the mass, and x the -vertical displacement from the position of equilibrium, the equation of -motion is of the form - - d^2x - M ---- = - Kx, (7) - dt^2 - -provided the inertia of the spring itself be neglected. This becomes -identical with (5) if we put [mu] = K/M; and the period is therefore -2[pi][root](M/K), the same for all amplitudes. The period is increased -by an increase of the mass M, and diminished by an increase in the -stiffness (K) of the spring. If c be the statical increase of length -which is produced by the gravity of the mass M, we have Kc = Mg, and the -period is 2[pi][root](c/g). - -[Illustration: FIG. 62.] - -The small oscillations of a simple pendulum in a vertical plane also -come under equation (5). According to the principles of S 13, the -horizontal motion of the bob is affected only by the horizontal -component of the force acting upon it. If the inclination of the string -to the vertical does not exceed a few degrees, the vertical displacement -of the particle is of the second order, so that the vertical -acceleration may be neglected, and the tension of the string may be -equated to the gravity mg of the particle. Hence if l be the length of -the string, and x the horizontal displacement of the bob from the -equilibrium position, the horizontal component of gravity is mgx/l, -whence - - d^2x gx - ---- = - ---, (8) - dt^2 l - -The motion is therefore simple-harmonic, of period [tau] = -2[pi][root](l/g). This indicates an experimental method of determining g -with considerable accuracy, using the formula g = 4[pi]^2l/[tau]^2. - - In the case of a repulsive force varying as the distance from the - origin, the equation of motion is of the type - - d^2x - ---- = [mu]x, (9) - dt^2 - - the solution of which is - - x = A e^(nt) + B e^(-nt), (10) - - where n = [root][mu]. Unless the initial conditions be adjusted so as - to make A = 0 exactly, x will ultimately increase indefinitely with t. - The position x = 0 is one of equilibrium, but it is unstable. This - applies to the inverted pendulum, with [mu] = g/l, but the equation - (9) is then only approximate, and the solution therefore only serves - to represent the initial stages of a motion in the neighbourhood of - the position of unstable equilibrium. - -In acoustics we meet with the case where a body is urged towards a fixed -point by a force varying as the distance, and is also acted upon by an -"extraneous" or "disturbing" force which is a given function of the -time. The most important case is where this function is simple-harmonic, -so that the equation (5) is replaced by - - d^2x - ---- + [mu]x = f cos ([sigma]1t + [alpha]), (11) - dt^2 - -where [sigma]1 is prescribed. A particular solution is - - f - x = ----------------- cos ([sigma]1t + [alpha]). (12) - [mu] - [sigma]1^2 - -This represents a _forced oscillation_ whose period 2[pi]/[sigma]1, -coincides with that of the disturbing force; and the phase agrees with -that of the force, or is opposed to it, according as [sigma]1^2 < or > [mu]; -i.e. according as the imposed period is greater or less than the natural -period 2[pi]/[root][mu]. The solution fails when the two periods agree -exactly; the formula (12) is then replaced by - - ft - x = ---------- sin ([sigma]1t + [alpha]), (13) - 2 [sigma]1 - -which represents a vibration of continually increasing amplitude. Since -the equation (12) is in practice generally only an approximation (as in -the case of the pendulum), this solution can only be accepted as a -representation of the initial stages of the forced oscillation. To -obtain the complete solution of (11) we must of course superpose the -free vibration (6) with its arbitrary constants in order to obtain a -complete representation of the most general motion consequent on -arbitrary initial conditions. - -[Illustration: FIG. 63.] - - A simple mechanical illustration is afforded by the pendulum. If the - point of suspension have an imposed simple vibration [xi] = a cos - [sigma]t in a horizontal line, the equation of small motion of the bob - is - - x - [xi] - m[:x] = -mg --------, - l - - or - - gx [xi] - [:x] + --- = ----. (14) - l l - - This is the same as if the point of suspension were fixed, and a - horizontal disturbing force mg[xi]/l were to act on the bob. The - difference of phase of the forced vibration in the two cases is - illustrated and explained in the annexed fig. 63, where the pendulum - virtually oscillates about C as a fixed point of suspension. This - illustration was given by T. Young in connexion with the kinetic - theory of the tides, where the same point arises. - - We may notice also the case of an attractive force varying inversely - as the square of the distance from the origin. If [mu] be the - acceleration at unit distance, we have - - du [mu] - u --- = - ---- (15) - dx x^2 - - whence - - 2[mu] - u^2 = ----- + C. (16) - x - - In the case of a particle falling directly towards the earth from rest - at a very great distance we have C = 0 and, by Newton's Law of - Gravitation, [mu]/a^2 = g, where a is the earth's radius. The deviation - of the earth's figure from sphericity, and the variation of g with - latitude, are here ignored. We find that the velocity with which the - particle would arrive at the earth's surface (x = a) is [root](2ga). - If we take as rough values a = 21 X 10^6 feet, g = 32 foot-second - units, we get a velocity of 36,500 feet, or about seven miles, per - second. If the particles start from rest at a finite distance c, we - have in (16), C = - 2[mu]/c, and therefore - - dx / / 2[mu](c - x) \ - -- = u = - / ( ------------- ), (17) - dt \/ \ cx / - - the minus sign indicating motion towards the origin. If we put x = c - cos^2 (1/2)[phi], we find - - c^(3/2) - t = ------------- ([phi] + sin [phi]), (18) - [root](8[mu]) - - no additive constant being necessary if t be reckoned from the instant - of starting, when [phi] = 0. The time t of reaching the origin ([phi] - = [pi]) is - - [pi] c^(3/2) - t1 = -------------. (19) - [root](8[mu]) - - This may be compared with the period of revolution in a circular orbit - of radius c about the same centre of force, viz. - 2[pi]c^(3/2)/[root][mu](S 14). We learn that if the orbital motion of - a planet, or a satellite, were arrested, the body would fall into the - sun, or into its primary, in the fraction 0.1768 of its actual - periodic time. Thus the moon would reach the earth in about five days. - It may be noticed that if the scales of x and t be properly adjusted, - the curve of positions in the present problem is the portion of a - cycloid extending from a vertex to a cusp. - -In any case of rectilinear motion, if we integrate both sides of the -equation - - du - mu -- = X, (20) - dx - -which is equivalent to (1), with respect to x between the limits x0, x1, -we obtain - _ - / x1 - (1/2) mu1^2 - (1/2) mu0^2 = | X dx. (21) - _/ x0 - -We recognize the right-hand member as the _work_ done by the force X on -the particle as the latter moves from the position x0 to the position -x1. If we construct a curve with x as abscissa and X as ordinate, this -work is represented, as in J. Watt's "indicator-diagram," by the area -cut off by the ordinates x = x0, x = x1. The product (1/2)mu^2 is called -the _kinetic energy_ of the particle, and the equation (21) is therefore -equivalent to the statement that the increment of the kinetic energy is -equal to the work done on the particle. If the force X be always the -same in the same position, the particle may be regarded as moving in a -certain invariable "field of force." The work which would have to be -supplied by other forces, extraneous to the field, in order to bring the -particle from rest in some standard position P0 to rest in any assigned -position P, will depend only on the position of P; it is called the -_statical_ or _potential energy_ of the particle with respect to the -field, in the position P. Denoting this by V, we have [delta]V - -X[delta]x = 0, whence - - dV - X = - --, (22) - dx - -The equation (21) may now be written - - (1/2) mu1^2 + V1 = (1/2) mu0^2 + V0, (23) - -which asserts that when no extraneous forces act the sum of the kinetic -and potential energies is constant. Thus in the case of a weight hanging -by a spiral spring the work required to increase the length by x is V = -[int 0 to x] Kxdx = (1/2)Kx^2, whence (1/2)Mu^2 + (1/2)Kx^2 = const., as -is easily verified from preceding results. It is easily seen that the -effect of extraneous forces will be to increase the sum of the kinetic -and potential energies by an amount equal to the work done by them. If -this amount be negative the sum in question is diminished by a -corresponding amount. It appears then that this sum is a measure of the -total capacity for doing work against extraneous resistances which the -particle possesses in virtue of its motion and its position; this is in -fact the origin of the term "energy." The product mv^2 had been called -by G. W. Leibnitz the "vis viva"; the name "energy" was substituted by -T. Young; finally the name "actual energy" was appropriated to the -expression (1/2)mv^2 by W. J. M. Rankine. - - The laws which regulate the resistance of a medium such as air to the - motion of bodies through it are only imperfectly known. We may briefly - notice the case of resistance varying as the square of the velocity, - which is mathematically simple. If the positive direction of x be - downwards, the equation of motion of a falling particle will be of the - form - - du - -- = g - ku^2; (24) - dt - - this shows that the velocity u will send asymptotically to a certain - limit V (called the _terminal velocity_) such that kV^2 = g. The - solution is - - gt V^2 gt - u = V tanh ---, x = --- log cosh ---, (25) - V g V - - if the particle start from rest in the position x = 0 at the instant t - = 0. In the case of a particle projected vertically upwards we have - - du - -- = -g - ku^2, (26) - dt - - the positive direction being now upwards. This leads to - - u u0 gt V^2 V^2 + u0^2 - tan^-1 --- = tan^-1 --- - ---, x = --- log ----------, (27) - V V V 2g V^2 + u^2 - - where u0 is the velocity of projection. The particle comes to rest - when - - V u0 V^2 / u0^2 \ - t = --- tan^-1 ---, x = --- log ( 1 + --- ). (28) - g V 2g \ V^2 / - - For small velocities the resistance of the air is more nearly - proportional to the first power of the velocity. The effect of forces - of this type on small vibratory motions may be investigated as - follows. The equation (5) when modified by the introduction of a - frictional term becomes - - [:x] = -[mu]x - k [.x]. (29) - - If k^2 < 4[mu] the solution is - - x = a e^{-t/[tau]} cos ([sigma]t + [epsilon]), (30) - - where - - [tau] = 2/k, [sigma] = [root]([mu] - (1/4)k^2), (31) - - and the constants a, [epsilon] are arbitrary. This may be described as - a simple harmonic oscillation whose amplitude diminishes - asymptotically to zero according to the law e^(-t/[tau]). The constant - [tau] is called the _modulus of decay_ of the oscillations; if it is - large compared with 2[pi]/[sigma] the effect of friction on the period - is of the second order of small quantities and may in general be - ignored. We have seen that a true simple-harmonic vibration may be - regarded as the orthogonal projection of uniform circular motion; it - was pointed out by P. G. Tait that a similar representation of the - type (30) is obtained if we replace the circle by an equiangular - spiral described, with a constant angular velocity about the pole, in - the direction of diminishing radius vector. When k^2 > 4[mu], the - solution of (29) is, in real form, - - x = a1 e^(-t/[tau]1) + a2 e^(-t/[tau]2), (32) - - where - - 1/[tau]1, 1/[tau]2 = (1/2)k [+-] [root]((1/4)k^2 - [mu]). (33) - - The body now passes once (at most) through its equilibrium position, - and the vibration is therefore styled _aperiodic_. - - To find the forced oscillation due to a periodic force we have - - [:x] + k[.x] + [mu]x = f cos ([sigma]1t + [epsilon]). (34) - - The solution is - - f - x = --- cos ([sigma]1t + [epsilon] - [epsilon]1), (35) - R - - provided - k[sigma]1 - R = {([mu] - [sigma]1^2)^2 + k^2[sigma]1^2}^(1/2), tan[epsilon]1 = -----------------. (36) - [mu] - [sigma]1^2 - - Hence the phase of the vibration lags behind that of the force by the - amount [epsilon]1, which lies between 0 and (1/2)[pi] or between - (1/2)[pi] and [pi], according as [sigma]1^2 <> [mu]. If the friction - be comparatively slight the amplitude is greatest when the imposed - period coincides with the free period, being then equal to - f/k[sigma]1, and therefore very great compared with that due to a - slowly varying force of the same average intensity. We have here, in - principle, the explanation of the phenomenon of "resonance" in - acoustics. The abnormal amplitude is greater, and is restricted to a - narrower range of frequency, the smaller the friction. For a complete - solution of (34) we must of course superpose the free vibration (30); - but owing to the factor e^(-t/[tau]) the influence of the initial - conditions gradually disappears. - -For purposes of mathematical treatment a force which produces a finite -change of velocity in a time too short to be appreciated is regarded as -infinitely great, and the time of action as infinitely short. The whole -effect is summed up in the value of the instantaneous impulse, which is -the time-integral of the force. Thus if an instantaneous impulse [xi] -changes the velocity of a mass m from u to u' we have - - mu'- mu = [xi]. (37) - -The effect of ordinary finite forces during the infinitely short -duration of this impulse is of course ignored. - -We may apply this to the theory of impact. If two masses m1, m2 moving -in the same straight line impinge, with the result that the velocities -are changed from u1, u2, to u1', u2', then, since the impulses on the -two bodies must be equal and opposite, the total momentum is unchanged, -i.e. - - m1u1' + m2u2' = m1u1 + m2u2. (38) - -The complete determination of the result of a collision under given -circumstances is not a matter of abstract dynamics alone, but requires -some auxiliary assumption. If we assume that there is no loss of -apparent kinetic energy we have also - - m1u1^2 + m2u2'^2 = m1u1^2 + m2u2^2. (39) - -Hence, and from (38), - - u2' - u1' = -(u2 - u1), (40) - -i.e. the relative velocity of the two bodies is reversed in direction, -but unaltered in magnitude. This appears to be the case very -approximately with steel or glass balls; generally, however, there is -some appreciable loss of apparent energy; this is accounted for by -vibrations produced in the balls and imperfect elasticity of the -materials. The usual empirical assumption is that - - u2' - u1' = -e(u2 - u1), (41) - -where e is a proper fraction which is constant for the same two bodies. -It follows from the formula S 15 (10) for the internal kinetic energy of -a system of particles that as a result of the impact this energy is -diminished by the amount - - m1m2 - (1/2)(1 - e^2) ------- (u1 - u2)^2. (42) - m1 + m2 - -The further theoretical discussion of the subject belongs to ELASTICITY. - -This is perhaps the most suitable place for a few remarks on the theory -of "dimensions." (See also UNITS, DIMENSIONS OF.) In any absolute system -of dynamical measurement the fundamental units are those of mass, length -and time; we may denote them by the symbols M, L, T, respectively. They -may be chosen quite arbitrarily, e.g. on the C.G.S. system they are the -gramme, centimetre and second. All other units are derived from these. -Thus the unit of velocity is that of a point describing the unit of -length in the unit of time; it may be denoted by LT^-1, this symbol -indicating that the magnitude of the unit in question varies directly as -the unit of length and inversely as the unit of time. The unit of -acceleration is the acceleration of a point which gains unit velocity in -unit time; it is accordingly denoted by LT^-2. The unit of momentum is -MLT^-1; the unit force generates unit momentum in unit time and is -therefore denoted by MLT^-2. The unit of work on the same principles is -ML^2T^-2, and it is to be noticed that this is identical with the unit of -kinetic energy. Some of these derivative units have special names -assigned to them; thus on the C.G.S. system the unit of force is called -the _dyne_, and the unit of work or energy the _erg_. The number which -expresses a physical quantity of any particular kind will of course vary -inversely as the magnitude of the corresponding unit. In any general -dynamical equation the dimensions of each term in the fundamental units -must be the same, for a change of units would otherwise alter the -various terms in different ratios. This principle is often useful as a -check on the accuracy of an equation. - - The theory of dimensions often enables us to forecast, to some extent, - the manner in which the magnitudes involved in any particular problem - will enter into the result. Thus, assuming that the period of a small - oscillation of a given pendulum at a given place is a definite - quantity, we see that it must vary as [root](l/g). For it can only - depend on the mass m of the bob, the length l of the string, and the - value of g at the place in question; and the above expression is the - only combination of these symbols whose dimensions are those of a - time, simply. Again, the time of falling from a distance a into a - given centre of force varying inversely as the square of the distance - will depend only on a and on the constant [mu] of equation (15). The - dimensions of [mu]/x^2 are those of an acceleration; hence the - dimensions of [mu] are L^3T^-2. Assuming that the time in question - varies as a^x[mu]^y, whose dimensions are L^(x + 3y)T^(-2y), we must - have x + 3y = 0, -2y = 1, so that the time of falling will vary as - a^(3/2)/[root][mu], in agreement with (19). - - The argument appears in a more demonstrative form in the theory of - "similar" systems, or (more precisely) of the similar motion of - similar systems. Thus, considering the equations - - d^2x [mu] d^2x' [mu]' - ---- = - ----, ------ = - -----, (43) - dt^2 x^2 dt'^2 x'^2 - - which refer to two particles falling independently into two distinct - centres of force, it is obvious that it is possible to have x in a - constant ratio to x', and t in a constant ratio to t', provided that - - x x' [mu] [mu]' - --- : ---- = ---- : -----, (44) - t^2 t'^2 x^2 x'^2 - - and that there is a suitable correspondence between the initial - conditions. The relation (44) is equivalent to - - x^(3/2) x'^(3/2) - t : t' = ---------- : -----------, (45) - [mu]^(1/2) [mu]'^(1/2) - - where x, x' are any two corresponding distances; e.g. they may be the - initial distances, both particles being supposed to start from rest. - The consideration of dimensions was introduced by J. B. Fourier (1822) - in connexion with the conduction of heat. - -[Illustration: FIG. 64.] - -S 13. _General Motion of a Particle._--Let P, Q be the positions of a -moving point at times t, t + [delta]t respectively. A vector [->OU] -drawn parallel to PQ, of length proportional to PQ/[delta]t on any -convenient scale, will represent the _mean velocity_ in the interval -[delta]t, i.e. a point moving with a constant velocity having the -magnitude and direction indicated by this vector would experience the -same resultant displacement [->PQ] in the same time. As [delta]t is -indefinitely diminished, the vector [->OU] will tend to a definite limit -[->OV]; this is adopted as the definition of the _velocity_ of the -moving point at the instant t. Obviously [->OV] is parallel to the -tangent to the path at P, and its magnitude is ds/dt, where s is the -arc. If we project [->OV] on the co-ordinate axes (rectangular or -oblique) in the usual manner, the projections u, v, w are called the -_component velocities_ parallel to the axes. If x, y, z be the -co-ordinates of P it is easily proved that - - dx dy dz - u = --, v = --, w = --. (1) - dt dt dt - -The momentum of a particle is the vector obtained by multiplying the -velocity by the mass m. The _impulse_ of a force in any infinitely small -interval of time [delta]t is the product of the force into [delta]t; it -is to be regarded as a vector. The total impulse in any finite interval -of time is the integral of the impulses corresponding to the -infinitesimal elements [delta]t into which the interval may be -subdivided; the summation of which the integral is the limit is of -course to be understood in the vectorial sense. - -Newton's Second Law asserts that change of momentum is equal to the -impulse; this is a statement as to equality of vectors and so implies -identity of direction as well as of magnitude. If X, Y, Z are the -components of force, then considering the changes in an infinitely short -time [delta]t we have, by projection on the co-ordinate axes, -[delta](mu) = X[delta]t, and so on, or - - du dv dw - m -- = X, m -- = Y, m -- = Z. (2) - dt dt dt - -For example, the path of a particle projected anyhow under gravity will -obviously be confined to the vertical plane through the initial -direction of motion. Taking this as the plane xy, with the axis of x -drawn horizontally, and that of y vertically upwards, we have X = 0, Y = --mg; so that - - d^2x d^2y - ---- = 0, ---- = -g. (3) - dt^2 dt^2 - -The solution is - - x = At + B, y = -(1/2) gt^2 + Ct + D. (4) - -If the initial values of x, y, [.x], [.y] are given, we have four -conditions to determine the four arbitrary constants A, B, C, D. Thus if -the particle start at time t = 0 from the origin, with the component -velocities u0, v0, we have - - x = u0t, y = v0t - (1/2) gt^2. (5) - -Eliminating t we have the equation of the path, viz. - - v0 gx^2 - y = -- x - ----. (6) - u0 2u^2 - -This is a parabola with vertical axis, of latus-rectum 2u0^2/g. The range -on a horizontal plane through O is got by putting y = 0, viz. it is -2u0v0/g. we denote the resultant velocity at any instant by [.s] we have - - [.s]^2 = [.x]^2 + [.y]^2 = [.s]0^2 - 2gy. (7) - -Another important example is that of a particle subject to an -acceleration which is directed always towards a fixed point O and is -proportional to the distance from O. The motion will evidently be in one -plane, which we take as the plane z = 0. If [mu] be the acceleration at -unit distance, the component accelerations parallel to axes of x and y -through O as origin will be -[mu]x, -[mu]y, whence - - d^2x d^2y - ---- = -[mu]x, ---- = - [mu]y. (8) - dt^2 dt^2 - -The solution is - - x = A cos nt + B sin nt, y = C cos nt + D sin nt, (9) - -where n = [root][mu]. If P be the initial position of the particle, we -may conveniently take OP as axis of x, and draw Oy parallel to the -direction of motion at P. If OP = a, and [.s]0 be the velocity at P, we -have, initially, x = a, y = 0, [.x] = 0, [.y] = [.s]0 whence - - x = a cos nt, y = b sin nt, (10) - -if b = [.s]0/n. The path is therefore an ellipse of which a, b are -conjugate semi-diameters, and is described in the period -2[pi]/[root][mu]; moreover, the velocity at any point P is equal to -[root][mu].OD, where OD is the semi-diameter conjugate to OP. This type -of motion is called _elliptic harmonic_. If the co-ordinate axes are the -principal axes of the ellipse, the angle nt in (10) is identical with -the "excentric angle." The motion of the bob of a "spherical pendulum," -i.e. a simple pendulum whose oscillations are not confined to one -vertical plane, is of this character, provided the extreme inclination -of the string to the vertical be small. The acceleration is towards the -vertical through the point of suspension, and is equal to gr/l, -approximately, if r denote distance from this vertical. Hence the path -is approximately an ellipse, and the period is 2[pi] [root](l/g). - -[Illustration: FIG. 65.] - - The above problem is identical with that of the oscillation of a - particle in a smooth spherical bowl, in the neighbourhood of the - lowest point. If the bowl has any other shape, the axes Ox, Oy may be - taken tangential to the lines of curvature at the lowest point O; the - equations of small motion then are - - d^2x x d^2y y - ---- = -g ------, ---- = -g ------, (11) - dt^2 [rho]1 dt^2 [rho]2 - - where [rho]1, [rho]2, are the principal radii of curvature at O. The - motion is therefore the resultant of two simple vibrations in - perpendicular directions, of periods 2[pi] [root]([rho]1/g), - 2[pi] [root]([rho]2/g). The circumstances are realized in "Blackburn's - pendulum," which consists of a weight P hanging from a point C of a - string ACB whose ends A, B are fixed. If E be the point in which the - line of the string meets AB, we have [rho]1 = CP, [rho]2 = EP. Many - contrivances for actually drawing the resulting curves have been - devised. - -[Illustration: FIG. 66.] - -It is sometimes convenient to resolve the accelerations in directions -having a more intrinsic relation to the path. Thus, in a plane path, let -P, Q be two consecutive positions, corresponding to the times t, t + -[delta]t; and let the normals at P, Q meet in C, making an angle -[delta][psi]. Let v (= [.s]) be the velocity at P, v + [delta]v that at -Q. In the time [delta]t the velocity parallel to the tangent at P -changes from v to v + [delta]v, ultimately, and the tangential -acceleration at P is therefore dv/dt or [:s]. Again, the velocity -parallel to the normal at P changes from 0 to v[delta][psi], ultimately, -so that the normal acceleration is v d[psi]/dt. Since - - dv dv ds dv d[psi] d[psi] ds v^2 - -- = -- -- = v --, v ------ = v ------ -- = -----, (12) - dt ds dt ds dt ds dt [rho] - -where [rho] is the radius of curvature of the path at P, the tangential -and normal accelerations are also expressed by v dv/ds and v^2/[rho], -respectively. Take, for example, the case of a particle moving on a -smooth curve in a vertical plane, under the action of gravity and the -pressure R of the curve. If the axes of x and y be drawn horizontal and -vertical (upwards), and if [psi] be the inclination of the tangent to -the horizontal, we have - - dv dy mv^2 - mv -- = - mg sin [psi] = - mg --, ----- = - mg cos [psi] + R. (13) - ds ds [rho] - -The former equation gives - - v^2 = C - 2gy, (14) - -and the latter then determines R. - - In the case of the pendulum the tension of the string takes the place - of the pressure of the curve. If l be the length of the string, [psi] - its inclination to the downward vertical, we have [delta]s = - l[delta][psi], so that v = ld[psi]/dt. The tangential resolution then - gives - - d^2[psi] - l -------- = - g sin [psi]. (15) - dt^2 - - If we multiply by 2d[psi]/dt and integrate, we obtain - - / d[psi]\^2 2g - ( ------ ) = --- cos [psi] + const., (16) - \ dt / l - - which is seen to be equivalent to (14). If the pendulum oscillate - between the limits [psi] = [+-][alpha], we have - - /[delta][psi]\^2 2g 4g - ( ------------ ) = --- (cos [psi] - cos [alpha]) = --- (sin^2 (1/2)[alpha] - sin^2 (1/2)[psi]); (17) - \ dt / l l - - and, putting sin (1/2)[psi] = sin (1/2)[alpha]. sin [phi], we find for - the period ([tau]) of a complete oscillation - - _(1/2)[pi] _(1/2)[pi] - / dt / l / d[phi] - [tau] = 4 | ------ d[phi] = 4 / --- . | ------------------------------------------ - _/0 d[phi] \/ g _/0 [root](1 - sin^2 (1/2)[alpha].sin^2 [phi]) - - / l - = 4 / ---.F1(sin (1/2)[alpha]), (18) - \/ g - - in the notation of elliptic integrals. The function F1 (sin [beta]) - was tabulated by A. M. Legendre for values of [beta] ranging from 0 deg. - to 90 deg. The following table gives the period, for various amplitudes - [alpha], in terms of that of oscillation in an infinitely small arc - [viz. 2[pi] [root](l/g)] as unit. - - +--------------+----------++--------------+----------+ - | [alpha]/[pi] | [tau] || [alpha]/[pi] | [tau] | - +--------------+----------++--------------+----------+ - | .1 | 1.0062 || .6 | 1.2817 | - | .2 | 1.0253 || .7 | 1.4283 | - | .3 | 1.0585 || .8 | 1.6551 | - | .4 | 1.1087 || .9 | 2.0724 | - | .5 | 1.1804 || 1.0 | [oo] | - +--------------+----------++--------------+----------+ - - The value of [tau] can also be obtained as an infinite series, by - expanding the integrand in (18) by the binomial theorem, and - integrating term by term. Thus - - / l / 1^2 1^2.3^2 \ - [tau] = 2[pi] / --- . ( 1 + --- sin^2 (1/2)[alpha] + ------- sin^4 (1/2)[alpha] + ... ). (19) - \/ g \ 2^2 2^2.4^2 / - - If [alpha] be small, an approximation (usually sufficient) is - - [tau] = 2[pi] [root](l/g).(1 + (1/16)[alpha]^2). - - In the extreme case of [alpha] = [pi], the equation (17) is - immediately integrable; thus the time from the lowest position is - - t = [root](l/g).log tan ((1/4)[pi] + (1/4)[psi]). (20) - - This becomes infinite for [psi] = [pi], showing that the pendulum only - tends asymptotically to the highest position. - - [Illustration: FIG. 67.] - - The variation of period with amplitude was at one time a hindrance to - the accurate performance of pendulum clocks, since the errors produced - are cumulative. It was therefore sought to replace the circular - pendulum by some other contrivance free from this defect. The equation - of motion of a particle in any smooth path is - - d^2s - ---- = -g sin [psi], (21) - dt^2 - - where [psi] is the inclination of the tangent to the horizontal. If - sin [psi] were accurately and not merely approximately proportional to - the arc s, say - - s = k sin [psi], (22) - - the equation (21) would assume the same form as S 12 (5). The motion - along the arc would then be accurately simple-harmonic, and the period - 2[pi][root](k/g) would be the same for all amplitudes. Now equation - (22) is the intrinsic equation of a cycloid; viz. the curve is that - traced by a point on the circumference of a circle of radius (1/4)k - which rolls on the under side of a horizontal straight line. Since the - evolute of a cycloid is an equal cycloid the object is attained by - means of two metal cheeks, having the form of the evolute near the - cusp, on which the string wraps itself alternately as the pendulum - swings. The device has long been abandoned, the difficulty being met - in other ways, but the problem, originally investigated by C. Huygens, - is important in the history of mathematics. - -The component accelerations of a point describing a tortuous curve, in -the directions of the tangent, the principal normal, and the binormal, -respectively, are found as follows. If [->OV], [->OV'] be vectors -representing the velocities at two consecutive points P, P' of the path, -the plane VOV' is ultimately parallel to the osculating plane of the -path at P; the resultant acceleration is therefore in the osculating -plane. Also, the projections of [->VV'] on OV and on a perpendicular to -OV in the plane VOV' are [delta]v and v[delta][epsilon], where -[delta][epsilon] is the angle between the directions of the tangents at -P, P'. Since [delta][epsilon] = [delta]s/[rho], where [delta]s = PP' = -v[delta]t and [rho] is the radius of principal curvature at P, the -component accelerations along the tangent and principal normal are dv/dt -and vd[epsilon]/dt, respectively, or vdv/ds and v^2/[rho]. For example, -if a particle moves on a smooth surface, under no forces except the -reaction of the surface, v is constant, and the principal normal to the -path will coincide with the normal to the surface. Hence the path is a -"geodesic" on the surface. - -If we resolve along the tangent to the path (whether plane or tortuous), -the equation of motion of a particle may be written - - dv - mv -- = [T], (23) - ds - -where [T] is the tangential component of the force. Integrating with -respect to s we find - _ - / s1 - (1/2) mv1^2 - (1/2) mv0^2 = | [T] ds; (24) - _/ s0 - -i.e. the increase of kinetic energy between any two positions is equal -to the work done by the forces. The result follows also from the -Cartesian equations (2); viz. we have - - m([.x][:x] + [.y][:y] + [.z][:z]) = X[.x] + Y[.y] + Z[.z], (25) - -whence, on integration with respect to t, - - _ - / - (1/2)m([.x]^2 + [.y]^2 + [.z]^2) = |(X[.x] + Y[.y] + Z[.z]) dt + const. - _/ - _ - / - = |(X dx + Y dy + Z dz) + const. (26) - _/ - -If the axes be rectangular, this has the same interpretation as (24). - -Suppose now that we have a constant field of force; i.e. the force -acting on the particle is always the same at the same place. The work -which must be done by forces extraneous to the field in order to bring -the particle from rest in some standard position A to rest in any other -position P will not necessarily be the same for all paths between A and -P. If it is different for different paths, then by bringing the particle -from A to P by one path, and back again from P to A by another, we might -secure a gain of work, and the process could be repeated indefinitely. -If the work required is the same for all paths between A and P, and -therefore zero for a closed circuit, the field is said to be -_conservative_. In this case the work required to bring the particle -from rest at A to rest at P is called the _potential energy_ of the -particle in the position P; we denote it by V. If PP' be a linear -element [delta]s drawn in any direction from P, and S be the force due -to the field, resolved in the direction PP', we have [delta]V = --S[delta]s or - - [dP]V - S = -----. (27) - [dP]s - -In particular, by taking PP' parallel to each of the (rectangular) -co-ordinate axes in succession, we find - - [dP]V [dP]V [dP]V - X = -----, Y = -----, Z = -----. (28) - [dP]x [dP]y [dP]z - -The equation (24) or (26) now gives - - (1/2) mv1^2 + V1 = (1/2) mv0^2 + V0; (29) - -i.e. the sum of the kinetic and potential energies is constant when no -work is done by extraneous forces. For example, if the field be that due -to gravity we have V = fmgdy = mgy + const., if the axis of y be drawn -vertically upwards; hence - - (1/2) mv^2 + mgy = const. (30) - -This applies to motion on a smooth curve, as well as to the free motion -of a projectile; cf. (7), (14). Again, in the case of a force Kr towards -O, where r denotes distance from O we have V = [int] Kr dr = (1/2)Kr^2 + -const., whence - - (1/2) mv^2 + (1/2) Kr^2 = const. (31) - -It has been seen that the orbit is in this case an ellipse; also that if -we put [mu] = K/m the velocity at any point P is v = [root][mu].OD, -where OD is the semi-diameter conjugate to OP. Hence (31) is consistent -with the known property of the ellipse that OP^2 + OD^2 is constant. - - The forms assumed by the dynamical equations when the axes of - reference are themselves in motion will be considered in S 21. At - present we take only the case where the rectangular axes Ox, Oy rotate - in their own plane, with angular velocity [omega] about Oz, which is - fixed. In the interval [delta]t the projections of the line joining - the origin to any point (x, y, z) on the directions of the co-ordinate - axes at time t are changed from x, y, z to (x + [delta]x) cos - [omega][delta]t - (y + [delta]y) sin [omega][delta]t, (x + [delta]x) - sin [omega][delta]t + (y + [delta]y) cos [omega][delta]t, z - respectively. Hence the component velocities parallel to the - instantaneous positions of the co-ordinate axes at time t are - - u = [.x] - [omega]y, v = [.y] + [omega]z, [omega] = [.z]. (32) - - In the same way we find that the component accelerations are - - [.u] - [omega]v, [.v] + [omega]u, [.omega]. (33) - - Hence if [omega] be constant the equations of motion take the forms - - m([:x] - 2[omega][.y] - [omega]^2[.x]) = X, m([:y] + 2[omega][.x] - [omega]^2y) = Y, m[:z] = Z. (34) - - These become identical with the equations of motion relative to fixed - axes provided we introduce a fictitious force m[omega]^2r acting - outwards from the axis of z, where r = [root](x^2 + y^2), and a second - fictitious force 2m[omega]v at right angles to the path, where v is - the component of the relative velocity parallel to the plane xy. The - former force is called by French writers the _force centrifuge - ordinaire_, and the latter the _force centrifuge composee_, or _force - de Coriolis_. As an application of (34) we may take the case of a - symmetrical Blackburn's pendulum hanging from a horizontal bar which - is made to rotate about a vertical axis half-way between the points - of attachment of the upper string. The equations of small motion are - then of the type - - [:x] - 2[omega][.y] - [omega]^2x = -p^2x, [:y] + 2[omega][.x] - [omega]^2y = -q^2y. (35) - - This is satisfied by - - [:x] = A cos ([sigma]t + [epsilon]), y = B sin ([sigma]t + [epsilon]), (36) - - provided - - ([sigma]^2 + [omega]^2 - p^2)A + 2[sigma][omega]B = 0, \ (37) - 2[sigma][omega]A + ([sigma]^2 + [omega]^2 - q^2)B = 0. / - - Eliminating the ratio A : B we have - - ([sigma]^2 + [omega]^2 - p^2)([sigma]^2 + [omega]^2 - q^2) - 4[sigma]^2[omega]^2 = 0. (38) - - It is easily proved that the roots of this quadratic in [sigma]^2 are - always real, and that they are moreover both positive unless [omega]^2 - lies between p^2 and q^2. The ratio B/A is determined in each case by - either of the equations (37); hence each root of the quadratic gives a - solution of the type (36), with two arbitrary constants A, [epsilon]. - Since the equations (35) are linear, these two solutions are to be - superposed. If the quadratic (38) has a negative root, the - trigonometrical functions in (36) are to be replaced by real - exponentials, and the position x = 0, y = 0 is unstable. This occurs - only when the period (2[pi]/[omega]) of revolution of the arm lies - between the two periods (2[pi]/p, 2[pi]/q) of oscillation when the arm - is fixed. - -S 14. _Central Forces. Hodograph._--The motion of a particle subject to -a force which passes always through a fixed point O is necessarily in a -plane orbit. For its investigation we require two equations; these may -be obtained in a variety of forms. - -Since the impulse of the force in any element of time [delta]t has zero -moment about O, the same will be true of the additional momentum -generated. Hence the moment of the momentum (considered as a localized -vector) about O will be constant. In symbols, if v be the velocity and p -the perpendicular from O to the tangent to the path, - - pv = h, (1) - -where h is a constant. If [delta]s be an element of the path, p[delta]s -is twice the area enclosed by [delta]s and the radii drawn to its -extremities from O. Hence if [delta]A be this area, we have [delta]A = -(1/2) p[delta]s = (1/2) h[delta]t, or - - dA - -- = (1/2)h. (2) - dt - -Hence equal areas are swept over by the radius vector in equal times. - -If P be the acceleration towards O, we have - - dv dr - v -- = -P --, (3) - ds ds - -since dr/ds is the cosine of the angle between the directions of r and -[delta]s. We will suppose that P is a function of r only; then -integrating (3) we find - _ - / - (1/2) v^2 = - | P dr + const., (4) - _/ - -which is recognized as the equation of energy. Combining this with (1) -we have - _ - h^2 / - --- = C - 2 | P dr, (5) - p^2 _/ - -which completely determines the path except as to its orientation with -respect to O. - -If the law of attraction be that of the inverse square of the distance, -we have P = [mu]/r^2, and - - h^2 2[mu] - --- = C + -----. (6) - p^2 [tau] - -Now in a conic whose focus is at O we have - - l 2 1 - --- = --- [+-] ---, (7) - p^2 r a - -where l is half the latus-rectum, a is half the major axis, and the -upper or lower sign is to be taken according as the conic is an ellipse -or hyperbola. In the intermediate case of the parabola we have a = [oo] -and the last term disappears. The equations (6) and (7) are identified -by putting - - l = h^2/[mu], a = [+-] [mu]/C. (8) - -Since - - h^2 / 2 1 \ - v^2 = --- = [mu]( --- [+-] --- ), (9) - p^2 \ r a / - -it appears that the orbit is an ellipse, parabola or hyperbola, -according as v^2 is less than, equal to, or greater than 2[mu]/r. Now it -appears from (6) that 2[mu]/r is the square of the velocity which would -be acquired by a particle falling from rest at infinity to the distance -r. Hence the character of the orbit depends on whether the velocity at -any point is less than, equal to, or greater than the _velocity from -infinity_, as it is called. In an elliptic orbit the area [pi]ab is -swept over in the time - - [pi]ab 2[pi]a^(3/2) - r = ------ = ------------, (10) - (1/2)h [root][mu] - -since h = [mu]^(1/2) l^(1/2) = [mu]^(1/2) ba^-(1/2) by (8). - - The converse problem, to determine the law of force under which a - given orbit can be described about a given pole, is solved by - differentiating (5) with respect to r; thus - - h^2 dp - P = ------. (11) - p^3 dr - - In the case of an ellipse described about the centre as pole we have - - a^2 b^2 - ------- = a^2 + b^2 - r^2; (12) - p^2 - - hence P = [mu]r, if [mu] = h^2/a^2b^2. This merely shows that a - particular ellipse may be described under the law of the direct - distance provided the circumstances of projection be suitably - adjusted. But since an ellipse can always be constructed with a given - centre so as to touch a given line at a given point, and to have a - given value of ab (= h/[root][mu]) we infer that the orbit will be - elliptic whatever the initial circumstances. Also the period is - 2[pi]ab/h = 2[pi]/[root][mu], as previously found. - - Again, in the equiangular spiral we have p = r sin[alpha], and - therefore P = [mu]/r^3, if [mu] = h^2/sin^2[alpha]. But since an - equiangular spiral having a given pole is completely determined by a - given point and a given tangent, this type of orbit is not a general - one for the law of the inverse cube. In order that the spiral may be - described it is necessary that the velocity of projection should be - adjusted to make h = [root][mu].sin[alpha]. Similarly, in the case of - a circle with the pole on the circumference we have p^2 = r^2/2a, P = - [mu]/r^5, if [mu] = 8h^2a^2; but this orbit is not a general one for - the law of the inverse fifth power. - -[Illustration: FIG. 68.] - -In astronomical and other investigations relating to central forces it -is often convenient to use polar co-ordinates with the centre of force -as pole. Let P, Q be the positions of a moving point at times t, t + -[delta]t, and write OP = r, OQ = r + [delta]r, [angle]POQ = -[delta][theta], O being any fixed origin. If u, v be the component -velocities at P along and perpendicular to OP (in the direction of -[theta] increasing), we have - - [delta]r dr r[delta][theta] d[theta] - u = lim.-------- = --, v = lim. --------------- = r --------. (13) - [delta]t dt [delta]t dt - -Again, the velocities parallel and perpendicular to OP change in the -time [delta]t from u, v to u - v[delta][theta], v + u[delta][theta], -ultimately. The component accelerations at P in these directions are -therefore - - du d[theta] d^2r /d[theta]\^2 \ - -- - v -------- = ---- - r ( -------- ), | - dt dt dt^2 \ dt / | - > (14) - dv d[theta] 1 d / d[theta] \ | - -- + u -------- = --- --- ( r^2 -------- ), | - dt dt r dt \ dt / / - -respectively. - -In the case of a central force, with O as pole, the transverse -acceleration vanishes, so that - - r^2d[theta]/dt = h, (15) - -where h is constant; this shows (again) that the radius vector sweeps -over equal areas in equal times. The radial resolution gives - - d^2r /d[theta]\^2 - ---- - r ( -------- ) = -P, (16) - dt^2 \ dt / - -where P, as before, denotes the acceleration towards O. If in this we -put r = 1/u, and eliminate t by means of (15), we obtain the general -differential equation of central orbits, viz. - - d^2u P - ---------- + u = -------. (17) - d[theta]^2 h^2 u^2 - - If, for example, the law be that of the inverse square, we have P = - [mu]u^2, and the solution is of the form - - [mu] - u = ------ {1 + e cos ([theta] - [alpha])}, (18) - h^2 - - where e, [alpha] are arbitrary constants. This is recognized as the - polar equation of a conic referred to the focus, the half latus-rectum - being h^2/[mu]. - - The law of the inverse cube P = [mu]u^3 is interesting by way of - contrast. The orbits may be divided into two classes according as h^2 - <> [mu], i.e. according as the transverse velocity (hu) is greater or - less than the velocity [root]([mu].u) appropriate to a circular orbit - at the same distance. In the former case the equation (17) takes the - form - - d^2u - ---------- + m^2u = 0, (19) - d[theta]^2 - - the solution of which is - - au = sin m ([theta] - [alpha]). (20) - - The orbit has therefore two asymptotes, inclined at an angle [pi]/m. - In the latter case the differential equation is of the form - - d^2u - ---------- = m^2u, (21) - d[theta]^2 - - so that - - u = A e^(m[theta]) + B e^(-m[theta]) (22) - - If A, B have the same sign, this is equivalent to - - au = cosh m[theta], (23) - - if the origin of [theta] be suitably adjusted; hence r has a maximum - value [alpha], and the particle ultimately approaches the pole - asymptotically by an infinite number of convolutions. If A, B have - opposite signs the form is - - au = sinh m[theta], (24) - - this has an asymptote parallel to [theta] = 0, but the path near the - origin has the same general form as in the case of (23). If A or B - vanish we have an equiangular spiral, and the velocity at infinity is - zero. In the critical case of h^2 = [mu], we have d^2u/d[theta]^2 = 0, - and - - u = A[theta] + B; (25) - - the orbit is therefore a "reciprocal spiral," except in the special - case of A = 0, when it is a circle. It will be seen that unless the - conditions be exactly adjusted for a circular orbit the particle will - either recede to infinity or approach the pole asymptotically. This - problem was investigated by R. Cotes (1682-1716), and the various - curves obtained arc known as _Coles's spirals_. - -A point on a central orbit where the radial velocity (dr/dt) vanishes is -called an _apse_, and the corresponding radius is called an _apse-line_. -If the force is always the same at the same distance any apse-line will -divide the orbit symmetrically, as is seen by imagining the velocity at -the apse to be reversed. It follows that the angle between successive -apse-lines is constant; it is called the _apsidal angle_ of the orbit. - -If in a central orbit the velocity is equal to the velocity from -infinity, we have, from (5), - _ - h^2 / [oo] - --- = 2 | P dr; (26) - p^2 _/ r - -this determines the form of the critical orbit, as it is called. If P = -[mu]/r^[n], its polar equation is - - r^m cos m[theta] = a^m, (27) - -where m = (1/2)(3 - n), except in the case n = 3, when the orbit is an -equiangular spiral. The case n = 2 gives the parabola as before. - - If we eliminate d[theta]/dt between (15) and (16) we obtain - - d^2r h^2 - ---- - --- = -P = -f(r), - dt^2 r^3 - - say. We may apply this to the investigation of the stability of a - circular orbit. Assuming that r = a + x, where x is small, we have, - approximately, - - d^2x h^2 / 3x\ - ---- - --- ( 1 - -- ) = -f(a) - xf'(a). - dt^2 r^3 \ a / - - Hence if h and a be connected by the relation h^2 = a^3f(a) proper to - a circular orbit, we have - - _ _ - d^2x | 3 | - ---- + | f'(a) + --- f(a)| x = 0. (28) - dt^2 |_ a _| - - If the coefficient of x be positive the variations of x are - simple-harmonic, and x can remain permanently small; the circular - orbit is then said to be stable. The condition for this may be written - _ _ - d | | - -- | a^3f(a) | > 0, (29) - da |_ _| - - i.e. the intensity of the force in the region for which r = a, nearly, - must diminish with increasing distance less rapidly than according to - the law of the inverse cube. Again, the half-period of x is - [pi]/sqrt[f'(a) + 3^{-1}f(a)], and since the angular velocity in the - orbit is h/a^2, approximately, the apsidal angle is, ultimately, - _ _ - / | f(a) | - [pi] / | --------------- |, (30) - \/ |_ af'(a) + 3f(a) _| - - or, in the case of f(a) = [mu]/r^n, [pi]/[root](3 - n). This is in - agreement with the known results for n = 2, n = -1. - - We have seen that under the law of the inverse square all finite - orbits are elliptical. The question presents itself whether there - then is any other law of force, giving a finite velocity from - infinity, under which all finite orbits are necessarily closed curves. - If this is the case, the apsidal angle must evidently be commensurable - with [pi], and since it cannot vary discontinuously the apsidal angle - in a nearly circular orbit must be constant. Equating the expression - (30) to [pi]/m, we find that f(a) = C/a^n, where n = 3 - m^2. The - force must therefore vary as a power of the distance, and n must be - less than 3. Moreover, the case n = 2 is the only one in which the - critical orbit (27) can be regarded as the limiting form of a closed - curve. Hence the only law of force which satisfies the conditions is - that of the inverse square. - -At the beginning of S 13 the velocity of a moving point P was -represented by a vector [->OV] drawn from a fixed origin O. The locus of -the point V is called the _hodograph_ (q.v.); and it appears that the -velocity of the point V along the hodograph represents in magnitude and -in direction the acceleration in the original orbit. Thus in the case of -a plane orbit, if v be the velocity of P, [psi] the inclination of the -direction of motion to some fixed direction, the polar co-ordinates of V -may be taken to be v, [psi]; hence the velocities of V along and -perpendicular to OV will be dv/dt and vd[psi]/dt. These expressions -therefore give the tangential and normal accelerations of P; cf. S 13 -(12). - -[Illustration: FIG. 69.] - - In the motion of a projectile under gravity the hodograph is a - vertical line described with constant velocity. In elliptic harmonic - motion the velocity of P is parallel and proportional to the - semi-diameter CD which is conjugate to the radius CP; the hodograph is - therefore an ellipse similar to the actual orbit. In the case of a - central orbit described under the law of the inverse square we have v - = h/SY = h. SZ/b^2, where S is the centre of force, SY is the - perpendicular to the tangent at P, and Z is the point where YS meets - the auxiliary circle again. Hence the hodograph is similar and - similarly situated to the locus of Z (the auxiliary circle) turned - about S through a right angle. This applies to an elliptic or - hyperbolic orbit; the case of the parabolic orbit may be examined - separately or treated as a limiting case. The annexed fig. 70 exhibits - the various cases, with the hodograph in its proper orientation. The - pole O of the hodograph is inside on or outside the circle, according - as the orbit is an ellipse, parabola or hyperbola. In any case of a - central orbit the hodograph (when turned through a right angle) is - similar and similarly situated to the "reciprocal polar" of the orbit - with respect to the centre of force. Thus for a circular orbit with - the centre of force at an excentric point, the hodograph is a conic - with the pole as focus. In the case of a particle oscillating under - gravity on a smooth cycloid from rest at the cusp the hodograph is a - circle through the pole, described with constant velocity. - -S 15. _Kinetics of a System of Discrete Particles._--The momenta of the -several particles constitute a system of localized vectors which, for -purposes of resolving and taking moments, may be reduced like a system -of forces in statics (S 8). Thus taking any point O as base, we have -first a _linear momentum_ whose components referred to rectangular axes -through O are - - [Sigma](m[.x]), [Sigma](m[.y]), [Sigma](m[.z]); (1) - -its representative vector is the same whatever point O be chosen. -Secondly, we have an _angular momentum_ whose components are - - [Sigma]{m(y[.z] - z[.y])}, [Sigma]{m(z[.x] - xz[.z])}, [Sigma]{m(x[.y] - y[.x])}, (2) - -these being the sums of the moments of the momenta of the several -particles about the respective axes. This is subject to the same -relations as a couple in statics; it may be represented by a vector -which will, however, in general vary with the position of O. - -The linear momentum is the same as if the whole mass were concentrated -at the centre of mass G, and endowed with the velocity of this point. -This follows at once from equation (8) of S 11, if we imagine the two -configurations of the system there referred to to be those corresponding -to the instants t, t + [delta]t. Thus - - __ / [->PP] \ __ [->GG'] - \ ( m.-------- ) = \ (m).--------. (3) - /__ \ [delta]t / /__ [delta]t - -Analytically we have - - d d[|x] - [Sigma](m[.x]) = --- [Sigma](mx) = [Sigma](m).-----. (4) - dt dt - -with two similar formulae. - -[Illustration: FIG. 70.] - -Again, if the instantaneous position of G be taken as base, the angular -momentum of the absolute motion is the same as the angular momentum of -the motion relative to G. For the velocity of a particle m at P may be -replaced by two components one of which (v) is identical in magnitude -and direction with the velocity of G, whilst the other (v) is the -velocity relative to G. The aggregate of the components mv of momentum -is equivalent to a single localized vector [Sigma](m).v in a line -through G, and has therefore zero moment about any axis through G; hence -in taking moments about such an axis we need only regard the velocities -relative to G. In symbols, we have - - / d[|z] d[|y]\ - [Sigma]{m(y[.z] - z[.y])} = [Sigma](m).( y ----- - z ----- ) + [Sigma]{m([eta][zeta] - [.zeta][eta])}. (5) - \ dt dt / - -since [Sigma](m[xi]) = 0, [Sigma](m[xi]) = 0, and so on, the notation -being as in S 11. This expresses that the moment of momentum about any -fixed axis (e.g. Ox) is equal to the moment of momentum of the motion -relative to G about a parallel axis through G, together with the moment -of momentum of the whole mass supposed concentrated at G and moving with -this point. If in (5) we make O coincide with the instantaneous position -of G, we have [|x], [|y], [|z] = 0, and the theorem follows. - -[Illustration: FIG. 71.] - -Finally, the rates of change of the components of the angular momentum -of the motion relative to G referred to G as a moving base, are equal to -the rates of change of the corresponding components of angular momentum -relative to a fixed base coincident with the instantaneous position of -G. For let G' be a consecutive position of G. At the instant t + -[delta]t the momenta of the system are equivalent to a linear momentum -represented by a localized vector [Sigma](m).(v + [delta]v) in a line -through G' tangential to the path of G', together with a certain angular -momentum. Now the moment of this localized vector with respect to any -axis through G is zero, to the first order of [delta]t, since the -perpendicular distance of G from the tangent line at G' is of the order -([delta]t)^2. Analytically we have from (5), - - d / d[|z]^2 d^2[|y] \ d - --- [Sigma] {m (y[.z] - z[.y])} = [Sigma](m).( y ------- - z ------- ) + --- [Sigma] {m([eta][zeta - [zeta][.eta])} (6) - dt \ dt^2 dt^2 / dt - -If we put x, y, z = 0, the theorem is proved as regards axes parallel to -Ox. - -Next consider the kinetic energy of the system. If from a fixed point O -we draw vectors [->OV1], [->OV2] to represent the velocities of the -several particles m1, m2 ..., and if we construct the vector - - - [Sigma](m.[->OV]) - [->OK] = ----------------- (7) - [Sigma](m) - -this will represent the velocity of the mass-centre, by (3). We find, -exactly as in the proof of Lagrange's First Theorem (S 11), that - - (1/2)[Sigma](m.OV^2) = (1/2)[Sigma](m).OK^2 + (1/2)[Sigma](m.KV^2); (8) - -i.e. the total kinetic energy is equal to the kinetic energy of the -whole mass supposed concentrated at G and moving with this point, -together with the kinetic energy of the motion relative to G. The latter -may be called the _internal kinetic energy_ of the system. Analytically -we have - _ _ - | /d[|x]\^2 /d[|y]\^2 /d[|z]\ | - (1/2)[Sigma]{m([.x]^2 + [.y]^2 + [.z]^2)} = (1/2)[Sigma](m).| ( ----- ) + ( ----- ) + ( ----- ) | - |_ \ dt / \ dt / \ dt / _| - - + (1/2)[Sigma] {m([zeta]^2 + [.eta]^2 + [zeta]^2)}. (9) - -There is also an analogue to Lagrange's Second Theorem, viz. - - [Sigma][Sigma] (m_p m_q.V_p V_q^2) - (1/2)[Sigma](m.KV^2) = (1/2) ---------------------------------- (10) - [Sigma]m - -which expresses the internal kinetic energy in terms of the relative -velocities of the several pairs of particles. This formula is due to -Mobius. - -The preceding theorems are purely kinematical. We have now to consider -the effect of the forces acting on the particles. These may be divided -into two categories; we have first, the _extraneous forces_ exerted on -the various particles from without, and, secondly, the mutual or -_internal forces_ between the various pairs of particles. It is assumed -that these latter are subject to the law of equality of action and -reaction. If the equations of motion of each particle be formed -separately, each such internal force will appear twice over, with -opposite signs for its components, viz. as affecting the motion of each -of the two particles between which it acts. The full working out is in -general difficult, the comparatively simple problem of "three bodies," -for instance, in gravitational astronomy being still unsolved, but some -general theorems can be formulated. - -The first of these may be called the _Principle of Linear Momentum_. If -there are no extraneous forces, the resultant linear momentum is -constant in every respect. For consider any two particles at P and Q, -acting on one another with equal and opposite forces in the line PQ. In -the time [delta]t a certain impulse is given to the first particle in -the direction (say) from P to Q, whilst an equal and opposite impulse is -given to the second in the direction from Q to P. Since these impulses -produce equal and opposite momenta in the two particles, the resultant -linear momentum of the system is unaltered. If extraneous forces act, it -is seen in like manner that the resultant linear momentum of the system -is in any given time modified by the geometric addition of the total -impulse of the extraneous forces. It follows, by the preceding kinematic -theory, that the mass-centre G of the system will move exactly as if the -whole mass were concentrated there and were acted on by the extraneous -forces applied parallel to their original directions. For example, the -mass-centre of a system free from extraneous force will describe a -straight line with constant velocity. Again, the mass-centre of a chain -of particles connected by strings, projected anyhow under gravity, will -describe a parabola. - -The second general result is the _Principle of Angular Momentum_. If -there are no extraneous forces, the moment of momentum about any fixed -axis is constant. For in time [delta]t the mutual action between two -particles at P and Q produces equal and opposite momenta in the line PQ, -and these will have equal and opposite moments about the fixed axis. If -extraneous forces act, the total angular momentum about any fixed axis -is in time [delta]t increased by the total extraneous impulse about that -axis. The kinematical relations above explained now lead to the -conclusion that in calculating the effect of extraneous forces in an -infinitely short time [delta]t we may take moments about an axis passing -through the instantaneous position of G exactly as if G were fixed; -moreover, the result will be the same whether in this process we employ -the true velocities of the particles or merely their velocities relative -to G. If there are no extraneous forces, or if the extraneous forces -have zero moment about any axis through G, the vector which represents -the resultant angular momentum relative to G is constant in every -respect. A plane through G perpendicular to this vector has a fixed -direction in space, and is called the _invariable plane_; it may -sometimes be conveniently used as a plane of reference. - - For example, if we have two particles connected by a string, the - invariable plane passes through the string, and if [omega] be the - angular velocity in this plane, the angular momentum relative to G is - - m1[omega]1r1.r1 + m2[omega]r2.r2 = (m1r1^2 + m2r2^2)[omega], - - where r1, r2 are the distances of m1, m2 from their mass-centre G. - Hence if the extraneous forces (e.g. gravity) have zero moment about - G, [omega] will be constant. Again, the tension R of the string is - given by - - m1m2 - R = m1[omega]^2r1 = ------- [omega]^2a, - m1 + m2 - - where a = r1 + r2. Also by (10) the internal kinetic energy is - - m1m2 - (1/2) ------- [omega]^2a^2. - m1 + m2 - -The increase of the kinetic energy of the system in any interval of time -will of course be equal to the total work done by all the forces acting -on the particles. In many questions relating to systems of discrete -particles the internal force R_pq (which we will reckon positive when -attractive) between any two particles m_p, m_q is a function only of the -distance r_pq between them. In this case the work done by the internal -forces will be represented by - _ - / - -[Sigma] | R_(pg) dr_(pq), - _/ - -when the summation includes every pair of particles, and each integral -is to be taken between the proper limits. If we write - _ - / - V = [Sigma] | R_(pq) dr_(pq), (11) - _/ - -when r_pq ranges from its value in some standard configuration A of the -system to its value in any other configuration P, it is plain that V -represents the work which would have to be done in order to bring the -system from rest in the configuration A to rest in the configuration P. -Hence V is a definite function of the configuration P; it is called the -_internal potential energy_. If T denote the kinetic energy, we may say -then that the sum T + V is in any interval of time increased by an -amount equal to the work done by the extraneous forces. In particular, -if there are no extraneous forces T + V is constant. Again, if some of -the extraneous forces are due to a conservative field of force, the work -which they do may be reckoned as a diminution of the potential energy -relative to the field as in S 13. - -S 16. _Kinetics of a Rigid Body. Fundamental Principles._--When we pass -from the consideration of discrete particles to that of continuous -distributions of matter, we require some physical postulate over and -above what is contained in the Laws of Motion, in their original -formulation. This additional postulate may be introduced under various -forms. One plan is to assume that any body whatever may be treated as if -it were composed of material particles, i.e. mathematical points endowed -with inertia coefficients, separated by finite intervals, and acting on -one another with forces in the lines joining them subject to the law of -equality of action and reaction. In the case of a rigid body we must -suppose that those forces adjust themselves so as to preserve the mutual -distances of the various particles unaltered. On this basis we can -predicate the principles of linear and angular momentum, as in S 15. - -An alternative procedure is to adopt the principle first formally -enunciated by J. Le R. d'Alembert and since known by his name. If x, y, -z be the rectangular co-ordinates of a mass-element m, the expressions -m[:x], m[:y], m[:z] must be equal to the components of the total force -on m, these forces being partly extraneous and partly forces exerted on -m by other mass-elements of the system. Hence (m[:x], m[:y], m[:z]) is -called the actual or _effective_ force on m. According to d'Alembert's -formulation, the extraneous forces together with the _effective forces -reversed_ fulfil the statical conditions of equilibrium. In other words, -the whole assemblage of effective forces is statically equivalent to the -extraneous forces. This leads, by the principles of S 8, to the -equations - - [Sigma](m[:x]) = X, [Sigma](m[:y]) = Y, [Sigma](m[:z]) = Z, \ - > (1) - [Sigma]{m(y[:z] - z[:y]) = L, [Sigma]{m(z[:x] - x[:z]) = M, [Sigma]{m(x[:y] - y[:x]) = N, / - -where (X, Y, Z) and (L, M, N) are the force--and couple--constituents of -the system of extraneous forces, referred to O as base, and the -summations extend over all the mass-elements of the system. These -equations may be written - - d d d - --- [Sigma](m[.x]) = X, --- [Sigma](m[.y]) = Y, --- [Sigma](m[.z]) = Z, \ - dt dt dt | } (2) - > (2) - d d d | - --- [Sigma]{m(y[.z] - z[.y]) = L, --- [Sigma]{m(z[.x]-x[.z]) = M, --- [Sigma]{m(x[.y] - y[.x]) = N, / - dt dt dt - -and so express that the rate of change of the linear momentum in any -fixed direction (e.g. that of Ox) is equal to the total extraneous force -in that direction, and that the rate of change of the angular momentum -about any fixed axis is equal to the moment of the extraneous forces -about that axis. If we integrate with respect to t between fixed limits, -we obtain the principles of linear and angular momentum in the form -previously given. Hence, whichever form of postulate we adopt, we are -led to the principles of linear and angular momentum, which form in fact -the basis of all our subsequent work. It is to be noticed that the -preceding statements are not intended to be restricted to rigid bodies; -they are assumed to hold for all material systems whatever. The peculiar -status of rigid bodies is that the principles in question are in most -cases sufficient for the complete determination of the motion, the -dynamical equations (1 or 2) being equal in number to the degrees of -freedom (six) of a rigid solid, whereas in cases where the freedom is -greater we have to invoke the aid of other supplementary physical -hypotheses (cf. ELASTICITY; HYDROMECHANICS). - -The increase of the kinetic energy of a rigid body in any interval of -time is equal to the work done by the extraneous forces acting on the -body. This is an immediate consequence of the fundamental postulate, in -either of the forms above stated, since the internal forces do on the -whole no work. The statement may be extended to a system of rigid -bodies, provided the mutual reactions consist of the stresses in -inextensible links, or the pressures between smooth surfaces, or the -reactions at rolling contacts (S 9). - -S 17. _Two-dimensional Problems._--In the case of rotation about a fixed -axis, the principles take a very simple form. The position of the body -is specified by a single co-ordinate, viz. the angle [theta] through -which some plane passing through the axis and fixed in the body has -turned from a standard position in space. Then d[theta]/dt, = [omega] -say, is the _angular velocity_ of the body. The angular momentum of a -particle m at a distance r from the axis is m[omega]r.r, and the total -angular momentum is [Sigma](mr^2).[omega], or I[omega], if I denote the -moment of inertia (S 11) about the axis. Hence if N be the moment of the -extraneous forces about the axis, we have - - d - --- (I[omega]) = N. (1) - dt - -This may be compared with the equation of rectilinear motion of a -particle, viz. d/dt.(Mu) = X; it shows that I measures the inertia of -the body as regards rotation, just as M measures its inertia as regards -translation. If N = 0, [omega] is constant. - -[Illustration: FIG. 72.] - -[Illustration: FIG. 73.] - - As a first example, suppose we have a flywheel free to rotate about a - horizontal axis, and that a weight m hangs by a vertical string from - the circumferences of an axle of radius b (fig. 72). Neglecting - frictional resistance we have, if R be the tension of the string, - - I[.omega] = Rb, m[.u] = mg - R, - - whence - mb^2 - b[.omega] = -------- (2) - 1 + mb^2 - - This gives the acceleration of m as modified by the inertia of the - wheel. - - A "compound pendulum" is a body of any form which is free to rotate - about a fixed horizontal axis, the only extraneous force (other than - the pressures of the axis) being that of gravity. If M be the total - mass, k the radius of gyration (S 11) about the axis, we have - - d / d[theta]\ - --- ( Mk^2 -------- ) = -Mgh sin [theta], (3) - dt \ dt / - - where [theta] is the angle which the plane containing the axis and the - centre of gravity G makes with the vertical, and h is the distance of - G from the axis. This coincides with the equation of motion of a - simple pendulum [S 13 (15)] of length l, provided l = k^2/h. The plane - of the diagram (fig. 73) is supposed to be a plane through G - perpendicular to the axis, which it meets in O. If we produce OG to P, - making OP = l, the point P is called the _centre of oscillation_; the - bob of a simple pendulum of length OP suspended from O will keep step - with the motion of P, if properly started. If [kappa] be the radius of - gyration about a parallel axis through G, we have k^2 = [kappa]^2 + h^2 - by S 11 (16), and therefore l = h + [kappa]^2/h, whence - - GO.GP = [kappa]^2. (4) - - This shows that if the body were swung from a parallel axis through P - the new centre of oscillation would be at O. For different parallel - axes, the period of a small oscillation varies as [root]l, or - [root](GO + OP); this is least, subject to the condition (4), when GO - = GP = [kappa]. The reciprocal relation between the centres of - suspension and oscillation is the basis of Kater's method of - determining g experimentally. A pendulum is constructed with two - parallel knife-edges as nearly as possible in the same plane with G, - the position of one of them being adjustable. If it could be arranged - that the period of a small oscillation should be exactly the same - about either edge, the two knife-edges would in general occupy the - positions of conjugate centres of suspension and oscillation; and the - distances between them would be the length l of the equivalent simple - pendulum. For if h1 + [kappa]^2/h1 = h2 + [kappa]^2/h2, then unless h1 = - h2, we must have [kappa]^2 = h1h2, l = h1 + h2. Exact equality of the - two observed periods ([tau]1, [tau]2, say) cannot of course be secured - in practice, and a modification is necessary. If we write l1 = h1 + - [kappa]^2/h1, l2 = h2 + [kappa]^2/h2, we find, on elimination of - [kappa], - - l1 + l2 l1 - l2 - (1/2) ------- + (1/2) ------- = 1, - h1 + h2 h1 - h2 - - whence - - 4[pi]^2 (1/2) ([tau]1^2 + [tau]2^2) (1/2) ([tau]1^2 - [tau]2^2) - ------- = --------------------------- + --------------------------- (5) - g h1 + h2 h1 - h2 - - The distance h1 + h2, which occurs in the first term on the right hand - can be measured directly. For the second term we require the values of - h1, h2 separately, but if [tau]1, [tau]2 are nearly equal whilst h1, - h2 are distinctly unequal this term will be relatively small, so that - an approximate knowledge of h1, h2 is sufficient. - - As a final example we may note the arrangement, often employed in - physical measurements, where a body performs small oscillations about - a vertical axis through its mass-centre G, under the influence of a - couple whose moment varies as the angle of rotation from the - equilibrium position. The equation of motion is of the type - - I[:theta] = -K[theta], (6) - - and the period is therefore [tau] = 2[pi][root](I/K). If by the - attachment of another body of known moment of inertia I', the period - is altered from [tau] to [tau]', we have [tau]' = 2[pi][root][(I + - I')/K]. We are thus enabled to determine both I and K, viz. - - I/I' = [tau]^2/([tau]'^2 - [tau]^2), K = 4[pi]^2[tau]^2I/([tau]'^2 - [tau]^2). (7) - - The couple may be due to the earth's magnetism, or to the torsion of - a suspending wire, or to a "bifilar" suspension. In the latter case, - the body hangs by two vertical threads of equal length l in a plane - through G. The motion being assumed to be small, the tensions of the - two strings may be taken to have their statical values Mgb/(a + b), - Mga/(a + b), where a, b are the distances of G from the two threads. - When the body is twisted through an angle [theta] the threads make - angles a[theta]/l, b[theta]/l with the vertical, and the moment of the - tensions about the vertical through G is accordingly -K[theta], where - K = M gab/l. - -For the determination of the motion it has only been necessary to use -one of the dynamical equations. The remaining equations serve to -determine the reactions of the rotating body on its bearings. Suppose, -for example, that there are no extraneous forces. Take rectangular axes, -of which Oz coincides with the axis of rotation. The angular velocity -being constant, the effective force on a particle m at a distance r from -Oz is m[omega]^2r towards this axis, and its components are accordingly --[omega]^2mx, -[omega]^2my, O. Since the reactions on the bearings must -be statically equivalent to the whole system of effective forces, they -will reduce to a force (X Y Z) at O and a couple (L M N) given by - - X = -[omega]^2[Sigma](mx) = -[omega]^2[Sigma](m)[|x], Y = -[omega]^2[Sigma](my) = -[omega]^2[Sigma](m)[|y], Z = 0, - - L = [omega]^2[Sigma](myz), M = -[omega]^2[Sigma](mzx), N = 0, (8) - - -where [|x], [|y] refer to the mass-centre G. The reactions do not -therefore reduce to a single force at O unless [Sigma](myz) = 0, -[Sigma](msx) = 0, i.e. unless the axis of rotation be a principal axis -of inertia (S 11) at O. In order that the force may vanish we must also -have x, y = 0, i.e. the mass-centre must lie in the axis of rotation. -These considerations are important in the "balancing" of machinery. We -note further that if a body be free to turn about a fixed point O, there -are three mutually perpendicular lines through this point about which it -can rotate steadily, without further constraint. The theory of principal -or "permanent" axes was first investigated from this point of view by J. -A. Segner (1755). The origin of the name "deviation moment" sometimes -applied to a product of inertia is also now apparent. - -[Illustration: FIG. 74.] - -Proceeding to the general motion of a rigid body in two dimensions we -may take as the three co-ordinates of the body the rectangular Cartesian -co-ordinates x, y of the mass-centre G and the angle [theta] through -which the body has turned from some standard position. The components of -linear momentum are then M[.x], M[.y], and the angular momentum relative -to G as base is I[.theta], where M is the mass and I the moment of -inertia about G. If the extraneous forces be reduced to a force (X, Y) -at G and a couple N, we have - - M[:x] = X, M[:y] = Y, I[:theta] = N. (9) - -If the extraneous forces have zero moment about G the angular velocity -[.theta] is constant. Thus a circular disk projected under gravity in a -vertical plane spins with constant angular velocity, whilst its centre -describes a parabola. - - We may apply the equations (9) to the case of a solid of revolution - rolling with its axis horizontal on a plane of inclination [alpha]. If - the axis of x be taken parallel to the slope of the plane, with x - increasing downwards, we have - - M[:x] = Mg sin [alpha] - F, 0 = Mg cos [alpha] - R, M[kappa]^2[:theta] = Fa (10) - - where [kappa] is the radius of gyration about the axis of symmetry, a - is the constant distance of G from the plane, and R, F are the normal - and tangential components of the reaction of the plane, as shown in - fig. 74. We have also the kinematical relation [.x] = a[.theta]. Hence - - a^2 [kappa]^2 - [:x] = --------------- g sin [alpha], R = Mg cos [alpha], F = --------------- Mg sin [alpha]. (11) - [kappa]^2 + a^2 [kappa]^2 + a^2 - - The acceleration of G is therefore less than in the case of - frictionless sliding in the ratio a^2/([kappa]^2 + a^2). For a - homogeneous sphere this ratio is 5/7, for a uniform circular cylinder - or disk 2/3, for a circular hoop or a thin cylindrical shell (1/2). - -The equation of energy for a rigid body has already been stated (in -effect) as a corollary from fundamental assumptions. It may also be -deduced from the principles of linear and angular momentum as embodied -in the equations (9). We have - - M([.x][:x] + [.y][:]y) + l[.theta][:theta] + X[.x] + Y[.y] + N[.theta], (12) - -whence, integrating with respect to t, - - (1/2) M([.x]^2 + [.y]^2) + (1/2)I[.theta]^2 = [int](X dx + Y dy + Nd[theta]) + const. (13) - -The left-hand side is the kinetic energy of the whole mass, supposed -concentrated at G and moving with this point, together with the kinetic -energy of the motion relative to G (S 15); and the right-hand member -represents the integral work done by the extraneous forces in the -successive infinitesimal displacements into which the motion may be -resolved. - -[Illustration: FIG. 75.] - - The formula (13) may be easily verified in the case of the compound - pendulum, or of the solid rolling down an incline. As another example, - suppose we have a circular cylinder whose mass-centre is at an - excentric point, rolling on a horizontal plane. This includes the case - of a compound pendulum in which the knife-edge is replaced by a - cylindrical pin. If [alpha] be the radius of the cylinder, h the - distance of G from its axis (O), [kappa] the radius of gyration about - a longitudinal axis through G, and [theta] the inclination of OG to - the vertical, the kinetic energy is 1/2M[kappa]^2[.theta]^2 + - (1/2)M.CG^2.[.theta]^2, by S 3, since the body is turning about the - line of contact (C) as instantaneous axis, and the potential energy - is--Mgh cos [theta]. The equation of energy is therefore - - (1/2) M([kappa]^2 + [alpha]^2 + h^2 - 2 ah cos [theta]) [.theta]^2 - Mgh cos [theta] - const. (14) - -Whenever, as in the preceding examples, a body or a system of bodies, is -subject to constraints which leave it virtually only one degree of -freedom, the equation of energy is sufficient for the complete -determination of the motion. If q be any variable co-ordinate defining -the position or (in the case of a system of bodies) the configuration, -the velocity of each particle at any instant will be proportional to -[.q], and the total kinetic energy may be expressed in the form -(1/2)A[.q]^2, where A is in general a function of q [cf. equation (14)]. -This coefficient A is called the coefficient of inertia, or the reduced -inertia of the system, referred to the co-ordinate q. - -[Illustration: FIG. 76.] - - Thus in the case of a railway truck travelling with velocity u the - kinetic energy is (1/2)(M + m[kappa]^2/[alpha]^2)u^2, where M is the - total mass, [alpha] the radius and [kappa] the radius of gyration of - each wheel, and m is the sum of the masses of the wheels; the reduced - inertia is therefore M + m[kappa]^2/[alpha]^2. Again, take the system - composed of the flywheel, connecting rod, and piston of a - steam-engine. We have here a limiting case of three-bar motion (S 3), - and the instantaneous centre J of the connecting-rod PQ will have the - position shown in the figure. The velocities of P and Q will be in the - ratio of JP to JQ, or OR to OQ; the velocity of the piston is - therefore y[.theta], where y = OR. Hence if, for simplicity, we - neglect the inertia of the connecting-rod, the kinetic energy will be - (1/2)(I + My^2)[.theta]^2, where I is the moment of inertia of the - flywheel, and M is the mass of the piston. The effect of the mass of - the piston is therefore to increase the apparent moment of inertia of - the flywheel by the variable amount My^2. If, on the other hand, we - take OP (= x) as our variable, the kinetic energy is 1/2(M + - I/y^2)[.x]^2. We may also say, therefore, that the effect of the - flywheel is to increase the apparent mass of the piston by the amount - I/y^2; this becomes infinite at the "dead-points" where the crank is - in line with the connecting-rod. - -If the system be "conservative," we have - - (1/2) Aq^2 + V = const., (15) - -where V is the potential energy. If we differentiate this with respect -to t, and divide out by [.q], we obtain - - dA dV - A[:q] + (1/2) -- q^2 + -- = 0 (16) - dq dq - -as the equation of motion of the system with the unknown reactions (if -any) eliminated. For equilibrium this must be satisfied by [.q] = O; -this requires that dV/dq = 0, i.e. the potential energy must be -"stationary." To examine the effect of a small disturbance from -equilibrium we put V = f(q), and write q = q0 + [eta], where q0 is a -root of f'(q0) = 0 and [eta] is small. Neglecting terms of the second -order in [eta] we have dV/dq = f'(q) = f"(q0).[eta], and the equation -(16) reduces to - - A[:eta] + f"(q0)[eta] = 0, (17) - -where A may be supposed to be constant and to have the value -corresponding to q = q0. Hence if f"(q0) > 0, i.e. if V is a minimum in -the configuration of equilibrium, the variation of [eta] is -simple-harmonic, and the period is 2[pi][root][A/f"(q0)]. This depends -only on the constitution of the system, whereas the amplitude and epoch -will vary with the initial circumstances. If f"(q0) < 0, the solution -of (17) will involve real exponentials, and [eta] will in general -increase until the neglect of the terms of the second order is no longer -justified. The configuration q = q0, is then unstable. - - As an example of the method, we may take the problem to which equation - (14) relates. If we differentiate, and divide by [theta], and retain - only the terms of the first order in [theta], we obtain - - {x^2 + (h - [alpha])^2} [:theta] + gh[theta] = 0, (18) - - as the equation of small oscillations about the position [theta] = 0. - The length of the equivalent simple pendulum is {[kappa]^2 + (h - - [alpha])^2}/h. - -The equations which express the change of motion (in two dimensions) due -to an instantaneous impulse are of the forms - - M(u'- u) = [xi], M([nu]' - [nu]) = [eta], I([omega]' - [omega]) = [nu]. (19) - -[Illustration: FIG. 77.] - -Here u', [nu]' are the values of the component velocities of G just -before, and u, [nu] their values just after, the impulse, whilst -[omega]', [omega] denote the corresponding angular velocities. Further, -[xi], [eta] are the time-integrals of the forces parallel to the -co-ordinate axes, and [nu] is the time-integral of their moment about G. -Suppose, for example, that a rigid lamina at rest, but free to move, is -struck by an instantaneous impulse F in a given line. Evidently G will -begin to move parallel to the line of F; let its initial velocity be u', -and let [omega]' be the initial angular velocity. Then Mu' = F, -I[omega]' = F.GP, where GP is the perpendicular from G to the line of F. -If PG be produced to any point C, the initial velocity of the point C of -the lamina will be - - u' - [omega]'.GC = (F/M).(I - GC.CP/[kappa]^2), - -where [kappa]^2 is the radius of gyration about G. The initial centre of -rotation will therefore be at C, provided GC.GP = [kappa]^2. If this -condition be satisfied there would be no impulsive reaction at C even if -this point were fixed. The point P is therefore called the _centre of -percussion_ for the axis at C. It will be noted that the relation -between C and P is the same as that which connects the centres of -suspension and oscillation in the compound pendulum. - -S 18. _Equations of Motion in Three Dimensions._--It was proved in S 7 -that a body moving about a fixed point O can be brought from its -position at time t to its position at time t + [delta]t by an -infinitesimal rotation [epsilon] about some axis through O; and the -limiting position of this axis, when [delta]t is infinitely small, was -called the "instantaneous axis." The limiting value of the ratio -[epsilon]/[delta]t is called the _angular velocity_ of the body; we -denote it by [omega]. If [xi], [eta], [zeta] are the components of -[epsilon] about rectangular co-ordinate axes through O, the limiting -values of [xi]/[delta]t, [eta]/[delta]t, [zeta]/[delta]t are called the -_component angular velocities_; we denote them by p, q, r. If l, m, n be -the direction-cosines of the instantaneous axis we have - - p = l[omega], q = m[omega], r = n[omega], (1) - p^2 + q^2 + r^2 = [omega]^2. (2) - -If we draw a vector OJ to represent the angular velocity, then J traces -out a certain curve in the body, called the _polhode_, and a certain -curve in space, called the _herpolhode_. The cones generated by the -instantaneous axis in the body and in space are called the polhode and -herpolhode cones, respectively; in the actual motion the former cone -rolls on the latter (S 7). - -[Illustration: FIG. 78.] - - The special case where both cones are right circular and [omega] is - constant is important in astronomy and also in mechanism (theory of - bevel wheels). The "precession of the equinoxes" is due to the fact - that the earth performs a motion of this kind about its centre, and - the whole class of such motions has therefore been termed - _precessional_. In fig. 78, which shows the various cases, OZ is the - axis of the fixed and OC that of the rolling cone, and J is the point - of contact of the polhode and herpolhode, which are of course both - circles. If [alpha]be the semi-angle of the rolling cone, [beta] the - constant inclination of OC to OZ, and [.psi] the angular velocity with - which the plane ZOC revolves about OZ, then, considering the velocity - of a point in OC at unit distance from O, we have - - [omega] sin [alpha] = [+-][.psi] sin [beta], (3) - - where the lower sign belongs to the third case. The earth's - precessional motion is of this latter type, the angles being [alpha] = - .0087", [beta] = 23 deg. 28'. - -If m be the mass of a particle at P, and PN the perpendicular to the -instantaneous axis, the kinetic energy T is given by - - 2T = [Sigma] {m([omega].PN)^2} = [omega]^2.[Sigma](m.PN^2) = I[omega]^2, (4) - -where I is the moment of inertia about the instantaneous axis. With the -same notation for moments and products of inertia as in S 11 (38), we -have - - I = Al^2 + Bm^2 + Cn^2 - 2Fmn - 2Gnl - 2Hlm, - -and therefore by (1), - - 2T = Ap^2 + Bq^2 + Cr^2 - 2Fqr - 2Grp - 2Hpq. (5) - -Again, if x, y, z be the co-ordinates of P, the component velocities of -m are - - qz - ry, rx - pz, py - qx, (6) - -by S 7 (5); hence, if [lambda], [mu], [nu] be now used to denote the -component angular momenta about the co-ordinate axes, we have [lambda] = -[Sigma][m(py - qx)y - m(rx - pz)z], with two similar formulae, or - - [dP]T \ - [lambda] = Ap - Hq - Gr= -----, | - [dP]p | - | - [dP]T | - [mu] = -Hp + Bq - Fr = -----, > (7) - [dP]q | - | - [dP]T | - [nu] = -Gp - Fq + Cr = -----. | - [dP]r / - -If the co-ordinate axes be taken to coincide with the principal axes of -inertia at O, at the instant under consideration, we have the simpler -formulae - - 2T = Ap^2 + Bq^2 + Cr^2, (8) - - [lambda] = Ap, [mu] = Bq, [nu] = Cr. (9) - -It is to be carefully noticed that the axis of resultant angular -momentum about O does not in general coincide with the instantaneous -axis of rotation. The relation between these axes may be expressed by -means of the momental ellipsoid at O. The equation of the latter, -referred to its principal axes, being as in S 11 (41), the co-ordinates -of the point J where it is met by the instantaneous axis are -proportional to p, q, r, and the direction-cosines of the normal at J -are therefore proportional to Ap, Bq, Cr, or [lambda], [mu], [nu]. The -axis of resultant angular momentum is therefore normal to the tangent -plane at J, and does not coincide with OJ unless the latter be a -principal axis. Again, if [Gamma] be the resultant angular momentum, so -that - - [lambda]^2 + [mu]^2 + [nu]^2 = [Gamma]^2, (10) - -the length of the perpendicular OH on the tangent plane at J is - - Ap p Bq q Cr r 2T [rho] - OH = ------- . -------[rho] + ------- . -------[rho] + ------- . -------[rho] = ------- . -------, (11) - [Gamma] [omega] [Gamma] [omega] [Gamma] [omega] [Gamma] [omega] - -where [rho] = OJ. This relation will be of use to us presently (S 19). - -The motion of a rigid body in the most general case may be specified by -means of the component velocities u, v, w of any point O of it which is -taken as base, and the component angular velocities p, q, r. The -component velocities of any point whose co-ordinates relative to O are -x, y, z are then - - u + qz - ry, v + rx - pz, w + py - qx (12) - -by S 7 (6). It is usually convenient to take as our base-point the -mass-centre of the body. In this case the kinetic energy is given by - - 2T = M0(u^2 + v^2 + w^2) + Ap^2 + Bq^2 + Cr^2 - 2Fqr - 2Grp - 2Hpg, (13) - -where M0 is the mass, and A, B, C, F, G, H are the moments and products -of inertia with respect to the mass-centre; cf. S 15 (9). - -The components [xi], [eta], [zeta] of linear momentum are - - [dP]T [dP]T [dP]T - [xi] = M0u = -----, [eta] = M0v = -----, [zeta] = M0w = -----, (14) - [dP]u [dP]v [dP]w - -whilst those of the relative angular momentum are given by (7). The -preceding formulae are sufficient for the treatment of instantaneous -impulses. Thus if an impulse ([xi], [eta], [zeta], [lambda], [mu], [nu]) -change the motion from (u, v, w, p, q, r) to (u', v', w', p', q', r') we -have - - M0(u'- u) = [xi], M0(v'- v) = [eta], M0(w'- w) = [zeta], \ - > (15) - A(p' - p) = [lambda], B(q'- q) = [mu], C(r'- r) = [nu], / - -where, for simplicity, the co-ordinate axes are supposed to coincide -with the principal axes at the mass-centre. Hence the change of kinetic -energy is - - T'- T = [xi] . (1/2)(u + u') + [eta] . (1/2)(v + v') + [zeta] . (1/2)(w + w'), - + [lambda] . (1/2)(p + p') + [mu] . (1/2)(q + q') + [nu] . (1/2)(r + r'). (16) - -The factors of [xi], [eta], [zeta], [lambda], [mu], [nu] on the -right-hand side are proportional to the constituents of a possible -infinitesimal displacement of the solid, and the whole expression is -proportional (on the same scale) to the work done by the given system of -impulsive forces in such a displacement. As in S 9 this must be equal to -the total work done in such a displacement by the several forces, -whatever they are, which make up the impulse. We are thus led to the -following statement: the change of kinetic energy due to any system of -impulsive forces is equal to the sum of the products of the several -forces into the semi-sum of the initial and final velocities of their -respective points of application, resolved in the directions of the -forces. Thus in the problem of fig. 77 the kinetic energy generated is -(1/2)M([kappa]^2 + Cq^2)[omega]'^2, if C be the instantaneous centre; -this is seen to be equal to (1/2)F.[omega]'.CP, where [omega]'.CP -represents the initial velocity of P. - -The equations of continuous motion of a solid are obtained by -substituting the values of [xi], [eta], [zeta], [lambda], [mu], [nu] -from (14) and (7) in the general equations - - d[xi] d[eta] d[zeta] \ - ----- = X, ------ = Y, ------- = Z, | - dt dt dt | - > (17) - d[lambda] d[mu] d[nu] | - --------- = L, ----- = M, ----- = N, | - dt dt dt / - -where (X, Y, Z, L, M, N) denotes the system of extraneous forces -referred (like the momenta) to the mass-centre as base, the co-ordinate -axes being of course fixed in direction. The resulting equations are not -as a rule easy of application, owing to the fact that the moments and -products of inertia A, B, C, F, G, H are not constants but vary in -consequence of the changing orientation of the body with respect to the -co-ordinate axes. - -[Illustration: FIG. 79.] - - An exception occurs, however, in the case of a solid which is - kinetically symmetrical (S 11) about the mass-centre, e.g. a uniform - sphere. The equations then take the forms - - M0[.u] = X, M0[.v] = Y, M0[.w] = Z, - C[.p] = L, C[.q] = M, C[.r] = N, (18) - - where C is the constant moment of inertia about any axis through the - mass-centre. Take, for example, the case of a sphere rolling on a - plane; and let the axes Ox, Oy be drawn through the centre parallel to - the plane, so that the equation of the latter is z = -a. We will - suppose that the extraneous forces consist of a known force (X, Y, Z) - at the centre, and of the reactions (F1, F2, R) at the point of - contact. Hence - - M0[.u] = X + F1, M0[.v] = Y + F2, 0 = Z + R, \ - C[.p] = F2a, C[.q] = -F1a, C[.r] = 0. / (19) - - The last equation shows that the angular velocity about the normal to - the plane is constant. Again, since the point of the sphere which is - in contact with the plane is instantaneously at rest, we have the - geometrical relations - - u + qa = 0, v + pa = 0, w = 0, (20) - - by (12). Eliminating p, q, we get - - (M0 + Ca^-2)[.u] = X, (M0 + Ca^-2)[.v] = Y. (21) - - The acceleration of the centre is therefore the same as if the plane - were smooth and the mass of the sphere were increased by C/[alpha]^2. - Thus the centre of a sphere rolling under gravity on a plane of - inclination a describes a parabola with an acceleration - - g sin [alpha]/(1 + C/Ma^2) - - parallel to the lines of greatest slope. - - Take next the case of a sphere rolling on a fixed spherical surface. - Let a be the radius of the rolling sphere, c that of the spherical - surface which is the locus of its centre, and let x, y, z be the - co-ordinates of this centre relative to axes through O, the centre of - the fixed sphere. If the only extraneous forces are the reactions (P, - Q, R) at the point of contact, we have - - M0[:x] = P, M0[.y] = Q, M0[:z] = R, \ - | - a a a > (22) - Cp = ---(yR - zQ), C[.q] = ---(zP - xR), C[.r] = ---(xQ - yP), | - c c c / - - the standard case being that where the rolling sphere is outside the - fixed surface. The opposite case is obtained by reversing the sign of - a. We have also the geometrical relations - - [.x] = (a/c)(qz - ry), [.y] = (a/c)(rx - pz), [.z] = (a/c)(py - gx), (23) - - If we eliminate P, Q, R from (22), the resulting equations are - integrable with respect to t; thus - - M0a M0a - p = - ---(y[.z] - z[.y]) + [alpha], q = - ---(z[.x] - x[.z]) + [beta], - Cc Cc - - M0a - r = - ---(x[.y] - y[.x]) + [gamma], (24) - Cc - - where [alpha], [beta], [gamma] are arbitrary constants. Substituting - in (23) we find - - / M0a^2\ a / M0a^2\ a - ( 1 + ----- )[.x] = ---([beta]z - [gamma]y), ( 1 + ----- )[.y] = ---([gamma]x - [alpha]z), - \ C / c \ C / c - - / M0a^2\ a - ( 1 + ----- )[.z] = ---([alpha]y - [beta]x). (25) - \ C / c - - Hence [alpha][.x] + [beta][.y] + [gamma][.z] = 0, or - - [alpha]x + [beta]y + [gamma]z = const.; (26) - - which shows that the centre of the rolling sphere describes a circle. - If the axis of z be taken normal to the plane of this circle we have - [alpha] = 0, [beta] = 0, and - - / M0a^2\ a / M0a^2\ a - ( 1 + ----- )[.x] = -[gamma]--- y, ( 1 + ------ )[.y] = [gamma]--- x. (27) - \ C / c \ C / c - - The solution of these equations is of the type - - x = b cos ([sigma][tau] + [epsilon]), y = b sin ([sigma][iota] + [epsilon]), (28) - - where b, [epsilon] are arbitrary, and - - [gamma]a/c - [sigma]= ----------- (29) - 1 + M0a^2/C - - The circle is described with the constant angular velocity [sigma]. - - When the gravity of the rolling sphere is to be taken into account the - preceding method is not in general convenient, unless the whole motion - of G is small. As an example of this latter type, suppose that a - sphere is placed on the highest point of a fixed sphere and set - spinning about the vertical diameter with the angular velocity n; it - will appear that under a certain condition the motion of G consequent - on a slight disturbance will be oscillatory. If Oz be drawn vertically - upwards, then in the beginning of the disturbed motion the quantities - x, y, p, q, P, Q will all be small. Hence, omitting terms of the - second order, we find - - M0[:x] = P, M0[.y] = Q, R = M0g, \ - > (30) - C[.p] = -(M0ga/c)y + aQ, C[.q] = (M0ga/c)x - aP, C[.r] = 0. / - - The last equation shows that the component r of the angular velocity - retains (to the first order) the constant value n. The geometrical - relations reduce to - - [.x] = aq - (na/c)y, [.y] = -ap + (na/c)x. (31) - - Eliminating p, g, P, Q, we obtain the equations - - (C + M0a^2)[:x] + (Cna/c)y - (M0ga^2/c)x = 0, } - (C + M0a^2)[:y] - (Cna/c)x - (M0ga^2/c)y = 0, } (32) - - which are both contained in - _ _ - | d^2 Cna d M0ga^2 | - |(C + M0a^2)---- - i --- --- - ------ | (x + iy) = 0. (33) - |_ dt^2 c dt c _| - - - This has two solutions of the type x + iy = [alpha]e^{i([sigma]t + - [epsilon])}, where [alpha], [epsilon] are arbitrary, and [sigma] is a - root of the quadratic - - (C + M0a^2)[sigma]^2 - (Cna/c)[sigma] + M0ga^2/c = 0. (34) - - If - - n^2 > (4Mgc/C) (1 + M0a^2/C), (35) - - both roots are real, and have the same sign as n. The motion of G then - consists of two superposed circular vibrations of the type - - x = [alpha] cos ([sigma]t + [epsilon]), y = [alpha] sin ([sigma]t + [epsilon]), (36) - - in each of which the direction of revolution is the same as that of - the initial spin of the sphere. It follows therefore that the original - position is stable provided the spin n exceed the limit defined by - (35). The case of a sphere spinning about a vertical axis at the - lowest point of a spherical bowl is obtained by reversing the signs of - [alpha] and c. It appears that this position is always stable. - - It is to be remarked, however, that in the first form of the problem - the stability above investigated is practically of a limited or - temporary kind. The slightest frictional forces--such as the - resistance of the air--even if they act in lines through the centre of - the rolling sphere, and so do not directly affect its angular - momentum, will cause the centre gradually to descend in an - ever-widening spiral path. - -S 19. _Free Motion of a Solid._--Before proceeding to further problems -of motion under extraneous forces it is convenient to investigate the -free motion of a solid relative to its mass-centre O, in the most -general case. This is the same as the motion about a fixed point under -the action of extraneous forces which have zero moment about that point. -The question was first discussed by Euler (1750); the geometrical -representation to be given is due to Poinsot (1851). - -The kinetic energy T of the motion relative to O will be constant. Now T -= (1/2)I[omega]^2, where [omega] is the angular velocity and I is the -moment of inertia about the instantaneous axis. If [rho] be the -radius-vector OJ of the momental ellipsoid - - Ax^2 + By^2 + Cz^2 = M[epsilon]^4 (1) - -drawn in the direction of the instantaneous axis, we have I = -M[epsilon]^4/[rho]^2 (S 11); hence [omega] varies as [rho]. The locus of -J may therefore be taken as the "polhode" (S 18). Again, the vector -which represents the angular momentum with respect to O will be constant -in every respect. We have seen (S 18) that this vector coincides in -direction with the perpendicular OH to the tangent plane of the momental -ellipsoid at J; also that - - 2T [rho] - OH = ------- . -------, (2) - [Gamma] [omega] - -where [Gamma] is the resultant angular momentum about O. Since [omega] -varies as [rho], it follows that OH is constant, and the tangent plane -at J is therefore fixed in space. The motion of the body relative to O -is therefore completely represented if we imagine the momental ellipsoid -at O to roll without sliding on a plane fixed in space, with an angular -velocity proportional at each instant to the radius-vector of the point -of contact. The fixed plane is parallel to the invariable plane at O, -and the line OH is called the _invariable line_. The trace of the point -of contact J on the fixed plane is the "herpolhode." - -If p, q, r be the component angular velocities about the principal axes -at O, we have - - (A^2p^2 + B^2q^2 + C^2r^2)/[Gamma]^2 = (Ap^2 + Bq^2 + Cr^2)/2T, (3) - -each side being in fact equal to unity. At a point on the polhode cone x -: y : z = p : q : r, and the equation of this cone is therefore - - / [Gamma]^2\ / [Gamma]^2\ / [Gamma]^2\ - A^2( 1 - -------- )x^2 + B^2( 1 - --------- )y^2 + C^2( 1 - --------- )z^2 = 0. (4) - \ 2AT / \ 2BT / \ 2CT / - -Since 2AT - [Gamma]^2 = B (A - B)q^2 + C(A - C)r^2, it appears that if A -> B > C the coefficient of x^2 in (4) is positive, that of z^2 is -negative, whilst that of y^2 is positive or negative according as 2BT <> -[Gamma]^2. Hence the polhode cone surrounds the axis of greatest or -least moment according as 2BT <> [Gamma]^2. In the critical case of 2BT -= [Gamma]^2 it breaks up into two planes through the axis of mean moment -(Oy). The herpolhode curve in the fixed plane is obviously confined -between two concentric circles which it alternately touches; it is not -in general a re-entrant curve. It has been shown by De Sparre that, -owing to the limitation imposed on the possible forms of the momental -ellipsoid by the relation B + C > A, the curve has no points of -inflexion. The invariable line OH describes another cone in the body, -called the _invariable cone_. At any point of this we have x : y : z = -Ap. Bq : Cr, and the equation is therefore - - / [Gamma]^2\ / [Gamma]^2\ / [Gamma]^2\ - ( 1 - --------- )x^2 + ( 1 - --------- )y^2 + ( 1 - --------- )z^2 = 0. (5) - \ 2AT / \ 2BT / \ 2CT / - -[Illustration: FIG. 80.] - -The signs of the coefficients follow the same rule as in the case of -(4). The possible forms of the invariable cone are indicated in fig. 80 -by means of the intersections with a concentric spherical surface. In -the critical case of 2BT = [Gamma]^2 the cone degenerates into two -planes. It appears that if the body be sightly disturbed from a state of -rotation about the principal axis of greatest or least moment, the -invariable cone will closely surround this axis, which will therefore -never deviate far from the invariable line. If, on the other hand, the -body be slightly disturbed from a state of rotation about the mean axis -a wide deviation will take place. Hence a rotation about the axis of -greatest or least moment is reckoned as stable, a rotation about the -mean axis as unstable. The question is greatly simplified when two of -the principal moments are equal, say A = B. The polhode and herpolhode -cones are then right circular, and the motion is "precessional" -according to the definition of S 18. If [alpha] be the inclination of -the instantaneous axis to the axis of symmetry, [beta] the inclination -of the latter axis to the invariable line, we have - - [Gamma] cos [beta] = C [omega] cos [alpha], [Gamma] sin [beta] = A [omega] sin [alpha], (6) - -whence - - A - tan [beta] = --- tan [alpha]. (7) - C - -[Illustration: FIG. 81.] - -Hence [beta] <> [alpha], and the circumstances are therefore those of -the first or second case in fig. 78, according as A <> C. If [psi] be -the rate at which the plane HOJ revolves about OH, we have - - sin [alpha] C cos [alpha] - [psi] = ----------- [omega] = ------------- [omega], (8) - sin [beta] A cos [beta] - -by S 18 (3). Also if [.chi] be the rate at which J describes the -polhode, we have [.psi] sin ([beta]-[alpha]) = [.chi] sin [beta], whence - - sin([alpha] - [beta]) - [.chi] = --------------------- [omega]. (9) - sin[alpha] - -If the instantaneous axis only deviate slightly from the axis of -symmetry the angles [alpha], [beta] are small, and [.chi] = (A - -C)A.[omega]; the instantaneous axis therefore completes its revolution -in the body in the period - - 2[pi] A - C - ------ = ----- [omega]. (10) - [.chi] A - - In the case of the earth it is inferred from the independent - phenomenon of luni-solar precession that (C - A)/A = .00313. Hence if - the earth's axis of rotation deviates slightly from the axis of - figure, it should describe a cone about the latter in 320 sidereal - days. This would cause a periodic variation in the latitude of any - place on the earth's surface, as determined by astronomical methods. - There appears to be evidence of a slight periodic variation of - latitude, but the period would seem to be about fourteen months. The - discrepancy is attributed to a defect of rigidity in the earth. The - phenomenon is known as the _Eulerian nutation_, since it is supposed - to come under the free rotations first discussed by Euler. - -S 20. _Motion of a Solid of Revolution._--In the case of a solid of -revolution, or (more generally) whenever there is kinetic symmetry about -an axis through the mass-centre, or through a fixed point O, a number -of interesting problems can be treated almost directly from first -principles. It frequently happens that the extraneous forces have zero -moment about the axis of symmetry, as e.g. in the case of the flywheel -of a gyroscope if we neglect the friction at the bearings. The angular -velocity (r) about this axis is then constant. For we have seen that r -is constant when there are no extraneous forces; and r is evidently not -affected by an instantaneous impulse which leaves the angular momentum -Cr, about the axis of symmetry, unaltered. And a continuous force may be -regarded as the limit of a succession of infinitesimal instantaneous -impulses. - -[Illustration: FIG. 82.] - - Suppose, for example, that a flywheel is rotating with angular - velocity n about its axis, which is (say) horizontal, and that this - axis is made to rotate with the angular velocity [psi] in the - horizontal plane. The components of angular momentum about the axis of - the flywheel and about the vertical will be Cn and A [psi] - respectively, where A is the moment of inertia about any axis through - the mass-centre (or through the fixed point O) perpendicular to that - of symmetry. If [->OK] be the vector representing the former component - at time t, the vector which represents it at time t + [delta]t will be - [->OK'], equal to [->OK] in magnitude and making with it an angle - [delta][psi]. Hence [->KK'] ( = Cn [delta][psi]) will represent the - change in this component due to the extraneous forces. Hence, so far - as this component is concerned, the extraneous forces must supply a - couple of moment Cn[.psi] in a vertical plane through the axis of the - flywheel. If this couple be absent, the axis will be tilted out of the - horizontal plane in such a sense that the direction of the spin n - approximates to that of the azimuthal rotation [.psi]. The remaining - constituent of the extraneous forces is a couple A[:psi] about the - vertical; this vanishes if [.psi] is constant. If the axis of the - flywheel make an angle [theta] with the vertical, it is seen in like - manner that the required couple in the vertical plane through the axis - is Cn sin [theta] [.psi]. This matter can be strikingly illustrated - with an ordinary gyroscope, e.g. by making the larger movable ring in - fig. 37 rotate about its vertical diameter. - -[Illustration: FIG. 83.] - -If the direction of the axis of kinetic symmetry be specified by means -of the angular co-ordinates [theta], [psi] of S 7, then considering the -component velocities of the point C in fig. 83, which are [.theta] and -sin [theta][.psi] along and perpendicular to the meridian ZC, we see -that the component angular velocities about the lines OA', OB' are -sin -[theta] [.psi] and [.theta] respectively. Hence if the principal moments -of inertia at O be A, A, C, and if n be the constant angular velocity -about the axis OC, the kinetic energy is given by - - 2T = A ([.theta]^2 + sin^2 [theta][.psi]^2) + Cn^2. (1) - -Again, the components of angular momentum about OC, OA' are Cn, -A sin -[theta] [.psi], and therefore the angular momentum ([mu], say) about OZ -is - - [mu] = A sin^2 [theta][.psi] + Cn cos [theta]. (2) - -We can hence deduce the condition of steady precessional motion in a -top. A solid of revolution is supposed to be free to turn about a fixed -point O on its axis of symmetry, its mass-centre G being in this axis at -a distance h from O. In fig. 83 OZ is supposed to be vertical, and OC is -the axis of the solid drawn in the direction OG. If [theta] is constant -the points C, A' will in time [delta]t come to positions C", A" such -that CC" = sin [theta] [delta][psi], A'A" = cos [theta] [delta][psi], -and the angular momentum about OB' will become Cn sin [theta] -[delta][psi] - A sin [theta] [.psi] . cos [theta] [delta][psi]. Equating -this to Mgh sin [theta] [delta]t, and dividing out by sin [theta], we -obtain - - A cos [theta] [.psi]^2 - Cn[.psi] + Mgh = 0, (3) - -as the condition in question. For given values of n and [theta] we have -two possible values of [.psi] provided n exceed a certain limit. With a -very rapid spin, or (more precisely) with Cn large in comparison with -[root](4AMgh cos [theta]), one value of [.psi] is small and the other -large, viz. the two values are Mgh/Cn and Cn/A cos [theta] -approximately. The absence of g from the latter expression indicates -that the circumstances of the rapid precession are very nearly those of -a free Eulerian rotation (S 19), gravity playing only a subordinate -part. - -[Illustration: FIG. 84.] - - Again, take the case of a circular disk rolling in steady motion on a - horizontal plane. The centre O of the disk is supposed to describe a - horizontal circle of radius c with the constant angular velocity - [.psi], whilst its plane preserves a constant inclination [theta] to - the horizontal. The components of the reaction of the horizontal lane - will be Mc[.psi]^2 at right angles to the tangent line at the point of - contact and Mg vertically upwards, and the moment of these about the - horizontal diameter of the disk, which corresponds to OB' in fig. 83, - is Mc[.psi]^2. [alpha] sin [theta] - Mg[alpha] cos [theta], where - [alpha] is the radius of the disk. Equating this to the rate of - increase of the angular momentum about OB', investigated as above, we - find - - / a \ a^2 - ( C + Ma^2 + A --- cos [theta] ) [.psi]^2 = Mg --- cot [theta], (4) - \ c / c - - where use has been made of the obvious relation n[alpha] = c[.psi]. If - c and [theta] be given this formula determines the value of [psi] for - which the motion will be steady. - -In the case of the top, the equation of energy and the condition of -constant angular momentum ([mu]) about the vertical OZ are sufficient to -determine the motion of the axis. Thus, we have - - (1/2)A ([.theta]^2 + sin^2 [theta][.psi]^2) + (1/2)Cn^2 + Mgh cos [theta] = const., (5) - - A sin^2 [theta][.psi] + [nu] cos [theta] = [mu], (6) - -where [nu] is written for Cn. From these [.psi] may be eliminated, and -on differentiating the resulting equation with respect to t we obtain - - ([mu] - [nu] cos [theta])([mu] cos [theta] - [nu]) - A[:theta] - -------------------------------------------------- - Mgh sin [theta] = 0. (7) - A sin^3 [theta] - -If we put [:theta] = 0 we get the condition of steady precessional -motion in a form equivalent to (3). To find the small oscillation about -a state of steady precession in which the axis makes a constant angle -[alpha] with the vertical, we write [theta] = [alpha] + [chi], and -neglect terms of the second order in [chi]. The result is of the form - - [:chi] + [sigma]^2[chi] = 0, (8) - -where - - [sigma]^2 = {([mu] - [nu] cos [alpha])^2 + 2([mu] - [nu] cos [alpha])([mu] cos [alpha] - [nu]) - cos [alpha] + ([mu] cos [alpha] - [nu])^2} / A^2 sin^4 [alpha]. (9) - -When [nu] is large we have, for the "slow" precession [sigma] = [nu]/A, -and for the "rapid" precession [sigma] = A/[nu] cos [alpha] = [.psi], -approximately. Further, on examining the small variation in [.psi], it -appears that in a slightly disturbed slow precession the motion of any -point of the axis consists of a rapid circular vibration superposed on -the steady precession, so that the resultant path has a trochoidal -character. This is a type of motion commonly observed in a top spun in -the ordinary way, although the successive undulations of the trochoid -may be too small to be easily observed. In a slightly disturbed rapid -precession the superposed vibration is elliptic-harmonic, with a period -equal to that of the precession itself. The ratio of the axes of the -ellipse is sec [alpha], the longer axis being in the plane of [theta]. -The result is that the axis of the top describes a circular cone about a -fixed line making a small angle with the vertical. This is, in fact, the -"invariable line" of the free Eulerian rotation with which (as already -remarked) we are here virtually concerned. For the more general -discussion of the motion of a top see GYROSCOPE. - -S 21. _Moving Axes of Reference._--For the more general treatment of the -kinetics of a rigid body it is usually convenient to adopt a system of -moving axes. In order that the moments and products of inertia with -respect to these axes may be constant, it is in general necessary to -suppose them fixed in the solid. - -We will assume for the present that the origin O is fixed. The moving -axes Ox, Oy, Oz form a rigid frame of reference whose motion at time t -may be specified by the three component angular velocities p, q, r. The -components of angular momentum about Ox, Oy, Oz will be denoted as usual -by [lambda], [mu], [nu]. Now consider a system of fixed axes Ox', Oy', -Oz' chosen so as to coincide at the instant t with the moving system Ox, -Oy, Oz. At the instant t + [delta]t, Ox, Oy, Oz will no longer coincide -with Ox', Oy', Oz'; in particular they will make with Ox' angles whose -cosines are, to the first order, 1, -r[delta]t, q[delta]t, respectively. -Hence the altered angular momentum about Ox' will be [lambda] + -[delta][lambda] + ([mu] + [delta][mu]) (-r[delta]t) + ([nu] + -[delta][nu]) q[delta]t. If L, M, N be the moments of the extraneous -forces about Ox, Oy, Oz this must be equal to [lambda] + L[delta]t. -Hence, and by symmetry, we obtain - - d[lambda] \ - --------- - r[nu] + q[nu] = L, | - dt | - | - d[mu] | - ----- - p[nu] + r[lanbda] = M, > (1) - dt | - | - d[nu] | - ----- - q[lambda] + p[nu] = N. | - dt / - -These equations are applicable to any dynamical system whatever. If we -now apply them to the case of a rigid body moving about a fixed point O, -and make Ox, Oy, Oz coincide with the principal axes of inertia at O, we -have [lambda], [mu], [nu] = Ap, Bq, Cr, whence - - dp \ - A -- - (B - C) qr = L, | - dt | - | - dq | - B -- - (C - A) rp = M, > (2) - dt | - | - dr | - C -- - (A - B) pq = N. | - dt / - -If we multiply these by p, q, r and add, we get - - d - --- . (1/2)(Ap^2 + Bq^2 + Cr^2) = Lp + Mq + Nr, (3) - dt - -which is (virtually) the equation of energy. - -As a first application of the equations (2) take the case of a solid -constrained to rotate with constant angular velocity [omega] about a -fixed axis (l, m, n). Since p, q, r are then constant, the requisite -constraining couple is - - L = (C - B) mn[omega]^2, M = (A - C) nl[omega]^2, N = (B - A) lm[omega]^2. (4) - -If we reverse the signs, we get the "centrifugal couple" exerted by the -solid on its bearings. This couple vanishes when the axis of rotation is -a principal axis at O, and in no other case (cf. S 17). - -If in (2) we put, L, M, N = O we get the case of free rotation; thus - - dp \ - A -- = (B - C) qr, | - dt | - | - dq | - B -- = (C - A) rp, > (5) - dt | - | - dr | - C -- = (A - B) pq. | - dt / - -These equations are due to Euler, with whom the conception of moving -axes, and the application to the problem of free rotation, originated. -If we multiply them by p, q, r, respectively, or again by Ap, Bq, Cr -respectively, and add, we verify that the expressions Ap^2 + Bq^2 + Cr^2 -and A^2p^2 + B^2q^2 + C^2r^2 are both constant. The former is, in fact, -equal to 2T, and the latter to [Gamma]^2, where T is the kinetic energy -and [Gamma] the resultant angular momentum. - - To complete the solution of (2) a third integral is required; this - involves in general the use of elliptic functions. The problem has - been the subject of numerous memoirs; we will here notice only the - form of solution given by Rueb (1834), and at a later period by G. - Kirchhoff (1875), If we write - _ - / [phi] d[phi] - u = | ------------, [Delta][phi] = [root](1 - k^2 sin^2 [phi]), - _/ 0 [Delta][phi] - - we have, in the notation of elliptic functions, [phi] = am u. If we - assume - - p = p0 cos am ([sigma]t + [epsilon]), q = q0sin am ([sigma]t + [epsilon]), - r = r0[Delta] am ([sigma]t + [epsilon]), (7) - - we find - - [sigma]p0 [sigma]q0 k^2[sigma]r0 - [.p] = - --------- qr, [.q] = --------- rp, [.r] = - ------------ pq. (8) - q0r0 r0p0 p0q0 - - Hence (5) will be satisfied, provided - - -[sigma]p0 B - C [sigma]q0 C - A -k^2[sigma]r0 A - B - ---------- = -----, --------- = -----, ------------- = -----. (9) - q0r0 A r0p0 B p0q0 C - - These equations, together with the arbitrary initial values of p, q, - r, determine the six constants which we have denoted by p0, q0, r0, - k^2, [sigma], [epsilon]. We will suppose that A > B > C. From the form - of the polhode curves referred to in S 19 it appears that the angular - velocity q about the axis of mean moment must vanish periodically. If - we adopt one of these epochs as the origin of t, we have [epsilon] = - 0, and p0, r0 will become identical with the initial values of p, r. - The conditions (9) then lead to - - A(A - C) (A - C)(B - C) A(A - B) p0^2 - q0^2 = -------- p0^2, [sigma]^2 = -------------- r0^2, k^2 = -------- . ----. (10) - B(B - C) AB C(B - C) r0^2 - - For a real solution we must have k^2 < 1, which is equivalent to 2BT > - [Gamma]^2. If the initial conditions are such as to make 2BT < - [Gamma]^2, we must interchange the forms of p and r in (7). In the - present case the instantaneous axis returns to its initial position in - the body whenever [phi] increases by 2[pi], i.e. whenever t increases - by 4K/[sigma], when K is the "complete" elliptic integral of the first - kind with respect to the modulus k. - - The elliptic functions degenerate into simpler forms when k^2 = 0 or - k^2 = 1. The former case arises when two of the principal moments are - equal; this has been sufficiently dealt with in S 19. If k^2 = 1, we - must have 2BT = [Gamma]^2. We have seen that the alternative 2BT <> - [Gamma]^2 determines whether the polhode cone surrounds the principal - axis of least or greatest moment. The case of 2BT = [Gamma]^2, - exactly, is therefore a critical case; it may be shown that the - instantaneous axis either coincides permanently with the axis of mean - moment or approaches it asymptotically. - -When the origin of the moving axes is also in motion with a velocity -whose components are u, v, w, the dynamical equations are - - d[xi] d[eta] d[zeta] - ----- - r[eta] + q[zeta] = X, ------ - p[zeta] - r[chi] = Y, ------- - q[chi] + p[eta] = Z, (11) - dt dt dt - - d[lambda] d[mu] \ - --------- - r[mu] + q[nu] - w[eta] + v[zeta] = L, ----- - p[nu] + r[lambda]- u[zeta] + w[xi] = M, | - dt dt | - > (12) - d[nu] | - ----- - q[lambda] + p[mu] - v[xi] + u[eta] = N. / - dt - -To prove these, we may take fixed axes O'x', O'y', O'z' coincident with -the moving axes at time t, and compare the linear and angular momenta -[xi] + [delta][xi], [eta] + [delta][eta], [zeta] + [delta][zeta], -[lambda] + [delta][lambda], [mu] + [delta][mu], [nu] + [delta][nu] -relative to the new position of the axes, Ox, Oy, Oz at time t + -[delta]t with the original momenta [xi], [eta], [zeta], [lambda], [mu], -[nu] relative to O'x', O'y', O'z' at time t. As in the case of (2), the -equations are applicable to any dynamical system whatever. If the moving -origin coincide always with the mass-centre, we have [xi], [eta], [zeta] -= M0u, M0v, M0w, where M0 is the total mass, and the equations simplify. - -When, in any problem, the values of u, v, w, p, q, r have been -determined as functions of t, it still remains to connect the moving -axes with some fixed frame of reference. It will be sufficient to take -the case of motion about a fixed point O; the angular co-ordinates -[theta], [phi], [psi] of Euler may then be used for the purpose. -Referring to fig. 36 we see that the angular velocities p, q, r of the -moving lines, OA, OB, OC about their instantaneous positions are - - p = [.theta] sin [phi] - sin [theta] cos [phi][.psi], \ - q = [.theta] cos [phi] + sin [theta] sin [phi][.psi], > (13) - r = [.phi] + cos [theta][.psi], / - -by S 7 (3), (4). If OA, OB, OC be principal axes of inertia of a solid, -and if A, B, C denote the corresponding moments of inertia, the kinetic -energy is given by - - 2T = A([.theta] sin [phi] - sin [theta] cos [phi][.psi])^2 \ - + B([.theta] cos [phi] + sin [theta] sin [theta][psi])^2 > (14) - + C([.phi] + cos [theta][.psi])^2. / - -If A = B this reduces to - - 2T = A([.theta]^2 + sin^2 [theta][.psi]^2) + C([.phi] + cos [theta][.psi])^2; (15) - -cf. S 20 (1). - -S 22. _Equations of Motion in Generalized Co-ordinates._--Suppose we -have a dynamical system composed of a finite number of material -particles or rigid bodies, whether free or constrained in any way, which -are subject to mutual forces and also to the action of any given -extraneous forces. The configuration of such a system can be completely -specified by means of a certain number (n) of independent quantities, -called the generalized co-ordinates of the system. These co-ordinates -may be chosen in an endless variety of ways, but their number is -determinate, and expresses the number of _degrees of freedom_ of the -system. We denote these co-ordinates by q1, q2, ... q_n. It is implied -in the above description of the system that the Cartesian co-ordinates -x, y, z of any particle of the system are known functions of the q's, -varying in form (of course) from particle to particle. Hence the kinetic -energy T is given by - - __ - 2T = \ {m([.x]^2 + [.y]^2 + [.z]^2)} - /__ - - = a11[.q]1^2 + a22[.q]2^2 + ... + 2a12[.q]1[.q]2 + ..., (1) - -where - _ _ - __ | { / [dP]x \^2 / [dP]y \^2 / [dP]z \^2} | \ - a_rr = \ | m { ( ------- ) + ( ------- ) + ( ------- ) } |, | - /__ |_ { \[dP]q_r/ \[dP]q_r/ \[dP]q_r/ } _| | - _ _ > (2) - __ | / [dP]x [dP]x [dP]y [dP]y [dP]z [dP]z \ | | - a_rs = \ | m ( ------- ------- + ------- ------- + ------- ------- ) | = a_sr. | - /__ |_ \[dP]q_r [dP]q_s [dP]q_r [dP]q_s [dP]q_r [dP]q_s/ _| / - -Thus T is expressed as a homogeneous quadratic function of the -quantities [.q]1, [.q]2, ... [.q]_n, which are called the _generalized -components of velocity_. The coefficients a_rr, a_rs are called the -coefficients of inertia; they are not in general constants, being -functions of the q's and so variable with the configuration. Again, If -(X, Y, Z) be the force on m, the work done in an infinitesimal change of -configuration is - - [Sigma](X[delta]x + Y[delta]y + Z[delta]z) = Q1[delta]q1 + Q2[delta]q2 + ... + Q_n[delta]q_n, (3) - -where - - / [dP]x [dP]y [dP]z \ - Q_r = [Sigma]( X------- + Y------- + Z------- ). (4) - \ [dP]q_r [dP]q_r [dP]q_r / - -The quantities Q_r are called the _generalized components of force_. - -The equations of motion of m being - - m[:x] = X, m[:y] = Y, m[:z] = Z, (5) - -we have - _ _ - __ | / [dP]x [dP]y [dP]z \ | - \ | m ( [:x]------- + [:y]------- + [:z]------- ) | = Q_r. (6) - /__ |_ \ [dP]q_r [dP]q_r [dP]q_r / _| - -Now - - [dP]x [dP]x [dP]x - [.x] = ------[.q]1 + ------[.q]2 + ... + -------[.q]_n, (7) - [dP]q1 [dP]q2 [dP]q_n - -whence - - [dP][.x] [dP]x - ---------- = -------. (8) - [dP][.q]_r [dP]q_r - -Also - - d / [dP]x \ [dP]^2x [dP]^2x [dP]^2x [dP]x - -- ( ------- ) = ------------[.q]1 + -------------[.q]2 + ... + --------------[.q]_r = --------. (9) - dt \[dP]q_r/ [dP]q1[dP]q_r [dP]q2[dP]q_r [dP]q_n[dP]q_r [dP]q_r - -Hence - - [dP]x d / [dP]x \ d / [dP]x \ d / [dP][.x] \ [dP][.x] - [:x]------- = ---( [.x]------- ) - [.x]---( ------- ) = ---( [.x]---------- ) - [.x]--------. (10) - [dP]q_r dt \ [dP]q_r/ dt \[dP]q_r/ dt \ [dP][.q]_r/ [dP]q_r - -By these and the similar transformations relating to y and z the -equation (6) takes the form - - d / [dP]T \ [dP]T - --- ( ---------- ) - ------ = Q_r. (11) - dt \[dP][.q]_r/ [dP]q_r - -If we put r = 1, 2, ... n in succession, we get the n independent -equations of motion of the system. These equations are due to Lagrange, -with whom indeed the first conception, as well as the establishment, of -a general dynamical method applicable to all systems whatever appears to -have originated. The above proof was given by Sir W. R. Hamilton (1835). -Lagrange's own proof will be found under DYNAMICS, S _Analytical_. In a -conservative system free from extraneous force we have - - [Sigma](X [delta]x + Y [delta]y + Z [delta]z) = -[delta]V, (12) - -where V is the potential energy. Hence - - [dP]V - Q_r = - -------, (13) - [dP]q_r - -and - - d / [dP]T \ [dP]T [dP]V - --- ( ---------- ) - ----- = - -------. (14) - dt \[dP][.q]_r/ Vq_r [dP]q_r - -If we imagine any given state of motion ([.q]1, [.q]2 ... [.q]_n) -through the configuration (q1, q2, ... q_n) to be generated -instantaneously from rest by the action of suitable impulsive forces, we -find on integrating (11) with respect to t over the infinitely short -duration of the impulse - - [dP]T - ---------- = Q_r', (15) - [dP][.q]_r - -where Q_r' is the time integral of Q_r and so represents a _generalized -component of impulse_. By an obvious analogy, the expressions -[dP]T/[dP][.q]_r may be called the _generalized components of momentum_; -they are usually denoted by p_r thus - - p_r = [dP]T/[dP][.q]_r = a_(1r)[.q]1 + a_(2r)[.q]2 + ... + a_(nr)[.q]_n. (16) - -Since T is a homogeneous quadratic function of the velocities [.q]1, -[.q]2, ... [.q]_n, we have - - [dP]T [dP]T [dP]T - 2T = ---------[.q]1 + ---------[.q]2 + ... + ----------[.q]_n = p1[.q]2 + p2[.q]2 + ... + p_n[.q]_n. (17) - [dP][.q]1 [dP][.q]2 [dP][.q]_n - -Hence - - dT - 2-- = [.p]1[.q]1 + [.p]2[.q]2 + ... [.p]_n[.q]_n \ - dt | - | - + [.p]1[:q]1 + [.p]2[:q]2 + ... + [.p]_n[:q]_n | - | - / [dP]T \ / [dP]T \ / [dP]T \ | - = ( --------- + Q1 ) [.q]1 + ( --------- + Q2 ) [.q]2 + ... + ( ---------- + Q_n )[.q]_n > (18) - \[dP][.q]1 / \[dP][.q]2 / \[dP][.q]_n / | - | - [dP]T [dP]T [dP]T | - + ---------[:q]1 + ---------[:q]2 + ... ----------[:q]_n | - [dP][.q]1 [dP][.q]2 [dP][.q]_n | - | - dT | - = -- + Q1[.q]1 + Q2[.q]2 + ... + Q_n[.q]_n, / - dt - -or - - dT - -- = Q1[.q]1 + Q2[.q]2 + ... + Q_n[.q]_n. (19) - dt - -This equation expresses that the kinetic energy is increasing at a rate -equal to that at which work is being done by the forces. In the case of -a conservative system free from extraneous force it becomes the equation -of energy - - d - -- (T + V) = 0, or T + V = const., (20) - dt - -in virtue of (13). - - As a first application of Lagrange's formula (11) we may form the - equations of motion of a particle in spherical polar co-ordinates. Let - r be the distance of a point P from a fixed origin O, [theta] the - angle which OP makes with a fixed direction OZ, [psi] the azimuth of - the plane ZOP relative to some fixed plane through OZ. The - displacements of P due to small variations of these co-ordinates are - [dP]r along OP, r [delta][theta] perpendicular to OP in the plane ZOP, - and r sin [theta] [delta][psi] perpendicular to this plane. The - component velocities in these directions are therefore [.r], - r[.theta], r sin [theta][.psi], and if m be the mass of a moving - particle at P we have - - 2T = m([.r]^2 + r^2[.theta]^2 + r^2 sin^2 [theta][.psi]^2). (21) - - Hence the formula (11) gives - - m([:r] - r[.theta]^2 - r sin^2 [theta][.psi]^2) = R, \ - | - d | - ---(mr^2[.theta]) - mr^2 . sin [theta] cos [theta][.psi]^2 = [Theta], > (22) - dt | - | - d | - ---(mr^2 sin^2 [theta][.psi]) = [Psi]. / - dt - - The quantities R, [Theta], [Psi] are the coefficients in the - expression R [delta]r + [Theta] [delta][theta] + [Psi] [delta][psi] - for the work done in an infinitely small displacement; viz. R is the - radial component of force, [Theta] is the moment about a line through - O perpendicular to the plane ZOP, and [Psi] is the moment about OZ. In - the case of the spherical pendulum we have r = l, [Theta] = - mgl sin - [theta], [Psi] = 0, if OZ be drawn vertically downwards, and therefore - - g \ - [:theta] - sin [theta] cos [theta][.psi]^2 = - --- sin [theta], | - l > (23) - | - sin^2 [theta][.psi] = h, / - - - where h is a constant. The latter equation expresses that the angular - momentum ml^2 sin^2 [theta][.psi] about the vertical OZ is constant. By - elimination of [.psi] we obtain - - g - [:theta] - h^2 cos^2 [theta] / sin^3[theta] = - --- sin [theta]. (24) - l - - If the particle describes a horizontal circle of angular radius - [alpha] with constant angular velocity [Omega], we have [.omega] = 0, - h = [Omega]^2 sin [alpha], and therefore - - g - [Omega]^2 = --- cos [alpha], (25) - l - - as is otherwise evident from the elementary theory of uniform circular - motion. To investigate the small oscillations about this state of - steady motion we write [theta] = [alpha] + [chi] in (24) and neglect - terms of the second order in [chi]. We find, after some reductions, - - [:chi] + (1 + 3 cos^2 [alpha]) [Omega]^2[chi] = 0; (26) - - this shows that the variation of [chi] is simple-harmonic, with the - period - - 2[pi]/[root](1 + 3 cos^2 [alpha]).[Omega] - - As regards the most general motion of a spherical pendulum, it is - obvious that a particle moving under gravity on a smooth sphere cannot - pass through the highest or lowest point unless it describes a - vertical circle. In all other cases there must be an upper and a lower - limit to the altitude. Again, a vertical plane passing through O and a - point where the motion is horizontal is evidently a plane of symmetry - as regards the path. Hence the path will be confined between two - horizontal circles which it touches alternately, and the direction of - motion is never horizontal except at these circles. In the case of - disturbed steady motion, just considered, these circles are nearly - coincident. When both are near the lowest point the horizontal - projection of the path is approximately an ellipse, as shown in S 13; - a closer investigation shows that the ellipse is to be regarded as - revolving about its centre with the angular velocity 2/3 ab[Omega]/l^2, - where a, b are the semi-axes. - - To apply the equations (11) to the case of the top we start with the - expression (15) of S 21 for the kinetic energy, the simplified form - (1) of S 20 being for the present purpose inadmissible, since it is - essential that the generalized co-ordinates employed should be - competent to specify the position of every particle. If [lambda], - [mu], [nu] be the components of momentum, we have - - [dP]T \ - [lambda]= ------------ = A[.theta], | - [dP][.theta] | - | - [dP]T | - [mu] = ---------- = A sin^2 [theta][.psi] + C([.phi] + cos [theta][.psi]) cos [theta], > (27) - [dP][.psi] | - | - [dP]T | - [nu] = ---------- = C ([.theta] + cos [theta][.psi]). / - [dP][.phi] - - The meaning of these quantities is easily recognized; thus [lambda] is - the angular momentum about a horizontal axis normal to the plane of - [theta], [mu] is the angular momentum about the vertical OZ, and [nu] - is the angular momentum about the axis of symmetry. If M be the total - mass, the potential energy is V = Mgh cos [theta], if OZ be drawn - vertically upwards. Hence the equations (11) become - - A[:theta] - A sin [theta] cos [theta][.psi]^2 + C([.phi] + cos [theta][.psi]) [.psi] sin [theta] = Mgh sin [theta], \ - d/dt . {A sin^2 [theta][.psi] + C([.phi] + cos [theta][.psi]) cos [theta]} = 0, > (28) - d/dt . {C([.phi] + cos [theta][.psi])} = 0, / - - of which the last two express the constancy of the momenta [mu], [nu]. - Hence - - A[:theta] - A sin [theta] cos [theta][.psi]^2 + [nu] sin [theta][.psi] = Mgh sin [theta], \ (29) - A sin^2 [theta][.psi] + [nu] cos [theta] = [mu]. / - - If we eliminate [.psi] we obtain the equation (7) of S 20. The theory - of disturbed precessional motion there outlined does not give a - convenient view of the oscillations of the axis about the vertical - position. If [theta] be small the equations (29) may be written - - [nu]^2- 4AMgh \ - [:theta] - [theta][.omega]^2 = - -------------[theta], > (30) - 4A^2 | - [theta]^2[.omega] = const., / - - where - - [nu] - [omega] = [psi] - ---- t. (31) - 2A - - Since [theta], [omega] are the polar co-ordinates (in a horizontal - plane) of a point on the axis of symmetry, relative to an initial line - which revolves with constant angular velocity [nu]/2A, we see by - comparison with S 14 (15) (16) that the motion of such a point will be - elliptic-harmonic superposed on a uniform rotation [nu]/2A, provided - [nu]^2 > 4AMgh. This gives (in essentials) the theory of the - "gyroscopic pendulum." - -S 23. _Stability of Equilibrium. Theory of Vibrations._--If, in a -conservative system, the configuration (q1, q2, ... q_n) be one of -equilibrium, the equations (14) of S 22 must be satisfied by [.q]1, -[.q]2 ... [.q]_n = 0, whence - - [dP]V / [dP]q_r = 0. (1) - -A necessary and sufficient condition of equilibrium is therefore that -the value of the potential energy should be stationary for infinitesimal -variations of the co-ordinates. If, further, V be a minimum, the -equilibrium is necessarily stable, as was shown by P. G. L. Dirichlet -(1846). In the motion consequent on any slight disturbance the total -energy T + V is constant, and since T is essentially positive it follows -that V can never exceed its equilibrium value by more than a slight -amount, depending on the energy of the disturbance. This implies, on the -present hypothesis, that there is an upper limit to the deviation of -each co-ordinate from its equilibrium value; moreover, this limit -diminishes indefinitely with the energy of the original disturbance. No -such simple proof is available to show without qualification that the -above condition is _necessary_. If, however, we recognize the existence -of dissipative forces called into play by any motion whatever of the -system, the conclusion can be drawn as follows. However slight these -forces may be, the total energy T + V must continually diminish so long -as the velocities [.q]1, [.q]2, ... [.q]_n differ from zero. Hence if -the system be started from rest in a configuration for which V is less -than in the equilibrium configuration considered, this quantity must -still further decrease (since T cannot be negative), and it is evident -that either the system will finally come to rest in some other -equilibrium configuration, or V will in the long run diminish -indefinitely. This argument is due to Lord Kelvin and P. G. Tait (1879). - -In discussing the small oscillations of a system about a configuration -of stable equilibrium it is convenient so to choose the generalized -cc-ordinates q1, q2, ... q_n that they shall vanish in the configuration -in question. The potential energy is then given with sufficient -approximation by an expression of the form - - 2V = c11q1^2 + c22q2^2 + ... + 2c12q1q2 + ..., (2) - -a constant term being irrelevant, and the terms of the first order being -absent since the equilibrium value of V is stationary. The coefficients -c_rr, c_rs are called _coefficients of stability_. We may further treat -the coefficients of inertia a_rr, a_rs of S 22 (1) as constants. The -Lagrangian equations of motion are then of the type - - a_(1r)[:q]1 + a_(2r)[:q]2 + ... + a_(nr)[:q]_n + c_(1r)q1 + c_(2r)q2 + ... + c_(nr)q_n = Q_r, (3) - -where Q_r now stands for a component of extraneous force. In a _free -oscillation_ we have Q1, Q2, ... Q_n = 0, and if we assume - - q_r = A_r e^(i[sigma]^t), (4) - -we obtain n equations of the type - - (c_(1r) - [sigma]^2a_(1r)) A1 + (c_(2r) - [sigma]^2a_(2r)) A2 + ... + (c_(nr) - [sigma]^2a_nr) A_n = 0. (5) - -Eliminating the n - 1 ratios A1 : A2 : ... : A_n we obtain the -determinantal equation - - [Delta]([sigma]^2) = 0, (6) - -where - - [Delta]([sigma]^2) = | c11 - [sigma]^2a11, c21 - [sigma]^2a21, ..., C_(n1) - [sigma]^2a_(nl) | - | c12 - [sigma]^2a12, c22 - [sigma]^2a22, ..., C_(n2) - [sigma]^2a_(n2) | - | . . ... . | - | . . ... . | (7) - | . . ... . | - | c_(1n) - [sigma]^2a{1n}, c_(2n) - [sigma]^2a_(2n), ..., C_(nn) - [sigma]^2a_(nn) | - -The quadratic expression for T is essentially positive, and the same -holds with regard to V in virtue of the assumed stability. It may be -shown algebraically that under these conditions the n roots of the above -equation in [sigma]^2 are all real and positive. For any particular root, -the equations (5) determine the ratios of the quantities A1, A2, ... -A_n, the absolute values being alone arbitrary; these quantities are in -fact proportional to the minors of any one row in the determinate -[Delta]([sigma]^2). By combining the solutions corresponding to a pair of -equal and opposite values of [sigma] we obtain a solution in real form: - - q_r = C_(a_r) cos ([sigma]t + [epsilon]), (8) - -where a1, a2 ... a_r are a determinate series of quantities having to -one another the above-mentioned ratios, whilst the constants C, -[epsilon] are arbitrary. This solution, taken by itself, represents a -motion in which each particle of the system (since its displacements -parallel to Cartesian co-ordinate axes are linear functions of the q's) -executes a simple vibration of period 2[pi]/[sigma]. The amplitudes of -oscillation of the various particles have definite ratios to one -another, and the phases are in agreement, the absolute amplitude -(depending on C) and the phase-constant ([epsilon]) being alone -arbitrary. A vibration of this character is called a _normal mode_ of -vibration of the system; the number n of such modes is equal to that of -the degrees of freedom possessed by the system. These statements require -some modification when two or more of the roots of the equation (6) are -equal. In the case of a multiple root the minors of [Delta]([sigma]^2) -all vanish, and the basis for the determination of the quantities a_r -disappears. Two or more normal modes then become to some extent -indeterminate, and elliptic vibrations of the individual particles are -possible. An example is furnished by the spherical pendulum (S 13). - -[Illustration: FIG. 85.] - - As an example of the method of determination of the normal modes we - may take the "double pendulum." A mass M hangs from a fixed point by a - string of length a, and a second mass m hangs from M by a string of - length b. For simplicity we will suppose that the motion is confined - to one vertical plane. If [theta], [phi] be the inclinations of the - two strings to the vertical, we have, approximately, - - 2T = Ma^2[.theta]^2 + m(a[.theta] + b[.psi])^2 \ (9) - 2V = Mga[theta]^2 + mg(a[theta]^2 + b[psi]^2). / - - The equations (3) take the forms - - a[:theta] + [mu]b[:phi] + g[theta] = 0, \ (10) - a[:theta] + b[:phi] + g[phi] = 0. / - - where [mu] = m/(M + m). Hence - - ([sigma]^2 - g/a)a[theta] + [mu][sigma]^2b[phi] = 0, \ (11) - [sigma]^2a[theta] + ([sigma]^2 - g/b)b[phi] = 0. / - - The frequency equation is therefore - - ([sigma]^2 - g/a)([sigma]^2 - g/b) - [mu][sigma]^4 = 0. (12) - - The roots of this quadratic in [sigma]^2 are easily seen to be real and - positive. If M be large compared with m, [mu] is small, and the roots - are g/a and g/b, approximately. In the normal mode corresponding to - the former root, M swings almost like the bob of a simple pendulum of - length a, being comparatively uninfluenced by the presence of m, - whilst m executes a "forced" vibration (S 12) of the corresponding - period. In the second mode, M is nearly at rest [as appears from the - second of equations (11)], whilst m swings almost like the bob of a - simple pendulum of length b. Whatever the ratio M/m, the two values of - [sigma]^2 can never be exactly equal, but they are approximately equal - if a, b are nearly equal and [mu] is very small. A curious phenomenon - is then to be observed; the motion of each particle, being made up (in - general) of two superposed simple vibrations of nearly equal period, - is seen to fluctuate greatly in extent, and if the amplitudes be equal - we have periods of approximate rest, as in the case of "beats" in - acoustics. The vibration then appears to be transferred alternately - from m to M at regular intervals. If, on the other hand, M is small - compared with m, [mu] is nearly equal to unity, and the roots of (12) - are [sigma]^2 = g/(a + b) and [sigma]^2 = mg/M.(a + b)/ab, - approximately. The former root makes [theta] = [phi], nearly; in the - corresponding normal mode m oscillates like the bob of a simple - pendulum of length a + b. In the second mode a[theta] + b[phi] = 0, - nearly, so that m is approximately at rest. The oscillation of M then - resembles that of a particle at a distance a from one end of a string - of length a + b fixed at the ends and subject to a tension mg. - -The motion of the system consequent on arbitrary initial conditions may -be obtained by superposition of the n normal modes with suitable -amplitudes and phases. We have then - - q_r = [alpha]_r[theta] + [alpha]_r'[theta]' + [alpha]_r"[theta]" + ..., (13) - -where - - [theta] = C cos ([sigma]t + [epsilon]), [theta]' - = C' cos ([sigma]'t + [epsilon]), [theta]" - = C" cos([sigma]"t + [epsilon]), ... (14) - -provided [sigma]^2, [sigma]'^2, [sigma]"^2, ... are the n roots of (6). -The coefficients of [theta], [theta]', [theta]", ... in (13) satisfy -the _conjugate_ or _orthogonal_ relations - - a11[alpha]1[alpha]1' + a22[alpha]2[alpha]2' + ... + a12([alpha]1[alpha]2' + [alpha]2[alpha]1') + ... = 0, (15) - c11[alpha]1[alpha]1' + c22[alpha]2[alpha]2' + ... + c12([alpha]1[alpha]2' + [alpha]2[alpha]1') + ... = 0, (16) - -provided the symbols [alpha]_r, [alpha]_r' correspond to two distinct -roots [sigma]^2, [sigma]'^2 of (6). To prove these relations, we replace -the symbols A1, A2, ... A_n in (5) by [alpha]1, [alpha]2, ... [alpha]_n -respectively, multiply the resulting equations by a'1, a'2, ... a'_n, in -order, and add. The result, owing to its symmetry, must still hold if we -interchange accented and unaccented Greek letters, and by comparison we -deduce (15) and (16), provided [sigma]^2 and [sigma]'^2 are unequal. The -actual determination of C, C', C", ... and [epsilon], [epsilon]', -[epsilon]", ... in terms of the initial conditions is as follows. If we -write - - C cos [epsilon] = H, -C sin [epsilon] = K, (17) - -we must have - - [alpha]_rH + [alpha]_r'H' + [alpha]_r"H" + ... = [q_r]0, \ (18) - [sigma][alpha]_rH + [sigma]'[alpha]_r'H' + [sigma]"[alpha]_r"H" + ... = [[.q]_r]0, / - -where the zero suffix indicates initial values. These equations can be -at once solved for H, H', H", ... and K, K', K", ... by means of the -orthogonal relations (15). - -By a suitable choice of the generalized co-ordinates it is possible to -reduce T and V simultaneously to sums of squares. The transformation is -in fact effected by the assumption (13), in virtue of the relations (15) -(16), and we may write - - 2T = a[.theta]^2 + a'[.theta]'^2 + a"[.theta]"^2 + ..., \ (19) - 2V = c[theta]^2 + c'[theta]'^2 + c"[theta]"^2 + .... / - -The new co-ordinates [theta], [theta]', [theta]" ... are called the -_normal_ co-ordinates of the system; in a normal mode of vibration one -of these varies alone. The physical characteristics of a normal mode are -that an impulse of a particular normal type generates an initial -velocity of that type only, and that a constant extraneous force of a -particular normal type maintains a displacement of that type only. The -normal modes are further distinguished by an important "stationary" -property, as regards the frequency. If we imagine the system reduced by -frictionless constraints to one degree of freedom, so that the -co-ordinates [theta], [theta]', [theta]", ... have prescribed ratios to -one another, we have, from (19), - - c[theta]^2 + c'[theta]'^2 = c"[theta]"^2 + ... - [sigma]^2 = ----------------------------------------------, (20) - a[theta]^2 + a'[theta]'^2 + a"[theta]"^2 + ... - -This shows that the value of [sigma]^2 for the constrained mode is -intermediate to the greatest and least of the values c/a, c'/a', -c"/a", ... proper to the several normal modes. Also that if the -constrained mode differs little from a normal mode of free vibration -(e.g. if [theta]', [theta]", ... are small compared with [theta]), the -change in the frequency is of the second order. This property can often -be utilized to estimate the frequency of the gravest normal mode of a -system, by means of an assumed approximate type, when the exact -determination would be difficult. It also appears that an estimate thus -obtained is necessarily too high. - -From another point of view it is easily recognized that the equations -(5) are exactly those to which we are led in the ordinary process of -finding the stationary values of the function - - V (q1, q2, ... q_n) - ------------------------, - T (q1, q2, ... q_n) - -where the denominator stands for the same homogeneous quadratic function -of the q's that T is for the [.q]'s. It is easy to construct in this -connexion a proof that the n values of [sigma]^2 are all real and -positive. - - The case of three degrees of freedom is instructive on account of the - geometrical analogies. With a view to these we may write - - 2T= a[.x]^2 + b[.y]^2 + c[.z]^2 + 2f[.y][.z] + 2g[.z][.x] + 2h[.x][.y], \ (21) - 2V = Ax^2 + By^2 + Cz^2 + 2Fyz + 2Gzx + 2Hxy. / - - It is obvious that the ratio - - V (x, y, z) - ----------- (22) - T (x, y, z) - - must have a least value, which is moreover positive, since the - numerator and denominator are both essentially positive. Denoting this - value by [sigma]1^2, we have - - Ax1 + Hy1 + Gz1 = [sigma]1^2(ax1 + hy1 + [dP]gz1), \ - Hx1 + By1 + Fz1 = [sigma]1^2(hx1 + by1 + fz1), > (23) - Gx1 + Fy1 + Cz1 = [sigma]1^2(gx1 + fy1 + cz1), / - - provided x1 : y1 : z1 be the corresponding values of the ratios x:y:z. - Again, the expression (22) will also have a least value when the - ratios x : y : z are subject to the condition - - [dP]V [dP]V [dP]V - x1 ----- + y1 ----- + z1 ----- = 0; (24) - [dP]x [dP]y [dP]z - - and if this be denoted by [sigma]2^2 we have a second system of - equations similar to (23). The remaining value [sigma]2^2 is the value - of (22) when x : y : z arc chosen so as to satisfy (24) and - - [dP]V [dP]V [dP]V - x2 ----- + y2 ----- + z2 ----- = 0 (25) - [dP]x [dP]y [dP]z - - The problem is identical with that of finding the common conjugate - diameters of the ellipsoids T(x, y, z) = const., V(x, y, z) = const. - If in (21) we imagine that x, y, z denote infinitesimal rotations of a - solid free to turn about a fixed point in a given field of force, it - appears that the three normal modes consist each of a rotation about - one of the three diameters aforesaid, and that the values of [sigma] - are proportional to the ratios of the lengths of corresponding - diameters of the two quadrics. - -We proceed to the _forced vibrations_ of the system. The typical case is -where the extraneous forces are of the simple-harmonic type cos -([sigma]t + [epsilon]); the most general law of variation with time can -be derived from this by superposition, in virtue of Fourier's theorem. -Analytically, it is convenient to put Q_r, equal to e^(i[sigma]^t) -multiplied by a complex coefficient; owing to the linearity of the -equations the factor e^(i[sigma]^t) will run through them all, and need -not always be exhibited. For a system of one degree of freedom we have - - a[:q] + cq = Q, (26) - -and therefore on the present supposition as to the nature of Q - - Q - q = --------------. (27) - c - [sigma]^2a - -This solution has been discussed to some extent in S 12, in connexion -with the forced oscillations of a pendulum. We may note further that -when [sigma] is small the displacement q has the "equilibrium value" -Q/c, the same as would be produced by a steady force equal to the -instantaneous value of the actual force, the inertia of the system being -inoperative. On the other hand, when [sigma]^2 is great q tends to the -value -Q/[sigma]^2a, the same as if the potential energy were ignored. -When there are n degrees of freedom we have from (3) - - (c_(1r) - [sigma]^2 a_(2r)) q1 + (c^2_r - [sigma]^2 a_(2r)) q2 + ... + (c_(nr) - [sigma]^2 a_(nr)) q_n = Qr, (28) - -and therefore - - [Delta]([sigma]^2).q_r = a_(1r)Q1 + a_(2r)Q2 + ... + a_(nr)Q_n, (29) - -where a_(1r), a_(2r), ... a_(nr) are the minors of the rth row of the -determinant (7). Every particle of the system executes in general a -simple vibration of the imposed period 2[pi]/[sigma], and all the -particles pass simultaneously through their equilibrium positions. The -amplitude becomes very great when [sigma]^2 approximates to a root of -(6), i.e. when the imposed period nearly coincides with one of the free -periods. Since a_(rs) = a_(sr), the coefficient of Q_s in the expression -for q_r is identical with that of Q_r in the expression for q_s. Various -important "reciprocal theorems" formulated by H. Helmholtz and Lord -Rayleigh are founded on this relation. Free vibrations must of course be -superposed on the forced vibrations given by (29) in order to obtain the -complete solution of the dynamical equations. - -In practice the vibrations of a system are more or less affected by -dissipative forces. In order to obtain at all events a qualitative -representation of these it is usual to introduce into the equations -frictional terms proportional to the velocities. Thus in the case of one -degree of freedom we have, in place of (26), - - a[:q] + b[.q] + cq = Q, (30) - -where a, b, c are positive. The solution of this has been sufficiently -discussed in S 12. In the case of multiple freedom, the equations of -small motion when modified by the introduction of terms proportional to -the velocities are of the type - - d [dP]T [dP]V - --- ---------- + B_(1r)[.q]1 + B_(2r)[.q]2 + ... + B_(nr)[.q]_n + ------- = Q_r (31) - dt [dP][.q]_r [dP]q_r - -If we put - - b_(rs) = b_(sr) = (1/2)[B_(rs) + B_(sr)], [beta]_(rs) = -[beta]_(sr) = (1/2)[B_(rs) - B_(sr)], (32) - -this may be written - - d [dP]T [dP]F [dP]V - --- --------- + ---------- + [beta]_(1r)[.q]1 + [beta]_(2r)[.q]2 + ... + [beta]_(nr)[.q]_r + ------- (33) - dt [dP][.q]_r [dP][.q]_r [dP]q_r - -provided - - 2F = b11[.q]1^2 + b22[.q]2^2 + ... + 2b12[.q]1[.q]2 + ... (34) - -The terms due to F in (33) are such as would arise from frictional -resistances proportional to the absolute velocities of the particles, or -to mutual forces of resistance proportional to the relative velocities; -they are therefore classed as _frictional_ or _dissipative_ forces. The -terms affected with the coefficients [beta]_(rs) on the other hand are -such as occur in "cyclic" systems with latent motion (DYNAMICS, S -_Analytical_); they are called the _gyrostatic terms_. If we multiply -(33) by [.q]_r and sum with respect to r from 1 to n, we obtain, in -virtue of the relations [beta]_(rs) = -[beta]_(sr), [beta]_(rr) = 0, - d - ---(T + V) = 2F + Q1[.q]1 + Q2[.q]2 + ... + Q_n[.q]_n. (35) - dt - -This shows that mechanical energy is lost at the rate 2F per unit time. -The function F is therefore called by Lord Rayleigh the _dissipation -function_. - -If we omit the gyrostatic terms, and write q_r = C_re^([lambda]t), we -find, for a free vibration, - - [a_(1r)[lambda]^2 + b_(1r)[lambda] + c_(1r)] C1 + [a_(2r)[lambda]^2 + b_(2r)[lambda] + c_(2r)] C2 + ... - + [a_(nr)[lambda]^2 + b_(nr)[lambda] + c_(nr)] C_n = 0. (36) - -This leads to a determinantal equation in [lambda] whose 2n roots are -either real and negative, or complex with negative real parts, on the -present hypothesis that the functions T, V, F are all essentially -positive. If we combine the solutions corresponding to a pair of -conjugate complex roots, we obtain, in real form, - - q_r = C[alpha]_re^(-t/[tau]) cos ([sigma]t + [epsilon] - [epsilon]_r), (37) - -where [sigma], [tau], [alpha]_r, [epsilon]_r are determined by the -constitution of the system, whilst C, [epsilon] are arbitrary, and -independent of r. The n formulae of this type represent a normal mode of -free vibration: the individual particles revolve as a rule in elliptic -orbits which gradually contract according to the law indicated by the -exponential factor. If the friction be relatively small, all the normal -modes are of this character, and unless two or more values of [sigma] -are nearly equal the elliptic orbits are very elongated. The effect of -friction on the period is moreover of the second order. - -In a forced vibration of e^(i[sigma]t) the variation of each co-ordinate -is simple-harmonic, with the prescribed period, but there is a -retardation of phase as compared with the force. If the friction be -small the amplitude becomes relatively very great if the imposed period -approximate to a free period. The validity of the "reciprocal theorems" -of Helmholtz and Lord Rayleigh, already referred to, is not affected by -frictional forces of the kind here considered. - - The most important applications of the theory of vibrations are to the - case of continuous systems such as strings, bars, membranes, plates, - columns of air, where the number of degrees of freedom is infinite. - The series of equations of the type (3) is then replaced by a single - linear partial differential equation, or by a set of two or three such - equations, according to the number of dependent variables. These - variables represent the whole assemblage of generalized co-ordinates - q_r; they are continuous functions of the independent variables x, y, - z whose range of variation corresponds to that of the index r, and of - t. For example, in a one-dimensional system such as a string or a bar, - we have one dependent variable, and two independent variables x and t. - To determine the free oscillations we assume a time factor - e^(i[sigma]t); the equations then become linear differential equations - between the dependent variables of the problem and the independent - variables x, or x, y, or x, y, z as the case may be. If the range of - the independent variable or variables is unlimited, the value of - [sigma] is at our disposal, and the solution gives us the laws of - wave-propagation (see WAVE). If, on the other hand, the body is - finite, certain terminal conditions have to be satisfied. These limit - the admissible values of [sigma], which are in general determined by - a transcendental equation corresponding to the determinantal equation - (6). - - Numerous examples of this procedure, and of the corresponding - treatment of forced oscillations, present themselves in theoretical - acoustics. It must suffice here to consider the small oscillations of - a chain hanging vertically from a fixed extremity. If x be measured - upwards from the lower end, the horizontal component of the tension P - at any point will be P[delta]y/[delta]x, approximately, if y denote - the lateral displacement. Hence, forming the equation of motion of a - mass-element, [rho][delta]x, we have - - [rho][delta]x.[:y] = [delta]P.([dP]y/[dP]x). (38) - - Neglecting the vertical acceleration we have P = g[rho]x, whence - - [dP]^2y [dP] / [dP]y \ - ------- = g ----- ( x ----- ). (39) - [dP]t^2 [dP]x \ [dP]x / - - Assuming that y varies as e^(i[sigma]t) we have - - [dP] / [dP]y \ - ----- ( x ----- ) + ky = 0 (40) - [dP]x \ [dP]x / - - provided k = [sigma]^2/g. The solution of (40) which is finite for x = - 0 is readily obtained in the form of a series, thus - - / kx k^2 x^2 \ - y = C ( 1 - -- + ------- - ... ) = CJ0(z), (41) - \ 1^2 1^2 2^2 / - - in the notation of Bessel's functions, if z^2 = 4kx. Since y must - vanish at the upper end (x = l), the admissible values of [sigma] are - determined by - - [sigma]^2 = gz^2/4l, J0(z) = 0. (42) - - The function J0(z) has been tabulated; its lower roots are given by - - z/[pi]= .7655, 1.7571, 2.7546,..., - - approximately, where the numbers tend to the form s - (1/4). The - frequency of the gravest mode is to that of a uniform bar in the ratio - .9815 That this ratio should be less than unity agrees with the theory - of "constrained types" already given. In the higher normal modes there - are nodes or points of rest (y = 0); thus in the second mode there is - a node at a distance .190l from the lower end. - - AUTHORITIES.--For indications as to the earlier history of the subject - see W. W. R. Ball, _Short Account of the History of Mathematics_; M. - Cantor, _Geschichte der Mathematik_ (Leipzig, 1880 ... ); J. Cox, - _Mechanics_ (Cambridge, 1904); E. Mach, _Die Mechanik in ihrer - Entwickelung_ (4th ed., Leipzig, 1901; Eng. trans.). Of the classical - treatises which have had a notable influence on the development of the - subject, and which may still be consulted with advantage, we may note - particularly, Sir I. Newton, _Philosophiae naturalis Principia - Mathematica_ (1st ed., London, 1687); J. L. Lagrange, _Mecanique - analytique_ (2nd ed., Paris, 1811-1815); P. S. Laplace, _Mecanique - celeste_ (Paris, 1799-1825); A. F. Mobius, _Lehrbuch der Statik_ - (Leipzig, 1837), and _Mechanik des Himmels_; L. Poinsot, _Elements de - statique_ (Paris, 1804), and _Theorie nouvelle de la rotation des - corps_ (Paris, 1834). - - Of the more recent general treatises we may mention Sir W. Thomson - (Lord Kelvin) and P. G. Tait, _Natural Philosophy_ (2nd ed., - Cambridge, 1879-1883); E. J. Routh, _Analytical Statics_ (2nd ed., - Cambridge, 1896), _Dynamics of a Particle_ (Cambridge, 1898), _Rigid - Dynamics_ (6th ed., Cambridge 1905); G. Minchin, _Statics_ (4th ed., - Oxford, 1888); A. E. H. Love, _Theoretical Mechanics_ (2nd ed., - Cambridge, 1909); A. G. Webster, _Dynamics of Particles_, &c. (1904); - E. T. Whittaker, _Analytical Dynamics_ (Cambridge, 1904); L. Arnal, - _Traite de mecanique_ (1888-1898); P. Appell, _Mecanique rationelle_ - (Paris, vols. i. and ii., 2nd ed., 1902 and 1904; vol. iii., 1st ed., - 1896); G. Kirchhoff, _Vorlesungen uber Mechanik_ (Leipzig, 1896); H. - Helmholtz, _Vorlesungen uber theoretische Physik_, vol. i. (Leipzig, - 1898); J. Somoff, _Theoretische Mechanik_ (Leipzig, 1878-1879). - - The literature of graphical statics and its technical applications is - very extensive. We may mention K. Culmann, _Graphische Statik_ (2nd - ed., Zurich, 1895); A. Foppl, _Technische Mechanik_, vol. ii. - (Leipzig, 1900); L. Henneberg, _Statik des starren Systems_ - (Darmstadt, 1886); M. Levy, _La statique graphique_ (2nd ed., Paris, - 1886-1888); H. Muller-Breslau, _Graphische Statik_ (3rd ed., Berlin, - 1901). Sir R. S. Ball's highly original investigations in kinematics - and dynamics were published in collected form under the title _Theory - of Screws_ (Cambridge, 1900). - - Detailed accounts of the developments of the various branches of the - subject from the beginning of the 19th century to the present time, - with full bibliographical references, are given in the fourth volume - (edited by Professor F. Klein) of the _Encyclopadie der mathematischen - Wissenschaften_ (Leipzig). There is a French translation of this work. - (See also DYNAMICS.) (H. Lb.) - - -II.--APPLIED MECHANICS[1] - -S 1. The practical application of mechanics may be divided into two -classes, according as the assemblages of material objects to which they -relate are intended to remain fixed or to move relatively to each -other--the former class being comprehended under the term "Theory of -Structures" and the latter under the term "Theory of Machines." - - -PART I.--OUTLINE OF THE THEORY OF STRUCTURES - - S 2. _Support of Structures._--Every structure, as a whole, is - maintained in equilibrium by the joint action of its own _weight_, of - the _external load_ or pressure applied to it from without and tending - to displace it, and of the _resistance_ of the material which supports - it. A structure is supported either by resting on the solid crust of - the earth, as buildings do, or by floating in a fluid, as ships do in - water and balloons in air. The principles of the support of a floating - structure form an important part of Hydromechanics (q.v.). The - principles of the support, as a whole, of a structure resting on the - land, are so far identical with those which regulate the equilibrium - and stability of the several parts of that structure that the only - principle which seems to require special mention here is one which - comprehends in one statement the power both of liquids and of loose - earth to support structures. This was first demonstrated in a paper - "On the Stability of Loose Earth," read to the Royal Society on the - 19th of June 1856 (Phil. _Trans._ 1856), as follows:-- - - Let E represent the weight of the portion of a horizontal stratum of - earth which is displaced by the foundation of a structure, S the - utmost weight of that structure consistently with the power of the - earth to resist displacement, [phi] the angle of repose of the earth; - then - - S /1 + sin[phi]\^2 - --- = ( ------------ ). - E \1 - sin[phi]/ - - To apply this to liquids [phi] must be made zero, and then S/E = 1, as - is well known. For a proof of this expression see Rankine's _Applied - Mechanics_, 17th ed., p. 219. - - S 3. _Composition of a Structure, and Connexion of its Pieces._--A - structure is composed of _pieces_,--such as the stones of a building - in masonry, the beams of a timber framework, the bars, plates and - bolts of an iron bridge. Those pieces are connected at their joints or - surfaces of mutual contact, either by simple pressure and friction (as - in masonry with moist mortar or without mortar), by pressure and - adhesion (as in masonry with cement or with hardened mortar, and - timber with glue), or by the resistance of _fastenings_ of different - kinds, whether made by means of the form of the joint (as dovetails, - notches, mortices and tenons) or by separate fastening pieces (as - trenails, pins, spikes, nails, holdfasts, screws, bolts, rivets, - hoops, straps and sockets.) - - S 4. _Stability, Stiffness and Strength._--A structure may be damaged - or destroyed in three ways:--first, by displacement of its pieces from - their proper positions relatively to each other or to the earth; - secondly by disfigurement of one or more of those pieces, owing to - their being unable to preserve their proper shapes under the pressures - to which they are subjected; thirdly, by _breaking_ of one or more of - those pieces. The power of resisting displacement constitutes - stability, the power of each piece to resist disfigurement is its - _stiffness_; and its power to resist breaking, its _strength_. - - S 5. _Conditions of Stability._--The principles of the stability of a - structure can be to a certain extent investigated independently of the - stiffness and strength, by assuming, in the first instance, that each - piece has strength sufficient to be safe against being broken, and - stiffness sufficient to prevent its being disfigured to an extent - inconsistent with the purposes of the structure, by the greatest - forces which are to be applied to it. The condition that each piece of - the structure is to be maintained in equilibrium by having its gross - load, consisting of its own weight and of the external pressure - applied to it, balanced by the _resistances_ or pressures exerted - between it and the contiguous pieces, furnishes the means of - determining the magnitude, position and direction of the resistances - required at each joint in order to produce equilibrium; and the - _conditions of stability_ are, first, that the _position_, and, - secondly, that the _direction_, of the resistance required at each - joint shall, under all the variations to which the load is subject, be - such as the joint is capable of exerting--conditions which are - fulfilled by suitably adjusting the figures and positions of the - joints, and the _ratios_ of the gross loads of the pieces. As for the - _magnitude_ of the resistance, it is limited by conditions, not of - stability, but of strength and stiffness. - - S 6. _Principle of Least Resistance._--Where more than one system of - resistances are alike capable of balancing the same system of loads - applied to a given structure, the _smallest_ of those alternative - systems, as was demonstrated by the Rev. Henry Moseley in his - _Mechanics of Engineering and Architecture_, is that which will - actually be exerted--because the resistances to displacement are the - effect of a strained state of the pieces, which strained state is the - effect of the load, and when the load is applied the strained state - and the resistances produced by it increase until the resistances - acquire just those magnitudes which are sufficient to balance the - load, after which they increase no further. - - This principle of least resistance renders determinate many problems - in the statics of structures which were formerly considered - indeterminate. - - S 7. _Relations between Polygons of Loads and of Resistances._--In a - structure in which each piece is supported at two joints only, the - well-known laws of statics show that the directions of the gross load - on each piece and of the two resistances by which it is supported must - lie in one plane, must either be parallel or meet in one point, and - must bear to each other, if not parallel, the proportions of the sides - of a triangle respectively parallel to their directions, and, if - parallel, such proportions that each of the three forces shall be - proportional to the distance between the other two,--all the three - distances being measured along one direction. - - [Illustration: FIG. 86.] - - Considering, in the first place, the case in which the load and the - two resistances by which each piece is balanced meet in one point, - which may be called the _centre of load_, there will be as many such - points of intersection, or centres of load, as there are pieces in the - structure; and the directions and positions of the resistances or - mutual pressures exerted between the pieces will be represented by the - sides of a polygon joining those points, as in fig. 86 where P1, P2, - P3, P4 represent the centres of load in a structure of four pieces, - and the sides of the _polygon of resistances_ P1 P2 P3 P4 represent - respectively the directions and positions of the resistances exerted - at the joints. Further, at any one of the centres of load let PL - represent the magnitude and direction of the gross load, and Pa, Pb - the two resistances by which the piece to which that load is applied - is supported; then will those three lines be respectively the diagonal - and sides of a parallelogram; or, what is the same thing, they will be - equal to the three sides of a triangle; and they must be in the same - plane, although the sides of the polygon of resistances may be in - different planes. - - [Illustration: FIG. 87.] - - According to a well-known principle of statics, because the loads or - external pressures P1L1, &c., balance each other, they must be - proportional to the sides of a closed polygon drawn respectively - parallel to their directions. In fig. 87 construct such a _polygon of - loads_ by drawing the lines L1, &c., parallel and proportional to, and - joined end to end in the order of, the gross loads on the pieces of - the structure. Then from the proportionality and parallelism of the - load and the two resistances applied to each piece of the structure to - the three sides of a triangle, there results the following theorem - (originally due to Rankine):-- - - _If from the angles of the polygon of loads there be drawn lines (R1, - R2, &c.), each of which is parallel to the resistance (as P1P2, &c.) - exerted at the joint between the pieces to which the two loads - represented by the contiguous sides of the polygon of loads (such as - L1, L2, &c.) are applied; then will all those lines meet in one point - (O), and their lengths, measured from that point to the angles of the - polygon, will represent the magnitudes of the resistances to which - they are respectively parallel._ - - When the load on one of the pieces is parallel to the resistances - which balance it, the polygon of resistances ceases to be closed, two - of the sides becoming parallel to each other and to the load in - question, and extending indefinitely. In the polygon of loads the - direction of a load sustained by parallel resistances traverses the - point O.[2] - - S 8. _How the Earth's Resistance is to be treated_.... When the - pressure exerted by a structure on the earth (to which the earth's - resistance is equal and opposite) consists either of one pressure, - which is necessarily the resultant of the weight of the structure and - of all the other forces applied to it, or of two or more parallel - vertical forces, whose amount can be determined at the outset of the - investigation, the resistance of the earth can be treated as one or - more upward loads applied to the structure. But in other cases the - earth is to be treated as _one of the pieces of the structure_, loaded - with a force equal and opposite in direction and position to the - resultant of the weight of the structure and of the other pressures - applied to it. - - S 9. _Partial Polygons of Resistance._--In a structure in which there - are pieces supported at more than two joints, let a polygon be - constructed of lines connecting the centres of load of any continuous - series of pieces. This may be called a _partial polygon of - resistances_. In considering its properties, the load at each centre - of load is to be held to _include_ the resistances of those joints - which are not comprehended in the partial polygon of resistances, to - which the theorem of S 7 will then apply in every respect. By - constructing several partial polygons, and computing the relations - between the loads and resistances which are determined by the - application of that theorem to each of them, with the aid, if - necessary, of Moseley's principle of the least resistance, the whole - of the relations amongst the loads and resistances may be found. - - S 10. _Line of Pressures--Centres and Line of Resistance._--The line - of pressures is a line to which the directions of all the resistances - in one polygon are tangents. The _centre of resistance_ at any joint - is the point where the line representing the total resistance exerted - at that joint intersects the joint. The _line of resistance_ is a line - traversing all the centres of resistance of a series of joints,--its - form, in the positions intermediate between the actual joints of the - structure, being determined by supposing the pieces and their loads to - be subdivided by the introduction of intermediate joints _ad - infinitum_, and finding the continuous line, curved or straight, in - which the intermediate centres of resistance are all situated, however - great their number. The difference between the line of resistance and - the line of pressures was first pointed out by Moseley. - - [Illustration: FIG. 88.] - - S 11.* The principles of the two preceding sections may be illustrated - by the consideration of a particular case of a buttress of blocks - forming a continuous series of pieces (fig. 88), where aa, bb, cc, dd - represent plane joints. Let the centre of pressure C at the first - joint aa be known, and also the pressure P acting at C in direction - and magnitude. Find R1 the resultant of this pressure, the weight of - the block aabb acting through its centre of gravity, and any other - external force which may be acting on the block, and produce its line - of action to cut the joint bb in C1. C1 is then the centre of pressure - for the joint bb, and R1 is the total force acting there. Repeating - this process for each block in succession there will be found the - centres of pressure C2, C3, &c., and also the resultant pressures R2, - R3, &c., acting at these respective centres. The centres of pressure - at the joints are also called _centres of resistance_, and the curve - passing through these points is called a _line of resistance_. Let all - the resultants acting at the several centres of resistance be produced - until they cut one another in a series of points so as to form an - unclosed polygon. This polygon is the _partial polygon of resistance_. - A curve tangential to all the sides of the polygon is the _line of - pressures_. - - S 12. _Stability of Position, and Stability of Friction._--The - resistances at the several joints having been determined by the - principles set forth in SS 6, 7, 8, 9 and 10, not only under the - ordinary load of the structure, but under all the variations to which - the load is subject as to amount and distribution, the joints are now - to be placed and shaped so that the pieces shall not suffer relative - displacement under any of those loads. The relative displacement of - the two pieces which abut against each other at a joint may take place - either by turning or by sliding. Safety against displacement by - turning is called _stability of position_; safety against displacement - by sliding, _stability of friction_. - - S 13. _Condition of Stability of Position._--If the materials of a - structure were infinitely stiff and strong, stability of position at - any joint would be insured simply by making the centre of resistance - fall within the joint under all possible variations of load. In order - to allow for the finite stiffness and strength of materials, the least - distance of the centre of resistance inward from the nearest edge of - the joint is made to bear a definite proportion to the depth of the - joint measured in the same direction, which proportion is fixed, - sometimes empirically, sometimes by theoretical deduction from the - laws of the strength of materials. That least distance is called by - Moseley the _modulus of stability_. The following are some of the - ratios of the modulus of stability to the depth of the joint which - occur in practice:-- - - Retaining walls, as designed by British engineers 1:8 - Retaining walls, as designed by French engineers 1:5 - Rectangular piers of bridges and other buildings, and - arch-stones 1:3 - Rectangular foundations, firm ground 1:3 - Rectangular foundations, very soft ground 1:2 - Rectangular foundations, intermediate kinds of ground 1:3 to 1:2 - Thin, hollow towers (such as furnace chimneys exposed - to high winds), square 1:6 - Thin, hollow towers, circular 1:4 - Frames of timber or metal, under their ordinary or - average distribution of load 1:3 - Frames of timber or metal, under the greatest - irregularities of load 1:3 - - In the case of the towers, the _depth of the joint_ is to be - understood to mean the _diameter of the tower_. - - [Illustration: FIG. 89.] - - S 14. _Condition of Stability of Friction._--If the resistance to be - exerted at a joint is always perpendicular to the surfaces which abut - at and form that joint, there is no tendency of the pieces to be - displaced by sliding. If the resistance be oblique, let JK (fig. 89) - be the joint, C its centre of resistance, CR a line representing the - resistance, CN a perpendicular to the joint at the centre of - resistance. The angle NCR is the _obliquity_ of the resistance. From R - draw RP parallel and RQ perpendicular to the joint; then, by the - principles of statics, the component of the resistance _normal_ to the - joint is-- - - CP = CR . cos PCR; - - and the component _tangential_ to the joint is-- - - CQ = CR . sin PCR = CP . tan PCR. - - If the joint be provided either with projections and recesses, such as - mortises and tenons, or with fastenings, such as pins or bolts, so as - to resist displacement by sliding, the question of the utmost amount - of the tangential resistance CQ which it is capable of exerting - depends on the _strength_ of such projections, recesses, or - fastenings; and belongs to the subject of strength, and not to that of - stability. In other cases the safety of the joint against displacement - by sliding depends on its power of exerting friction, and that power - depends on the law, known by experiment, that the friction between two - surfaces bears a constant ratio, depending on the nature of the - surfaces, to the force by which they are pressed together. In order - that the surfaces which abut at the joint JK may be pressed together, - the resistance required by the conditions of equilibrium CR, must be a - _thrust_ and not a _pull_; and in that case the force by which the - surfaces are pressed together is equal and opposite to the normal - component CP of the resistance. The condition of stability of friction - is that the tangential component CQ of the resistance required shall - not exceed the friction due to the normal component; that is, that - - CQ [/>] f . CP, - - where f denotes the _coefficient of friction_ for the surfaces in - question. The angle whose tangent is the coefficient of friction is - called _the angle of repose_, and is expressed symbolically by-- - - [phi] = tan^-1 f. - - Now CQ = CP . tan PCR; - - consequently the condition of stability of friction is fulfilled if - the angle PCR is not greater than [phi]; that is to say, if _the - obliquity of the resistance required at the joint does not exceed the - angle of repose_; and this condition ought to be fulfilled under all - possible variations of the load. - - It is chiefly in masonry and earthwork that stability of friction is - relied on. - - S 15. _Stability of Friction in Earth._--The grains of a mass of loose - earth are to be regarded as so many separate pieces abutting against - each other at joints in all possible positions, and depending for - their stability on friction. To determine whether a mass of earth is - stable at a given point, conceive that point to be traversed by planes - in all possible positions, and determine which position gives the - greatest obliquity to the total pressure exerted between the portions - of the mass which abut against each other at the plane. The condition - of stability is that this obliquity shall not exceed the angle of - repose of the earth. The consequences of this principle are developed - in a paper, "On the Stability of Loose Earth," already cited in S 2. - - S 16. _Parallel Projections of Figures._--If any figure be referred to - a system of co-ordinates, rectangular or oblique, and if a second - figure be constructed by means of a second system of co-ordinates, - rectangular or oblique, and either agreeing with or differing from the - first system in rectangularity or obliquity, but so related to the - co-ordinates of the first figure that for each point in the first - figure there shall be a corresponding point in the second figure, the - lengths of whose co-ordinates shall bear respectively to the three - corresponding co-ordinates of the corresponding point in the first - figure three ratios which are the same for every pair of corresponding - points in the two figures, these corresponding figures are called - _parallel projections_ of each other. The properties of parallel - projections of most importance to the subject of the present article - are the following:-- - - (1) A parallel projection of a straight line is a straight line. - - (2) A parallel projection of a plane is a plane. - - (3) A parallel projection of a straight line or a plane surface - divided in a given ratio is a straight line or a plane surface divided - in the same ratio. - - (4) A parallel projection of a pair of equal and parallel straight - lines, or plain surfaces, is a pair of equal and parallel straight lines, - or plane surfaces; whence it follows - - (5) That a parallel projection of a parallelogram is a parallelogram, - and - - (6) That a parallel projection of a parallelepiped is a parallelepiped. - - (7) A parallel projection of a pair of solids having a given ratio - is a pair of solids having the same ratio. - - Though not essential for the purposes of the present article, the - following consequence will serve to illustrate the principle of - parallel projections:-- - - (8) A parallel projection of a curve, or of a surface of a given - algebraical order, is a curve or a surface of the same order. - - For example, all ellipsoids referred to co-ordinates parallel to any - three conjugate diameters are parallel projections of each other and - of a sphere referred to rectangular co-ordinates. - - S 17. _Parallel Projections of Systems of Forces._--If a balanced - system of forces be represented by a system of lines, then will every - parallel projection of that system of lines represent a balanced - system of forces. - - For the condition of equilibrium of forces not parallel is that they - shall be represented in direction and magnitude by the sides and - diagonals of certain parallelograms, and of parallel forces that they - shall divide certain straight lines in certain ratios; and the - parallel projection of a parallelogram is a parallelogram, and that of - a straight line divided in a given ratio is a straight line divided in - the same ratio. - - The resultant of a parallel projection of any system of forces is the - projection of their resultant; and the centre of gravity of a parallel - projection of a solid is the projection of the centre of gravity of - the first solid. - - S 18. _Principle of the Transformation of Structures._--Here we have - the following theorem: If a structure of a given figure have stability - of position under a system of forces represented by a given system of - lines, then will any structure whose figure is a parallel projection - of that of the first structure have stability of position under a - system of forces represented by the corresponding projection of the - first system of lines. - - For in the second structure the weights, external pressures, and - resistances will balance each other as in the first structure; the - weights of the pieces and all other parallel systems of forces will - have the same ratios as in the first structure; and the several - centres of resistance will divide the depths of the joints in the same - proportions as in the first structure. - - If the first structure have stability of friction, the second - structure will have stability of friction also, so long as the effect - of the projection is not to increase the obliquity of the resistance - at any joint beyond the angle of repose. - - The lines representing the forces in the second figure show their - _relative_ directions and magnitudes. To find their _absolute_ - directions and magnitudes, a vertical line is to be drawn in the first - figure, of such a length as to represent the weight of a particular - portion of the structure. Then will the projection of that line in the - projected figure indicate the vertical direction, and represent the - weight of the part of the second structure corresponding to the - before-mentioned portion of the first structure. - - The foregoing "principle of the transformation of structures" was - first announced, though in a somewhat less comprehensive form, to the - Royal Society on the 6th of March 1856. It is useful in practice, by - enabling the engineer easily to deduce the conditions of equilibrium - and stability of structures of complex and unsymmetrical figures from - those of structures of simple and symmetrical figures. By its aid, for - example, the whole of the properties of elliptical arches, whether - square or skew, whether level or sloping in their span, are at once - deduced by projection from those of symmetrical circular arches, and - the properties of ellipsoidal and elliptic-conoidal domes from those - of hemispherical and circular-conoidal domes; and the figures of - arches fitted to resist the thrust of earth, which is less - horizontally than vertically in a certain given ratio, can be deduced - by a projection from those of arches fitted to resist the thrust of a - liquid, which is of equal intensity, horizontally and vertically. - - S 19. _Conditions of Stiffness and Strength._--After the arrangement - of the pieces of a structure and the size and figure of their joints - or surfaces of contact have been determined so as to fulfil the - conditions of _stability_,--conditions which depend mainly on the - position and direction of the _resultant_ or _total_ load on each - piece, and the _relative_ magnitude of the loads on the different - pieces--the dimensions of each piece singly have to be adjusted so as - to fulfil the conditions of _stiffness_ and _strength_--conditions - which depend not only on the _absolute_ magnitude of the load on each - piece, and of the resistances by which it is balanced, but also on the - _mode of distribution_ of the load over the piece, and of the - resistances over the joints. - - The effect of the pressures applied to a piece, consisting of the load - and the supporting resistances, is to force the piece into a state of - _strain_ or disfigurement, which increases until the elasticity, or - resistance to strain, of the material causes it to exert a _stress_, - or effort to recover its figure, equal and opposite to the system of - applied pressures. The condition of _stiffness_ is that the strain or - disfigurement shall not be greater than is consistent with the - purposes of the structure; and the condition of _strength_ is that the - stress shall be within the limits of that which the material can bear - with safety against breaking. The ratio in which the utmost stress - before breaking exceeds the safe working stress is called the _factor - of safety_, and is determined empirically. It varies from three to - twelve for various materials and structures. (See STRENGTH OF - MATERIALS.) - - - PART II. THEORY OF MACHINES - - S 20. _Parts of a Machine: Frame and Mechanism._--The parts of a - machine may be distinguished into two principal divisions,--the frame, - or fixed parts, and the _mechanism_, or moving parts. The frame is a - structure which supports the pieces of the mechanism, and to a certain - extent determines the nature of their motions. - - The form and arrangement of the pieces of the frame depend upon the - arrangement and the motions of the mechanism; the dimensions of the - pieces of the frame required in order to give it stability and - strength are determined from the pressures applied to it by means of - the mechanism. It appears therefore that in general the mechanism is - to be designed first and the frame afterwards, and that the designing - of the frame is regulated by the principles of the stability of - structures and of the strength and stiffness of materials,--care being - taken to adapt the frame to the most severe load which can be thrown - upon it at any period of the action of the mechanism. - - Each independent piece of the mechanism also is a structure, and its - dimensions are to be adapted, according to the principles of the - strength and stiffness of materials, to the most severe load to which - it can be subjected during the action of the machine. - - S 21. _Definition and Division of the Theory of Machines._--From what - has been said in the last section it appears that the department of - the art of designing machines which has reference to the stability of - the frame and to the stiffness and strength of the frame and mechanism - is a branch of the art of construction. It is therefore to be - separated from the _theory of machines_, properly speaking, which has - reference to the action of machines considered as moving. In the - action of a machine the following three things take place:-- - - _Firstly_, Some natural source of energy communicates motion and force - to a piece or pieces of the mechanism, called the _receiver of power_ - or _prime mover_. - - _Secondly_, The motion and force are transmitted from the prime mover - through the _train of mechanism_ to the _working piece_ or _pieces_, - and during that transmission the motion and force are modified in - amount and direction, so as to be rendered suitable for the purpose to - which they are to be applied. - - _Thirdly_, The working piece or pieces by their motion, or by their - motion and force combined, produce some useful effect. - - Such are the phenomena of the action of a machine, arranged in the - order of _causation_. But in studying or treating of the theory of - machines, the order of _simplicity_ is the best; and in this order the - first branch of the subject is the modification of motion and force by - the train of mechanism; the next is the effect or purpose of the - machine; and the last, or most complex, is the action of the prime - mover. - - The modification of motion and the modification of force take place - together, and are connected by certain laws; but in the study of the - theory of machines, as well as in that of pure mechanics, much - advantage has been gained in point of clearness and simplicity by - first considering alone the principles of the modification of motion, - which are founded upon what is now known as Kinematics, and afterwards - considering the principles of the combined modification of motion and - force, which are founded both on geometry and on the laws of dynamics. - The separation of kinematics from dynamics is due mainly to G. Monge, - Ampere and R. Willis. - - The theory of machines in the present article will be considered under - the following heads:-- - - I. PURE MECHANISM, or APPLIED KINEMATICS; being the theory of machines - considered simply as modifying motion. - - II. APPLIED DYNAMICS; being the theory of machines considered as - modifying both motion and force. - - - CHAP. I. ON PURE MECHANISM - - S 22. _Division of the Subject._--Proceeding in the order of - simplicity, the subject of Pure Mechanism, or Applied Kinematics, may - be thus divided:-- - - _Division 1._--Motion of a point. - - _Division 2._--Motion of the surface of a fluid. - - _Division 3._--Motion of a rigid solid. - - _Division 4._--Motions of a pair of connected pieces, or of an - "elementary combination" in mechanism. - - _Division 5._--Motions of trains of pieces of mechanism. - - _Division 6._--Motions of sets of more than two connected pieces, or of - "aggregate combinations." - - A point is the boundary of a line, which is the boundary of a surface, - which is the boundary of a volume. Points, lines and surfaces have no - independent existence, and consequently those divisions of this - chapter which relate to their motions are only preliminary to the - subsequent divisions, which relate to the motions of bodies. - - - _Division 1. Motion of a Point._ - - S 23. _Comparative Motion._--The comparative motion of two points is - the relation which exists between their motions, without having regard - to their absolute amounts. It consists of two elements,--the _velocity - ratio_, which is the ratio of any two magnitudes bearing to each other - the proportions of the respective velocities of the two points at a - given instant, and the _directional relation_, which is the relation - borne to each other by the respective directions of the motions of the - two points at the same given instant. - - It is obvious that the motions of a pair of points may be varied in - any manner, whether by direct or by lateral deviation, and yet that - their _comparative motion_ may remain constant, in consequence of the - deviations taking place in the same proportions, in the same - directions and at the same instants for both points. - - Robert Willis (1800-1875) has the merit of having been the first to - simplify considerably the theory of pure mechanism, by pointing out - that that branch of mechanics relates wholly to comparative motions. - - The comparative motion of two points at a given instant is capable of - being completely expressed by one of Sir William Hamilton's - Quaternions,--the "tensor" expressing the velocity ratio, and the - "versor" the directional relation. - - Graphical methods of analysis founded on this way of representing - velocity and acceleration were developed by R. H. Smith in a paper - communicated to the Royal Society of Edinburgh in 1885, and - illustrations of the method will be found below. - - - _Division 2. Motion of the Surface of a Fluid Mass._ - - S 24. _General Principle._--A mass of fluid is used in mechanism to - transmit motion and force between two or more movable portions (called - _pistons_ or _plungers_) of the solid envelope or vessel in which the - fluid is contained; and, when such transmission is the sole action, or - the only appreciable action of the fluid mass, its volume is either - absolutely constant, by reason of its temperature and pressure being - maintained constant, or not sensibly varied. - - Let a represent the area of the section of a piston made by a plane - perpendicular to its direction of motion, and v its velocity, which is - to be considered as positive when outward, and negative when inward. - Then the variation of the cubic contents of the vessel in a unit of - time by reason of the motion of one piston is va. The condition that - the volume of the fluid mass shall remain unchanged requires that - there shall be more than one piston, and that the velocities and areas - of the pistons shall be connected by the equation-- - - [Sigma].va = 0. (1) - - S 25. _Comparative Motion of Two Pistons._--If there be but two - pistons, whose areas are a1 and a2, and their velocities v1 and v2, - their comparative motion is expressed by the equation-- - - v2/v1 = -a1/a2; (2) - - that is to say, their velocities are opposite as to inwardness and - outwardness and inversely proportional to their areas. - - S 26. _Applications: Hydraulic Press: Pneumatic - Power-Transmitter._--In the hydraulic press the vessel consists of two - cylinders, viz. the pump-barrel and the press-barrel, each having its - piston, and of a passage connecting them having a valve opening - towards the press-barrel. The action of the enclosed water in - transmitting motion takes place during the inward stroke of the - pump-plunger, when the above-mentioned valve is open; and at that time - the press-plunger moves outwards with a velocity which is less than - the inward velocity of the pump-plunger, in the same ratio that the - area of the pump-plunger is less than the area of the press-plunger. - (See HYDRAULICS.) - - In the pneumatic power-transmitter the motion of one piston is - transmitted to another at a distance by means of a mass of air - contained in two cylinders and an intervening tube. When the pressure - and temperature of the air can be maintained constant, this machine - fulfils equation (2), like the hydraulic press. The amount and effect - of the variations of pressure and temperature undergone by the air - depend on the principles of the mechanical action of heat, or - THERMODYNAMICS (q.v.), and are foreign to the subject of pure - mechanism. - - - _Division 3. Motion of a Rigid Solid._ - - S 27. _Motions Classed._--In problems of mechanism, each solid piece - of the machine is supposed to be so stiff and strong as not to undergo - any sensible change of figure or dimensions by the forces applied to - it--a supposition which is realized in practice if the machine is - skilfully designed. - - This being the case, the various possible motions of a rigid solid - body may all be classed under the following heads: (1) _Shifting or - Translation_; (2) _Turning or Rotation_; (3) _Motions compounded of - Shifting and Turning_. - - The most common forms for the paths of the points of a piece of - mechanism, whose motion is simple shifting, are the straight line and - the circle. - - Shifting in a straight line is regulated either by straight fixed - guides, in contact with which the moving piece slides, or by - combinations of link-work, called _parallel motions_, which will be - described in the sequel. Shifting in a straight line is usually - _reciprocating_; that is to say, the piece, after shifting through a - certain distance, returns to its original position by reversing its - motion. - - Circular shifting is regulated by attaching two or more points of the - shifting piece to ends of equal and parallel rotating cranks, or by - combinations of wheel-work to be afterwards described. As an example - of circular shifting may be cited the motion of the coupling rod, by - which the parallel and equal cranks upon two or more axles of a - locomotive engine are connected and made to rotate simultaneously. The - coupling rod remains always parallel to itself, and all its points - describe equal and similar circles relatively to the frame of the - engine, and move in parallel directions with equal velocities at the - same instant. - - S 28. _Rotation about a Fixed Axis: Lever, Wheel and Axle._--The fixed - axis of a turning body is a line fixed relatively to the body and - relatively to the fixed space in which the body turns. In mechanism it - is usually the central line either of a rotating shaft or axle having - journals, gudgeons, or pivots turning in fixed bearings, or of a fixed - spindle or dead centre round which a rotating bush turns; but it may - sometimes be entirely beyond the limits of the turning body. For - example, if a sliding piece moves in circular fixed guides, that piece - rotates about an ideal fixed axis traversing the centre of those - guides. - - Let the angular velocity of the rotation be denoted by [alpha] = - d[theta]/dt, then the linear velocity of any point A at the distance r - from the axis is [alpha]r; and the path of that point is a circle of - the radius r described about the axis. - - This is the principle of the modification of motion by the lever, - which consists of a rigid body turning about a fixed axis called a - fulcrum, and having two points at the same or different distances from - that axis, and in the same or different directions, one of which - receives motion and the other transmits motion, modified in direction - and velocity according to the above law. - - In the wheel and axle, motion is received and transmitted by two - cylindrical surfaces of different radii described about their common - fixed axis of turning, their velocity-ratio being that of their radii. - - [Illustration: FIG. 90.] - - S 29. _Velocity Ratio of Components of Motion._--As the distance - between any two points in a rigid body is invariable, the projections - of their velocities upon the line joining them must be equal. Hence it - follows that, if A in fig. 90 be a point in a rigid body CD, rotating - round the fixed axis F, the component of the velocity of A in any - direction AP parallel to the plane of rotation is equal to the total - velocity of the point m, found by letting fall Fm perpendicular to AP; - that is to say, is equal to - - [alpha].Fm. - - Hence also the ratio of the components of the velocities of two points - A and B in the directions AP and BW respectively, both in the plane of - rotation, is equal to the ratio of the perpendiculars Fm and Fn. - - S 30. _Instantaneous Axis of a Cylinder rolling on a Cylinder._--Let a - cylinder bbb, whose axis of figure is B and angular velocity [gamma], - roll on a fixed cylinder [alpha][alpha][alpha], whose axis of figure - is A, either outside (as in fig. 91), when the rolling will be towards - the same hand as the rotation, or inside (as in fig. 92), when the - rolling will be towards the opposite hand; and at a given instant let - T be the line of contact of the two cylindrical surfaces, which is at - their common intersection with the plane AB traversing the two axes of - figure. - - The line T on the surface bbb has for the instant no velocity in a - direction perpendicular to AB; because for the instant it touches, - without sliding, the line T on the fixed surface aaa. - - The line T on the surface bbb has also for the instant no velocity in - the plane AB; for it has just ceased to move towards the fixed surface - aaa, and is just about to begin to move away from that surface. - - The line of contact T, therefore, on the surface of the cylinder bbb, - is _for the instant_ at rest, and is the "instantaneous axis" about - which the cylinder bbb turns, together with any body rigidly attached - to that cylinder. - - [Illustration: FIG. 91.] - - [Illustration: FIG. 92.] - - To find, then, the direction and velocity at the given instant of any - point P, either in or rigidly attached to the rolling cylinder T, draw - the plane PT; the direction of motion of P will be perpendicular to - that plane, and towards the right or left hand according to the - direction of the rotation of bbb; and the velocity of P will be - - v_P = [gamma].PT, (3) - - PT denoting the perpendicular distance of P from T. The path of P is a - curve of the kind called _epitrochoids_. If P is in the circumference - of bbb, that path becomes an _epicycloid_. - - The velocity of any point in the axis of figure B is - - v_B = [gamma].TB; (4) - - and the path of such a point is a circle described about A with the - radius AB, being for outside rolling the sum, and for inside rolling - the difference, of the radii of the cylinders. - - Let [alpha] denote the angular velocity with which the _plane of axes_ - AB rotates about the fixed axis A. Then it is evident that - - v_B = [alpha].AB, (5) - - and consequently that - - [alpha] = [gamma].TB/AB. (6) - - For internal rolling, as in fig. 92, AB is to be treated as negative, - which will give a negative value to [alpha], indicating that in this - case the rotation of AB round A is contrary to that of the cylinder - bbb. - - The angular velocity of the rolling cylinder, _relatively to the plane - of axes_ AB, is obviously given by the equation-- - - [beta] = [gamma] - [alpha] \ - >, (7) - whence [beta] = [gamma].TA/AB / - - care being taken to attend to the sign of [alpha], so that when that - is negative the arithmetical values of [gamma] and [alpha] are to be - added in order to give that of [beta]. - - The whole of the foregoing reasonings are applicable, not merely when - aaa and bbb are actual cylinders, but also when they are the - osculating cylinders of a pair of cylindroidal surfaces of varying - curvature, A and B being the axes of curvature of the parts of those - surfaces which are in contact for the instant under consideration. - - [Illustration: FIG. 93.] - - S 31. _Instantaneous Axis of a Cone rolling on a Cone._--Let Oaa (fig. - 93) be a fixed cone, OA its axis, Obb a cone rolling on it, OB the - axis of the rolling cone, OT the line of contact of the two cones at - the instant under consideration. By reasoning similar to that of S 30, - it appears that OT is the instantaneous axis of rotation of the - rolling cone. - - Let [gamma] denote the total angular velocity of the rotation of the - cone B about the instantaneous axis, [beta] its angular velocity about - the axis OB _relatively_ to the plane AOB, and [alpha] the angular - velocity with which the plane AOB turns round the axis OA. It is - required to find the ratios of those angular velocities. - - _Solution._--In OT take any point E, from which draw EC parallel to - OA, and ED parallel to OB, so as to construct the parallelogram OCED. - Then - - OD : OC : OE :: [alpha] : [beta] : [gamma]. (8) - - Or because of the proportionality of the sides of triangles to the - sines of the opposite angles, - - sin TOB : sin TOA : sin AOB :: [alpha] : [beta] : [gamma], (8 A) - - that is to say, the angular velocity about each axis is proportional - to the sine of the angle between the other two. - - _Demonstration._--From C draw CF perpendicular to OA, and CG - perpendicular to OE - - area ECO - Then CF = 2 X --------, - CE - - area ECO - and CG = 2 X --------; - OE - - :. CG : CF :: CE = OD : OE. - - Let v_c denote the linear velocity of the point C. Then - - v_c = [alpha] . CF = [gamma].CG - :. [gamma] : [alpha] :: CF : CG :: OE : OD, - - which is one part of the solution above stated. From E draw EH - perpendicular to OB, and EK to OA. Then it can be shown as before that - - EK : EH :: OC : OD. - - Let v_E be the linear velocity of the point E _fixed in the plane of - axes_ AOB. Then - - v_K = [alpha] . EK. - - Now, as the line of contact OT is for the instant at rest on the - rolling cone as well as on the fixed cone, the linear velocity of the - point E fixed to the plane AOB relatively to the rolling cone is the - same with its velocity relatively to the fixed cone. That is to say, - - [beta].EH = v_E = [alpha].EK; - - therefore - - [alpha] : [beta] :: EH : EK :: OD : OC, - - which is the remainder of the solution. - - The path of a point P in or attached to the rolling cone is a - spherical epitrochoid traced on the surface of a sphere of the radius - OP. From P draw PQ perpendicular to the instantaneous axis. Then the - motion of P is perpendicular to the plane OPQ, and its velocity is - - v_P = [gamma].PQ. (9) - - The whole of the foregoing reasonings are applicable, not merely when - A and B are actual regular cones, but also when they are the - osculating regular cones of a pair of irregular conical surfaces, - having a common apex at O. - - S 32. _Screw-like or Helical Motion._--Since any displacement in a - plane can be represented in general by a rotation, it follows that the - only combination of translation and rotation, in which a complex - movement which is not a mere rotation is produced, occurs when there - is a translation _perpendicular to the plane and parallel to the axis_ - of rotation. - - [Illustration: FIG. 94.] - - Such a complex motion is called _screw-like_ or _helical_ motion; for - each point in the body describes a _helix_ or _screw_ round the axis - of rotation, fixed or instantaneous as the case may be. To cause a - body to move in this manner it is usually made of a helical or - screw-like figure, and moves in a guide of a corresponding figure. - Helical motion and screws adapted to it are said to be right- or - left-handed according to the appearance presented by the rotation to - an observer looking towards the direction of the translation. Thus the - screw G in fig. 94 is right-handed. - - The translation of a body in helical motion is called its _advance_. - Let v_x denote the velocity of advance at a given instant, which of - course is common to all the particles of the body; [alpha] the angular - velocity of the rotation at the same instant; 2[pi] = 6.2832 nearly, - the circumference of a circle of the radius unity. Then - - T = 2[pi]/[alpha] (10) - - is the time of one turn at the rate [alpha]; and - - p = v_x T = 2[pi]v_x/[alpha] (11) - - is the _pitch_ or _advance per turn_--a length which expresses the - _comparative motion_ of the translation and the rotation. - - The pitch of a screw is the distance, measured parallel to its axis, - between two successive turns of the same _thread_ or helical - projection. - - Let r denote the perpendicular distance of a point in a body moving - helically from the axis. Then - - v_r = [alpha]r (12) - - is the component of the velocity of that point in a plane - perpendicular to the axis, and its total velocity is - - v = [root](v_x^2 + v_r^2). (13) - - The ratio of the two components of that velocity is - - v_x/v_r = p/2[pi]r = tan [theta]. (14) - - where [theta] denotes the angle made by the helical path of the point - with a plane perpendicular to the axis. - - - _Division 4. Elementary Combinations in Mechanism_ - - S 33. _Definitions._--An _elementary combination_ in mechanism - consists of two pieces whose kinds of motion are determined by their - connexion with the frame, and their comparative motion by their - connexion with each other--that connexion being effected either by - direct contact of the pieces, or by a connecting piece, which is not - connected with the frame, and whose motion depends entirely on the - motions of the pieces which it connects. - - The piece whose motion is the cause is called the _driver_; the piece - whose motion is the effect, the _follower_. - - The connexion of each of those two pieces with the frame is in general - such as to determine the path of every point in it. In the - investigation, therefore, of the comparative motion of the driver and - follower, in an elementary combination, it is unnecessary to consider - relations of angular direction, which are already fixed by the - connexion of each piece with the frame; so that the inquiry is - confined to the determination of the velocity ratio, and of the - directional relation, so far only as it expresses the connexion - between _forward_ and _backward_ movements of the driver and follower. - When a continuous motion of the driver produces a continuous motion of - the follower, forward or backward, and a reciprocating motion a motion - reciprocating at the same instant, the directional relation is said to - be _constant_. When a continuous motion produces a reciprocating - motion, or vice versa, or when a reciprocating motion produces a - motion not reciprocating at the same instant, the directional relation - is said to be _variable_. - - The _line of action_ or _of connexion_ of the driver and follower is a - line traversing a pair of points in the driver and follower - respectively, which are so connected that the component of their - velocity relatively to each other, resolved along the line of - connexion, is null. There may be several or an indefinite number of - lines of connexion, or there may be but one; and a line of connexion - may connect either the same pair of points or a succession of - different pairs. - - S 34. _General Principle._--From the definition of a line of connexion - it follows that _the components of the velocities of a pair of - connected points along their line of connexion are equal_. And from - this, and from the property of a rigid body, already stated in S 29, - it follows, that _the components along a line of connexion of all the - points traversed by that line, whether in the driver or in the - follower, are equal_; and consequently, _that the velocities of any - pair of points traversed by a line of connexion are to each other - inversely as the cosines, or directly as the secants, of the angles - made by the paths of those points with the line of connexion_. - - The general principle stated above in different forms serves to solve - every problem in which--the mode of connexion of a pair of pieces - being given--it is required to find their comparative motion at a - given instant, or vice versa. - - [Illustration: FIG. 95.] - - S 35. _Application to a Pair of Shifting Pieces._--In fig. 95, let - P1P2 be the line of connexion of a pair of pieces, each of which has a - motion of translation or shifting. Through any point T in that line - draw TV1, TV2, respectively parallel to the simultaneous direction of - motion of the pieces; through any other point A in the line of - connexion draw a plane perpendicular to that line, cutting TV1, TV2 in - V1, V2; then, velocity of piece 1 : velocity of piece 2 :: TV1 : TV2. - Also TA represents the equal components of the velocities of the - pieces parallel to their line of connexion, and the line V1V2 - represents their velocity relatively to each other. - - S 36. _Application to a Pair of Turning Pieces._--Let [alpha]1, - [alpha]2 be the angular velocities of a pair of turning pieces; - [theta]1, [theta]2 the angles which their line of connexion makes with - their respective planes of rotation; r1, r2 the common perpendiculars - let fall from the line of connexion upon the respective axes of - rotation of the pieces. Then the equal components, along the line of - connexion, of the velocities of the points where those perpendiculars - meet that line are-- - - [alpha]1r1 cos [theta]1 = [alpha]2r2 cos [theta]2; - - consequently, the comparative motion of the pieces is given by the - equation - - [alpha]2 r1 cos [theta]1 - -------- = ---------------. (15) - [alpha]1 r2 cos [theta]2 - - S 37. _Application to a Shifting Piece and a Turning Piece._--Let a - shifting piece be connected with a turning piece, and at a given - instant let [alpha]1 be the angular velocity of the turning piece, r1 - the common perpendicular of its axis of rotation and the line of - connexion, [theta]1 the angle made by the line of connexion with the - plane of rotation, [theta]2 the angle made by the line of connexion - with the direction of motion of the shifting piece, v2 the linear - velocity of that piece. Then - - [alpha]1r1 cos [theta]1 = v2 cos [theta]2; (16) - - which equation expresses the comparative motion of the two pieces. - - S 38. _Classification of Elementary Combinations in Mechanism._--The - first systematic classification of elementary combinations in - mechanism was that founded by Monge, and fully developed by Lanz and - Betancourt, which has been generally received, and has been adopted in - most treatises on applied mechanics. But that classification is - founded on the absolute instead of the comparative motions of the - pieces, and is, for that reason, defective, as Willis pointed out in - his admirable treatise _On the Principles of Mechanism_. - - Willis's classification is founded, in the first place, on comparative - motion, as expressed by velocity ratio and directional relation, and - in the second place, on the mode of connexion of the driver and - follower. He divides the elementary combinations in mechanism into - three classes, of which the characters are as follows:-- - - Class A: Directional relation constant; velocity ratio constant. - - Class B: Directional relation constant; velocity ratio varying. - - Class C: Directional relation changing periodically; velocity ratio - constant or varying. - - Each of those classes is subdivided by Willis into five divisions, of - which the characters are as follows:-- - - Division A: Connexion by rolling contact. - " B: " " sliding contact. - " C: " " wrapping connectors. - " D: " " link-work. - " E: " " reduplication. - - In the Reuleaux system of analysis of mechanisms the principle of - comparative motion is generalized, and mechanisms apparently very - diverse in character are shown to be founded on the same sequence of - elementary combinations forming a kinematic chain. A short description - of this system is given in S 80, but in the present article the - principle of Willis's classification is followed mainly. The - arrangement is, however, modified by taking the _mode of connexion_ as - the basis of the primary classification, and by removing the subject - of connexion by reduplication to the section of aggregate - combinations. This modified arrangement is adopted as being better - suited than the original arrangement to the limits of an article in an - encyclopaedia; but it is not disputed that the original arrangement - may be the best for a separate treatise. - - S 39. _Rolling Contact: Smooth Wheels and Racks._--In order that two - pieces may move in rolling contact, it is necessary that each pair of - points in the two pieces which touch each other should at the instant - of contact be moving in the same direction with the same velocity. In - the case of two _shifting_ pieces this would involve equal and - parallel velocities for all the points of each piece, so that there - could be no rolling, and, in fact, the two pieces would move like one; - hence, in the case of rolling contact, either one or both of the - pieces must rotate. - - The direction of motion of a point in a turning piece being - perpendicular to a plane passing through its axis, the condition that - each pair of points in contact with each other must move in the same - direction leads to the following consequences:-- - - I. That, when both pieces rotate, their axes, and all their points of - contact, lie in the same plane. - - II. That, when one piece rotates, and the other shifts, the axis of - the rotating piece, and all the points of contact, lie in a plane - perpendicular to the direction of motion of the shifting piece. - - The condition that the velocity of each pair of points of contact must - be equal leads to the following consequences:-- - - III. That the angular velocities of a pair of turning pieces in - rolling contact must be inversely as the perpendicular distances of - any pair of points of contact from the respective axes. - - IV. That the linear velocity of a shifting piece in rolling contact - with a turning piece is equal to the product of the angular velocity - of the turning piece by the perpendicular distance from its axis to a - pair of points of contact. - - The _line of contact_ is that line in which the points of contact are - all situated. Respecting this line, the above Principles III. and IV. - lead to the following conclusions:-- - - V. That for a pair of turning pieces with parallel axes, and for a - turning piece and a shifting piece, the line of contact is straight, - and parallel to the axes or axis; and hence that the rolling surfaces - are either plane or cylindrical (the term "cylindrical" including all - surfaces generated by the motion of a straight line parallel to - itself). - - VI. That for a pair of turning pieces with intersecting axes the line - of contact is also straight, and traverses the point of intersection - of the axes; and hence that the rolling surfaces are conical, with a - common apex (the term "conical" including all surfaces generated by - the motion of a straight line which traverses a fixed point). - - Turning pieces in rolling contact are called _smooth_ or _toothless - wheels_. Shifting pieces in rolling contact with turning pieces may be - called _smooth_ or _toothless racks_. - - VII. In a pair of pieces in rolling contact every straight line - traversing the line of contact is a line of connexion. - - S 40. _Cylindrical Wheels and Smooth Racks._--In designing cylindrical - wheels and smooth racks, and determining their comparative motion, it - is sufficient to consider a section of the pair of pieces made by a - plane perpendicular to the axis or axes. - - The points where axes intersect the plane of section are called - _centres_; the point where the line of contact intersects it, the - _point of contact_, or _pitch-point_; and the wheels are described as - _circular_, _elliptical_, &c., according to the forms of their - sections made by that plane. - - When the point of contact of two wheels lies between their centres, - they are said to be in _outside gearing_; when beyond their centres, - in _inside gearing_, because the rolling surface of the larger wheel - must in this case be turned inward or towards its centre. - - From Principle III. of S 39 it appears that the angular velocity-ratio - of a pair of wheels is the inverse ratio of the distances of the point - of contact from the centres respectively. - - [Illustration: FIG. 96.] - - For outside gearing that ratio is _negative_, because the wheels turn - contrary ways; for inside gearing it is _positive_, because they turn - the same way. - - If the velocity ratio is to be constant, as in Willis's Class A, the - wheels must be circular; and this is the most common form for wheels. - - If the velocity ratio is to be variable, as in Willis's Class B, the - figures of the wheels are a pair of _rolling curves_, subject to the - condition that the distance between their _poles_ (which are the - centres of rotation) shall be constant. - - The following is the geometrical relation which must exist between - such a pair of curves:-- - - Let C1, C2 (fig. 96) be the poles of a pair of rolling curves; T1, T2 - any pair of points of contact; U1, U2 any other pair of points of - contact. Then, for every possible pair of points of contact, the two - following equations must be simultaneously fulfilled:-- - - Sum of radii, C1U1 + C2U2 = C1T1 + C2T2 = constant; - arc, T2U2 = T1U1. (17) - - A condition equivalent to the above, and necessarily connected with - it, is, that at each pair of points of contact the inclinations of the - curves to their radii-vectores shall be equal and contrary; or, - denoting by r1, r2 the radii-vectores at any given pair of points of - contact, and s the length of the equal arcs measured from a certain - fixed pair of points of contact-- - - dr2/ds = -dr1/ds; (18) - - which is the differential equation of a pair of rolling curves whose - poles are at a constant distance apart. - - For full details as to rolling curves, see Willis's work, already - mentioned, and Clerk Maxwell's paper on Rolling Curves, _Trans. Roy. - Soc. Edin._, 1849. - - A rack, to work with a circular wheel, must be straight. To work with - a wheel of any other figure, its section must be a rolling curve, - subject to the condition that the perpendicular distance from the pole - or centre of the wheel to a straight line parallel to the direction of - the motion of the rack shall be constant. Let r1 be the radius-vector - of a point of contact on the wheel, x2 the ordinate from the straight - line before mentioned to the corresponding point of contact on the - rack. Then - - dx2/ds = -dr1/ds (19) - - is the differential equation of the pair of rolling curves. - - To illustrate this subject, it may be mentioned that an ellipse - rotating about one focus rolls completely round in outside gearing - with an equal and similar ellipse also rotating about one focus, the - distance between the axes of rotation being equal to the major axis of - the ellipses, and the velocity ratio varying from (1 + - eccentricity)/(1 - eccentricity) to (1 - eccentricity)/(1 + - eccentricity); an hyperbola rotating about its further focus rolls in - inside gearing, through a limited arc, with an equal and similar - hyperbola rotating about its nearer focus, the distance between the - axes of rotation being equal to the axis of the hyperbolas, and the - velocity ratio varying between (eccentricity + 1)/(eccentricity - 1) - and unity; and a parabola rotating about its focus rolls with an equal - and similar parabola, shifting parallel to its directrix. - - [Illustration: FIG. 97.] - - S 41. _Conical or Bevel and Disk Wheels._--From Principles III. and - VI. of S 39 it appears that the angular velocities of a pair of wheels - whose axes meet in a point are to each other inversely as the sines of - the angles which the axes of the wheels make with the line of contact. - Hence we have the following construction (figs. 97 and 98).--Let O be - the apex or point of intersection of the two axes OC1, OC2. The - angular velocity ratio being given, it is required to find the line of - contact. On OC1, OC2 take lengths OA1, OA2, respectively proportional - to the angular velocities of the pieces on whose axes they are taken. - Complete the parallelogram OA1EA2; the diagonal OET will be the line - of contact required. - - When the velocity ratio is variable, the line of contact will shift - its position in the plane C1OC2, and the wheels will be cones, with - eccentric or irregular bases. In every case which occurs in practice, - however, the velocity ratio is constant; the line of contact is - constant in position, and the rolling surfaces of the wheels are - regular circular cones (when they are called _bevel wheels_); or one - of a pair of wheels may have a flat disk for its rolling surface, as - W2 in fig. 98, in which case it is a _disk wheel_. The rolling - surfaces of actual wheels consist of frusta or zones of the complete - cones or disks, as shown by W1, W2 in figs. 97 and 98. - - [Illustration: FIG. 98.] - - S 42. _Sliding Contact (lateral): Skew-Bevel Wheels._--An hyperboloid - of revolution is a surface resembling a sheaf or a dice box, generated - by the rotation of a straight line round an axis from which it is at a - constant distance, and to which it is inclined at a constant angle. If - two such hyperboloids E, F, equal or unequal, be placed in the closest - possible contact, as in fig. 99, they will touch each other along one - of the generating straight lines of each, which will form their line - of contact, and will be inclined to the axes AG, BH in opposite - directions. The axes will not be parallel, nor will they intersect - each other. - - [Illustration: FIG. 99.] - - The motion of two such hyperboloids, turning in contact with each - other, has hitherto been classed amongst cases of rolling contact; but - that classification is not strictly correct, for, although the - component velocities of a pair of points of contact in a direction at - right angles to the line of contact are equal, still, as the axes are - parallel neither to each other nor to the line of contact, the - velocities of a pair of points of contact have components along the - line of contact which are unequal, and their difference constitutes a - _lateral sliding_. - - The directions and positions of the axes being given, and the required - angular velocity ratio, the following construction serves to determine - the line of contact, by whose rotation round the two axes respectively - the hyperboloids are generated:-- - - [Illustration: FIG. 100.] - - In fig. 100, let B1C1, B2C2 be the two axes; B1B2 their common - perpendicular. Through any point O in this common perpendicular draw - OA1 parallel to B1C1 and OA2 parallel to B2C2; make those lines - proportional to the angular velocities about the axes to which they - are respectively parallel; complete the parallelogram OA1EA2, and draw - the diagonal OE; divide B1B2 in D into two parts, _inversely_ - proportional to the angular velocities about the axes which they - respectively adjoin; through D parallel to OE draw DT. This will be - the line of contact. - - A pair of thin frusta of a pair of hyperboloids are used in practice - to communicate motion between a pair of axes neither parallel nor - intersecting, and are called _skew-bevel wheels_. - - In skew-bevel wheels the properties of a line of connexion are not - possessed by every line traversing the line of contact, but only by - every line traversing the line of contact at right angles. - - If the velocity ratio to be communicated were variable, the point D - would alter its position, and the line DT its direction, at different - periods of the motion, and the wheels would be hyperboloids of an - eccentric or irregular cross-section; but forms of this kind are not - used in practice. - - S 43. _Sliding Contact (circular): Grooved Wheels._--As the adhesion - or friction between a pair of smooth wheels is seldom sufficient to - prevent their slipping on each other, contrivances are used to - increase their mutual hold. One of those consists in forming the rim - of each wheel into a series of alternate ridges and grooves parallel - to the plane of rotation; it is applicable to cylindrical and bevel - wheels, but not to skew-bevel wheels. The comparative motion of a pair - of wheels so ridged and grooved is the same as that of a pair of - smooth wheels in rolling contact, whose cylindrical or conical - surfaces lie midway between the tops of the ridges and bottoms of the - grooves, and those ideal smooth surfaces are called the _pitch - surfaces_ of the wheels. - - The relative motion of the faces of contact of the ridges and grooves - is a _rotatory sliding_ or _grinding_ motion, about the line of - contact of the pitch-surfaces as an instantaneous axis. - - Grooved wheels have hitherto been but little used. - - S 44. _Sliding Contact (direct): Teeth of Wheels, their Number and - Pitch._--The ordinary method of connecting a pair of wheels, or a - wheel and a rack, and the only method which ensures the exact - maintenance of a given numerical velocity ratio, is by means of a - series of alternate ridges and hollows parallel or nearly parallel to - the successive lines of contact of the ideal smooth wheels whose - velocity ratio would be the same with that of the toothed wheels. The - ridges are called _teeth_; the hollows, _spaces_. The teeth of the - driver push those of the follower before them, and in so doing - sliding takes place between them in a direction across their lines of - contact. - - The _pitch-surfaces_ of a pair of toothed wheels are the ideal smooth - surfaces which would have the same comparative motion by rolling - contact that the actual wheels have by the sliding contact of their - teeth. The _pitch-circles_ of a pair of circular toothed wheels are - sections of their pitch-surfaces, made for _spur-wheels_ (that is, for - wheels whose axes are parallel) by a plane at right angles to the - axes, and for bevel wheels by a sphere described about the common - apex. For a pair of skew-bevel wheels the pitch-circles are a pair of - contiguous rectangular sections of the pitch-surfaces. The - _pitch-point_ is the point of contact of the pitch-circles. - - The pitch-surface of a wheel lies intermediate between the points of - the teeth and the bottoms of the hollows between them. That part of - the acting surface of a tooth which projects beyond the pitch-surface - is called the _face_; that part which lies within the pitch-surface, - the _flank_. - - Teeth, when not otherwise specified, are understood to be made in one - piece with the wheel, the material being generally cast-iron, brass or - bronze. Separate teeth, fixed into mortises in the rim of the wheel, - are called _cogs_. A _pinion_ is a small toothed wheel; a _trundle_ is - a pinion with cylindrical _staves_ for teeth. - - The radius of the pitch-circle of a wheel is called the _geometrical - radius_; a circle touching the ends of the teeth is called the - _addendum circle_, and its radius the _real radius_; the difference - between these radii, being the projection of the teeth beyond the - pitch-surface, is called the _addendum_. - - The distance, measured along the pitch-circle, from the face of one - tooth to the face of the next, is called the _pitch_. The pitch and - the number of teeth in wheels are regulated by the following - principles:-- - - I. In wheels which rotate continuously for one revolution or more, it - is obviously necessary _that the pitch should be an aliquot part of - the circumference_. - - In wheels which reciprocate without performing a complete revolution - this condition is not necessary. Such wheels are called _sectors_. - - II. In order that a pair of wheels, or a wheel and a rack, may work - correctly together, it is in all cases essential _that the pitch - should be the same in each_. - - III. Hence, in any pair of circular wheels which work together, the - numbers of teeth in a complete circumference are directly as the radii - and inversely as the angular velocities. - - IV. Hence also, in any pair of circular wheels which rotate - continuously for one revolution or more, the ratio of the numbers of - teeth and its reciprocal the angular velocity ratio must be - expressible in whole numbers. - - From this principle arise problems of a kind which will be referred to - in treating of _Trains of Mechanism_. - - V. Let n, N be the respective numbers of teeth in a pair of wheels, N - being the greater. Let t, T be a pair of teeth in the smaller and - larger wheel respectively, which at a particular instant work - together. It is required to find, first, how many pairs of teeth must - pass the line of contact of the pitch-surfaces before t and T work - together again (let this number be called a); and, secondly, with how - many different teeth of the larger wheel the tooth t will work at - different times (let this number be called b); thirdly, with how many - different teeth of the smaller wheel the tooth T will work at - different times (let this be called c). - - CASE 1. If n is a divisor of N, - - a = N; b = N/n; c = 1. (20) - - CASE 2. If the greatest common divisor of N and n be d, a number less - than n, so that n = md, N = Md; then - - a = mN = Mn = Mmd; b = M; c = m. (21) - - CASE 3. If N and n be prime to each other, - - a = nN; b = N; c = n. (22) - - It is considered desirable by millwrights, with a view to the - preservation of the uniformity of shape of the teeth of a pair of - wheels, that each given tooth in one wheel should work with as many - different teeth in the other wheel as possible. They therefore study - that the numbers of teeth in each pair of wheels which work together - shall either be prime to each other, or shall have their greatest - common divisor as small as is consistent with a velocity ratio suited - for the purposes of the machine. - - S 45. _Sliding Contact: Forms of the Teeth of Spur-wheels and - Racks._--A line of connexion of two pieces in sliding contact is a - line perpendicular to their surfaces at a point where they touch. - Bearing this in mind, the principle of the comparative motion of a - pair of teeth belonging to a pair of spur-wheels, or to a spur-wheel - and a rack, is found by applying the principles stated generally in SS - 36 and 37 to the case of parallel axes for a pair of spur-wheels, and - to the case of an axis perpendicular to the direction of shifting for - a wheel and a rack. - - In fig. 101, let C1, C2 be the centres of a pair of spur-wheels; - B1IB1', B2IB2' portions of their pitch-circles, touching at I, the - pitch-point. Let the wheel 1 be the driver, and the wheel 2 the - follower. - - [Illustration: FIG. 101.] - - Let D1TB1A1, D2TB2A2 be the positions, at a given instant, of the - acting surfaces of a pair of teeth in the driver and follower - respectively, touching each other at T; the line of connexion of those - teeth is P1P2, perpendicular to their surfaces at T. Let C1P1, C2P2 be - perpendiculars let fall from the centres of the wheels on the line of - contact. Then, by S 36, the angular velocity-ratio is - - [alpha]2/[alpha]1 = C1P1/C2P2. (23) - - The following principles regulate the forms of the teeth and their - relative motions:-- - - I. The angular velocity ratio due to the sliding contact of the teeth - will be the same with that due to the rolling contact of the - pitch-circles, if the line of connexion of the teeth cuts the line of - centres at the pitch-point. - - For, let P1P2 cut the line of centres at I; then, by similar - triangles, - - [alpha]1 : [alpha]2 :: C2P2 : C1P1 :: IC2 :: IC1; (24) - - which is also the angular velocity ratio due to the rolling contact of - the circles B1IB1', B2IB2'. - - This principle determines the _forms_ of all teeth of spur-wheels. It - also determines the forms of the teeth of straight racks, if one of - the centres be removed, and a straight line EIE', parallel to the - direction of motion of the rack, and perpendicular to C1IC2, be - substituted for a pitch-circle. - - II. The component of the velocity of the point of contact of the teeth - T along the line of connexion is - - [alpha]1.C1P1 = [alpha]2.C2P2. (25) - - III. The relative velocity perpendicular to P1P2 of the teeth at their - point of contact--that is, their _velocity of sliding_ on each - other--is found by supposing one of the wheels, such as 1, to be - fixed, the line of centres C1C2 to rotate backwards round C1 with the - angular velocity [alpha]1, and the wheel 2 to rotate round C2 as - before, with the angular velocity [alpha]2 relatively to the line of - centres C1C2, so as to have the same motion as if its pitch-circle - _rolled_ on the pitch-circle of the first wheel. Thus the _relative_ - motion of the wheels is unchanged; but 1 is considered as fixed, and 2 - has the total motion, that is, a rotation about the instantaneous axis - I, with the angular velocity [alpha]1 + [alpha]2. Hence the _velocity - of sliding_ is that due to this rotation about I, with the radius IT; - that is to say, its value is - - ([alpha]1 + [alpha]2).IT; (26) - - so that it is greater the farther the point of contact is from the - line of centres; and at the instant when that point passes the line of - centres, and coincides with the _pitch-point_, the velocity of sliding - is null, and the action of the teeth is, for the instant, that of - rolling contact. - - IV. The _path of contact_ is the line traversing the various positions - of the point T. If the line of connexion preserves always the same - position, the path of contact coincides with it, and is straight; in - other cases the path of contact is curved. - - It is divided by the pitch-point I into two parts--the _arc_ or _line - of approach_ described by T in approaching the line of centres, and - the _arc_ or _line of recess_ described by T after having passed the - line of centres. - - During the _approach_, the _flank_ D1B1 of the driving tooth drives - the face D2B2 of the following tooth, and the teeth are sliding - _towards_ each other. During the _recess_ (in which the position of - the teeth is exemplified in the figure by curves marked with accented - letters), the _face_ B1'A1' of the driving tooth drives the _flank_ - B2'A2' of the following tooth, and the teeth are sliding _from_ each - other. - - The path of contact is bounded where the approach commences by the - addendum-circle of the follower, and where the recess terminates by - the addendum-circle of the driver. The length of the path of contact - should be such that there shall always be at least one pair of teeth - in contact; and it is better still to make it so long that there shall - always be at least two pairs of teeth in contact. - - V. The _obliquity_ of the action of the teeth is the angle EIT = IC1, - P1 = IC2P2. - - In practice it is found desirable that the mean value of the obliquity - of action during the contact of teeth should not exceed 15 deg., nor - the maximum value 30 deg. - - It is unnecessary to give separate figures and demonstrations for - inside gearing. The only modification required in the formulae is, - that in equation (26) the _difference_ of the angular velocities - should be substituted for their sum. - - S 46. _Involute Teeth._--The simplest form of tooth which fulfils the - conditions of S 45 is obtained in the following manner (see fig. 102). - Let C1, C2 be the centres of two wheels, B1IB1', B2IB2' their - pitch-circles, I the pitch-point; let the obliquity of action of the - teeth be constant, so that the same straight line P1IP2 shall - represent at once the constant line of connexion of teeth and the path - of contact. Draw C1P1, C2P2 perpendicular to P1IP2, and with those - lines as radii describe about the centres of the wheels the circles - D1D1', D2D2', called _base-circles_. It is evident that the radii of - the base-circles bear to each other the same proportions as the radii - of the pitch-circles, and also that - - C1P1 = IC1 . cos obliquity \ (27) - C2P2 = IC2 . cos obliquity / - - (The obliquity which is found to answer best in practice is about - 14(1/2) deg.; its cosine is about 31/22, and its sine about (1/4). - These values though not absolutely exact, are near enough to the truth - for practical purposes.) - - [Illustration: FIG. 102.] - - Suppose the base-circles to be a pair of circular pulleys connected by - means of a cord whose course from pulley to pulley is P1IP2. As the - line of connexion of those pulleys is the same as that of the proposed - teeth, they will rotate with the required velocity ratio. Now, suppose - a tracing point T to be fixed to the cord, so as to be carried along - the path of contact P1IP2, that point will trace on a plane rotating - along with the wheel 1 part of the involute of the base-circle D1D1', - and on a plane rotating along with the wheel 2 part of the involute of - the base-circle D2D2'; and the two curves so traced will always touch - each other in the required point of contact T, and will therefore - fulfil the condition required by Principle I. of S 45. - - Consequently, one of the forms suitable for the teeth of wheels is the - involute of a circle; and the obliquity of the action of such teeth is - the angle whose cosine is the ratio of the radius of their base-circle - to that of the pitch-circle of the wheel. - - All involute teeth of the same pitch work smoothly together. - - To find the length of the path of contact on either side of the - pitch-point I, it is to be observed that the distance between the - fronts of two successive teeth, as measured along P1IP2, is less than - the pitch in the ratio of cos obliquity : I; and consequently that, if - distances equal to the pitch be marked off either way from I towards - P1 and P2 respectively, as the extremities of the path of contact, and - if, according to Principle IV. of S 45, the addendum-circles be - described through the points so found, there will always be at least - two pairs of teeth in action at once. In practice it is usual to make - the path of contact somewhat longer, viz. about 2.4 times the pitch; - and with this length of path, and the obliquity already mentioned of - 14(1/2) deg., the addendum is about 3.1 of the pitch. - - The teeth of a _rack_, to work correctly with wheels having involute - teeth, should have plane surfaces perpendicular to the line of - connexion, and consequently making with the direction of motion of the - rack angles equal to the complement of the obliquity of action. - - S 47. _Teeth for a given Path of Contact: Sang's Method._--In the - preceding section the form of the teeth is found by assuming a figure - for the path of contact, viz. the straight line. Any other convenient - figure may be assumed for the path of contact, and the corresponding - forms of the teeth found by determining what curves a point T, moving - along the assumed path of contact, will trace on two disks rotating - round the centres of the wheels with angular velocities bearing that - relation to the component velocity of T along TI, which is given by - Principle II. of S 45, and by equation (25). This method of finding - the forms of the teeth of wheels forms the subject of an elaborate and - most interesting treatise by Edward Sang. - - All wheels having teeth of the same pitch, traced from the same path - of contact, work correctly together, and are said to belong to the - same set. - - [Illustration: FIG. 103.] - - S 48. _Teeth traced by Rolling Curves._--If any curve R (fig. 103) be - rolled on the inside of the pitch-circle BB of a wheel, it appears, - from S 30, that the instantaneous axis of the rolling curve at any - instant will be at the point I, where it touches the pitch-circle for - the moment, and that consequently the line AT, traced by a - tracing-point T, fixed to the rolling curve upon the plane of the - wheel, will be everywhere perpendicular to the straight line TI; so - that the traced curve AT will be suitable for the flank of a tooth, in - which T is the point of contact corresponding to the position I of the - pitch-point. If the same rolling curve R, with the same tracing-point - T, be rolled on the _outside_ of any other pitch-circle, it will have - the _face_ of a tooth suitable to work with the _flank_ AT. - - In like manner, if either the same or any other rolling curve R' be - rolled the opposite way, on the _outside_ of the pitch-circle BB, so - that the tracing point T' shall start from A, it will trace the face - AT' of a tooth suitable to work with a _flank_ traced by rolling the - same curve R' with the same tracing-point T' _inside_ any other - pitch-circle. - - The figure of the _path of contact_ is that traced on a fixed plane by - the tracing-point, when the rolling curve is rotated in such a manner - as always to touch a fixed straight line EIE (or E'I'E', as the case - may be) at a fixed point I (or I'). - - If the same rolling curve and tracing-point be used to trace both the - faces and the flanks of the teeth of a number of wheels of different - sizes but of the same pitch, all those wheels will work correctly - together, and will form a _set_. The teeth of a _rack_, of the same - set, are traced by rolling the rolling curve on both sides of a - straight line. - - The teeth of wheels of any figure, as well as of circular wheels, may - be traced by rolling curves on their pitch-surfaces; and all teeth of - the same pitch, traced by the same rolling curve with the same - tracing-point, will work together correctly if their pitch-surfaces - are in rolling contact. - - [Illustration: FIG. 104.] - - S 49. _Epicycloidal Teeth._--The most convenient rolling curve is the - circle. The path of contact which it traces is identical with itself; - and the flanks of the teeth are internal and their faces external - epicycloids for wheels, and both flanks and faces are cycloids for a - rack. - - For a pitch-circle of twice the radius of the rolling or _describing_ - circle (as it is called) the internal epicycloid is a straight line, - being, in fact, a diameter of the pitch-circle, so that the flanks of - the teeth for such a pitch-circle are planes radiating from the axis. - For a smaller pitch-circle the flanks would be convex and _in-curved_ - or _under-cut_, which would be inconvenient; therefore the smallest - wheel of a set should have its pitch-circle of twice the radius of the - describing circle, so that the flanks may be either straight or - concave. - - In fig. 104 let BB' be part of the pitch-circle of a wheel with - epicycloidal teeth; CIC' the line of centres; I the pitch-point; EIE' - a straight tangent to the pitch-circle at that point; R the internal - and R' the equal external describing circles, so placed as to touch - the pitch-circle and each other at I. Let DID' be the path of contact, - consisting of the arc of approach DI and the arc of recess ID'. In - order that there may always be at least two pairs of teeth in action, - each of those arcs should be equal to the pitch. - - The obliquity of the action in passing the line of centres is nothing; - the maximum obliquity is the angle EID = E'ID; and the mean obliquity - is one-half of that angle. - - It appears from experience that the mean obliquity should not exceed - 15 deg.; therefore the maximum obliquity should be about 30 deg.; - therefore the equal arcs DI and ID' should each be one-sixth of a - circumference; therefore the circumference of the describing circle - should be _six times the pitch_. - - It follows that the smallest pinion of a set in which pinion the - flanks are straight should have twelve teeth. - - S 50. _Nearly Epicycloidal Teeth: Willis's Method._--To facilitate the - drawing of epicycloidal teeth in practice, Willis showed how to - approximate to their figure by means of two circular arcs--one - concave, for the flank, and the other convex, for the face--and each - having for its radius the _mean_ radius of curvature of the - epicycloidal arc. Willis's formulae are founded on the following - properties of epicycloids:-- - - Let R be the radius of the pitch-circle; r that of the describing - circle; [theta] the angle made by the normal TI to the epicycloid at a - given point T, with a tangent to the circle at I--that is, the - obliquity of the action at T. - - Then the radius of curvature of the epicycloid at T is-- - - R - r \ - For an internal epicycloid, [rho] = 4r sin [theta]------ | - R - 2r | - > (28) - R + r | - For an external epicycloid, [rho]' = 4r sin [theta]------ | - R + 2r / - - Also, to find the position of the centres of curvature relatively to - the pitch-circle, we have, denoting the chord of the describing circle - TI by c, c = 2r sin [theta]; and therefore - - R \ - For the flank, [rho] - c = 2r sin [theta]------ | - R - 2r | - > (29) - R | - For the face, [rho]' - c = 2r sin [theta]------ | - R + 2r / - - - For the proportions approved of by Willis, sin [theta] = (1/4) nearly; - r = p (the pitch) nearly; c = (1/2)p nearly; and, if N be the number - of teeth in the wheel, r/R = 6/N nearly; therefore, approximately, - - [rho] - c = p/2 . N/N - 12 \ (30) - [rho]' - c = p/2 . N/N + 12 / - - [Illustration: FIG. 105.] - - Hence the following construction (fig. 105). Let BB be part of the - pitch-circle, and a the point where a tooth is to cross it. Set off ab - = ac - (1/2)p. Draw radii bd, ce; draw fb, cg, making angles of - 75(1/2) deg. with those radii. Make bf = p' - c, cg = p - c. From f, - with the radius fa, draw the circular arc ah; from g, with the radius - ga, draw the circular arc ak. Then ah is the face and ak the flank of - the tooth required. - - To facilitate the application of this rule, Willis published tables of - [rho] - c and [rho]' - c, and invented an instrument called the - "odontograph." - - S 51. _Trundles and Pin-Wheels._--If a wheel or trundle have - cylindrical pins or staves for teeth, the faces of the teeth of a - wheel suitable for driving it are described by first tracing external - epicycloids, by rolling the pitch-circle of the pin-wheel or trundle - on the pitch-circle of the driving-wheel, with the centre of a stave - for a tracing-point, and then drawing curves parallel to, and within - the epicycloids, at a distance from them equal to the radius of a - stave. Trundles having only six staves will work with large wheels. - - S 52. _Backs of Teeth and Spaces._--Toothed wheels being in general - intended to rotate either way, the _backs_ of the teeth are made - similar to the fronts. The _space_ between two teeth, measured on the - pitch-circle, is made about (1/6)th part wider than the thickness of - the tooth on the pitch-circle--that is to say, - - Thickness of tooth = 5/11 pitch; - Width of space = 6/11 pitch. - - The difference of 1/11 of the pitch is called the _back-lash_. The - clearance allowed between the points of teeth and the bottoms of the - spaces between the teeth of the other wheel is about one-tenth of the - pitch. - - S 53. _Stepped and Helical Teeth._--R. J. Hooke invented the making of - the fronts of teeth in a series of steps with a view to increase the - smoothness of action. A wheel thus formed resembles in shape a series - of equal and similar toothed disks placed side by side, with the teeth - of each a little behind those of the preceding disk. He also invented, - with the same object, teeth whose fronts, instead of being parallel to - the line of contact of the pitch-circles, cross it obliquely, so as to - be of a screw-like or helical form. In wheel-work of this kind the - contact of each pair of teeth commences at the foremost end of the - helical front, and terminates at the aftermost end; and the helix is - of such a pitch that the contact of one pair of teeth shall not - terminate until that of the next pair has commenced. - - Stepped and helical teeth have the desired effect of increasing the - smoothness of motion, but they require more difficult and expensive - workmanship than common teeth; and helical teeth are, besides, open to - the objection that they exert a laterally oblique pressure, which - tends to increase resistance, and unduly strain the machinery. - - S 54. _Teeth of Bevel-Wheels._--The acting surfaces of the teeth of - bevel-wheels are of the conical kind, generated by the motion of a - line passing through the common apex of the pitch-cones, while its - extremity is carried round the outlines of the cross section of the - teeth made by a sphere described about that apex. - - [Illustration: FIG. 106.] - - The operations of describing the exact figures of the teeth of - bevel-wheels, whether by involutes or by rolling curves, are in every - respect analogous to those for describing the figures of the teeth of - spur-wheels, except that in the case of bevel-wheels all those - operations are to be performed on the surface of a sphere described - about the apex instead of on a plane, substituting _poles_ for - _centres_, and _great circles_ for _straight lines_. - - In consideration of the practical difficulty, especially in the case - of large wheels, of obtaining an accurate spherical surface, and of - drawing upon it when obtained, the following approximate method, - proposed originally by Tredgold, is generally used:-- - - Let O (fig. 106) be the common apex of a pair of bevel-wheels; OB1I, - OB2I their pitch cones; OC1, OC2 their axes; OI their line of contact. - Perpendicular to OI draw A1IA2, cutting the axes in A1, A2; make the - outer rims of the patterns and of the wheels portions of the cones - A1B1I, A2B2I, of which the narrow zones occupied by the teeth will be - sufficiently near to a spherical surface described about O for - practical purposes. To find the figures of the teeth, draw on a flat - surface circular arcs ID1, ID2, with the radii A1I, A2I; those arcs - will be the _developments_ of arcs of the pitch-circles B1I, B2I, when - the conical surfaces A1B1I, A2B2I are spread out flat. Describe the - figures of teeth for the developed arcs as for a pair of spur-wheels; - then wrap the developed arcs on the cones, so as to make them coincide - with the pitch-circles, and trace the teeth on the conical surfaces. - - S 55. _Teeth of Skew-Bevel Wheels._--The crests of the teeth of a - skew-bevel wheel are parallel to the generating straight line of the - hyperboloidal pitch-surface; and the transverse sections of the teeth - at a given pitch-circle are similar to those of the teeth of a - bevel-wheel whose pitch surface is a cone touching the hyperboloidal - surface at the given circle. - - S 56. _Cams._--A _cam_ is a single tooth, either rotating continuously - or oscillating, and driving a sliding or turning piece either - constantly or at intervals. All the principles which have been stated - in S 45 as being applicable to teeth are applicable to cams; but in - designing cams it is not usual to determine or take into consideration - the form of the ideal pitch-surface, which would give the same - comparative motion by rolling contact that the cam gives by sliding - contact. - - S 57. _Screws._--The figure of a screw is that of a convex or concave - cylinder, with one or more helical projections, called _threads_, - winding round it. Convex and concave screws are distinguished - technically by the respective names of _male_ and _female_; a short - concave screw is called a _nut_; and when a _screw_ is spoken of - without qualification a _convex_ screw is usually understood. - - The relation between the _advance_ and the _rotation_, which compose - the motion of a screw working in contact with a fixed screw or helical - guide, has already been demonstrated in S 32; and the same relation - exists between the magnitudes of the rotation of a screw about a fixed - axis and the advance of a shifting nut in which it rotates. The - advance of the nut takes place in the opposite direction to that of - the advance of the screw in the case in which the nut is fixed. The - _pitch_ or _axial pitch_ of a screw has the meaning assigned to it in - that section, viz. the distance, measured parallel to the axis, - between the corresponding points in two successive turns of the _same - thread_. If, therefore, the screw has several equidistant threads, the - true pitch is equal to the _divided axial pitch_, as measured between - two adjacent threads, multiplied by the number of threads. - - If a helix be described round the screw, crossing each turn of the - thread at right angles, the distance between two corresponding points - on two successive turns of the same thread, measured along this - _normal helix_, may be called the _normal pitch_; and when the screw - has more than one thread the normal pitch from thread to thread may be - called the _normal divided pitch_. - - The distance from thread to thread, measured on a circle described - about the axis of the screw, called the pitch-circle, may be called - the _circumferential pitch_; for a screw of one thread it is one - circumference; for a screw of n threads, (one circumference)/n. - - Let r denote the radius of the pitch circle; - n the number of threads; - [theta] the obliquity of the threads to the pitch circle, and of the - normal helix to the axis; - - P_a \ / pitch - P_a > the axial < - --- = p_a | | - n / \ divided pitch; - - P_n \ / pitch - P_n > the normal < - --- = p_n | | - n / \ divided pitch; - - P_c the circumferential pitch; - - then - - 2[pi]r \ - p_c = p_a cot [theta] = p_n cos [theta] = ------, | - n | - | - 2[pi]r tan [theta] | - p_a = p_n sec [theta] = p_c tan [theta] = ------------------, > (31) - n | - | - 2[pi]r sin [theta] | - p_n = p_c sin [theta] = p_a cos [theta] = ------------------, | - n / - - If a screw rotates, the number of threads which pass a fixed point in - one revolution is the number of threads in the screw. - - A pair of convex screws, each rotating about its axis, are used as an - elementary combination to transmit motion by the sliding contact of - their threads. Such screws are commonly called _endless screws_. At - the point of contact of the screws their threads must be parallel; and - their line of connexion is the common perpendicular to the acting - surfaces of the threads at their point of contact. Hence the following - principles:-- - - I. If the screws are both right-handed or both left-handed, the angle - between the directions of their axes is the sum of their obliquities; - if one is right-handed and the other left-handed, that angle is the - difference of their obliquities. - - II. The normal pitch for a screw of one thread, and the normal divided - pitch for a screw of more than one thread, must be the same in each - screw. - - III. The angular velocities of the screws are inversely as their - numbers of threads. - - Hooke's wheels with oblique or helical teeth are in fact screws of - many threads, and of large diameters as compared with their lengths. - - The ordinary position of a pair of endless screws is with their axes - at right angles to each other. When one is of considerably greater - diameter than the other, the larger is commonly called in practice a - _wheel_, the name _screw_ being applied to the smaller only; but they - are nevertheless both screws in fact. - - To make the teeth of a pair of endless screws fit correctly and work - smoothly, a hardened steel screw is made of the figure of the smaller - screw, with its thread or threads notched so as to form a cutting - tool; the larger screw, or "wheel," is cast approximately of the - required figure; the larger screw and the steel screw are fitted up in - their proper relative position, and made to rotate in contact with - each other by turning the steel screw, which cuts the threads of the - larger screw to their true figure. - - [Illustration: FIG. 107.] - - S 58. _Coupling of Parallel Axes--Oldham's Coupling._--A _coupling_ is - a mode of connecting a pair of shafts so that they shall rotate in the - same direction with the same mean angular velocity. If the axes of the - shafts are in the same straight line, the coupling consists in so - connecting their contiguous ends that they shall rotate as one piece; - but if the axes are not in the same straight line combinations of - mechanism are required. A coupling for parallel shafts which acts by - _sliding contact_ was invented by Oldham, and is represented in fig. - 107. C1, C2 are the axes of the two parallel shafts; D1, D2 two disks - facing each other, fixed on the ends of the two shafts respectively; - E1E1 a bar sliding in a diametral groove in the face of D1; E2E2 a bar - sliding in a diametral groove in the face of D2: those bars are fixed - together at A, so as to form a rigid cross. The angular velocities of - the two disks and of the cross are all equal at every instant; the - middle point of the cross, at A, revolves in the dotted circle - described upon the line of centres C1C2 as a diameter twice for each - turn of the disks and cross; the instantaneous axis of rotation of the - cross at any instant is at I, the point in the circle C1C2 - diametrically opposite to A. - - Oldham's coupling may be used with advantage where the axes of the - shafts are intended to be as nearly in the same straight line as is - possible, but where there is some doubt as to the practibility or - permanency of their exact continuity. - - S 59. _Wrapping Connectors--Belts, Cords and Chains._--Flat belts of - leather or of gutta percha, round cords of catgut, hemp or other - material, and metal chains are used as wrapping connectors to transmit - rotatory motion between pairs of pulleys and drums. - - _Belts_ (the most frequently used of all wrapping connectors) require - nearly cylindrical pulleys. A belt tends to move towards that part of - a pulley whose radius is greatest; pulleys for belts, therefore, are - slightly swelled in the middle, in order that the belt may remain on - the pulley, unless forcibly shifted. A belt when in motion is shifted - off a pulley, or from one pulley on to another of equal size alongside - of it, by pressing against that part of the belt which is moving - _towards_ the pulley. - - _Cords_ require either cylindrical drums with ledges or grooved - pulleys. - - _Chains_ require pulleys or drums, grooved, notched and toothed, so as - to fit the links of the chain. - - Wrapping connectors for communicating continuous motion are endless. - - Wrapping connectors for communicating reciprocating motion have - usually their ends made fast to the pulleys or drums which they - connect, and which in this case may be sectors. - - [Illustration: FIG. 108.] - - The line of connexion of two pieces connected by a wrapping connector - is the centre line of the belt, cord or chain; and the comparative - motions of the pieces are determined by the principles of S 36 if both - pieces turn, and of S 37 if one turns and the other shifts, in which - latter case the motion must be reciprocating. - - The _pitch-line_ of a pulley or drum is a curve to which the line of - connexion is always a tangent--that is to say, it is a curve parallel - to the acting surface of the pulley or drum, and distant from it by - half the thickness of the wrapping connector. - - Pulleys and drums for communicating a constant velocity ratio are - circular. The _effective radius_, or radius of the pitch-circle of a - circular pulley or drum, is equal to the real radius added to half the - thickness of the connector. The angular velocities of a pair of - connected circular pulleys or drums are inversely as the effective - radii. - - A _crossed_ belt, as in fig. 108, A, reverses the direction of the - rotation communicated; an _uncrossed_ belt, as in fig. 108, B, - preserves that direction. - - The _length_ L of an endless belt connecting a pair of pulleys whose - effective radii are r1, r2, with parallel axes whose distance apart is - c, is given by the following formulae, in each of which the first - term, containing the radical, expresses the length of the straight - parts of the belt, and the remainder of the formula the length of the - curved parts. - - For a crossed belt:-- - - / r1 + r2 \ - L = 2[root][c^2 - (r1 + r2)^2] + (r1 + r2)( [pi] - 2 sin^-1 ------- ); (32 A) - \ c / - and for an uncrossed belt:-- - - r1 - r2 - L = 2[root][c^2 - (r1 - r2)^2] + [pi](r1 + r2 + 2(r1 - r2) sin^-1 -------; (32 B) - c - in which r1 is the greater radius, and r2 the less. - - When the axes of a pair of pulleys are not parallel, the pulleys - should be so placed that the part of the belt which is _approaching_ - each pulley shall be in the plane of the pulley. - - S 60. _Speed-Cones._--A pair of speed-cones (fig. 109) is a - contrivance for varying and adjusting the velocity ratio communicated - between a pair of parallel shafts by means of a belt. The speed-cones - are either continuous cones or conoids, as A, B, whose velocity ratio - can be varied gradually while they are in motion by shifting the belt, - or sets of pulleys whose radii vary by steps, as C, D, in which case - the velocity ratio can be changed by shifting the belt from one pair - of pulleys to another. - - [Illustration: FIG. 109.] - - In order that the belt may fit accurately in every possible position - on a pair of speed-cones, the quantity L must be constant, in - equations (32 A) or (32 B), according as the belt is crossed or - uncrossed. - - For a _crossed_ belt, as in A and C, fig. 109, L depends solely on c - and on r1 + r2. Now c is constant because the axes are parallel; - therefore the _sum of the radii_ of the pitch-circles connected in - every position of the belt is to be constant. That condition is - fulfilled by a pair of continuous cones generated by the revolution of - two straight lines inclined opposite ways to their respective axes at - equal angles. - - For an uncrossed belt, the quantity L in equation (32 B) is to be made - constant. The exact fulfilment of this condition requires the solution - of a transcendental equation; but it may be fulfilled with accuracy - sufficient for practical purposes by using, instead of (32 B) the - following _approximate_ equation:-- - - L nearly = 2c + [pi](r1 + r2) + (r1 - r2)^2/c. (33) - - The following is the most convenient practical rule for the - application of this equation:-- - - Let the speed-cones be equal and similar conoids, as in B, fig. 109, - but with their large and small ends turned opposite ways. Let r1 be - the radius of the large end of each, r2 that of the small end, r0 that - of the middle; and let v be the _sagitta_, measured perpendicular to - the axes, of the arc by whose revolution each of the conoids is - generated, or, in other words, the _bulging_ of the conoids in the - middle of their length. Then - - v = r0 - (r1 + r2)/2 = (r1 - r2)^2/2[pi]c. (34) - - 2[pi] = 6.2832; but 6 may be used in most practical cases without - sensible error. - - The radii at the middle and end being thus determined, make the - generating curve an arc either of a circle or of a parabola. - - S 61. _Linkwork in General._--The pieces which are connected by - linkwork, if they rotate or oscillate, are usually called _cranks_, - _beams_ and levers. The _link_ by which they are connected is a rigid - rod or bar, which may be straight or of any other figure; the straight - figure being the most favourable to strength, is always used when - there is no special reason to the contrary. The link is known by - various names in various circumstances, such as _coupling-rod_, - _connecting-rod_, _crank-rod_, _eccentric-rod_, &c. It is attached to - the pieces which it connects by two pins, about which it is free to - turn. The effect of the link is to maintain the distance between the - axes of those pins invariable; hence the common perpendicular of the - axes of the pins is _the line of connexion_, and its extremities may - be called the _connected points_. In a turning piece, the - perpendicular let fall from its connected point upon its axis of - rotation is the _arm_ or _crank-arm_. - - The axes of rotation of a pair of turning pieces connected by a link - are almost always parallel, and perpendicular to the line of connexion - in which case the angular velocity ratio at any instant is the - reciprocal of the ratio of the common perpendiculars let fall from the - line of connexion upon the respective axes of rotation. - - If at any instant the direction of one of the crank-arms coincides - with the line of connexion, the common perpendicular of the line of - connexion and the axis of that crank-arm vanishes, and the directional - relation of the motions becomes indeterminate. The position of the - connected point of the crank-arm in question at such an instant is - called a _dead-point_. The velocity of the other connected point at - such an instant is null, unless it also reaches a dead-point at the - same instant, so that the line of connexion is in the plane of the two - axes of rotation, in which case the velocity ratio is indeterminate. - Examples of dead-points, and of the means of preventing the - inconvenience which they tend to occasion, will appear in the sequel. - - S 62. _Coupling of Parallel Axes._--Two or more parallel shafts (such - as those of a locomotive engine, with two or more pairs of driving - wheels) are made to rotate with constantly equal angular velocities by - having equal cranks, which are maintained parallel by a coupling-rod - of such a length that the line of connexion is equal to the distance - between the axes. The cranks pass their dead-points simultaneously. To - obviate the unsteadiness of motion which this tends to cause, the - shafts are provided with a second set of cranks at right angles to the - first, connected by means of a similar coupling-rod, so that one set - of cranks pass their dead points at the instant when the other set are - farthest from theirs. - - S 63. _Comparative Motion of Connected Points._--As the link is a - rigid body, it is obvious that its action in communicating motion may - be determined by finding the comparative motion of the connected - points, and this is often the most convenient method of proceeding. - - If a connected point belongs to a turning piece, the direction of its - motion at a given instant is perpendicular to the plane containing the - axis and crank-arm of the piece. If a connected point belongs to a - shifting piece, the direction of its motion at any instant is given, - and a plane can be drawn perpendicular to that direction. - - The line of intersection of the planes perpendicular to the paths of - the two connected points at a given instant is the _instantaneous axis - of the link_ at that instant; and the _velocities of the connected - points are directly as their distances from that axis_. - - [Illustration: FIG. 110.] - - In drawing on a plane surface, the two planes perpendicular to the - paths of the connected points are represented by two lines (being - their sections by a plane normal to them), and the instantaneous axis - by a point (fig. 110); and, should the length of the two lines render - it impracticable to produce them until they actually intersect, the - velocity ratio of the connected points may be found by the principle - that it is equal to the ratio of the segments which a line parallel to - the line of connexion cuts off from any two lines drawn from a given - point, perpendicular respectively to the paths of the connected - points. - - To illustrate this by one example. Let C1 be the axis, and T1 the - connected point of the beam of a steam-engine; T1T2 the connecting or - crank-rod; T2 the other connected point, and the centre of the - crank-pin; C2 the axis of the crank and its shaft. Let v1 denote the - velocity of T1 at any given instant; v2 that of T2. To find the ratio - of these velocities, produce C1T1, C2T2 till they intersect in K; K is - the instantaneous axis of the connecting rod, and the velocity ratio - is - - v1 : v2 :: KT1 : KT2. (35) - - Should K be inconveniently far off, draw any triangle with its sides - respectively parallel to C1T1, C2T2 and T1T2; the ratio of the two - sides first mentioned will be the velocity ratio required. For - example, draw C2A parallel to C1T1, cutting T1T2 in A; then - - v1 : v2 :: C2A : C2T2. (36) - - S 64. _Eccentric._--An eccentric circular disk fixed on a shaft, and - used to give a reciprocating motion to a rod, is in effect a crank-pin - of sufficiently large diameter to surround the shaft, and so to avoid - the weakening of the shaft which would arise from bending it so as to - form an ordinary crank. The centre of the eccentric is its connected - point; and its eccentricity, or the distance from that centre to the - axis of the shaft, is its crank-arm. - - An eccentric may be made capable of having its eccentricity altered by - means of an adjusting screw, so as to vary the extent of the - reciprocating motion which it communicates. - - S 65. _Reciprocating Pieces--Stroke--Dead-Points._--The distance - between the extremities of the path of the connected point in a - reciprocating piece (such as the piston of a steam-engine) is called - the _stroke_ or _length of stroke_ of that piece. When it is connected - with a continuously turning piece (such as the crank of a - steam-engine) the ends of the stroke of the reciprocating piece - correspond to the _dead-points_ of the path of the connected point of - the turning piece, where the line of connexion is continuous with or - coincides with the crank-arm. - - Let S be the length of stroke of the reciprocating piece, L the length - of the line of connexion, and R the crank-arm of the continuously - turning piece. Then, if the two ends of the stroke be in one straight - line with the axis of the crank, - - S = 2R; (37) - - and if these ends be not in one straight line with that axis, then S, - L - R, and L + R, are the three sides of a triangle, having the angle - opposite S at that axis; so that, if [theta] be the supplement of the - arc between the dead-points, - - S^2 = 2(L^2 + R^2) - 2(L^2 - R^2) cos [theta], \ - | - 2L^2 + 2R^2 - S^2 > (38) - cos [theta] = ----------------- | - 2(L^2 - R^2) / - - [Illustration: FIG. 111.] - - S 66. _Coupling of Intersecting Axes--Hooke's Universal - Joint._--Intersecting axes are coupled by a contrivance of Hooke's, - known as the "universal joint," which belongs to the class of linkwork - (see fig. 111). Let O be the point of intersection of the axes OC1, - OC2, and [theta] their angle of inclination to each other. The pair of - shafts C1, C2 terminate in a pair of forks F1, F2 in bearings at the - extremities of which turn the gudgeons at the ends of the arms of a - rectangular cross, having its centre at O. This cross is the link; the - connected points are the centres of the bearings F1, F2. At each - instant each of those points moves at right angles to the central - plane of its shaft and fork, therefore the line of intersection of the - central planes of the two forks at any instant is the instantaneous - axis of the cross, and the _velocity ratio_ of the points F1, F2 - (which, as the forks are equal, is also the _angular velocity ratio_ - of the shafts) is equal to the ratio of the distances of those points - from that instantaneous axis. The _mean_ value of that velocity ratio - is that of equality, for each successive _quarter-turn_ is made by - both shafts in the same time; but its actual value fluctuates between - the limits:-- - - [alpha]2 1 \ - -------- = ----------- when F1 is the plane of OC1C2 | - [alpha]1 cos [theta] | - > (39) - [alpha]2 | - and -------- = cos [theta] when F2 is in that plane. | - [alpha]1 / - - Its value at intermediate instants is given by the following - equations: let [phi]1, [phi]2 be the angles respectively made by the - central planes of the forks and shafts with the plane OC1C2 at a given - instant; then - - cos [theta] = tan [phi]1 tan [phi]2, \ - | - [alpha]2 d[phi]2 tan [phi]1 + cot [phi]1 > (40) - --------- = - ------- = -----------------------. | - [alpha]1 d[phi]1 tan [phi]2 + cot [phi]2 / - - S 67. _Intermittent Linkwork--Click and Ratchet._--A click acting upon - a ratchet-wheel or rack, which it pushes or pulls through a certain - arc at each forward stroke and leaves at rest at each backward stroke, - is an example of intermittent linkwork. During the forward stroke the - action of the click is governed by the principles of linkwork; during - the backward stroke that action ceases. A _catch_ or _pall_, turning - on a fixed axis, prevents the ratchet-wheel or rack from reversing its - motion. - - - _Division 5.--Trains of Mechanism._ - - S 68. _General Principles.--A train of mechanism_ consists of a series - of pieces each of which is follower to that which drives it and driver - to that which follows it. - - The comparative motion of the first driver and last follower is - obtained by combining the proportions expressing by their terms the - velocity ratios and by their signs the directional relations of the - several elementary combinations of which the train consists. - - S 69. _Trains of Wheelwork._--Let A1, A2, A3, &c., A_(m-1), A_m denote - a series of axes, and [alpha]1, [alpha]2, [alpha]3, &c., - [alpha]_(m-1), [alpha]_m their angular velocities. Let the axis A1 - carry a wheel of N1 teeth, driving a wheel of n2 teeth on the axis A2, - which carries also a wheel of N2 teeth, driving a wheel of n3 teeth on - the axis A3, and so on; the numbers of teeth in drivers being denoted - by N's, and in followers by n's, and the axes to which the wheels are - fixed being denoted by numbers. Then the resulting velocity ratio is - denoted by - - [alpha]_m [alpha]2 [alpha]3 [alpha]_m N1 . N2 ... &c. ... N_(m-1) - --------- = -------- . -------- . &c. ... ------------- = ---------------------------; (41) - [alpha]1 [alpha]1 [alpha]2 [alpha]_(m-1) n2 . n3 ... &c. ... n_m - - that is to say, the velocity ratio of the last and first axes is the - ratio of the product of the numbers of teeth in the drivers to the - product of the numbers of teeth in the followers. - - Supposing all the wheels to be in outside gearing, then, as each - elementary combination reverses the direction of rotation, and as the - number of elementary combinations m - 1 is one less than the number - of axes m, it is evident that if m is odd the direction of rotation is - preserved, and if even reversed. - - It is often a question of importance to determine the number of teeth - in a train of wheels best suited for giving a determinate velocity - ratio to two axes. It was shown by Young that, to do this with the - _least total number of teeth_, the velocity ratio of each elementary - combination should approximate as nearly as possible to 3.59. This - would in many cases give too many axes; and, as a useful practical - rule, it may be laid down that from 3 to 6 ought to be the limit of - the velocity ratio of an elementary combination in wheel-work. The - smallest number of teeth in a pinion for epicycloidal teeth ought to - be _twelve_ (see S 49)--but it is better, for smoothness of motion, - not to go below _fifteen_; and for involute teeth the smallest number - is about _twenty-four_. - - Let B/C be the velocity ratio required, reduced to its least terms, - and let B be greater than C. If B/C is not greater than 6, and C lies - between the prescribed minimum number of teeth (which may be called t) - and its double 2t, then one pair of wheels will answer the purpose, - and B and C will themselves be the numbers required. Should B and C be - inconveniently large, they are, if possible, to be resolved into - factors, and those factors (or if they are too small, multiples of - them) used for the number of teeth. Should B or C, or both, be at once - inconveniently large and prime, then, instead of the exact ratio B/C - some ratio approximating to that ratio, and capable of resolution into - convenient factors, is to be found by the method of continued - fractions. - - Should B/C be greater than 6, the best number of elementary - combinations m - 1 will lie between - - (log B - log C) log B - log C - --------------- and -------------. - log 6 log 3 - - Then, if possible, B and C themselves are to be resolved each into m - - 1 factors (counting 1 as a factor), which factors, or multiples of - them, shall be not less than t nor greater than 6t; or if B and C - contain inconveniently large prime factors, an approximate velocity - ratio, found by the method of continued fractions, is to be - substituted for B/C as before. - - So far as the resultant velocity ratio is concerned, the _order_ of - the drivers N and of the followers n is immaterial: but to secure - equable wear of the teeth, as explained in S 44, the wheels ought to - be so arranged that, for each elementary combination, the greatest - common divisor of N and n shall be either 1, or as small as possible. - - S 70. _Double Hooke's Coupling._--It has been shown in S 66 that the - velocity ratio of a pair of shafts coupled by a universal joint - fluctuates between the limits cos [theta] and 1/cos [theta]. Hence one - or both of the shafts must have a vibratory and unsteady motion, - injurious to the mechanism and framework. To obviate this evil a short - intermediate shaft is introduced, making equal angles with the first - and last shaft, coupled with each of them by a Hooke's joint, and - having its own two forks in the same plane. Let [alpha]1, [alpha]2, - [alpha]3 be the angular velocities of the first, intermediate, and - last shaft in this _train of two Hooke's couplings_. Then, from the - principles of S 60 it is evident that at each instant - [alpha]2/[alpha]1 = [alpha]2/[alpha]3, and consequently that [alpha]3 - = [alpha]1; so that the fluctuations of angular velocity ratio caused - by the first coupling are exactly neutralized by the second, and the - first and last shafts have equal angular velocities at each instant. - - S 71. _Converging and Diverging Trains of Mechanism._--Two or more - trains of mechanism may converge into one--as when the two pistons of - a pair of steam-engines, each through its own connecting-rod, act upon - one crank-shaft. One train of mechanism may _diverge_ into two or - more--as when a single shaft, driven by a prime mover, carries several - pulleys, each of which drives a different machine. The principles of - comparative motion in such converging and diverging trains are the - same as in simple trains. - - - _Division 6.--Aggregate Combinations._ - - S 72. _General Principles._--Willis designated as "aggregate - combinations" those assemblages of pieces of mechanism in which the - motion of one follower is the _resultant_ of component motions - impressed on it by more than one driver. Two classes of aggregate - combinations may be distinguished which, though not different in their - actual nature, differ in the _data_ which they present to the - designer, and in the method of solution to be followed in questions - respecting them. - - Class I. comprises those cases in which a piece A is not carried - directly by the frame C, but by another piece B, _relatively_ to which - the motion of A is given--the motion of the piece B relatively to the - frame C being also given. Then the motion of A relatively to the frame - C is the _resultant_ of the motion of A relatively to B and of B - relatively to C; and that resultant is to be found by the principles - already explained in Division 3 of this Chapter SS 27-32. - - Class II. comprises those cases in which the motions of three points - in one follower are determined by their connexions with two or with - three different drivers. - - This classification is founded on the kinds of problems arising from - the combinations. Willis adopts another classification founded on the - _objects_ of the combinations, which objects he divides into two - classes, viz. (1) to produce _aggregate velocity_, or a velocity which - is the resultant of two or more components in the same path, and (2) - to produce _an aggregate path_--that is, to make a given point in a - rigid body move in an assigned path by communicating certain motions - to other points in that body. - - It is seldom that one of these effects is produced without at the same - time producing the other; but the classification of Willis depends - upon which of those two effects, even supposing them to occur - together, is the practical object of the mechanism. - - [Illustration: FIG. 112.] - - S 73. _Differential Windlass._--The axis C (fig. 112) carries a larger - barrel AE and a smaller barrel DB, rotating as one piece with the - angular velocity [alpha]1 in the direction AE. The pulley or _sheave_ - FG has a weight W hung to its centre. A cord has one end made fast to - and wrapped round the barrel AE; it passes from A under the sheave FG, - and has the other end wrapped round and made fast to the barrel BD. - Required the relation between the velocity of translation v2 of W and - the angular velocity [alpha]1 of the _differential barrel_. - - In this case v2 is an _aggregate velocity_, produced by the joint - action of the two drivers AE and BD, transmitted by wrapping - connectors to FG, and combined by that sheave so as to act on the - follower W, whose motion is the same with that of the centre of FG. - - The velocity of the point F is [alpha]1.AC, _upward_ motion being - considered positive. The velocity of the point G is -[alpha]1.CB, - _downward_ motion being negative. Hence the instantaneous axis of the - sheave FG is in the diameter FG, at the distance - - FG AC - BC - --- . ------- - 2 AC + BC - - from the centre towards G; the angular velocity of the sheave is - - AC + BC - [alpha]2 = [alpha]1 . -------; - FG - - and, consequently, the velocity of its centre is - - FG AC - BC [alpha]1(AC - BC) - v2 = [alpha]2 . --- . ------- = -----------------, (42) - 2 AC + BC 2 - - or the _mean between the velocities of the two vertical parts of the - cord_. - - If the cord be fixed to the framework at the point B, instead of being - wound on a barrel, the velocity of W is half that of AF. - - A case containing several sheaves is called a _block_. A _fall-block_ - is attached to a fixed point; a _running-block_ is movable to and from - a fall-block, with which it is connected by two or more plies of a - rope. The whole combination constitutes a _tackle_ or _purchase_. (See - PULLEYS for practical applications of these principles.) - - S 74. _Differential Screw._--On the same axis let there be two screws - of the respective pitches p1 and p2, made in one piece, and rotating - with the angular velocity [alpha]. Let this piece be called B. Let the - first screw turn in a fixed nut C, and the second in a sliding nut A. - The velocity of advance of B relatively to C is (according to S 32) - [alpha]p1, and of A relatively to B (according to S 57) -[alpha]p2; - hence the velocity of A relatively to C is - - [alpha](p1 - p2), (46) - - being the same with the velocity of advance of a screw of the pitch p1 - - p2. This combination, called _Hunter's_ or the _differential screw_, - combines the strength of a large thread with the slowness of motion - due to a small one. - - S 75. _Epicyclic Trains._--The term _epicyclic train_ is used by - Willis to denote a train of wheels carried by an arm, and having - certain rotations relatively to that arm, which itself rotates. The - arm may either be driven by the wheels or assist in driving them. The - comparative motions of the wheels and of the arm, and the _aggregate - paths_ traced by points in the wheels, are determined by the - principles of the composition of rotations, and of the description of - rolling curves, explained in SS 30, 31. - - S 76. _Link Motion._--A slide valve operated by a link motion receives - an aggregate motion from the mechanism driving it. (See STEAM-ENGINE - for a description of this and other types of mechanism of this class.) - - [Illustration: FIG. 113.] - - S 77. _Parallel Motions._--A _parallel motion_ is a combination of - turning pieces in mechanism designed to guide the motion of a - reciprocating piece either exactly or approximately in a straight - line, so as to avoid the friction which arises from the use of - straight guides for that purpose. - - Fig. 113 represents an exact parallel motion, first proposed, it is - believed, by Scott Russell. The arm CD turns on the axis C, and is - jointed at D to the middle of the bar ADB, whose length is double of - that of CD, and one of whose ends B is jointed to a slider, sliding in - straight guides along the line CB. Draw BE perpendicular to CB, - cutting CD produced in E, then E is the instantaneous axis of the bar - ADB; and the direction of motion of A is at every instant - perpendicular to EA--that is, along the straight line ACa. While the - stroke of A is ACa, extending to equal distances on either side of C, - and equal to twice the chord of the arc Dd, the stroke of B is only - equal to twice the sagitta; and thus A is guided through a - comparatively long stroke by the sliding of B through a comparatively - short stroke, and by rotatory motions at the joints C, D, B. - - [Illustration: FIG. 114.] - - [Illustration: FIG. 115.] - - S 78.* An example of an approximate straight-line motion composed of - three bars fixed to a frame is shown in fig. 114. It is due to P. L. - Tchebichev of St Petersburg. The links AB and CD are equal in length - and are centred respectively at A and C. The ends D and B are joined - by a link DB. If the respective lengths are made in the proportions AC - : CD : DB = 1 : 1.3 : 0.4 the middle point P of DB will describe an - approximately straight line parallel to AC within limits of length - about equal to AC. C. N. Peaucellier, a French engineer officer, was - the first, in 1864, to invent a linkwork with which an exact straight - line could be drawn. The linkwork is shown in fig. 115, from which it - will be seen that it consists of a rhombus of four equal bars ABCD, - jointed at opposite corners with two equal bars BE and DE. The seventh - link AF is equal in length to halt the distance EA when the mechanism - is in its central position. The points E and F are fixed. It can be - proved that the point C always moves in a straight line at right - angles to the line EF. The more general property of the mechanism - corresponding to proportions between the lengths FA and EF other than - that of equality is that the curve described by the point C is the - inverse of the curve described by A. There are other arrangements of - bars giving straight-line motions, and these arrangements together - with the general properties of mechanisms of this kind are discussed - in _How to Draw a Straight Line_ by A. B. Kempe (London, 1877). - - [Illustration: FIG. 116.] - - [Illustration: FIG. 117.] - - S 79.* _The Pantograph._--If a parallelogram of links (fig. 116), be - fixed at any one point a in any one of the links produced in either - direction, and if any straight line be drawn from this point to cut - the links in the points b and c, then the points a, b, c will be in a - straight line for all positions of the mechanism, and if the point b - be guided in any curve whatever, the point c will trace a similar - curve to a scale enlarged in the ratio ab : ac. This property of the - parallelogram is utilized in the construction of the pantograph, an - instrument used for obtaining a copy of a map or drawing on a - different scale. Professor J. J. Sylvester discovered that this - property of the parallelogram is not confined to points lying in one - line with the fixed point. Thus if b (fig. 117) be any point on the - link CD, and if a point c be taken on the link DE such that the - triangles CbD and DcE are similar and similarly situated with regard - to their respective links, then the ratio of the distances ab and ac - is constant, and the angle bac is constant for all positions of the - mechanism; so that, if b is guided in any curve, the point c will - describe a similar curve turned through an angle bac, the scales of - the curves being in the ratio ab to ac. Sylvester called an instrument - based on this property a plagiograph or a skew pantograph. - - The combination of the parallelogram with a straight-line motion, for - guiding one of the points in a straight line, is illustrated in Watt's - parallel motion for steam-engines. (See STEAM-ENGINE.) - - S 80.* _The Reuleaux System of Analysis._--If two pieces, A and B, - (fig. 118) are jointed together by a pin, the pin being fixed, say, to - A, the only relative motion possible between the pieces is one of - turning about the axis of the pin. Whatever motion the pair of pieces - may have as a whole each separate piece shares in common, and this - common motion in no way affects the relative motion of A and B. The - motion of one piece is said to be completely constrained relatively to - the other piece. Again, the pieces A and B (fig. 119) are paired - together as a slide, and the only relative motion possible between - them now is that of sliding, and therefore the motion of one - relatively to the other is completely constrained. The pieces may be - paired together as a screw and nut, in which case the relative motion - is compounded of turning with sliding. - - [Illustration: FIG. 118.] - - [Illustration: FIG. 119.] - - These combinations of pieces are known individually as _kinematic - pairs of elements_, or briefly _kinematic pairs_. The three pairs - mentioned above have each the peculiarity that contact between the two - pieces forming the pair is distributed over a surface. Kinematic pairs - which have surface contact are classified as _lower pairs_. Kinematic - pairs in which contact takes place along a line only are classified as - _higher pairs_. A pair of spur wheels in gear is an example of a - higher pair, because the wheels have contact between their teeth along - lines only. - - A _kinematic link_ of the simplest form is made by joining up the - halves of two kinematic pairs by means of a rigid link. Thus if A1B1 - represent a turning pair, and A2B2 a second turning pair, the rigid - link formed by joining B1 to B2 is a kinematic link. Four links of - this kind are shown in fig. 120 joined up to form a _closed kinematic - chain_. - - [Illustration: FIG. 120.] - - In order that a kinematic chain may be made the basis of a mechanism, - every point in any link of it must be completely constrained with - regard to every other link. Thus in fig. 120 the motion of a point a - in the link A1A2 is completely constrained with regard to the link - B1B4 by the turning pair A1B1, and it can be proved that the motion of - a relatively to the non-adjacent link A3A4 is completely constrained, - and therefore the four-bar chain, as it is called, can be and is used - as the basis of many mechanisms. Another way of considering the - question of constraint is to imagine any one link of the chain fixed; - then, however the chain be moved, the path of a point, as a, will - always remain the same. In a five-bar chain, if a is a point in a link - non-adjacent to a fixed link, its path is indeterminate. Still another - way of stating the matter is to say that, if any one link in the chain - be fixed, any point in the chain must have only one degree of freedom. - In a five-bar chain a point, as a, in a link non-adjacent to the fixed - link has two degrees of freedom and the chain cannot therefore be used - for a mechanism. These principles may be applied to examine any - possible combination of links forming a kinematic chain in order to - test its suitability for use as a mechanism. Compound chains are - formed by the superposition of two or more simple chains, and in these - more complex chains links will be found carrying three, or even more, - halves of kinematic pairs. The Joy valve gear mechanism is a good - example of a compound kinematic chain. - - [Illustration: FIG. 121.] - - A chain built up of three turning pairs and one sliding pair, and - known as the _slider crank chain_, is shown in fig. 121. It will be - seen that the piece A1 can only slide relatively to the piece B1, and - these two pieces therefore form the sliding pair. The piece A1 carries - the pin B4, which is one half of the turning pair A4 B4. The piece A1 - together with the pin B4 therefore form a kinematic link A1B4. The - other links of the chain are, B1A2, B2B3, A3A4. In order to convert a - chain into a mechanism it is necessary to fix one link in it. Any one - of the links may be fixed. It follows therefore that there are as many - possible mechanisms as there are links in the chain. For example, - there is a well-known mechanism corresponding to the fixing of three - of the four links of the slider crank chain (fig. 121). If the link d - is fixed the chain at once becomes the mechanism of the ordinary steam - engine; if the link e is fixed the mechanism obtained is that of the - oscillating cylinder steam engine; if the link c is fixed the - mechanism becomes either the Whitworth quick-return motion or the - slot-bar motion, depending upon the proportion between the lengths of - the links c and e. These different mechanisms are called _inversions_ - of the slider crank chain. What was the fixed framework of the - mechanism in one case becomes a moving link in an inversion. - - The Reuleaux system, therefore, consists essentially of the analysis - of every mechanism into a kinematic chain, and since each link of the - chain may be the fixed frame of a mechanism quite diverse mechanisms - are found to be merely inversions of the same kinematic chain. Franz - Reuleaux's _Kinematics of Machinery_, translated by Sir A. B. W. - Kennedy (London, 1876), is the book in which the system is set forth - in all its completeness. In _Mechanics of Machinery_, by Sir A. B. W. - Kennedy (London, 1886), the system was used for the first time in an - English textbook, and now it has found its way into most modern - textbooks relating to the subject of mechanism. - - S 81.* _Centrodes, Instantaneous Centres, Velocity Image, Velocity - Diagram._--Problems concerning the relative motion of the several - parts of a kinematic chain may be considered in two ways, in addition - to the way hitherto used in this article and based on the principle of - S 34. The first is by the method of instantaneous centres, already - exemplified in S 63, and rolling centroids, developed by Reuleaux in - connexion with his method of analysis. The second is by means of - Professor R. H. Smith's method already referred to in S 23. - - _Method 1._--By reference to S 30 it will be seen that the motion of a - cylinder rolling on a fixed cylinder is one of rotation about an - instantaneous axis T, and that the velocity both as regards direction - and magnitude is the same as if the rolling piece B were for the - instant turning about a fixed axis coincident with the instantaneous - axis. If the rolling cylinder B and its path A now be assumed to - receive a common plane motion, what was before the velocity of the - point P becomes the velocity of P relatively to the cylinder A, since - the motion of B relatively to A still takes place about the - instantaneous axis T. If B stops rolling, then the two cylinders - continue to move as though they were parts of a rigid body. Notice - that the shape of either rolling curve (fig. 91 or 92) may be found by - considering each fixed in turn and then tracing out the locus of the - instantaneous axis. These rolling cylinders are sometimes called - axodes, and a section of an axode in a plane parallel to the plane of - motion is called a centrode. The axode is hence the locus of the - instantaneous axis, whilst the centrode is the locus of the - instantaneous centre in any plane parallel to the plane of motion. - There is no restriction on the shape of these rolling axodes; they may - have any shape consistent with rolling (that is, no slipping is - permitted), and the relative velocity of a point P is still found by - considering it with regard to the instantaneous centre. - - Reuleaux has shown that the relative motion of any pair of - non-adjacent links of a kinematic chain is determined by the rolling - together of two ideal cylindrical surfaces (cylindrical being used - here in the general sense), each of which may be assumed to be formed - by the extension of the material of the link to which it corresponds. - These surfaces have contact at the instantaneous axis, which is now - called the instantaneous axis of the two links concerned. To find the - form of these surfaces corresponding to a particular pair of - non-adjacent links, consider each link of the pair fixed in turn, then - the locus of the instantaneous axis is the axode corresponding to the - fixed link, or, considering a plane of motion only, the locus of the - instantaneous centre is the centrode corresponding to the fixed link. - - To find the instantaneous centre for a particular link corresponding - to any given configuration of the kinematic chain, it is only - necessary to know the direction of motion of any two points in the - link, since lines through these points respectively at right angles to - their directions of motion intersect in the instantaneous centre. - - [Illustration: FIG. 122.] - - To illustrate this principle, consider the four-bar chain shown in - fig. 122 made up of the four links, a, b, c, d. Let a be the fixed - link, and consider the link c. Its extremities are moving respectively - in directions at right angles to the links b and d; hence produce the - links b and d to meet in the point O_(ac). This point is the - instantaneous centre of the motion of the link c relatively to the - fixed link a, a fact indicated by the suffix ac placed after the - letter O. The process being repeated for different values of the angle - [theta] the curve through the several points Oac is the centroid which - may be imagined as formed by an extension of the material of the link - a. To find the corresponding centroid for the link c, fix c and repeat - the process. Again, imagine d fixed, then the instantaneous centre - O_(bd) of b with regard to d is found by producing the links c and a - to intersect in O_(bd), and the shapes of the centroids belonging - respectively to the links b and d can be found as before. The axis - about which a pair of adjacent links turn is a permanent axis, and is - of course the axis of the pin which forms the point. Adding the - centres corresponding to these several axes to the figure, it will be - seen that there are six centres in connexion with the four-bar chain - of which four are permanent and two are instantaneous or virtual - centres; and, further, that whatever be the configuration of the chain - these centres group themselves into three sets of three, each set - lying on a straight line. This peculiarity is not an accident or a - special property of the four-bar chain, but is an illustration of a - general law regarding the subject discovered by Aronhold and Sir A. B. - W. Kennedy independently, which may be thus stated: If any three - bodies, a, b, c, have plane motion their three virtual centres, - O_(ab), O_(bc), O_(ac), are three points on one straight line. A proof - of this will be found in _The Mechanics of Machinery_ quoted above. - Having obtained the set of instantaneous centres for a chain, suppose - a is the fixed link of the chain and c any other link; then O_(ac) is - the instantaneous centre of the two links and may be considered for - the instant as the trace of an axis fixed to an extension of the link - a about which c is turning, and thus problems of instantaneous - velocity concerning the link c are solved as though the link c were - merely rotating for the instant about a fixed axis coincident with the - instantaneous axis. - - [Illustration: FIG. 123.] - - [Illustration: FIG. 124.] - - _Method 2._--The second method is based upon the vector representation - of velocity, and may be illustrated by applying it to the four-bar - chain. Let AD (fig. 123) be the fixed link. Consider the link BC, and - let it be required to find the velocity of the point B having given - the velocity of the point C. The principle upon which the solution is - based is that the only motion which B can have relatively to an axis - through C fixed to the link CD is one of turning about C. Choose any - pole O (fig. 124). From this pole set out Oc to represent the velocity - of the point C. The direction of this must be at right angles to the - line CD, because this is the only direction possible to the point C. - If the link BC moves without turning, Oc will also represent the - velocity of the point B; but, if the link is turning, B can only move - about the axis C, and its direction of motion is therefore at right - angles to the line CB. Hence set out the possible direction of B's - motion in the velocity diagram, namely cb1, at right angles to CB. But - the point B must also move at right angles to AB in the case under - consideration. Hence draw a line through O in the velocity diagram at - right angles to AB to cut cb1 in b. Then Ob is the velocity of the - point b in magnitude and direction, and cb is the tangential velocity - of B relatively to C. Moreover, whatever be the actual magnitudes of - the velocities, the instantaneous velocity ratio of the points C and B - is given by the ratio Oc/Ob. - - A most important property of the diagram (figs. 123 and 124) is the - following: If points X and x are taken dividing the link BC and the - tangential velocity cb, so that cx:xb = CX:XB, then Ox represents the - velocity of the point X in magnitude and direction. The line cb has - been called the _velocity image_ of the rod, since it may be looked - upon as a scale drawing of the rod turned through 90 deg. from the - actual rod. Or, put in another way, if the link CB is drawn to scale - on the new length cb in the velocity diagram (fig. 124), then a vector - drawn from O to any point on the new drawing of the rod will represent - the velocity of that point of the actual rod in magnitude and - direction. It will be understood that there is a new velocity diagram - for every new configuration of the mechanism, and that in each new - diagram the image of the rod will be different in scale. Following the - method indicated above for a kinematic chain in general, there will be - obtained a velocity diagram similar to that of fig. 124 for each - configuration of the mechanism, a diagram in which the velocity of the - several points in the chain utilized for drawing the diagram will - appear to the same scale, all radiating from the pole O. The lines - joining the ends of these several velocities are the several - tangential velocities, each being the velocity image of a link in the - chain. These several images are not to the same scale, so that - although the images may be considered to form collectively an image of - the chain itself, the several members of this chain-image are to - different scales in any one velocity diagram, and thus the chain-image - is distorted from the actual proportions of the mechanism which it - represents. - - [Illustration: FIG. 125.] - - S 82.* _Acceleration Diagram. Acceleration Image._--Although it is - possible to obtain the acceleration of points in a kinematic chain - with one link fixed by methods which utilize the instantaneous centres - of the chain, the vector method more readily lends itself to this - purpose. It should be understood that the instantaneous centre - considered in the preceding paragraphs is available only for - estimating relative velocities; it cannot be used in a similar manner - for questions regarding acceleration. That is to say, although the - instantaneous centre is a centre of no velocity for the instant, it is - not a centre of no acceleration, and in fact the centre of no - acceleration is in general a quite different point. The general - principle on which the method of drawing an acceleration diagram - depends is that if a link CB (fig. 125) have plane motion and the - acceleration of any point C be given in magnitude and direction, the - acceleration of any other point B is the vector sum of the - acceleration of C, the radial acceleration of B about C and the - tangential acceleration of B about C. Let A be any origin, and let Ac - represent the acceleration of the point C, ct the radial acceleration - of B about C which must be in a direction parallel to BC, and tb the - tangential acceleration of B about C, which must of course be at right - angles to ct; then the vector sum of these three magnitudes is Ab, and - this vector represents the acceleration of the point B. The directions - of the radial and tangential accelerations of the point B are always - known when the position of the link is assigned, since these are to be - drawn respectively parallel to and at right angles to the link itself. - The magnitude of the radial acceleration is given by the expression - v^2/BC, v being the velocity of the point B about the point C. This - velocity can always be found from the velocity diagram of the chain of - which the link forms a part. If dw/dt is the angular acceleration of - the link, dw/dt X CB is the tangential acceleration of the point B - about the point C. Generally this tangential acceleration is unknown - in magnitude, and it becomes part of the problem to find it. An - important property of the diagram is that if points X and x are taken - dividing the link CB and the whole acceleration of B about C, namely, - cb in the same ratio, then Ax represents the acceleration of the point - X in magnitude and direction; cb is called the acceleration image of - the rod. In applying this principle to the drawing of an acceleration - diagram for a mechanism, the velocity diagram of the mechanism must be - first drawn in order to afford the means of calculating the several - radial accelerations of the links. Then assuming that the acceleration - of one point of a particular link of the mechanism is known together - with the corresponding configuration of the mechanism, the two vectors - Ac and ct can be drawn. The direction of tb, the third vector in the - diagram, is also known, so that the problem is reduced to the - condition that b is somewhere on the line tb. Then other conditions - consequent upon the fact that the link forms part of a kinematic chain - operate to enable b to be fixed. These methods are set forth and - exemplified in _Graphics_, by R. H. Smith (London, 1889). Examples, - completely worked out, of velocity and acceleration diagrams for the - slider crank chain, the four-bar chain, and the mechanism of the Joy - valve gear will be found in ch. ix. of _Valves and Valve Gear - Mechanism_, by W. E. Dalby (London, 1906). - - - CHAPTER II. ON APPLIED DYNAMICS. - - S 83. _Laws of Motion._--The action of a machine in transmitting - _force_ and _motion_ simultaneously, or performing _work_, is - governed, in common with the phenomena of moving bodies in general, by - two "laws of motion." - - - _Division 1. Balanced Forces in Machines of Uniform Velocity._ - - S 84. _Application of Force to Mechanism._--Forces are applied in - units of weight; and the unit most commonly employed in Britain is the - _pound avoirdupois_. The action of a force applied to a body is always - in reality distributed over some definite space, either a volume of - three dimensions or a surface of two. An example of a force - distributed throughout a volume is the _weight_ of the body itself, - which acts on every particle, however small. The _pressure_ exerted - between two bodies at their surface of contact, or between the two - parts of one body on either side of an ideal surface of separation, is - an example of a force distributed over a surface. The mode of - distribution of a force applied to a solid body requires to be - considered when its stiffness and strength are treated of; but, in - questions respecting the action of a force upon a rigid body - considered as a whole, the _resultant_ of the distributed force, - determined according to the principles of statics, and considered as - acting in a _single line_ and applied at a _single point_, may, for - the occasion, be substituted for the force as really distributed. - Thus, the weight of each separate piece in a machine is treated as - acting wholly at its _centre of gravity_, and each pressure applied to - it as acting at a point called the _centre of pressure_ of the surface - to which the pressure is really applied. - - S 85. _Forces applied to Mechanism Classed._--If [theta] be the - _obliquity_ of a force F applied to a piece of a machine--that is, the - angle made by the direction of the force with the direction of motion - of its point of application--then by the principles of statics, F may - be resolved into two rectangular components, viz.:-- - - Along the direction of motion, P = F cos [theta] \ (49) - Across the direction of motion, Q = F sin [theta] / - - If the component along the direction of motion acts with the motion, - it is called an _effort_; if _against_ the motion, a _resistance_. The - component _across_ the direction of motion is a _lateral pressure_; - the unbalanced lateral pressure on any piece, or part of a piece, is - _deflecting force_. A lateral pressure may increase resistance by - causing friction; the friction so caused acts against the motion, and - is a resistance, but the lateral pressure causing it is not a - resistance. Resistances are distinguished into _useful_ and - _prejudicial_, according as they arise from the useful effect produced - by the machine or from other causes. - - S 86. _Work._--_Work_ consists in moving against resistance. The work - is said to be _performed_, and the resistance _overcome_. Work is - measured by the product of the resistance into the distance through - which its point of application is moved. The _unit of work_ commonly - used in Britain is a resistance of one pound overcome through a - distance of one foot, and is called a _foot-pound_. - - Work is distinguished into _useful work_ and _prejudicial_ or _lost - work_, according as it is performed in producing the useful effect of - the machine, or in overcoming prejudicial resistance. - - S 87. _Energy: Potential Energy._--_Energy_ means _capacity for - performing work_. The _energy of an effort_, or _potential energy_, is - measured by the product of the effort into the distance through which - its point of application is _capable_ of being moved. The unit of - energy is the same with the unit of work. - - When the point of application of an effort _has been moved_ through a - given distance, energy is said to have been _exerted_ to an amount - expressed by the product of the effort into the distance through which - its point of application has been moved. - - S 88. _Variable Effort and Resistance._--If an effort has different - magnitudes during different portions of the motion of its point of - application through a given distance, let each different magnitude of - the effort P be multiplied by the length [Delta]s of the corresponding - portion of the path of the point of application; the sum - - [Sigma] . P[Delta]s (50) - - is the whole energy exerted. If the effort varies by insensible - gradations, the energy exerted is the integral or limit towards which - that sum approaches continually as the divisions of the path are made - smaller and more numerous, and is expressed by - - [int]P ds. (51) - - Similar processes are applicable to the finding of the work performed - in overcoming a varying resistance. - - The work done by a machine can be actually measured by means of a - dynamometer (q.v.). - - S 89. _Principle of the Equality of Energy and Work._--From the first - law of motion it follows that in a machine whose pieces move with - uniform velocities the efforts and resistances must balance each - other. Now from the laws of statics it is known that, in order that a - system of forces applied to a system of connected points may be in - equilibrium, it is necessary that the sum formed by putting together - the products of the forces by the respective distances through which - their points of application are capable of moving simultaneously, each - along the direction of the force applied to it, shall be - zero,--products being considered positive or negative according as the - direction of the forces and the possible motions of their points of - application are the same or opposite. - - In other words, the sum of the negative products is equal to the sum - of the positive products. This principle, applied to a machine whose - parts move with uniform velocities, is equivalent to saying that in - any given interval of time _the energy exerted is equal to the work - performed_. - - The symbolical expression of this law is as follows: let efforts be - applied to one or any number of points of a machine; let any one of - these efforts be represented by P, and the distance traversed by its - point of application in a given interval of time by ds; let - resistances be overcome at one or any number of points of the same - machine; let any one of these resistances be denoted by R, and the - distance traversed by its point of application in the given interval - of time by ds'; then - - [Sigma] . P ds = [Sigma] . R ds'. (52) - - The lengths ds, ds' are proportional to the velocities of the points - to whose paths they belong, and the proportions of those velocities to - each other are deducible from the construction of the machine by the - principles of pure mechanism explained in Chapter I. - - S 90. _Static Equilibrium of Mechanisms._--The principle stated in the - preceding section, namely, that the energy exerted is equal to the - work performed, enables the ratio of the components of the forces - acting in the respective directions of motion at two points of a - mechanism, one being the point of application of the effort, and the - other the point of application of the resistance, to be readily found. - Removing the summation signs in equation (52) in order to restrict its - application to two points and dividing by the common time interval - during which the respective small displacements ds and ds' were made, - it becomes P ds/dt = R ds'/dt, that is, Pv = Rv', which shows that the - force ratio is the inverse of the velocity ratio. It follows at once - that any method which may be available for the determination of the - velocity ratio is equally available for the determination of the force - ratio, it being clearly understood that the forces involved are the - components of the actual forces resolved in the direction of motion - of the points. The relation between the effort and the resistance may - be found by means of this principle for all kinds of mechanisms, when - the friction produced by the components of the forces across the - direction of motion of the two points is neglected. Consider the - following example:-- - - [Illustration: FIG. 126.] - - A four-bar chain having the configuration shown in fig. 126 supports a - load P at the point x. What load is required at the point y to - maintain the configuration shown, both loads being supposed to act - vertically? Find the instantaneous centre O_(bd), and resolve each - load in the respective directions of motion of the points x and y; - thus there are obtained the components P cos [theta] and R cos [phi]. - Let the mechanism have a small motion; then, for the instant, the link - b is turning about its instantaneous centre O_(bd), and, if [omega] is - its instantaneous angular velocity, the velocity of the point x is - [omega]r, and the velocity of the point y is [omega]s. Hence, by the - principle just stated, P cos [theta] X [omega]r = R cos [phi] X - [omega]s. But, p and q being respectively the perpendiculars to the - lines of action of the forces, this equation reduces to P_p = R_q, - which shows that the ratio of the two forces may be found by taking - moments about the instantaneous centre of the link on which they act. - - The forces P and R may, however, act on different links. The general - problem may then be thus stated: Given a mechanism of which r is the - fixed link, and s and t any other two links, given also a force f_s, - acting on the link s, to find the force f_t acting in a given - direction on the link t, which will keep the mechanism in static - equilibrium. The graphic solution of this problem may be effected - thus:-- - - (1) Find the three virtual centres O_(rs), O_(rt), O_(st), which - must be three points in a line. - - (2) Resolve f_s into two components, one of which, namely, f_q, - passes through O_(rs) and may be neglected, and the other f_p passes - through O_(st). - - (3) Find the point M, where f_p joins the given direction of f_t, - and resolve f_p into two components, of which one is in the - direction MO_(rt), and may be neglected because it passes through - O_(rt), and the other is in the given direction of f_t and is - therefore the force required. - - [Illustration: FIG. 127.] - - This statement of the problem and the solution is due to Sir A. B. W. - Kennedy, and is given in ch. 8 of his _Mechanics of Machinery_. - Another general solution of the problem is given in the _Proc. Lond. - Math. Soc._ (1878-1879), by the same author. An example of the method - of solution stated above, and taken from the _Mechanics of Machinery_, - is illustrated by the mechanism fig. 127, which is an epicyclic train - of three wheels with the first wheel r fixed. Let it be required to - find the vertical force which must act at the pitch radius of the last - wheel t to balance exactly a force f_s acting vertically downwards on - the arm at the point indicated in the figure. The two links concerned - are the last wheel t and the arm s, the wheel r being the fixed link - of the mechanism. The virtual centres O_(rs), O_(st) are at the - respective axes of the wheels r and t, and the centre O_(rt) divides - the line through these two points externally in the ratio of the train - of wheels. The figure sufficiently indicates the various steps of the - solution. - - The relation between the effort and the resistance in a machine to - include the effect of friction at the joints has been investigated in - a paper by Professor Fleeming Jenkin, "On the application of graphic - methods to the determination of the efficiency of machinery" (_Trans. - Roy. Soc. Ed._, vol. 28). It is shown that a machine may at any - instant be represented by a frame of links the stresses in which are - identical with the pressures at the joints of the mechanism. This - self-strained frame is called the _dynamic frame_ of the machine. The - driving and resisting efforts are represented by elastic links in the - dynamic frame, and when the frame with its elastic links is drawn the - stresses in the several members of it may be determined by means of - reciprocal figures. Incidentally the method gives the pressures at - every joint of the mechanism. - - S 91. _Efficiency._--The _efficiency_ of a machine is the ratio of the - _useful_ work to the _total_ work--that is, to the energy exerted--and - is represented by - - [Sigma].R_u ds' [Sigma].R_u ds' [Sigma].R_u ds' U - --------------- = --------------------------------- = --------------- = ---. (53) - [Sigma].R ds' [Sigma].R_u ds' + [Sigma].R_p ds' [Sigma].P ds E - - R_u being taken to represent useful and R_p prejudicial resistances. - The more nearly the efficiency of a machine approaches to unity the - better is the machine. - - S 92. _Power and Effect._--The _power_ of a machine is the energy - exerted, and the _effect_ the useful work performed, in some interval - of time of definite length, such as a second, an hour, or a day. - - The unit of power, called conventionally a horse-power, is 550 - foot-pounds per second, or 33,000 foot-pounds per minute, or 1,980,000 - foot-pounds per hour. - - S 93. _Modulus of a Machine._--In the investigation of the properties - of a machine, the useful resistances to be overcome and the useful - work to be performed are usually given. The prejudicial resistances - arc generally functions of the useful resistances of the weights of - the pieces of the mechanism, and of their form and arrangement; and, - having been determined, they serve for the computation of the _lost_ - work, which, being added to the useful work, gives the expenditure of - energy required. The result of this investigation, expressed in the - form of an equation between this energy and the useful work, is called - by Moseley the _modulus_ of the machine. The general form of the - modulus may be expressed thus-- - - E = U + [phi](U, A) + [psi](A), (54) - - where A denotes some quantity or set of quantities depending on the - form, arrangement, weight and other properties of the mechanism. - Moseley, however, has pointed out that in most cases this equation - takes the much more simple form of - - E = (1 + A)U + B, (55) - - where A and B are _constants_, depending on the form, arrangement and - weight of the mechanism. The efficiency corresponding to the last - equation is - - U 1 - --- = -----------. (56) - E 1 + A + B/U - - S 94. _Trains of Mechanism._--In applying the preceding principles to - a train of mechanism, it may either be treated as a whole, or it may - be considered in sections consisting of single pieces, or of any - convenient portion of the train--each section being treated as a - machine, driven by the effort applied to it and energy exerted upon it - through its line of connexion with the preceding section, performing - useful work by driving the following section, and losing work by - overcoming its own prejudicial resistances. It is evident that _the - efficiency of the whole train is the product of the efficiencies of - its sections_. - - S 95. _Rotating Pieces: Couples of Forces._--It is often convenient to - express the energy exerted upon and the work performed by a turning - piece in a machine in terms of the _moment_ of the _couples of forces_ - acting on it, and of the angular velocity. The ordinary British unit - of moment is a _foot-pound_; but it is to be remembered that this is a - foot-pound of a different sort from the unit of energy and work. - - If a force be applied to a turning piece in a line not passing through - its axis, the axis will press against its bearings with an equal and - parallel force, and the equal and opposite reaction of the bearings - will constitute, together with the first-mentioned force, a couple - whose arm is the perpendicular distance from the axis to the line of - action of the first force. - - A couple is said to be _right_ or _left handed_ with reference to the - observer, according to the direction in which it tends to turn the - body, and is a _driving_ couple or a _resisting_ couple according as - its tendency is with or against that of the actual rotation. - - Let dt be an interval of time, [alpha] the angular velocity of the - piece; then [alpha]dt is the angle through which it turns in the - interval dt, and ds = vdt = r[alpha]dt is the distance through which - the point of application of the force moves. Let P represent an - effort, so that Pr is a driving couple, then - - P ds = Pv dt = Pr[alpha] dt = M[alpha] dt (57) - - is the energy exerted by the couple M in the interval dt; and a - similar equation gives the work performed in overcoming a resisting - couple. When several couples act on one piece, the resultant of their - moments is to be multiplied by the common angular velocity of the - whole piece. - - S 96. _Reduction of Forces to a given Point, and of Couples to the - Axis of a given Piece._--In computations respecting machines it is - often convenient to substitute for a force applied to a given point, - or a couple applied to a given piece, the _equivalent_ force or couple - applied to some other point or piece; that is to say, the force or - couple, which, if applied to the other point or piece, would exert - equal energy or employ equal work. The principles of this reduction - are that the ratio of the given to the equivalent force is the - reciprocal of the ratio of the velocities of their points of - application, and the ratio of the given to the equivalent couple is - the reciprocal of the ratio of the angular velocities of the pieces to - which they are applied. - - These velocity ratios are known by the construction of the mechanism, - and are independent of the absolute speed. - - S 97. _Balanced Lateral Pressure of Guides and Bearings._--The most - important part of the lateral pressure on a piece of mechanism is the - reaction of its guides, if it is a sliding piece, or of the bearings - of its axis, if it is a turning piece; and the balanced portion of - this reaction is equal and opposite to the resultant of all the other - forces applied to the piece, its own weight included. There may be or - may not be an unbalanced component in this pressure, due to the - deviated motion. Its laws will be considered in the sequel. - - S 98. _Friction. Unguents._--The most important kind of resistance in - machines is the _friction_ or _rubbing resistance_ of surfaces which - slide over each other. The _direction_ of the resistance of friction - is opposite to that in which the sliding takes place. Its _magnitude_ - is the product of the _normal pressure_ or force which presses the - rubbing surfaces together in a direction perpendicular to themselves - into a specific constant already mentioned in S 14, as the - _coefficient of friction_, which depends on the nature and condition - of the surfaces of the unguent, if any, with which they are covered. - The _total pressure_ exerted between the rubbing surfaces is the - resultant of the normal pressure and of the friction, and its - _obliquity_, or inclination to the common perpendicular of the - surfaces, is the _angle of repose_ formerly mentioned in S 14, whose - tangent is the coefficient of friction. Thus, let N be the normal - pressure, R the friction, T the total pressure, f the coefficient of - friction, and [phi] the angle of repose; then - - f = tan [phi] \ (58) - R = fN = N tan [phi] = T sin [phi] / - - Experiments on friction have been made by Coulomb, Samuel Vince, John - Rennie, James Wood, D. Rankine and others. The most complete and - elaborate experiments are those of Morin, published in his _Notions - fondamentales de mecanique_, and republished in Britain in the works - of Moseley and Gordon. - - The experiments of Beauchamp Tower ("Report of Friction Experiments," - _Proc. Inst. Mech. Eng._, 1883) showed that when oil is supplied to a - journal by means of an oil bath the coefficient of friction varies - nearly inversely as the load on the bearing, thus making the product - of the load on the bearing and the coefficient of friction a constant. - Mr Tower's experiments were carried out at nearly constant - temperature. The more recent experiments of Lasche (_Zeitsch, Verein - Deutsche Ingen._, 1902, 46, 1881) show that the product of the - coefficient of friction, the load on the bearing, and the temperature - is approximately constant. For further information on this point and - on Osborne Reynolds's theory of lubrication see BEARINGS and - LUBRICATION. - - S 99. _Work of Friction. Moment of Friction._--The work performed in a - unit of time in overcoming the friction of a pair of surfaces is the - product of the friction by the velocity of sliding of the surfaces - over each other, if that is the same throughout the whole extent of - the rubbing surfaces. If that velocity is different for different - portions of the rubbing surfaces, the velocity of each portion is to - be multiplied by the friction of that portion, and the results summed - or integrated. - - When the relative motion of the rubbing surfaces is one of rotation, - the work of friction in a unit of time, for a portion of the rubbing - surfaces at a given distance from the axis of rotation, may be found - by multiplying together the friction of that portion, its distance - from the axis, and the angular velocity. The product of the force of - friction by the distance at which it acts from the axis of rotation is - called the _moment of friction_. The total moment of friction of a - pair of rotating rubbing surfaces is the sum or integral of the - moments of friction of their several portions. - - To express this symbolically, let du represent the area of a portion - of a pair of rubbing surfaces at a distance r from the axis of their - relative rotation; p the intensity of the normal pressure at du per - unit of area; and f the coefficient of friction. Then the moment of - friction of du is fprdu; - - the total moment of friction is f [integral] pr.du; \ - and the work performed in a unit cf time in overcoming friction, > (59) - when the angular velocity is [alpha], is [alpha]f [int] pr.du. / - - It is evident that the moment of friction, and the work lost by being - performed in overcoming friction, are less in a rotating piece as the - bearings are of smaller radius. But a limit is put to the diminution - of the radii of journals and pivots by the conditions of durability - and of proper lubrication, and also by conditions of strength and - stiffness. - - S 100. _Total Pressure between Journal and Bearing._--A single piece - rotating with a uniform velocity has four mutually balanced forces - applied to it: (l) the effort exerted on it by the piece which drives - it; (2) the resistance of the piece which follows it--which may be - considered for the purposes of the present question as useful - resistance; (3) its weight; and (4) the reaction of its own - cylindrical bearings. There are given the following data:-- - - The direction of the effort. - The direction of the useful resistance. - The weight of the piece and the direction in which it acts. - The magnitude of the useful resistance. - The radius of the bearing r. - The angle of repose [phi], corresponding to the friction of the - journal on the bearing. - - And there are required the following:-- - - The direction of the reaction of the bearing. - The magnitude of that reaction. - The magnitude of the effort. - - Let the useful resistance and the weight of the piece be compounded by - the principles of statics into one force, and let this be called _the - given force_. - - [Illustration: FIG. 128.] - - The directions of the effort and of the given force are either - parallel or meet in a point. If they are parallel, the direction of - the reaction of the bearing is also parallel to them; if they meet in - a point, the direction of the reaction traverses the same point. - - Also, let AAA, fig. 128, be a section of the bearing, and C its axis; - then the direction of the reaction, at the point where it intersects - the circle AAA, must make the angle [phi] with the radius of that - circle; that is to say, it must be a line such as PT touching the - smaller circle BB, whose radius is r . sin [phi]. The side on which it - touches that circle is determined by the fact that the obliquity of - the reaction is such as to oppose the rotation. - - Thus is determined the direction of the reaction of the bearing; and - the magnitude of that reaction and of the effort are then found by the - principles of the equilibrium of three forces already stated in S 7. - - The work lost in overcoming the friction of the bearing is the same as - that which would be performed in overcoming at the circumference of - the small circle BB a resistance equal to the whole pressure between - the journal and bearing. - - In order to diminish that pressure to the smallest possible amount, - the effort, and the resultant of the useful resistance, and the weight - of the piece (called above the "given force") ought to be opposed to - each other as directly as is practicable consistently with the - purposes of the machine. - - An investigation of the forces acting on a bearing and journal - lubricated by an oil bath will be found in a paper by Osborne Reynolds - in the _Phil. Trans._ pt. i. (1886). (See also BEARINGS.) - - S 101. _Friction of Pivots and Collars._--When a shaft is acted upon - by a force tending to shift it lengthways, that force must be balanced - by the reaction of a bearing against a _pivot_ at the end of the - shaft; or, if that be impossible, against one or more _collars_, or - rings _projecting_ from the body of the shaft. The bearing of the - pivot is called a _step_ or _footstep_. Pivots require great hardness, - and are usually made of steel. The _flat_ pivot is a cylinder of steel - having a plane circular end as a rubbing surface. Let N be the total - pressure sustained by a flat pivot of the radius r; if that pressure - be uniformly distributed, which is the case when the rubbing surfaces - of the pivot and its step are both true planes, the _intensity_ of the - pressure is - - p = N/[pi]r^2; (60) - - and, introducing this value into equation 59, the _moment of friction - of the flat pivot_ is found to be - - (2/3)fNr (61) - - or two-thirds of that of a cylindrical journal of the same radius - under the same normal pressure. - - The friction of a _conical_ pivot exceeds that of a flat pivot of the - same radius, and under the same pressure, in the proportion of the - side of the cone to the radius of its base. - - The moment of friction of a _collar_ is given by the formula-- - - r^3 - r'^3 - (2/3)fN ----------, (62) - r^2 - r'^2 - - where r is the external and r' the internal radius. - - [Illustration: FIG. 129.] - - In the _cup and ball_ pivot the end of the shaft and the step present - two recesses facing each other, into which art fitted two shallow cups - of steel or hard bronze. Between the concave spherical surfaces of - those cups is placed a steel ball, being either a complete sphere or a - lens having convex surfaces of a somewhat less radius than the concave - surfaces of the cups. The moment of friction of this pivot is at first - almost inappreciable from the extreme smallness of the radius of the - circles of contact of the ball and cups, but, as they wear, that - radius and the moment of friction increase. - - It appears that the rapidity with which a rubbing surface wears away - is proportional to the friction and to the velocity jointly, or nearly - so. Hence the pivots already mentioned wear unequally at different - points, and tend to alter their figures. Schiele has invented a pivot - which preserves its original figure by wearing equally at all points - in a direction parallel to its axis. The following are the principles - on which this equality of wear depends:-- - - The rapidity of wear of a surface measured in an _oblique_ direction - is to the rapidity of wear measured normally as the secant of the - obliquity is to unity. Let OX (fig. 129) be the axis of a pivot, and - let RPC be a portion of a curve such that at any point P the secant of - the obliquity to the normal of the curve of a line parallel to the - axis is inversely proportional to the ordinate PY, to which the - velocity of P is proportional. The rotation of that curve round OX - will generate the form of pivot required. Now let PT be a tangent to - the curve at P, cutting OX in T; PT = PY X _secant obliquity_, and - this is to be a constant quantity; hence the curve is that known as - the _tractory_ of the straight line OX, in which PT = OR = constant. - This curve is described by having a fixed straight edge parallel to - OX, along which slides a slider carrying a pin whose centre is T. On - that pin turns an arm, carrying at a point P a tracing-point, pencil - or pen. Should the pen have a nib of two jaws, like those of an - ordinary drawing-pen, the plane of the jaws must pass through PT. - Then, while T is slid along the axis from O towards X, P will be drawn - after it from R towards C along the tractory. This curve, being an - asymptote to its axis, is capable of being indefinitely prolonged - towards X; but in designing pivots it should stop before the angle PTY - becomes less than the angle of repose of the rubbing surfaces, - otherwise the pivot will be liable to stick in its bearing. The moment - of friction of "Schiele's anti-friction pivot," as it is called, is - equal to that of a cylindrical journal of the radius OR = PT the - constant tangent, under the same pressure. - - Records of experiments on the friction of a pivot bearing will be - found in the _Proc. Inst. Mech. Eng._ (1891), and on the friction of a - collar bearing ib. May 1888. - - S 102. _Friction of Teeth._--Let N be the normal pressure exerted - between a pair of teeth of a pair of wheels; s the total distance - through which they slide upon each other; n the number of pairs of - teeth which pass the plane of axis in a unit of time; then - - nfNs (63) - - is the work lost in unity of time by the friction of the teeth. The - sliding s is composed of two parts, which take place during the - approach and recess respectively. Let those be denoted by s1 and s2, - so that s = s1 + s2. In S 45 the _velocity_ of sliding at any instant - has been given, viz. u = c ([alpha]1 + [alpha]2), where u is that - velocity, c the distance T1 at any instant from the point of contact - of the teeth to the pitch-point, and [alpha]1, [alpha]2 the respective - angular velocities of the wheels. - - Let v be the common velocity of the two pitch-circles, r1, r2, their - radii; then the above equation becomes - - / 1 1 \ - u = cv ( --- + --- ). - \r1 r2 / - - To apply this to involute teeth, let c1 be the length of the approach, - c2 that of the recess, u1, the _mean_ volocity of sliding during the - approach, u2 that during the recess; then - - c1v / 1 1 \ c2v / 1 1 \ - u1 = --- ( --- + --- ); u2 = --- ( --- + --- ) - 2 \r1 r2 / 2 \r1 r2 / - - also, let [theta] be the obliquity of the action; then the times - occupied by the approach and recess are respectively - - c1 c2 - -------------, -------------; - v cos [theta] v cos [theta] - - giving, finally, for the length of sliding between each pair of teeth, - - c1^2 + c2^2 / 1 1 \ - s = s1 + s2 = ------------- ( --- + --- ) (64) - 2 cos [theta] \r1 r2 / - - which, substituted in equation (63), gives the work lost in a unit of - time by the friction of involute teeth. This result, which is exact - for involute teeth, is approximately true for teeth of any figure. - - For inside gearing, if r1 be the less radius and r2 the greater, 1/r1 - - 1/r2 is to be substituted for 1/r1 + 1/r2. - - S 103. _Friction of Cords and Belts._--A flexible band, such as a - cord, rope, belt or strap, may be used either to exert an effort or a - resistance upon a pulley round which it wraps. In either case the - tangential force, whether effort or resistance, exerted between the - band and the pulley is their mutual friction, caused by and - proportional to the normal pressure between them. - - Let T1 be the tension of the free part of the band at that side - _towards_ which it tends to draw the pulley, or _from_ which the - pulley tends to draw it; T2 the tension of the free part at the other - side; T the tension of the band at any intermediate point of its arc - of contact with the pulley; [theta] the ratio of the length of that - arc to the radius of the pulley; d[theta] the ratio of an indefinitely - small element of that arc to the radius; F = T1 - T2 the total - friction between the band and the pulley; dF the elementary portion of - that friction due to the elementary arc d[theta]; f the coefficient of - friction between the materials of the band and pulley. - - Then, according to a well-known principle in statics, the normal - pressure at the elementary arc d[theta] is Td[theta], T being the mean - tension of the band at that elementary arc; consequently the friction - on that arc is dF = fTd[theta]. Now that friction is also the - difference between the tensions of the band at the two ends of the - elementary arc, or dT = dF = fTd[theta]; which equation, being - integrated throughout the entire arc of contact, gives the following - formulae:-- - - T1 \ - hyp log. -- = f^[theta] | - T2 | - | - T1 > (65) - -- = ef^[theta] | - T2 | - | - F = T1 - T2 = T1(1 - e - f^[theta]) = T2(ef^[theta] - 1) / - - When a belt connecting a pair of pulleys has the tensions of its two - sides originally equal, the pulleys being at rest, and when the - pulleys are next set in motion, so that one of them drives the other - by means of the belt, it is found that the advancing side of the belt - is exactly as much tightened as the returning side is slackened, so - that the _mean_ tension remains unchanged. Its value is given by this - formula-- - - T1 + T2 ef^[theta] + 1 - ------- = ----------------- (66) - 2 2(ef^[theta] - 1) - - which is useful in determining the original tension required to enable - a belt to transmit a given force between two pulleys. - - The equations 65 and 66 are applicable to a kind of _brake_ called a - _friction-strap_, used to stop or moderate the velocity of machines by - being tightened round a pulley. The strap is usually of iron, and the - pulley of hard wood. - - Let [alpha] denote the arc of contact expressed in _turns and - fractions of a turn_; then - - [theta] = 6.2832a \ (67) - ef^[theta] = number whose common logarithm is 2.7288fa / - - See also DYNAMOMETER for illustrations of the use of what are - essentially friction-straps of different forms for the measurement of - the brake horse-power of an engine or motor. - - S 104. _Stiffness of Ropes._--Ropes offer a resistance to being bent, - and, when bent, to being straightened again, which arises from the - mutual friction of their fibres. It increases with the sectional area - of the rope, and is inversely proportional to the radius of the curve - into which it is bent. - - The _work lost_ in pulling a given length of rope over a pulley is - found by multiplying the length of the rope in feet by its stiffness - in pounds, that stiffness being the excess of the tension at the - leading side of the rope above that at the following side, which is - necessary to bend it into a curve fitting the pulley, and then to - straighten it again. - - The following empirical formulae for the stiffness of hempen ropes - have been deduced by Morin from the experiments of Coulomb:-- - - Let F be the stiffness in pounds avoirdupois; d the diameter of the - rope in inches, n = 48d^2 for white ropes and 35d^2 for tarred ropes; - r the _effective_ radius of the pulley in inches; T the tension in - pounds. Then - - n \ - For white ropes, F = --- (0.0012 + 0.001026n + 0.0012T) | - r | - > (68) - n | - For tarred ropes, F = --- (0.006 + 0.001392n + 0.00168T) | - r / - - S 105. _Friction-Couplings._--Friction is useful as a means of - communicating motion where sudden changes either of force or velocity - take place, because, being limited in amount, it may be so adjusted as - to limit the forces which strain the pieces of the mechanism within - the bounds of safety. Amongst contrivances for effecting this object - are _friction-cones_. A rotating shaft carries upon a cylindrical - portion of its figure a wheel or pulley turning loosely on it, and - consequently capable of remaining at rest when the shaft is in motion. - This pulley has fixed to one side, and concentric with it, a short - frustum of a hollow cone. At a small distance from the pulley the - shaft carries a short frustum of a solid cone accurately turned to fit - the hollow cone. This frustum is made always to turn along with the - shaft by being fitted on a square portion of it, or by means of a rib - and groove, or otherwise, but is capable of a slight longitudinal - motion, so as to be pressed into, or withdrawn from, the hollow cone - by means of a lever. When the cones are pressed together or engaged, - their friction causes the pulley to rotate along with the shaft; when - they are disengaged, the pulley is free to stand still. The angle made - by the sides of the cones with the axis should not be less than the - angle of repose. In the _friction-clutch_, a pulley loose on a shaft - has a hoop or gland made to embrace it more or less tightly by means - of a screw; this hoop has short projecting arms or ears. A fork or - _clutch_ rotates along with the shaft, and is capable of being moved - longitudinally by a handle. When the clutch is moved towards the hoop, - its arms catch those of the hoop, and cause the hoop to rotate and to - communicate its rotation to the pulley by friction. There are many - other contrivances of the same class, but the two just mentioned may - serve for examples. - - S 106. _Heat of Friction: Unguents._--The work lost in friction is - employed in producing heat. This fact is very obvious, and has been - known from a remote period; but the _exact_ determination of the - proportion of the work lost to the heat produced, and the experimental - proof that that proportion is the same under all circumstances and - with all materials, solid, liquid and gaseous, are comparatively - recent achievements of J. P. Joule. The quantity of work which - produces a British unit of heat (or so much heat as elevates the - temperature of one pound of pure water, at or near ordinary - atmospheric temperatures, by 1 deg. F.) is 772 foot-pounds. This - constant, now designated as "Joule's equivalent," is the principal - experimental datum of the science of thermodynamics. - - A more recent determination (_Phil. Trans._, 1897), by Osborne - Reynolds and W. M. Moorby, gives 778 as the mean value of Joule's - equivalent through the range of 32 deg. to 212 deg. F. See also the - papers of Rowland in the _Proc. Amer. Acad._ (1879), and Griffiths, - _Phil. Trans._ (1893). - - The heat produced by friction, when moderate in amount, is useful in - softening and liquefying thick unguents; but when excessive it is - prejudicial, by decomposing the unguents, and sometimes even by - softening the metal of the bearings, and raising their temperature so - high as to set fire to neighbouring combustible matters. - - Excessive heating is prevented by a constant and copious supply of a - good unguent. The elevation of temperature produced by the friction of - a journal is sometimes used as an experimental test of the quality of - unguents. For modern methods of forced lubrication see BEARINGS. - - S 107. _Rolling Resistance._--By the rolling of two surfaces over each - other without sliding a resistance is caused which is called sometimes - "rolling friction," but more correctly _rolling resistance_. It is of - the nature of a _couple_, resisting rotation. Its _moment_ is found by - multiplying the normal pressure between the rolling surfaces by an - _arm_, whose length depends on the nature of the rolling surfaces, and - the work lost in a unit of time in overcoming it is the product of its - moment by the _angular velocity_ of the rolling surfaces relatively to - each other. The following are approximate values of the arm in - decimals of a foot:-- - - Oak upon oak 0.006 (Coulomb). - Lignum vitae on oak 0.004 " - Cast iron on cast iron 0.002 (Tredgold). - - S 108. _Reciprocating Forces: Stored and Restored Energy._--When a - force acts on a machine alternately as an effort and as a resistance, - it may be called a _reciprocating force_. Of this kind is the weight - of any piece in the mechanism whose centre of gravity alternately - rises and falls; for during the rise of the centre of gravity that - weight acts as a resistance, and energy is employed in lifting it to - an amount expressed by the product of the weight into the vertical - height of its rise; and during the fall of the centre of gravity the - weight acts as an effort, and exerts in assisting to perform the work - of the machine an amount of energy exactly equal to that which had - previously been employed in lifting it. Thus that amount of energy is - not lost, but has its operation deferred; and it is said to be - _stored_ when the weight is lifted, and _restored_ when it falls. - - In a machine of which each piece is to move with a uniform velocity, - if the effort and the resistance be constant, the weight of each piece - must be balanced on its axis, so that it may produce lateral pressure - only, and not act as a reciprocating force. But if the effort and the - resistance be alternately in excess, the uniformity of speed may still - be preserved by so adjusting some moving weight in the mechanism that - when the effort is in excess it may be lifted, and so balance and - employ the excess of effort, and that when the resistance is in excess - it may fall, and so balance and overcome the excess of - resistance--thus _storing_ the periodical excess of energy and - _restoring_ that energy to perform the periodical excess of work. - - Other forces besides gravity may be used as reciprocating forces for - storing and restoring energy--for example, the elasticity of a spring - or of a mass of air. - - In most of the delusive machines commonly called "perpetual motions," - of which so many are patented in each year, and which are expected by - their inventors to perform work without receiving energy, the - fundamental fallacy consists in an expectation that some reciprocating - force shall restore more energy than it has been the means of storing. - - - _Division 2. Deflecting Forces._ - - S 109. _Deflecting Force for Translation in a Curved Path._--In - machinery, deflecting force is supplied by the tenacity of some piece, - such as a crank, which guides the deflected body in its curved path, - and is _unbalanced_, being employed in producing deflexion, and not in - balancing another force. - - S 110. _Centrifugal Force of a Rotating Body._--_The centrifugal force - exerted by a rotating body on its axis of rotation is the same in - magnitude as if the mass of the body were concentrated at its centre - of gravity, and acts in a plane passing through the axis of rotation - and the centre of gravity of the body._ - - The particles of a rotating body exert centrifugal forces on each - other, which strain the body, and tend to tear it asunder, but these - forces balance each other, and do not affect the resultant centrifugal - force exerted on the axis of rotation.[3] - - _If the axis of rotation traverses the centre of gravity of the body, - the centrifugal force exerted on that axis is nothing._ - - Hence, unless there be some reason to the contrary, each piece of a - machine should be balanced on its axis of rotation; otherwise the - centrifugal force will cause strains, vibration and increased - friction, and a tendency of the shafts to jump out of their bearings. - - S 111. _Centrifugal Couples of a Rotating Body._--Besides the tendency - (if any) of the combined centrifugal forces of the particles of a - rotating body to _shift_ the axis of rotation, they may also tend to - _turn_ it out of its original direction. The latter tendency is called - _a centrifugal couple_, and vanishes for rotation about a principal - axis. - - It is essential to the steady motion of every rapidly rotating piece - in a machine that its axis of rotation should not merely traverse its - centre of gravity, but should be a permanent axis; for otherwise the - centrifugal couples will increase friction, produce oscillation of the - shaft and tend to make it leave its bearings. - - The principles of this and the preceding section are those which - regulate the adjustment of the weight and position of the - counterpoises which are placed between the spokes of the - driving-wheels of locomotive engines. - - [Illustration: (From _Balancing of Engines_, by permission of Edward - Arnold.) - - FIG. 130.] - - S 112.* _Method of computing the position and magnitudes of balance - weights which must be added to a given system of arbitrarily chosen - rotating masses in order to make the common axis of rotation a - permanent axis._--The method here briefly explained is taken from a - paper by W. E. Dalby, "The Balancing of Engines with special reference - to Marine Work," _Trans. Inst. Nav. Arch._ (1899). Let the weight - (fig. 130), attached to a truly turned disk, be rotated by the shaft - OX, and conceive that the shaft is held in a bearing at one point, O. - The force required to constrain the weight to move in a circle, that - is the deviating force, produces an equal and opposite reaction on the - shaft, whose amount F is equal to the centrifugal force Wa^2r/g lb., - where r is the radius of the mass centre of the weight, and a is its - angular velocity in radians per second. Transferring this force to the - point O, it is equivalent to, (1) a force at O equal and parallel to - F, and, (2) a centrifugal couple of Fa foot-pounds. In order that OX - may be a permanent axis it is necessary that there should be a - sufficient number of weights attached to the shaft and so distributed - that when each is referred to the point O - - (1) [Sigma]F = 0 \ (a) - (2) [Sigma]Fa = 0 / - - The plane through O to which the shaft is perpendicular is called the - _reference plane_, because all the transferred forces act in that - plane at the point O. The plane through the radius of the weight - containing the axis OX is called the _axial plane_ because it contains - the forces forming the couple due to the transference of F to the - reference plane. Substituting the values of F in (a) the two - conditions become - - a^2 - (1) (W1r1 + W2r2 + W3r3 + ...)--- = 0 - g - (b) - a^2 - (2) (W1a1r1 + W2a2r2 + ... )--- = 0 - g - - In order that these conditions may obtain, the quantities in the - brackets must be zero, since the factor a^2/g is not zero. Hence - finally the conditions which must be satisfied by the system of - weights in order that the axis of rotation may be a permanent axis is - - (1) (W1r1 + W2r2 + W3r3) = 0 - (2) (W1a1r1 + W2a2r2 + W3a3r3) = 0 (c) - - It must be remembered that these are all directed quantities, and that - their respective sums are to be taken by drawing vector polygons. In - drawing these polygons the magnitude of the vector of the type Wr is - the product Wr, and the direction of the vector is from the shaft - outwards towards the weight W, parallel to the radius r. For the - vector representing a couple of the type War, if the masses are all on - the same side of the reference plane, the direction of drawing is from - the axis outwards; if the masses are some on one side of the reference - plane and some on the other side, the direction of drawing is from the - axis outwards towards the weight for all masses on the one side, and - from the mass inwards towards the axis for all weights on the other - side, drawing always parallel to the direction defined by the radius - r. The magnitude of the vector is the product War. The conditions (c) - may thus be expressed: first, that the sum of the vectors Wr must form - a closed polygon, and, second, that the sum of the vectors War must - form a closed polygon. The general problem in practice is, given a - system of weights attached to a shaft, to find the respective weights - and positions of two balance weights or counterpoises which must be - added to the system in order to make the shaft a permanent axis, the - planes in which the balance weights are to revolve also being given. - To solve this the reference plane must be chosen so that it coincides - with the plane of revolution of one of the as yet unknown balance - weights. The balance weight in this plane has therefore no couple - corresponding to it. Hence by drawing a couple polygon for the given - weights the vector which is required to close the polygon is at once - found and from it the magnitude and position of the balance weight - which must be added to the system to balance the couples follow at - once. Then, transferring the product Wr corresponding with this - balance weight to the reference plane, proceed to draw the force - polygon. The vector required to close it will determine the second - balance weight, the work may be checked by taking the reference plane - to coincide with the plane of revolution of the second balance weight - and then re-determining them, or by taking a reference plane anywhere - and including the two balance weights trying if condition (c) is - satisfied. - - When a weight is reciprocated, the equal and opposite force required - for its acceleration at any instant appears as an unbalanced force on - the frame of the machine to which the weight belongs. In the - particular case, where the motion is of the kind known as "simple - harmonic" the disturbing force on the frame due to the reciprocation - of the weight is equal to the component of the centrifugal force in - the line of stroke due to a weight equal to the reciprocated weight - supposed concentrated at the crank pin. Using this principle the - method of finding the balance weights to be added to a given system of - reciprocating weights in order to produce a system of forces on the - frame continuously in equilibrium is exactly the same as that just - explained for a system of revolving weights, because for the purpose - of finding the balance weights each reciprocating weight may be - supposed attached to the crank pin which operates it, thus forming an - equivalent revolving system. The balance weights found as part of the - equivalent revolving system when reciprocated by their respective - crank pins form the balance weights for the given reciprocating - system. These conditions may be exactly realized by a system of - weights reciprocated by slotted bars, the crank shaft driving the - slotted bars rotating uniformly. In practice reciprocation is usually - effected through a connecting rod, as in the case of steam engines. In - balancing the mechanism of a steam engine it is often sufficiently - accurate to consider the motion of the pistons as simple harmonic, and - the effect on the framework of the acceleration of the connecting rod - may be approximately allowed for by distributing the weight of the rod - between the crank pin and the piston inversely as the centre of - gravity of the rod divides the distance between the centre of the - cross head pin and the centre of the crank pin. The moving parts of - the engine are then divided into two complete and independent systems, - namely, one system of revolving weights consisting of crank pins, - crank arms, &c., attached to and revolving with the crank shaft, and a - second system of reciprocating weights consisting of the pistons, - cross-heads, &c., supposed to be moving each in its line of stroke - with simple harmonic motion. The balance weights are to be separately - calculated for each system, the one set being added to the crank shaft - as revolving weights, and the second set being included with the - reciprocating weights and operated by a properly placed crank on the - crank shaft. Balance weights added in this way to a set of - reciprocating weights are sometimes called bob-weights. In the case of - locomotives the balance weights required to balance the pistons are - added as revolving weights to the crank shaft system, and in fact are - generally combined with the weights required to balance the revolving - system so as to form one weight, the counterpoise referred to in the - preceding section, which is seen between the spokes of the wheels of a - locomotive. Although this method balances the pistons in the - horizontal plane, and thus allows the pull of the engine on the train - to be exerted without the variation due to the reciprocation of the - pistons, yet the force balanced horizontally is introduced vertically - and appears as a variation of pressure on the rail. In practice about - two-thirds of the reciprocating weight is balanced in order to keep - this variation of rail pressure within safe limits. The assumption - that the pistons of an engine move with simple harmonic motion is - increasingly erroneous as the ratio of the length of the crank r, to - the length of the connecting rod l increases. A more accurate though - still approximate expression for the force on the frame due to the - acceleration of the piston whose weight is W is given by - - W / r \ - --- [omega]^2 r ( cos [theta] + --- cos 2[theta] ) - g \ l / - - The conditions regulating the balancing of a system of weights - reciprocating under the action of accelerating forces given by the - above expression are investigated in a paper by Otto Schlick, "On - Balancing of Steam Engines," _Trans, Inst. Nav. Arch._ (1900), and in - a paper by W. E. Dalby, "On the Balancing of the Reciprocating Parts - of Engines, including the Effect of the Connecting Rod" (ibid., 1901). - A still more accurate expression than the above is obtained by - expansion in a Fourier series, regarding which and its bearing on - balancing engines see a paper by J. H. Macalpine, "A Solution of the - Vibration Problem" (ibid., 1901). The whole subject is dealt with in a - treatise, _The Balancing of Engines_, by W. E. Dalby (London, 1906). - Most of the original papers on this subject of engine balancing are to - be found in the _Transactions_ of the Institution of Naval Architects. - - S 113.* _Centrifugal Whirling of Shafts._--When a system of revolving - masses is balanced so that the conditions of the preceding section are - fulfilled, the centre of gravity of the system lies on the axis of - revolution. If there is the slightest displacement of the centre of - gravity of the system from the axis of revolution a force acts on the - shaft tending to deflect it, and varies as the deflexion and as the - square of the speed. If the shaft is therefore to revolve stably, this - force must be balanced at any instant by the elastic resistance of the - shaft to deflexion. To take a simple case, suppose a shaft, supported - on two bearings to carry a disk of weight W at its centre, and let the - centre of gravity of the disk be at a distance e from the axis of - rotation, this small distance being due to imperfections of material - or faulty construction. Neglecting the mass of the shaft itself, when - the shaft rotates with an angular velocity a, the centrifugal force - Wa^2e/g will act upon the shaft and cause its axis to deflect from the - axis of rotation a distance, y say. The elastic resistance evoked by - this deflexion is proportional to the deflexion, so that if c is a - constant depending upon the form, material and method of support of - the shaft, the following equality must hold if the shaft is to rotate - stably at the stated speed-- - - W - ---(y + e)a^2 = cy, - g - - from which y = Wa^2e/(gc - Wa^2). - - This expression shows that as a increases y increases until when Wa^2 = - gc, y becomes infinitely large. The corresponding value of a, namely - [root]gc/W, is called the _critical velocity_ of the shaft, and is the - speed at which the shaft ceases to rotate stably and at which - centrifugal whirling begins. The general problem is to find the value - of a corresponding to all kinds of loadings on shafts supported in any - manner. The question was investigated by Rankine in an article in the - _Engineer_ (April 9, 1869). Professor A. G. Greenhill treated the - problem of the centrifugal whirling of an unloaded shaft with - different supporting conditions in a paper "On the Strength of - Shafting exposed both to torsion and to end thrust," _Proc. Inst. - Mech. Eng._ (1883). Professor S. Dunkerley ("On the Whirling and - Vibration of Shafts," _Phil. Trans._, 1894) investigated the question - for the cases of loaded and unloaded shafts, and, owing to the - complication arising from the application of the general theory to the - cases of loaded shafts, devised empirical formulae for the critical - speeds of shafts loaded with heavy pulleys, based generally upon the - following assumption, which is stated for the case of a shaft carrying - one pulley: If N1, N2 be the separate speeds of whirl of the shaft and - pulley on the assumption that the effect of one is neglected when that - of the other is under consideration, then the resulting speed of whirl - due to both causes combined may be taken to be of the form N1N2 - [root][(N^21 + N1^2)] where N means revolutions per minute. This form is - extended to include the cases of several pulleys on the same shaft. - The interesting and important part of the investigation is that a - number of experiments were made on small shafts arranged in different - ways and loaded in different ways, and the speed at which whirling - actually occurred was compared with the speed calculated from formulae - of the general type indicated above. The agreement between the - observed and calculated values of the critical speeds was in most - cases quite remarkable. In a paper by Dr C. Chree, "The Whirling and - Transverse Vibrations of Rotating Shafts," _Proc. Phys. Soc. Lon._, - vol. 19 (1904); also _Phil. Mag._, vol. 7 (1904), the question is - investigated from a new mathematical point of view, and expressions - for the whirling of loaded shafts are obtained without the necessity - of any assumption of the kind stated above. An elementary presentation - of the problem from a practical point of view will be found in _Steam - Turbines_, by Dr A. Stodola (London, 1905). - - [Illustration: FIG. 131.] - - S 114. _Revolving Pendulum. Governors._--In fig. 131 AO represents an - upright axis or spindle; B a weight called a _bob_, suspended by rod - OB from a horizontal axis at O, carried by the vertical axis. When the - spindle is at rest the bob hangs close to it; when the spindle - rotates, the bob, being made to revolve round it, diverges until the - resultant of the centrifugal force and the weight of the bob is a - force acting at O in the direction OB, and then it revolves steadily - in a circle. This combination is called a _revolving_, _centrifugal_, - or _conical pendulum_. Revolving pendulums are usually constructed - with _pairs_ of rods and bobs, as OB, Ob, hung at opposite sides of - the spindle, that the centrifugal forces exerted at the point O may - balance each other. - - In finding the position in which the bob will revolve with a given - angular velocity, a, for most practical cases connected with machinery - the mass of the rod may be considered as insensible compared with that - of the bob. Let the bob be a sphere, and from the centre of that - sphere draw BH = y perpendicular to OA. Let OH = z; let W be the - weight of the bob, F its centrifugal force. Then the condition of its - steady revolution is W : F :: z : y; that is to say, y/z = F/W = - ya^2/g; consequently - - z = g/[alpha]^2 (69) - - Or, if n = [alpha] 2[pi] = [alpha]/6.2832 be the number of turns or - fractions of a turn in a second, - - g 0.8165 ft. 9.79771 in. \ - z = ---------- = ---------- = ----------- > (70) - 4[pi]^2n^2 n^2 n^2 / - - z is called the _altitude of the pendulum_. - - [Illustration: FIG. 132.] - - If the rod of a revolving pendulum be jointed, as in fig. 132, not to - a point in the vertical axis, but to the end of a projecting arm C, - the position in which the bob will revolve will be the same as if the - rod were jointed to the point O, where its prolongation cuts the - vertical axis. - - A revolving pendulum is an essential part of most of the contrivances - called _governors_, for regulating the speed of prime movers, for - further particulars of which see STEAM ENGINE. - - - _Division 3. Working of Machines of Varying Velocity._ - - S 115. _General Principles._--In order that the velocity of every - piece of a machine may be uniform, it is necessary that the forces - acting on each piece should be always exactly balanced. Also, in order - that the forces acting on each piece of a machine may be always - exactly balanced, it is necessary that the velocity of that piece - should be uniform. - - An excess of the effort exerted on any piece, above that which is - necessary to balance the resistance, is accompanied with acceleration; - a deficiency of the effort, with retardation. - - When a machine is being started from a state of rest, and brought by - degrees up to its proper speed, the effort must be in excess; when it - is being retarded for the purpose of stopping it, the resistance must - be in excess. - - An excess of effort above resistance involves an excess of energy - exerted above work performed; that excess of energy is employed in - producing acceleration. - - An excess of resistance above effort involves an excess of work - performed above energy expended; that excess of work is performed by - means of the retardation of the machinery. - - When a machine undergoes alternate acceleration and retardation, so - that at certain instants of time, occurring at the end of intervals - called _periods_ or _cycles_, it returns to its original speed, then - in each of those periods or cycles the alternate excesses of energy - and of work neutralize each other; and at the end of each cycle the - principle of the equality of energy and work stated in S 87, with all - its consequences, is verified exactly as in the case of machines of - uniform speed. - - At intermediate instants, however, other principles have also to be - taken into account, which are deduced from the second law of motion, - as applied to _direct deviation_, or acceleration and retardation. - - S 116. _Energy of Acceleration and Work of Retardation for a Shifting - Body._--Let w be the weight of a body which has a motion of - translation in any path, and in the course of the interval of time - [Delta]t let its velocity be increased at a uniform rate of - acceleration from v1 to v2. The rate of acceleration will be - - dv/dt = const. = (v2 - v1)[Delta]t; - - and to produce this acceleration a uniform effort will be required, - expressed by - - P = w(v2 - v1)g[Delta]t (71) - - (The product wv/g of the mass of a body by its velocity is called its - _momentum_; so that the effort required is found by dividing the - increase of momentum by the time in which it is produced.) - - To find the _energy_ which has to be exerted to produce the - acceleration from v1 to v2, it is to be observed that the _distance_ - through which the effort P acts during the acceleration is - - [Delta]s = (v2 + v1)[Delta]t/2; - - consequently, the _energy of acceleration_ is - - P[Delta]s = w(v2 - v1) (v2 + v1)/2g = w(v2^2 - v1^2)2g, (72) - - being proportional to the increase in the square of the velocity, and - _independent of the time_. - - In order to produce a _retardation_ from the greater velocity v2 to - the less velocity v1, it is necessary to apply to the body a - _resistance_ connected with the retardation and the time by an - equation identical in every respect with equation (71), except by the - substitution of a resistance for an effort; and in overcoming that - resistance the body _performs work_ to an amount determined by - equation (72), putting Rds for Pas. - - S 117. _Energy Stored and Restored by Deviations of Velocity._--Thus a - body alternately accelerated and retarded, so as to be brought back to - its original speed, performs work during its retardation exactly equal - in amount to the energy exerted upon it during its acceleration; so - that that energy may be considered as _stored_ during the - acceleration, and _restored_ during the retardation, in a manner - analogous to the operation of a reciprocating force (S 108). - - Let there be given the mean velocity V = (1/2)(v2 + v1) of a body whose - weight is w, and let it be required to determine the fluctuation of - velocity v2 - v1, and the extreme velocities v1, v2, which that body - must have, in order alternately to store and restore an amount of - energy E. By equation (72) we have - - E = w(v2^2 - v1^2)'2g - - which, being divided by V = (1/2)(v2 + v1), gives - - E/V = w(v2 - v1)/g; - - and consequently - - v2 - v1 = gE/Vw (73) - - The ratio of this fluctuation to the mean velocity, sometimes called - the unsteadiness of the motion of the body, is - - (v2 - v1)V = gE/V^2w. (74) - - S 118. _Actual Energy of a Shifting Body._--The energy which must be - exerted on a body of the weight w, to accelerate it from a state of - rest up to a given velocity of translation v, and the equal amount of - work which that body is capable of performing by overcoming resistance - while being retarded from the same velocity of translation v to a - state of rest, is - - wv^2/2g. (75) - - This is called the _actual energy_ of the motion of the body, and is - half the quantity which in some treatises is called vis viva. - - The energy stored or restored, as the case may be, by the deviations - of velocity of a body or a system of bodies, is the amount by which - the actual energy is increased or diminished. - - S 119. _Principle of the Conservation of Energy in Machines._--The - following principle, expressing the general law of the action of - machines with a velocity uniform or varying, includes the law of the - equality of energy and work stated in S 89 for machines of uniform - speed. - - _In any given interval during the working of a machine, the energy - exerted added to the energy restored is equal to the energy stored - added to the work performed._ - - S 120. _Actual Energy of Circular Translation--Moment of - Inertia._--Let a small body of the weight w undergo translation in a - circular path of the radius [rho], with the angular velocity of - deflexion [alpha], so that the common linear velocity of all its - particles is v = [alpha][rho]. Then the actual energy of that body is - - wv^2/2g = w[alpha]^2p^2/2g. (76) - - By comparing this with the expression for the centrifugal force - (w[alpha]^2p/g), it appears that the actual energy of a revolving body - is equal to the potential energy Fp/2 due to the action of the - deflecting force along one-half of the radius of curvature of the path - of the body. - - The product wp^2/g, by which the half-square of the angular velocity is - multiplied, is called the _moment of inertia_ of the revolving body. - - S 121. _Flywheels._--A flywheel is a rotating piece in a machine, - generally shaped like a wheel (that is to say, consisting of a rim - with spokes), and suited to store and restore energy by the periodical - variations in its angular velocity. - - The principles according to which variations of angular velocity store - and restore energy are the same as those of S 117, only substituting - _moment of inertia_ for _mass_, and _angular_ for _linear_ velocity. - - Let W be the weight of a flywheel, R its radius of gyration, a2 its - maximum, a1 its minimum, and A = (1/2)([alpha]2 + [alpha]1) its mean - angular velocity. Let - - I/S = ([alpha]2 - [alpha]2)/A - - denote the _unsteadiness_ of the motion of the flywheel; the - denominator S of this fraction is called the _steadiness_. Let e - denote the quantity by which the energy exerted in each cycle of the - working of the machine alternately exceeds and falls short of the work - performed, and which has consequently to be alternately stored by - acceleration and restored by retardation of the flywheel. The value of - this _periodical excess_ is-- - - e = R^2W ([alpha]2^2 - [alpha]1^2), 2g, (77) - - from which, dividing both sides by A^2, we obtain the following - equations:-- - - e/A^2 = R^2 W/gS \ - >. (78) - R^2 WA^2/2g = Se/2 / - - The latter of these equations may be thus expressed in words: _The - actual energy due to the rotation of the fly, with its mean angular - velocity, is equal to one-half of the periodical excess of energy - multiplied by the steadiness._ - - In ordinary machinery S = about 32; in machinery for fine purposes S = - from 50 to 60; and when great steadiness is required S = from 100 to - 150. - - The periodical excess e may arise either from variations in the effort - exerted by the prime mover, or from variations in the resistance of - the work, or from both these causes combined. When but one flywheel is - used, it should be placed in as direct connexion as possible with that - part of the mechanism where the greatest amount of the periodical - excess originates; but when it originates at two or more points, it is - best to have a flywheel in connexion with each of these points. For - example, in a machine-work, the steam-engine, which is the prime mover - of the various tools, has a flywheel on the crank-shaft to store and - restore the periodical excess of energy arising from the variations in - the effort exerted by the connecting-rod upon the crank; and each of - the slotting machines, punching machines, riveting machines, and other - tools has a flywheel of its own to store and restore energy, so as to - enable the very different resistances opposed to those tools at - different times to be overcome without too great unsteadiness of - motion. For tools performing useful work at intervals, and having only - their own friction to overcome during the intermediate intervals, e - should be assumed equal to the whole work performed at each separate - operation. - - S 122. _Brakes._--A brake is an apparatus for stopping and diminishing - the velocity of a machine by friction, such as the friction-strap - already referred to in S 103. To find the distance s through which a - brake, exerting the friction F, must rub in order to stop a machine - having the total actual energy E at the moment when the brake begins - to act, reduce, by the principles of S 96, the various efforts and - other resistances of the machine which act at the same time with the - friction of the brake to the rubbing surface of the brake, and let R - be their resultant--positive if _resistance_, _negative_ if effort - preponderates. Then - - s = E/(F + R). (79) - - S 123. _Energy distributed between two Bodies: Projection and - Propulsion._--Hitherto the effort by which a machine is moved has been - treated as a force exerted between a movable body and a fixed body, so - that the whole energy exerted by it is employed upon the movable body, - and none upon the fixed body. This conception is sensibly realized in - practice when one of the two bodies between which the effort acts is - either so heavy as compared with the other, or has so great a - resistance opposed to its motion, that it may, without sensible error, - be treated as fixed. But there are cases in which the motions of both - bodies are appreciable, and must be taken into account--such as the - projection of projectiles, where the velocity of the _recoil_ or - backward motion of the gun bears an appreciable proportion to the - forward motion of the projectile; and such as the propulsion of - vessels, where the velocity of the water thrown backward by the - paddle, screw or other propeller bears a very considerable proportion - to the velocity of the water moved forwards and sideways by the ship. - In cases of this kind the energy exerted by the effort is - _distributed_ between the two bodies between which the effort is - exerted in shares proportional to the velocities of the two bodies - during the action of the effort; and those velocities are to each - other directly as the portions of the effort unbalanced by resistance - on the respective bodies, and inversely as the weights of the bodies. - - To express this symbolically, let W1, W2 be the weights of the bodies; - P the effort exerted between them; S the distance through which it - acts; R1, R2 the resistances opposed to the effort overcome by W1, W2 - respectively; E1, E2 the shares of the whole energy E exerted upon W1, - W2 respectively. Then - - E : E1 : E2 \ - W2(P - R1) + W1(P - R2) P - R1 P - R2 | - :: ----------------------- : ------ : ------ >. (80) - W1W2 W1 W2 / - - If R1 = R2, which is the case when the resistance, as well as the - effort, arises from the mutual actions of the two bodies, the above - becomes, - - E : E1 : E2 \ - :: W1 + W2 : W2 : W1 /, (81) - - that is to say, the energy is exerted on the bodies in shares - inversely proportional to their weights; and they receive - accelerations inversely proportional to their weights, according to - the principle of dynamics, already quoted in a note to S 110, that the - mutual actions of a system of bodies do not affect the motion of their - common centre of gravity. - - For example, if the weight of a gun be 160 times that of its ball - 160/161 of the energy exerted by the powder in exploding will be - employed in propelling the ball, and 1/161 in producing the recoil of - the gun, provided the gun up to the instant of the ball's quitting the - muzzle meets with no resistance to its recoil except the friction of - the ball. - - S 124. _Centre of Percussion._--It is obviously desirable that the - deviations or changes of motion of oscillating pieces in machinery - should, as far as possible, be effected by forces applied at their - centres of percussion. - - If the deviation be a _translation_--that is, an equal change of - motion of all the particles of the body--the centre of percussion is - obviously the centre of gravity itself; and, according to the second - law of motion, if dv be the deviation of velocity to be produced in - the interval dt, and W the weight of the body, then - - W dv - P = --- . -- (82) - g dt - - is the unbalanced effort required. - - If the deviation be a rotation about an axis traversing the centre of - gravity, there is no centre of percussion; for such a deviation can - only be produced by a _couple_ of forces, and not by any single force. - Let d[alpha] be the deviation of angular velocity to be produced in - the interval dt, and I the moment of the inertia of the body about an - axis through its centre of gravity; then (1/2)Id([alpha]^2) = I[alpha] - d[alpha] is the variation of the body's actual energy. Let M be the - moment of the unbalanced couple required to produce the deviation; - then by equation 57, S 104, the energy exerted by this couple in the - interval dt is M[alpha] dt, which, being equated to the variation of - energy, gives - - d[alpha] R^2W d[alpha] - M = I-------- = ---- . --------. (83) - dt g dt - - R is called the radius of gyration of the body with regard to an axis - through its centre of gravity. - - [Illustration: FIG. 133.] - - Now (fig. 133) let the required deviation be a rotation of the body BB - about an axis O, not traversing the centre of gravity G, d[alpha] - being, as before, the deviation of angular velocity to be produced in - the interval dt. A rotation with the angular velocity [alpha] about an - axis O may be considered as compounded of a rotation with the same - angular velocity about an axis drawn through G parallel to O and a - translation with the velocity [alpha]. OG, OG being the perpendicular - distance between the two axes. Hence the required deviation may be - regarded as compounded of a deviation of translation dv = OG.d[alpha], - to produce which there would be required, according to equation (82), - a force applied at G perpendicular to the plane OG-- - - W d[alpha] - P = --- . OG . -------- (84) - g dt - - and a deviation d[alpha] of rotation about an axis drawn through G - parallel to O, to produce which there would be required a couple of - the moment M given by equation (83). According to the principles of - statics, the resultant of the force P, applied at G perpendicular to - the plane OG, and the couple M is a force equal and parallel to P, but - applied at a distance GC from G, in the prolongation of the - perpendicular OG, whose value is - - GC = M/P = R^2/OG. (85) - - Thus is determined the position of the centre of percussion C, - corresponding to the axis of rotation O. It is obvious from this - equation that, for an axis of rotation parallel to O traversing C, the - centre of percussion is at the point where the perpendicular OG meets - O. - - S 125.* _To find the moment of inertia of a body about an axis through - its centre of gravity experimentally._--Suspend the body from any - conveniently selected axis O (fig. 48) and hang near it a small plumb - bob. Adjust the length of the plumb-line until it and the body - oscillate together in unison. The length of the plumb-line, measured - from its point of suspension to the centre of the bob, is for all - practical purposes equal to the length OC, C being therefore the - centre of percussion corresponding to the selected axis O. From - equation (85) - - R^2 = CG X OG = (OC - OG)OG. - - The position of G can be found experimentally; hence OG is known, and - the quantity R^2 can be calculated, from which and the ascertained - weight W of the body the moment of inertia about an axis through G, - namely, W/g X R^2, can be computed. - - [Illustration: FIG. 134.] - - S 126.* _To find the force competent to produce the instantaneous - acceleration of any link of a mechanism._--In many practical problems - it is necessary to know the magnitude and position of the forces - acting to produce the accelerations of the several links of a - mechanism. For a given link, this force is the resultant of all the - accelerating forces distributed through the substance of the material - of the link required to produce the requisite acceleration of each - particle, and the determination of this force depends upon the - principles of the two preceding sections. The investigation of the - distribution of the forces through the material and the stress - consequently produced belongs to the subject of the STRENGTH OF - MATERIALS (q.v.). Let BK (fig. 134) be any link moving in any manner - in a plane, and let G be its centre of gravity. Then its motion may be - analysed into (1) a translation of its centre of gravity; and (2) a - rotation about an axis through its centre of gravity perpendicular to - its plane of motion. Let [alpha] be the acceleration of the centre of - gravity and let A be the angular acceleration about the axis through - the centre of gravity; then the force required to produce the - translation of the centre of gravity is F = W[alpha]/g, and the couple - required to produce the angular acceleration about the centre of - gravity is M = IA/g, W and I being respectively the weight and the - moment of inertia of the link about the axis through the centre of - gravity. The couple M may be produced by shifting the force F parallel - to itself through a distance x. such that Fx = M. When the link forms - part of a mechanism the respective accelerations of two points in the - link can be determined by means of the velocity and acceleration - diagrams described in S 82, it being understood that the motion of one - link in the mechanism is prescribed, for instance, in the - steam-engine's mechanism that the crank shall revolve uniformly. Let - the acceleration of the two points B and K therefore be supposed - known. The problem is now to find the acceleration [alpha] and A. Take - any pole O (fig. 49), and set out Ob equal to the acceleration of B - and Ok equal to the acceleration of K. Join bk and take the point g so - that KG: GB = kg : gb. Og is then the acceleration of the centre of - gravity and the force F can therefore be immediately calculated. To - find the angular acceleration A, draw kt, bt respectively parallel to - and at right angles to the link KB. Then tb represents the angular - acceleration of the point B relatively to the point K and hence tb/KB - is the value of A, the angular acceleration of the link. Its moment of - inertia about G can be found experimentally by the method explained in - S 125, and then the value of the couple M can be computed. The value - of x is found immediately from the quotient M/F. Hence the magnitude F - and the position of F relatively to the centre of gravity of the link, - necessary to give rise to the couple M, are known, and this force is - therefore the resultant force required. - - [Illustration: FIG. 135.] - - S 127.* _Alternative construction for finding the position of F - relatively to the centre of gravity of the link._--Let B and K be any - two points in the link which for greater generality are taken in fig. - 135, so that the centre of gravity G is not in the line joining them. - First find the value of R experimentally. Then produce the given - directions of acceleration of B and K to meet in O; draw a circle - through the three points B, K and O; produce the line joining O and G - to cut the circle in Y; and take a point Z on the line OY so that YG X - GZ = R^2. Then Z is a point in the line of action of the force F. This - useful theorem is due to G. T. Bennett, of Emmanuel College, - Cambridge. A proof of it and three corollaries are given in appendix 4 - of the second edition of Dalby's _Balancing of Engines_ (London, - 1906). It is to be noticed that only the directions of the - accelerations of two points are required to find the point Z. - - For an example of the application of the principles of the two - preceding sections to a practical problem see _Valve and Valve Gear - Mechanisms_, by W. E. Dalby (London, 1906), where the inertia stresses - brought upon the several links of a Joy valve gear, belonging to an - express passenger engine of the Lancashire & Yorkshire railway, are - investigated for an engine-speed of 68 m. an hour. - - [Illustration: FIG. 136.] - - S 128.* _The Connecting Rod Problem._--A particular problem of - practical importance is the determination of the force producing the - motion of the connecting rod of a steam-engine mechanism of the usual - type. The methods of the two preceding sections may be used when the - acceleration of two points in the rod are known. In this problem it is - usually assumed that the crank pin K (fig. 136) moves with uniform - velocity, so that if [alpha] is its angular velocity and r its radius, - the acceleration is [alpha]^2r in a direction along the crank arm from - the crank pin to the centre of the shaft. Thus the acceleration of one - point K is known completely. The acceleration of a second point, - usually taken at the centre of the crosshead pin, can be found by the - principles of S 82, but several special geometrical constructions have - been devised for this purpose, notably the construction of Klein,[4] - discovered also independently by Kirsch.[5] But probably the most - convenient is the construction due to G. T. Bennett[6] which is as - follows: Let OK be the crank and KB the connecting rod. On the - connecting rod take a point L such that KL X KB = KO^2. Then, the crank - standing at any angle with the line of stroke, draw LP at right angles - to the connecting rod, PN at right angles to the line of stroke OB and - NA at right angles to the connecting rod; then AO is the acceleration - of the point B to the scale on which KO represents the acceleration of - the point K. The proof of this construction is given in _The Balancing - of Engines_. - - The finding of F may be continued thus: join AK, then AK is the - acceleration image of the rod, OKA being the acceleration diagram. - Through G, the centre of gravity of the rod, draw Gg parallel to the - line of stroke, thus dividing the image at g in the proportion that - the connecting rod is divided by G. Hence Og represents the - acceleration of the centre of gravity and, the weight of the - connecting rod being ascertained, F can be immediately calculated. To - find a point in its line of action, take a point Q on the rod such - that KG X GQ = R^2, R having been determined experimentally by the - method of S 125; join G with O and through Q draw a line parallel to - BO to cut GO in Z. Z is a point in the line of action of the resultant - force F; hence through Z draw a line parallel to Og. The force F acts - in this line, and thus the problem is completely solved. The above - construction for Z is a corollary of the general theorem given in S - 127. - - S 129. _Impact._ Impact or collision is a pressure of short duration - exerted between two bodies. - - The effects of impact are sometimes an alteration of the distribution - of actual energy between the two bodies, and always a loss of a - portion of that energy, depending on the imperfection of the - elasticity of the bodies, in permanently altering their figures, and - producing heat. The determination of the distribution of the actual - energy after collision and of the loss of energy is effected by means - of the following principles:-- - - I. The motion of the common centre of gravity of the two bodies is - unchanged by the collision. - - II. The loss of energy consists of a certain proportion of that part - of the actual energy of the bodies which is due to their motion - relatively to their common centre of gravity. - - Unless there is some special reason for using impact in machines, it - ought to be avoided, on account not only of the waste of energy which - it causes, but from the damage which it occasions to the frame and - mechanism. (W. J. M. R.; W. E. D.) - - -FOOTNOTES: - - [1] In view of the great authority of the author, the late Professor - Macquorn Rankine, it has been thought desirable to retain the greater - part of this article as it appeared in the 9th edition of the - _Encyclopaedia Britannica_. Considerable additions, however, have - been introduced in order to indicate subsequent developments of the - subject; the new sections are numbered continuously with the old, but - are distinguished by an asterisk. Also, two short chapters which - concluded the original article have been omitted--ch. iii., "On - Purposes and Effects of Machines," which was really a classification - of machines, because the classification of Franz Reuleaux is now - usually followed, and ch. iv., "Applied Energetics, or Theory of - Prime Movers," because its subject matter is now treated in various - special articles, e.g. Hydraulics, Steam Engine, Gas Engine, Oil - Engine, and fully developed in Rankine's The Steam Engine and Other - Prime Movers (London, 1902). (Ed. _E.B._) - - [2] Since the relation discussed in S 7 was enunciated by Rankine, an - enormous development has taken place in the subject of Graphic - Statics, the first comprehensive textbook on the subject being _Die - Graphische Statik_ by K. Culmann, published at Zurich in 1866. Many - of the graphical methods therein given have now passed into the - textbooks usually studied by engineers. One of the most beautiful - graphical constructions regularly used by engineers and known as "the - method of reciprocal figures" is that for finding the loads supported - by the several members of a braced structure, having given a system - of external loads. The method was discovered by Clerk Maxwell, and - the complete theory is discussed and exemplified in a paper "On - Reciprocal Figures, Frames and Diagrams of Forces," _Trans. Roy. Soc. - Ed._, vol. xxvi. (1870). Professor M. W. Crofton read a paper on - "Stress-Diagrams in Warren and Lattice Girders" at the meeting of the - Mathematical Society (April 13, 1871), and Professor O. Henrici - illustrated the subject by a simple and ingenious notation. The - application of the method of reciprocal figures was facilitated by a - system of notation published in _Economics of Construction in - relation to framed Structures_, by Robert H. Bow (London, 1873). A - notable work on the general subject is that of Luigi Cremona, - translated from the Italian by Professor T. H. Beare (Oxford, 1890), - and a discussion of the subject of reciprocal figures from the - special point of view of the engineering student is given in _Vectors - and Rotors_ by Henrici and Turner (London, 1903). See also above - under "_Theoretical Mechanics_," Part 1. S 5. - - [3] This is a particular case of a more general principle, that _the - motion of the centre of gravity of a body is not affected by the - mutual actions of its parts_. - - [4] J. F. Klein, "New Constructions of the Force of Inertia of - Connecting Rods and Couplers and Constructions of the Pressures on - their Pins," _Journ. Franklin Inst._, vol. 132 (Sept. and Oct., - 1891). - - [5] Prof. Kirsch, "Uber die graphische Bestimmung der - Kolbenbeschleunigung," _Zeitsch. Verein deutsche Ingen_. (1890), p. - 1320. - - [6] Dalby, _The Balancing of Engines_ (London, 1906), app. 1. - - - - -MECHANICVILLE, a village of Saratoga county, New York, U.S.A., on the -west bank of the Hudson River, about 20 m. N. of Albany; on the Delaware -& Hudson and Boston & Maine railways. Pop. (1900), 4695 (702 -foreign-born); (1905, state census), 5877; (1910) 6,634. It lies partly -within Stillwater and partly within Half-Moon townships, in the -bottom-lands at the mouth of the Anthony Kill, about 1-1/2 m. S. of the -mouth of the Hoosick River. On the north and south are hills reaching a -maximum height of 200 ft. There is ample water power, and there are -manufactures of paper, sash and blinds, fibre, &c. From a dam here power -is derived for the General Electric Company at Schenectady. The first -settlement in this vicinity was made in what is now Half-Moon township -about 1680. Mechanicville (originally called Burrow) was chartered by -the county court in 1859, and incorporated as a village in 1870. It was -the birthplace of Colonel Ephraim Elmer Ellsworth (1837-1861), the first -Federal officer to lose his life in the Civil War. - - - - -MECHITHARISTS, a congregation of Armenian monks in communion with the -Church of Rome. The founder, Mechithar, was born at Sebaste in Armenia, -1676. He entered a monastery, but under the influence of Western -missionaries he became possessed with the idea of propagating Western -ideas and culture in Armenia, and of converting the Armenian Church from -its monophysitism and uniting it to the Latin Church. Mechithar set out -for Rome in 1695 to make his ecclesiastical studies there, but he was -compelled by illness to abandon the journey and return to Armenia. In -1696 he was ordained priest and for four years worked among his people. -In 1700 he went to Constantinople and began to gather disciples around -him. Mechithar formally joined the Latin Church, and in 1701, with -sixteen companions, he formed a definitely religious institute of which -he became the superior. Their Uniat propaganda encountered the -opposition of the Armenians and they were compelled to move to the -Morea, at that time Venetian territory, and there built a monastery, -1706. On the outbreak of hostilities between the Turks and Venetians -they migrated to Venice, and the island of St Lazzaro was bestowed on -them, 1717. This has since been the headquarters of the congregation, -and here Mechithar died in 1749, leaving his institute firmly -established. The rule followed at first was that attributed to St -Anthony; but when they settled in the West modifications from the -Benedictine rule were introduced, and the Mechitharists are numbered -among the lesser orders affiliated to the Benedictines. They have ever -been faithful to their founder's programme. Their work has been -fourfold: (1) they have brought out editions of important patristic -works, some Armenian, others translated into Armenian from Greek and -Syriac originals no longer extant; (2) they print and circulate Armenian -literature among the Armenians, and thereby exercise a powerful -educational influence; (3) they carry on schools both in Europe and -Asia, in which Uniat Armenian boys receive a good secondary education; -(4) they work as Uniat missioners in Armenia. The congregation is -divided into two branches, the head houses being at St Lazzaro and -Vienna. They have fifteen establishments in various places in Asia Minor -and Europe. There are some 150 monks, all Armenians; they use the -Armenian language and rite in the liturgy. - - See _Vita del servo di Dio Mechitar_ (Venice, 1901); E. Bore, - _Saint-Lazare_ (1835); Max Heimbucher, _Orden u. Kongregationen_ - (1907) I. S 37; and the articles in Wetzer u. Welte, _Kirchenlexicon_ - (ed. 2) and Herzog, _Realencyklopadie_ (ed. 3), also articles by - Sargisean, a Mechitharist, in _Rivista storica benedettina_ (1906), - "La Congregazione Mechitarista." (E. C. B.) - - - - -MECKLENBURG, a territory in northern Germany, on the Baltic Sea, -extending from 53 deg. 4' to 54 deg. 22' N. and from 10 deg. 35' to 13 -deg. 57' E., unequally divided into the two grand duchies of -Mecklenburg-Schwerin and Mecklenburg-Strelitz. - -MECKLENBURG-SCHWERIN is bounded N. by the Baltic Sea, W. by the -principality of Ratzeburg and Schleswig-Holstein, S. by Brandenburg and -Hanover, and E. by Pomerania and Mecklenburg-Strelitz. It embraces the -duchies of Schwerin and Gustrow, the district of Rostock, the -principality of Schwerin, and the barony of Wismar, besides several -small enclaves (Ahrensberg, Rosson, Tretzeband, &c.) in the adjacent -territories. Its area is 5080 sq. m. Pop. (1905), 625,045. - -MECKLENBURG-STRELITZ consists of two detached parts, the duchy of -Strelitz on the E. of Mecklenburg-Schwerin, and the principality of -Ratzeburg on the W. The first is bounded by Mecklenburg-Schwerin, -Pomerania and Brandenburg, the second by Mecklenburg-Schwerin, -Lauenburg, and the territory of the free town of Lubeck. Their joint -area is 1130 sq. m. Pop. (1905), 103,451. - - Mecklenburg lies wholly within the great North-European plain, and its - flat surface is interrupted only by one range of low hills, - intersecting the country from south-east to north-west, and forming - the watershed between the Baltic Sea and the Elbe. Its highest point, - the Helpter Berg, is 587 ft. above sea-level. The coast-line runs for - 65 m. along the Baltic (without including indentations), for the most - part in flat sandy stretches covered with dunes. The chief inlets are - Wismar Bay, the Salzhaff, and the roads of Warnemunde. The rivers are - numerous though small; most of them are affluents of the Elbe, which - traverses a small portion of Mecklenburg. Several are navigable, and - the facilities for inland water traffic are increased by canals. Lakes - are numerous; about four hundred, covering an area of 500 sq. m., are - reckoned in the two duchies. The largest is Lake Muritz, 52 sq. m. in - extent. The climate resembles that of Great Britain, but the winters - are generally more severe; the mean annual temperature is 48 deg. F., - and the annual rainfall is about 28 in. Although there are long - stretches of marshy moorland along the coast, the soil is on the whole - productive. About 57% of the total area of Mecklenburg-Schwerin - consists of cultivated land, 18% of forest, and 13% of heath and - pasture. In Mecklenburg-Strelitz the corresponding figures are 47, 21 - and 10%. Agriculture is by far the most important industry in both - duchies. The chief crops are rye, oats, wheat, potatoes and hay. - Smaller areas are devoted to maize, buckwheat, pease, rape, hemp, - flax, hops and tobacco. The extensive pastures support large herds of - sheep and cattle, including a noteworthy breed of merino sheep. The - horses of Mecklenburg are of a fine sturdy quality and highly - esteemed. Red deer, wild swine and various other game are found in the - forests. The industrial establishments include a few iron-foundries, - wool-spinning mills, carriage and machine factories, dyeworks, - tanneries, brick-fields, soap-works, breweries, distilleries, numerous - limekilns and tar-boiling works, tobacco and cigar factories, and - numerous mills of various kinds. Mining is insignificant, though a - fair variety of minerals is represented in the district. Amber is - found on and near the Baltic coast. Rostock, Warnemunde and Wismar are - the principal commercial centres. The chief exports are grain and - other agricultural produce, live stock, spirits, wood and wool; the - chief imports are colonial produce, iron, coal, salt, wine, beer and - tobacco. The horse and wool markets of Mecklenburg are largely - attended by buyers from various parts of Germany. Fishing is carried - on extensively in the numerous inland lakes. - - In 1907 the grand dukes of both duchies promised a constitution to - their subjects. The duchies had always been under a government of - feudal character, the grand dukes having the executive entirely in - their hands (though acting through ministers), while the duchies - shared a diet (_Landtag_), meeting for a short session each year, and - at other times represented by a committee, and consisting of the - proprietors of knights' estates (_Ritterguter_), known as the - _Ritterschaft_, and the _Landschaft_ or burgomasters of certain towns. - Mecklenburg-Schwerin returns six members to the Reichstag and - Mecklenburg-Strelitz one member. - - In Mecklenburg-Schwerin the chief towns are Rostock (with a - university), Schwerin, and Wismar the capital. The capital of - Mecklenburg-Strelitz is Neu-Strelitz. The peasantry of Mecklenburg - retain traces of their Slavonic origin, especially in speech, but - their peculiarities have been much modified by amalgamation with - German colonists. The townspeople and nobility are almost wholly of - Saxon strain. The slowness of the increase in population is chiefly - accounted for by emigration. - -_History._--The Teutonic peoples, who in the time of Tacitus occupied -the region now known as Mecklenburg, were succeeded in the 6th century -by some Slavonic tribes, one of these being the Obotrites, whose chief -fortress was Michilenburg, the modern Mecklenburg, near Wismar; hence -the name of the country. Though partly subdued by Charlemagne towards -the close of the 8th century, they soon regained their independence, and -until the 10th century no serious effort was made by their Christian -neighbours to subject them. Then the German king, Henry the Fowler, -reduced the Slavs of Mecklenburg to obedience and introduced -Christianity among them. During the period of weakness through which the -German kingdom passed under the later Ottos, however, they wrenched -themselves free from this bondage; the 11th and the early part of the -12th century saw the ebb and flow of the tide of conquest, and then came -the effective subjugation of Mecklenburg by Henry the Lion, duke of -Saxony. The Obotrite prince Niklot was killed in battle in 1160 whilst -resisting the Saxons, but his son Pribislaus (d. 1178) submitted to -Henry the Lion, married his daughter to the son of the duke, embraced -Christianity, and was permitted to retain his office. His descendants -and successors, the present grand dukes of Mecklenburg, are the only -ruling princes of Slavonic origin in Germany. Henry the Lion introduced -German settlers and restored the bishoprics of Ratzeburg and Schwerin; -in 1170 the emperor Frederick I. made Pribislaus a prince of the empire. -From 1214 to 1227 Mecklenburg was under the supremacy of Denmark; then, -in 1229, after it had been regained by the Germans, there took place the -first of the many divisions of territory which with subsequent reunions -constitute much of its complicated history. At this time the country was -divided between four princes, grandsons of duke Henry Borwin, who had -died two years previously. But in less than a century the families of -two of these princes became extinct, and after dividing into three -branches a third family suffered the same fate in 1436. There then -remained only the line ruling in Mecklenburg proper, and the princes of -this family, in addition to inheriting the lands of their dead kinsmen, -made many additions to their territory, including the counties of -Schwerin and of Strelitz. In 1352 the two princes of this family made a -division of their lands, Stargard being separated from the rest of the -country to form a principality for John (d. 1393), but on the extinction -of his line in 1471 the whole of Mecklenburg was again united under a -single ruler. One member of this family, Albert (c. 1338-1412), was king -of Sweden from 1364 to 1389. In 1348 the emperor Charles IV. had raised -Mecklenburg to the rank of a duchy, and in 1418 the university of -Rostock was founded. - -The troubles which arose from the rivalry and jealousy of two or more -joint rulers incited the prelates, the nobles and the burghers to form a -union among themselves, and the results of this are still visible in the -existence of the _Landesunion_ for the whole country which was -established in 1523. About the same time the teaching of Luther and the -reformers was welcomed in Mecklenburg, although Duke Albert (d. 1547) -soon reverted to the Catholic faith; in 1549 Lutheranism was recognized -as the state religion; a little later the churches and schools were -reformed and most of the monasteries were suppressed. A division of the -land which took place in 1555 was of short duration, but a more -important one was effected in 1611, although Duke John Albert I. (d. -1576) had introduced the principle of primogeniture and had forbidden -all further divisions of territory. By this partition John Albert's -grandson Adolphus Frederick I. (d. 1658) received Schwerin, and another -grandson John Albert II. (d. 1636) received Gustrow. The town of -Rostock "with its university and high court of justice" was declared to -be common property, while the Diet or _Landtag_ also retained its joint -character, its meetings being held alternately at Sternberg and at -Malchin. - -During the early part of the Thirty Years' War the dukes of -Mecklenburg-Schwerin and Mecklenburg-Gustrow were on the Protestant -side, but about 1627 they submitted to the emperor Ferdinand II. This -did not prevent Ferdinand from promising their land to Wallenstein, who, -having driven out the dukes, was invested with the duchies in 1629 and -ruled them until 1631. In this year the former rulers were restored by -Gustavus Adolphus of Sweden, and in 1635 they came to terms with the -emperor and signed the peace of Prague, but their land continued to be -ravaged by both sides until the conclusion of the war. In 1648 by the -Treaty of Westphalia, Wismar and some other parts of Mecklenburg were -surrendered to Sweden, the recompense assigned to the duchies including -the secularized bishoprics of Schwerin and of Ratzeburg. The sufferings -of the peasants in Mecklenburg during the Thirty Years' War were not -exceeded by those of their class in any other part of Germany; most of -them were reduced to a state of serfdom and in some cases whole villages -vanished. Christian Louis who ruled Mecklenburg-Schwerin from 1658 until -his death in 1692 was, like his father Adolphus Frederick, frequently at -variance with the estates of the land and with members of his family. He -was a Roman Catholic and a supporter of Louis XIV., and his country -suffered severely during the wars waged by France and her allies in -Germany. - -In June 1692 when Christian Louis died in exile and without sons, a -dispute arose about the succession to his duchy between his brother -Adolphus Frederick and his nephew Frederick William. The emperor and the -rulers of Sweden and of Brandenburg took part in this struggle which was -intensified when, three years later, on the death of Duke Gustavus -Adolphus, the family ruling over Mecklenburg-Gustrow became extinct. At -length the partition Treaty of Hamburg was signed on the 8th of March -1701, and a new division of the country was made. Mecklenburg was -divided between the two claimants, the shares given to each being -represented by the existing duchies of Mecklenburg-Schwerin, the part -which fell to Frederick William, and Mecklenburg-Strelitz, the share of -Adolphus Frederick. At the same time the principle of primogeniture was -again asserted, and the right of summoning the joint _Landtag_ was -reserved to the ruler of Mecklenburg-Schwerin. - -Mecklenburg-Schwerin began its existence by a series of constitutional -struggles between the duke and the nobles. The heavy debt incurred by -Duke Charles Leopold (d. 1747), who had joined Russia in a war against -Sweden, brought matters to a crisis; the emperor Charles VI. interfered -and in 1728 the imperial court of justice declared the duke incapable of -governing and his brother Christian Louis was appointed administrator of -the duchy. Under this prince, who became ruler _de jure_ in 1747, there -was signed in April 1755 the convention of Rostock by which a new -constitution was framed for the duchy. By this instrument all power was -in the hands of the duke, the nobles and the upper classes generally, -the lower classes being entirely unrepresented. During the Seven Years' -War Duke Frederick (d. 1785) took up a hostile attitude towards -Frederick the Great, and in consequence Mecklenburg was occupied by -Prussian troops, but in other ways his rule was beneficial to the -country. In the early years of the French revolutionary wars Duke -Frederick Francis I. (1756-1837) remained neutral, and in 1803 he -regained Wismar from Sweden, but in 1806 his land was overrun by the -French and in 1808 he joined the Confederation of the Rhine. He was the -first member of the confederation to abandon Napoleon, to whose armies -he had sent a contingent, and in 1813-1814 he fought against France. In -1815 he joined the Germanic Confederation (Bund) and took the title of -grand duke. In 1819 serfdom was abolished in his dominions. During the -movement of 1848 the duchy witnessed a considerable agitation in favour -of a more liberal constitution, but in the subsequent reaction all the -concessions which had been made to the democracy were withdrawn and -further restrictive measures were introduced in 1851 and 1852. - -Mecklenburg-Strelitz adopted the constitution of the sister duchy by an -act of September 1755. In 1806 it was spared the infliction of a French -occupation through the good offices of the king of Bavaria; in 1808 its -duke, Charles (d. 1816), joined the confederation of the Rhine, but in -1813 he withdrew therefrom. Having been a member of the alliance against -Napoleon he joined the Germanic confederation in 1815 and assumed the -title of grand duke. - -In 1866 both the grand dukes of Mecklenburg joined the North German -confederation and the _Zollverein_, and began to pass more and more -under the influence of Prussia, who in the war with Austria had been -aided by the soldiers of Mecklenburg-Schwerin. In the Franco-German War -also Prussia received valuable assistance from Mecklenburg, Duke -Frederick Francis II. (1823-1883), an ardent advocate of German unity, -holding a high command in her armies. In 1871 the two grand duchies -became states of the German Empire. There was now a renewal of the -agitation for a more democratic constitution, and the German Reichstag -gave some countenance to this movement. In 1897 Frederick Francis IV. -(b. 1882) succeeded his father Frederick Francis III. (1851-1897) as -grand duke of Mecklenburg-Schwerin, and in 1904 Adolphus Frederick (b. -1848) a son of the grand duke Frederick William (1819-1904) and his wife -Augusta Carolina, daughter of Adolphus Frederick, duke of Cambridge, -became grand duke of Mecklenburg-Strelitz. The grand dukes still style -themselves princes of the Wends. - - See F. A. Rudloff, _Pragmatisches Handbuch der mecklenburgischen - Geschichte_ (Schwerin, 1780-1822); C. C. F. von Lutzow, _Versuch einer - pragmatischen Geschichte von Mecklenburg_ (Berlin, 1827-1835); - _Mecklenburgische Geschichte in Einzeldarstellungen_, edited by R. - Beltz, C. Beyer, W. P. Graff and others; C. Hegel, _Geschichte der - mecklenburgischen Landstande bis 1555_ (Rostock, 1856); A. Mayer, - _Geschichte des Grossherzogtums Mecklenburg-Strelitz 1816-1890_ (New - Strelitz, 1890); Tolzien, _Die Grossherzoge von Mecklenburg-Schwerin_ - (Wismar, 1904); Lehsten, _Der Adel Mecklenburgs seit dem - landesgrundgesetslichen Erbvergleich_ (Rostock, 1864); the - _Mecklenburgisches Urkundenbuch_ in 21 vols. (Schwerin, 1873-1903); - the _Jahrbucher des Vereins fur mecklenburgische Geschichte und - Altertumskunde_ (Schwerin, 1836 fol.); and W. Raabe, _Mecklenburgische - Vaterlandskunde_ (Wismar, 1894-1896); von Hirschfeld, _Friedrich Franz - II., Grossherzog von Mecklenburg-Schwerin und seine Vorganger_ - (Leipzig, 1891); Volz, _Friedrich Franz II._ (Wismar, 1893); C. - Schroder, _Friedrich Franz III._ (Schwerin, 1898); Bartold, _Friedrich - Wilhelm, Grossherzog von Mecklenburg-Strelitz und Augusta Carolina_ - (New Strelitz, 1893); and H. Sachsse, _Mecklenburgische Urkunden und - Daten_ (Rostock, 1900). - - - - - - -End of the Project Gutenberg EBook of Encyclopaedia Britannica, 11th -Edition, Volume 17, Slice 8, by Various - -*** END OF THIS PROJECT GUTENBERG EBOOK ENCYCLOPAEDIA BRITANNICA *** - -***** This file should be named 42473.txt or 42473.zip ***** -This and all associated files of various formats will be found in: - http://www.gutenberg.org/4/2/4/7/42473/ - -Produced by Marius Masi, Don Kretz and the Online -Distributed Proofreading Team at http://www.pgdp.net - - -Updated editions will replace the previous one--the old editions -will be renamed. - -Creating the works from public domain print editions 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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