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diff --git a/old/67166-0.txt b/old/67166-0.txt deleted file mode 100644 index a7a3928..0000000 --- a/old/67166-0.txt +++ /dev/null @@ -1,8615 +0,0 @@ -The Project Gutenberg eBook of Self-Help Mechanical Drawing, by -Nehemiah Hawkins - -This eBook is for the use of anyone anywhere in the United States and -most other parts of the world at no cost and with almost no restrictions -whatsoever. You may copy it, give it away or re-use it under the terms -of the Project Gutenberg License included with this eBook or online at -www.gutenberg.org. If you are not located in the United States, you -will have to check the laws of the country where you are located before -using this eBook. - -Title: Self-Help Mechanical Drawing - An Educational Treatise - -Author: Nehemiah Hawkins - -Release Date: January 15, 2022 [eBook #67166] - -Language: English - -Produced by: deaurider, Harry Lamé and the Online Distributed - Proofreading Team at https://www.pgdp.net (This file was - produced from images generously made available by The - Internet Archive) - -*** START OF THE PROJECT GUTENBERG EBOOK SELF-HELP MECHANICAL -DRAWING *** - - - - Transcriber’s Notes - - Text printed in italics in the source document has been transcribed - _between underscores_, bold face text =between equal signs=. Small - capitals have been replaced with ALL CAPITALS. - - More Transcriber’s Notes may be found at the end of this text. - - -[Illustration] - - -[Illustration] - - - - - SELF-HELP - - MECHANICAL DRAWING - - AN EDUCATIONAL TREATISE - -[Illustration] - - - - - “_LEARN TO DO A THING BY DOING IT._”--_OLD PROVERB_ - - SELF-HELP - MECHANICAL DRAWING - - _AN EDUCATIONAL TREATISE_ - - [Illustration] - - BY - N. HAWKINS, M. E. - - _Author of Handbook of Calculations, etc._ - - _New York_: THEO. AUDEL & CO., Publishers - 1902 - - - _Copyrighted - by - Theo. Audel & Co. - New York - 1902_ - - - _This work - is - most kindly and - respectfully dedicated to - THE COMING MAN - who at the present time - is undoubtedly devoting - a goodly share of - his spare time to - the study of - drawing._ - - -[Illustration] - - -[Illustration: INTRODUCTION] - - - - -Preface. - - -_It is because of a personal and practical experience of the advantage -to be gained by the possession of a knowledge of drawing, that the -author is prompted to undertake the rather pleasant task of producing a -self-help book relating to the subject._ - -_Since the days of youthful endeavor, the author has passed through an -extended experience of mechanical life, and scarcely ever without chalk, -pencil or instrument in hand, to illustrate by sketch or drawing, the -tools to be employed, or to picture the finished product; accordingly, -throughout this work, words of explanation and the drawings will go -together to aid the diligent student._ - -_It has been said by an eminent writer, that “one workman is superior to -another--other circumstances being the same--directly in proportion to -his knowledge in drawing, and those who are ignorant of it must in many -respects be subservient to others who have obtained that knowledge.”_ - -_It has been also said that no man is fitted to be foreman of a shop who -cannot draw, and it is generally true that no one will be appointed to -that position, except temporarily, who does not possess some knowledge -of the art, either “freehand” or instrumental._ - -_It is a question how far a good working knowledge of drawing can be -attained without a teacher; it is true that but few have become -proficient without such aid, but it is equally true that “self-help” has -been the key note to all advancement._ - -_The author received personal instruction in several ways and times, at -home, in school, in an architect’s office, and under an experienced -mechanical engineer, but it was in the early morning hours of a bright -summer time--lang syne--that he made his first serious attempt to master -the art of mechanical drawing. It was a struggle and a battle to hold -himself down to “the board” to the finish, but it was a victory--one, -won over slothfulness and impatience, and of such a nature as to warrant -the use of the term “self-help” to the encouragement of others._ - -_In conclusion two sentiments may be added; if a good working knowledge -of drawing is “worth the while” then, 1, the student should be -thoroughly in earnest in acquiring it; 2, he should be willing to take -sufficient time and give much hard study to gain the skill necessary for -success._ - -_This persistence is not irksome. It carries its own reward, and the -results are definite and sure._ - - “_One step and then another, and the longest walk is ended; - One stitch and then another, and the largest rent is mended. - One brick upon another, and the highest wall is made; - One flake upon another, and the deepest snow is laid._” - - - - -Introduction. - - -Drawing is one of the arts; art relates to something to be done, and art -in the industrial and mechanical sense aims chiefly at utility, and is -governed by exact rules; hence mechanical drawing--so-called--tends -first to be useful and helpful, and second to accuracy in execution, -including most minute details; it aspires to the perfection of nature in -adaptability of the means to the end. - -Drawing constitutes a universal language, to acquire which is a matter -of importance, for by its use one is able to illustrate the form and -dimensions of an object, device, or utility, in very much less time, and -far more clearly, than by a verbal description. - -To a person who may not be able perfectly to understand the language of -a country, to be able to draw is an aid and a safeguard; to use the -words of Sir Joshua Reynolds, “the pencil speaks the language of every -land.” - -In extensive iron works and metal-working establishments the designer -and draughtsman is always in demand. His services are indispensable and -his position is a highly responsible one. It becomes his special -province to design improvements, to furnish sketches and to make -finished drawings; to calculate strains, strength, power, motion, -weight, friction and durability. All this and much more is the -professional draughtsman’s work. - -In “directory” classification, he who accomplishes such comprehensive -results as above described is termed a “Draughtsman,” but the word has -as wide a meaning as “Engineer,” which takes in civil, mechanical, -naval, sanitary, steam and other engineering specialists. So, in -drafting, it includes the office boy employed in making blue prints, it -embraces the copyists, tracers and assistants, as well as the head -draughtsman and chief engineer. - -Consequently the range is wide, and the line hard to draw between -draughtsmen who work with their hands, and those who work with their -brains. It may be added that the best men are too frequently -undervalued, owing to the unavoidable difficulty in distinguishing the -difference in true worth, between the two widely separated classes. - -It may be remarked that they only draw well who draw intelligently; -aptness in this, as in many other virtues, is a matter of slow growth, -“here a line and there a line”--it’s the proper direction, not the rate -of progress, that counts in the end. - -There are several methods of drawing--1, Free-hand; 2, Instrumental; 3, -Geometrical; 4. Perspective. In the first the work, also termed -sketching, is executed by pencil, pen, crayon, or even paint-brush; in -the second the result is attained by the use of rule, tee-square, -drawing pen, etc.; this method is also denominated mechanical drawing, -and suggests the title of this volume. - -The great usefulness, not to say necessity, of readiness in executing -accurately, drawings “to scale,” is emphasized by the fact that now, -more than ever, is all machinery designed, and it may almost be said, is -“built,” in the draughting room--this is a valuable hint relating to -“reading” drawings. - -It is wise, as well as easy, to begin at the beginning of things; thus, -it is altogether the good part to mount a ladder by the first and second -rounds rather than to attempt it by taking the third, sixth, ninth, -etc.--especially are first and second rounds the very best to start -upon; “Chalk-work,” is the first subject introduced, next, that of -“Free-hand.” These are the first steps leading upward in this most -agreeable attainment--skill in illustrating and designing of objects, -tools, and utilities. - -A single word of advice before introducing the elementary work connected -with mechanical drawing: if the student should experience difficulty in -mastering the diagrams and curves abounding in this book, let him -consult an experienced draughtsman or teacher, who, by a few strokes of -a lead pencil, can easily make them plain; that knowledge--which cannot -be printed or self-taught--termed _the Craftsman’s Art_, is communicated -largely by personal telling and showing, from man to man; in drawing, -this help should be thankfully availed of, when necessity arises. - - NOTE.--Sketching is often in demand because there is no time for - finished or careful drawings, and the one who can draw a few lines in - a moment to let a sudden necessity be known, is the man of the hour. - All candidates for First Class Engineer’s Certificates in marine - service in the navy have to undergo an examination in rough drawing; - this is intended not so much as a proof of the applicant possessing - the capability of a draughtsman, but in the event of any injury to the - engines in his charge, so that he may be able to send to his - Superintendent a rough drawing of the particular part, properly - dimensioned, so that it could be worked from, and time saved on the - arrival of the ship at the port where the repairs are to be done. - -[Illustration] - - -[Illustration: PLAN OF THE WORK] - - “_No matter how thorough our education may have been at the first, - rules and formulas will slip from the memory, and every day’s - experience gives additional evidence of the truth of the old adage - that_ ‘THE KEY THAT RESTS, RUSTS.’”--SIMPSON BALLARD. - - - - -The Plan of the Work. - - -The purpose or scope of this work may be briefly stated: It is to aid -the aspiring student in making the first advance towards a thorough and -useful knowledge of drawing in its several divisions, as elsewhere -defined. - -The method to be followed in presenting the subject will be the natural -order clearly and simply defined, as “from the less to the greater.” - -The first subject to be discussed comes under the heading of chalk-work, -_i. e._, such drawings as can be executed on a blackboard, a floor, or -even on more primitive surfaces, such as a smooth stone or board. - -This is indeed a lowly beginning, but the author is quite confident it -will awaken as much interest as any part of the book--even in the most -experienced in the art of drawing, as to them it will revive the -ambitions and first crude attempts made in the golden days of their -youth. - -Let it be clearly understood by all, and especially by those who wish to -learn drawing, that the study of this delightful art does not require -any special qualification. We need only ask one question: Have you -learned to write? If so, be assured you may learn to draw, but to all -the same rule applies, first the elements of the art, and afterwards the -more advanced study. - -It is not expected that all should exhibit a decided taste for drawing, -for the possession of this is rather a gift of nature than the result of -education; but a knowledge of principles and a certain amount of -executive ability may be obtained by every one of average capacity, and -whatever the natural power may be, it will be increased and developed by -exercise; if the progress is steady and continuous and in the right -direction, success is sure to crown the work. - -The second division of the book will be free-hand drawing, _i. e._, that -which is executed without instruments. Nothing to instruct has been -spared in this important step in the path of advancement. - -The illustrations accompanying the two opening sections have been made -designedly elementary, for there are many who have a taste for drawing -and who have a desire to learn, who from place of residence or other -circumstances have not the opportunity of receiving the assistance of a -master. To such this book presents itself as a friend directing to the -right road, talking, reasoning, and explaining by the way. - -The “chalk-work” and “free-hand” sections of the book relate to the -foundations upon which all must rest who seek the aid to be derived from -the art; hence, the following pages are written with a view to encourage -all, and those who are prepared to follow the directions given in them -may look forward to the possession of sufficient drawing power to add to -their usefulness in after life. - -At this point of attainment there arises a need to know the meaning of -many words and phrases used by draughtsmen; these are grouped -alphabetically from A to Z under the heading of,---- - -Useful Terms and Definitions: Memorizing these few pages will be of -benefit, as an intimate knowledge of the language of the drawing office -stamps a man as worthy of a hearing, and assures attention to anything -which he may write or say pertaining to the art. - -After the Definitions the subject explained will be the Instruments and -Materials used in mechanical drawings; following in due course appear -Geometrical and Mechanical Drawing, Gearing, Linear Perspective, -Projection, Shading, Tracing, Lettering, Drawing Office Rules, Reading -Drawings, Useful Tables and a General Index, to which the student is -referred. A careful reading is requested to the following helpful note. - -Grateful acknowledgment is made to George Perrott, Esq., M. E., for -practical and technical assistance throughout the work, and to Theo. -Lucas, Engineer, for text and illustrations in the portions of the book -relating to Linear Perspective and Projection. - - NOTE.--In MACHINERY Prof. Chas. H. Benjamin says, referring to - drawing, under heading “How and what to study,” “... I have so far - said nothing about drawing, for I do not think it of much use to learn - that, until you know what you want of it. All this time that you have - been studying mechanism you should have had a sketch book or pad of - note paper, and made free hand sketches of mechanical movements which - interested you and of various machine details. You should accustom - yourself to use drawing as a means of expressing ideas, just as you - use written words, so that it becomes a second nature to you to sketch - anything you wish to remember or describe. If you work from - blue-prints in the shop, or if you can borrow some to study, this will - help you to understand how a drawing is made. You can get some drawing - instruments at any time and begin to practice on drawing straight - lines and circles, so as to become familiar with the instruments. And - here it will be of great benefit to you if you can attend an evening - drawing school for one night in the week at least. - - “When you have become sufficiently familiar with the principles of - drawing, a book on mechanism will tell you how to draw gear teeth and - cams, and how to design various link motions. Make up your own - problems from what you see in the shop and make your drawing a means - to an end and not the principal thing; it is of little use to be able - to make a nice drawing unless you know what to draw and why. - - “Drawing is a convenient tool as an aid in expressing to others the - ideas which you wish to convey; in all cases take the problems and the - ideas from your every-day work and that which is around you; your - success will depend upon the close connection which you keep at all - times between your acquired knowledge and your practical work.” - - - - -General List of Contents. - - - INTRODUCTION, 1-24 - - CHALK WORK, 25-38 - - PRELIMINARY TERMS AND DEFINITIONS, 39-52 - - FREEHAND DRAWING, 53-78 - - GEOMETRICAL DRAWING, 79-100 - - DRAWING MATERIALS AND INSTRUMENTS, 101-134 - - MECHANICAL DRAWING, 135-188 - - PENCILING, 139-147 - - PROJECTION, 148-164 - - INKING IN DRAWINGS, 167-170 - - LETTERING DRAWINGS, 171-175 - - DIMENSIONING DRAWINGS, 176-179 - - SHADING DRAWINGS, 180-181 - - SECTION LINING AND COLORS, 182-185 - - REPRODUCING DRAWINGS, 186-188 - - DRAWING OFFICE RULES, 189-195 - - GEARING, 197-208 - - DESIGNING GEARS, 209-216 - - WORKING DRAWINGS, 219-227 - - READING WORKING DRAWINGS, 228-230 - - PATENT OFFICE RULES FOR DRAWINGS, 231-236 - - USEFUL HINTS AND “POINTS,” 237-244 - - LINEAR PERSPECTIVE, 245-265 - - PERSONAL, BY THE EDITOR, 281 - - USEFUL TABLES, 269-280 - - REFERENCE INDEX, 283 - - -[Illustration: CHALK-WORK] - - - The peculiarity of all art is that it cannot - be communicated in writing alone, craft is a - term which is synonymous with art; a craft - requires manual dexterity which cannot be - taught in books. - - - - -Chalk Work. - - -The blackboard has been well called the great weapon of the modern -educator; this is especially true in reference to instruction in an art -dealing with lines, curves and figures. - -[Illustration: Fig. 6.] - -Many a man can chalk out on a blackboard, or on a piece of sheet-iron, -or on the floor, just what he wants to show, and make his meaning very -plain; hence, in every workshop, and many other places, a blackboard is -more than useful, and it has been said that no draughting office is -complete without one. - -Fig. 6 represents a chalk-crayon. - -[Illustration: Fig. 7.] - -[Illustration: Fig. 8.] - -Figs. 7 and 8 need no explanation, as they represent two forms of the -well-known blackboard. - -Chalk lines have this advantage--they are easily altered or rubbed out -when not needed any longer. The work executed upon a blackboard is -mostly done by hand, without aid from instruments; a few tools, however, -are useful--such as, 1, large wooden blackboard compasses holding a -crayon, which are made and sold by the trade in size twelve inches to -thirty inches in length; 2, a straight-edge; and 3, some crayons. With -the compasses circles and part of the circle can be made, and with the -straight-edge the larger lines can be drawn. - -These instruments are shown on page 29, and are, 1, compasses, for -holding chalk for making circles; 2, a tee-square; 3, a straight-edge; -4, a protractor for measuring angles; 5, a triangle 60° and 30°; 6, a -brass holder for crayons. - -Blackboard Drawing.--The use of a blackboard comes principally and -properly under the head of free-hand drawing, but its importance is such -that a separate division of the volume is assigned to it. - -[Illustration] - -Thus, chalk-work may be considered the first lesson in “free-hand,” as -all the examples can also be most profitably practiced with pencil and -paper. - -Very rapid drawing upon the board should not be encouraged, as it is -likely not to be accurate enough; again, the board should be entirely -free from grease. Cloths, sponges or chamois skin rubbers may be used to -erase or change the chalk marks. Vertical lines should be drawn from -above downward; short lines should be drawn with the fingers alone, -those somewhat longer with the hand, using the wrist-joint; the still -longer lines with the forearm, using the elbow-joint; those longer yet -with the whole arm, using the shoulder-joint; lines should always be -drawn with a uniform motion, slow enough for the eye to follow. - -Practice in chalk-work should alternate with sketching in a sketchbook -and with geometrical drawing--to be hereafter described. The student -should practice a short time on the board, at least once a week; large -sizes are the most profitable for the representations to be made; when -drawing in different directions the hand should be turned, not the paper -or board; the hand should never be allowed to obstruct the sight, hence -the hand and fingers should be held in a position of freedom--with -fingers not nearer than 1¹⁄₂ or 2 inches from the board. - - NOTE.--The first lesson of any kind the author received in drawing was - to make a straight line; this was effected by holding the pencil - nearly erect and guiding it along by the aid of the little finger held - pressed against the edge of a board; this was a useful item of - knowledge, as proved by passing years. - - A well-known artist, in telling his early experience, said: “The first - thing I was taught was to draw a line, divide it, erect a - perpendicular from its center, and afterwards to divide the angle made - by the perpendicular.” In answer to a question asking how long he was - kept at the lines, he replied, “about two months--or a month or two,” - indicating that even the longer time would have been well spent in - learning to draw a straight line. - - -PREPARATORY PRACTICE IN DRAWING. - -Every visible object is bounded by lines which enable the observer to -determine its shape. If these lines are straight or curved, the shape of -the object is regular; if broken, the shape of the object is irregular. - -The elements, then, of form are lines, straight, curved, or broken, and -these, therefore, furnish the beginning of all instruction in free-hand -or mechanical drawing. - - -PERPENDICULAR LINES. - -Fig. 15 shows six lines--upright and perpendicular, with points or -“dots” indicated at the top and bottom of each line; to draw these, -proceed thus: - -[Illustration: Fig. 15.] - -[Illustration: Fig. 16.] - -The learner should stand with his right shoulder opposite the board, and -the weight of the hand and the arm should be allowed to fall naturally; -now, make on the board two points, one being six inches above the other, -these being merely “dots,” shown at the ends of the lines, figs. 15, -etc., and made with two motions; the line between the points should now -be drawn not too quickly from the upper to the lower point; three -movements of the hand and arm complete the line; to draw the other five -lines the movements have simply to be repeated. - -If the student pronounces to himself “one,” “two,” “three,” at each -motion, it will be helpful; in this exercise, fig. 15, the aim is to -make six lines, each line being parallel to the first. Again, in the -example, it is intended that the lower point should be made first, next -the upper, and lastly the line drawn from the upper to the lower point, -but the order may be reversed; at _one_ the upper point, at _two_ the -lower, at _three_ the stroke upwards to complete the line. - - -HORIZONTAL LINES. - -To make these as shown in fig. 16, proceed as follows: With the word -_one_ make a point, with _two_ another point six inches at the left, -with _three_ draw a straight line from the left point to the right. All -added lines should be parallel: for practice, reverse the process thus, -_one_, make a “point,” at _two_ another point at the right, at _three_ -draw line to the left. - -The student will note that the two motions--at the words _one_ and -_two_--are to fix the positions of the ends of the lines; this practice -will be found useful in the most advanced examples and an item of -elementary practice never to be forgotten--like the help to be derived -by the first round of a ladder. - -[Illustration: Fig. 17.] - -[Illustration: Fig. 18.] - - -OBLIQUE LINES. - -In drawing oblique straight lines as shown in fig. 17, at the word _one_ -let the student make the lower point; at the word _two_ the upper, a -little to the right of the lower; at the word _three_ draw a line -quickly from the upper to the lower point. In pronouncing the words -_one_, _two_, _three_, let the student make the additional parallel -lines. - -As shown in fig. 18, at the word _one_ make the lower point; at the word -_two_ the upper point, a little _to the left_; at the word _three_ draw -a line rapidly from the upper to the lower point, and “timing” the -process by repeating _one_, _two_, _three_, make the additional parallel -lines. - -[Illustration: Fig. 19.] - - -BROKEN LINES. - -A broken line is composed of two or more straight lines at angles to -each other (see fig. 19). To draw them begin (saying) _one_, make a -point; _two_ a point below at the left; _three_, a point above at the -left; _four_, draw a line from the left hand point to the lower point; -at the word _five_, from the lower point to the upper right hand point. -For practice draw numerous lines in the same way, keeping them parallel -to each other, as shown in fig. 20. - -[Illustration: Fig. 20.] - -[Illustration: Fig. 21.] - -[Illustration: Fig. 22.] - -[Illustration: Fig. 24.] - -[Illustration: Fig. 23.] - -[Illustration: Fig. 25.] - -In example, fig. 21, the arrangement of the points is changed--let the -student draw at the words, as follows: _One_, a point; _two_, a point -above at the left; _three_, a point below at the left; _four_, draw from -the point at the left to the upper point; _five_, from the upper point -to the lower right hand point; continue to add parallel lines to -complete the figure as shown. - -Figs. 22 and 23 are given as examples to practice, making first the -points and then the connecting lines and afterward the parallel lines to -complete the figures. - - -CURVED LINES. - -To draw curved lines, as shown in fig. 24. At the word _one_, point; at -the word _two_, point three inches directly above; _three_, at the same -distance above again make a point; now draw a curve as shown, joining -the middle point and the upper point; now draw the curve as shown below -it; finally complete figure as shown. - -[Illustration: Fig. 26.] - -[Illustration: Fig. 27.] - -[Illustration: Fig. 28.] - -[Illustration: Fig. 29.] - -[Illustration: Fig. 30.] - -Figs. 25 to 30 are to be practiced, making first the points and then -connecting them by the curves to complete the figures. - -When two or more students are working together, with each having a -blackboard, the counting may be in concert--or a teacher could count for -a class. In these line examples care should be used in making them of -uniform length. There is a difference to be noted between a crooked line -and a broken line, the latter being a straight line and the former -deviating from it. - -Square chalk crayons are the best for hand work, as lines of an even or -uniform width can be drawn with them. - -A very fine effect is produced by using two thicknesses of chalk, one -being double the thickness of the other; the heavy lines being used on -the shade side of objects will produce a good effect, giving thickness -and body to the object. - -Round chalk crayons are used in the compasses to draw circles, but hand -lines drawn with them are not so neat as those produced with the -square-shaped chalk. - -To obliterate or remove the construction, or false lines made on the -blackboard, a wooden handle two inches in diameter with a cone end 3 or -4 inches long, covered with chamois skin or soft cloth tightly wrapped -round the cone and fastened with a tack or drawing pin, makes the best -implement to erase lines not required, the point of the cone will remove -these without destroying the lines or curves which meet them. - -Sponges, chamois skin or cloth rubbers are used to rub out the chalk -drawings and clean the blackboard. - -The best height for a diagram on the blackboard is not higher than the -head, nor lower than the elbow. - -Horizontal lines should be made from the left to the right; the body and -arm being moved with the hand, and kept in the same relative position -with it, will steady the hand. - -Curved lines to the left should be drawn first, enabling the eye to take -in not only the curve in process of formation but that already made. - -Passing the crayon in the hand, over the intended curve previous to -marking it, will guide the eye and give confidence to the hand in -chalking the curve. - -A proper distance from the blackboard is essential, the face being about -two feet away from it. - -Draw with the whole arm extended from the shoulder-joint, not from the -elbow or wrist. - -[Illustration: Fig. 31.] - -[Illustration: “There are more ways than one of telling things: by -speech, by writing, by printing, also by pictures and drawings.” -Knowles] - - -[Illustration: TERMS DEFINITIONS] - -[Illustration: ELECTRA.] - - - - -Preliminary Terms and Definitions. - - -Like all the arts, drawing has a nomenclature of its own, and nothing -can be more helpful to the beginner than to know the name of things -relating to the art of drawing. This is a language almost peculiar to -itself, and used daily and hourly by many thousands of superintendents, -foremen and master mechanics, as well as by owners, designers and -draughtsmen, hence its introduction at this early stage. - - =ALTITUDE.=--This is the elevation of an object above its base, or the - perpendicular distance between the top and bottom of a figure. - -[Illustration: Fig. 33.] - -[Illustration: Fig. 34.] - -[Illustration: Fig. 35.] - -[Illustration: Fig. 36.] - -[Illustration: Fig. 37.] - - =ANGLE= is the difference in the direction of two lines which meet or - tend to meet. The lines are called _the sides_ and the point of - meeting, the _vertex_ of the lines. - - To make an angle apparent, the two lines must meet in a point, as _A - B_ and _A C_, which meet in the point _A_, as shown in fig. 33. - - Angles are measured by degrees. - - A _Degree_ is one of the three hundred and sixty equal parts of the - space about a point in a plane. - - Angles are distinguished in respect to magnitude by the terms Right, - Acute and Obtuse Angles. - - A _Right Angle_ is that formed by one line meeting another, so as to - make equal angles with that other. - - The lines forming a right angle are _perpendicular to each other_. - - An _Acute Angle_ is less than a right angle. See Fig. 35. - - An _Obtuse Angle_ is greater than a right angle. See Fig. 36. - - Obtuse and acute angles are also called _oblique angles_; and lines - which are neither parallel nor perpendicular to each other are called - _oblique lines_. - - The _Vertex_ or _Apex_ of an angle is the point in which the including - lines meet. - - An angle is commonly designated by a letter at its vertex; but when - two or more angles have their vertices at the same point, they cannot - be thus distinguished. - - For example, when the three lines _A B_, _A C_, and _A D_ in fig. 37 - meet in the common point _A_, we designate either of the angles - formed, by three letters, placing that at the vertex between those at - the opposite extremities of the including lines. Thus, we say, the - angle _B A C_, etc. - - =APEX.=--The summit or highest point of an object. - - =ARC.=--See circle. - - =AXIS OF A SOLID.=--An imaginary straight line passing through its - center. - - =AXIS OF A FIGURE.=--A straight line passing through the center of a - figure, and dividing it into two equal parts. - - =BASE.=--The base of a solid figure is that on which it stands--the - lowest part. - - =BISECT.=--To divide into two equal parts. - - =BISECTOR.=--A line which bisects. - -[Illustration: Fig. 38.] - - =CIRCLE.=--A _Circle_ is a plane figure bounded by one uniformly - curved line, all of the points in which are at the same distance from - a certain point within, called the _Center_. - - The _Circumference_ of a circle is the curved line that bounds it. - - The _Diameter_ of a circle is a line passing through its center, and - terminating at both ends in the circumference, as _A C B_. - - The _Radius_ of a circle is a line extending from its center to any - point in the circumference. It is one-half of the diameter. All the - diameters of a circle are equal, as are also all the radii _C D_, _C - B_ and _C A_. - - An _Arc_ of a circle is any portion of the circumference, as _B D_ and - _A D_. - - _Semi-Circle._--Half a circle formed by bisecting it with a diameter, - as _A C B_. Fig. 38. - - An angle having its vertex at the center of a circle is measured by - the arc intercepted by its sides. Thus, the arc _A D_ measures the - angle _A C D_, and in general, to compare different angles, we have - but to compare the arcs, included by their sides, of the equal circles - having their centers at the vertices of the angles. - - =CIRCUMSCRIBE.=--To draw a line of figures about or outside, such as a - circle drawn around a square touching its corners or angles. - - _Inscribe._--To draw a line or figure inside or on the interior, such - as a circle drawn within a square touching its sides. - - =CONCAVE.=--Curving inwardly. - - =CONE.=--A solid body or figure having a circle for its base, and its - top terminated in a point or vertex. - - =CONSTRUCTION.=--The making of any object. - - =CONTOUR.=--The outline of the general appearance of an object. - - =CONVERGENCE.=--Lines extending towards a common point. - - =CONVEX.=--Rising or swelling into a round form--the opposite to - concave. - - =CORNER.=--The point of meeting of the edges of a solid, or the two - sides of a plane figure. - - =CROSS-HATCHES.=--In free-hand drawing the use of lines crossing each - other to produce light and shade effects. - - =CURVE.=--A line of which no part is straight. - - _Reversed Curve._--One whose curvature is first in one direction and - then in the opposite direction. - - _Spiral Curve._--A plain curve which winds about and recedes, - according to some law, from its point of beginning, which is called - its center. - - =CYLINDER.=--A solid bounded by a curved surface and by two opposite - faces called bases; the bases may be any curved figures and give the - name to the cylinder; thus a circular cylinder is one whose bases are - circles. - - =CYLINDRICAL.=--Having the general form of a cylinder. - - =DEGREE.=--The 360th part of a circle. - - =DESCRIBE.=--To make or draw a curved line; to draw a plan. - - =DESIGN.=--Any arrangement or combination to produce desired results - in industry or art. To delineate a form or figure by drawing the - outline--a sketch. - - =DEVELOP.=--To unroll or lay out. - - =DIAGONAL.=--A right line drawn from angle to angle of a quadrilateral - or many angled figure and dividing it into two parts. - - =DIAMETER.=--A right line passing through the center of a circle or - other round figure terminated by the curve and dividing the figure - symmetrically into two equal parts. - - =EDGE.=--The intersection of any two surfaces. - - =ELEVATION.=--The term elevation, vertical projection and _front - view_--applied to drawings--all have the same meaning. - - =FACE.=--One of the plane surfaces of a solid; it may be bounded by - straight or curved edges. - - =FINISHING.=--Completing a drawing whose lines have been determined by - erasing unnecessary lines and strengthening and accentuating where - this is needed. - - =FORESHORTENING.=--Apparent decrease in length, owing to objects being - viewed obliquely; thus a wheel, when seen obliquely, instead of - appearing round, presents the appearance of an ellipse. - - =FREE-HAND.=--Executed by the hand unaided by instruments. - - =GENERATED.=--Produced by. - - =GEOMETRIC.=--According to geometry. - - =HALF-TINT.=--The shading produced by means of parallel equidistant - lines. - - =HEMISPHERE.=--Half a sphere obtained by bisecting a sphere by a - plane. - - =HORIZONTAL.=--Parallel to the surface of smooth water. In drawing, a - line drawn parallel to the top and bottom of the sheet is called - horizontal. - - =INSCRIBE.=--See circumscribe--its opposite. - - =INSTRUMENTAL.=--By the use of instruments. - - =LINE.=--A line has length, only, as A C; a right line is a straight - line, the shortest line that can be drawn between two points, A----C. - - _Straight._ One which has the same direction throughout its entire - length. - - _Curved._ One no part of which is straight. - - _Broken._ One composed of different successive straight lines. - - _Mixed._ One of straight and curved lines. - - _Center._ A line used to indicate the center of an object. - - _Construction._ A working line used to obtain required lines. - - _Dotted._ A line composed of short dashes. - - - - - - - - _Dash._ A line composed of long dashes. -- -- -- - - _Dot_ and _Dash_. A line composed of dots and dashes alternating. -- · - -- · -- - - _Dimension._ A line upon which a dimension is placed. - - _Full._ An unbroken line, usually representing a visible edge. ------ - - _Shadow._ A line about twice as wide as the ordinary full line. - - A straight line is often called simply a line, and a curved line a - curve. - - =LONGITUDINAL.=--In the direction of the length of an object. - - =MODEL.=--A form used for study. - - =OBLIQUE.=--Neither horizontal nor vertical. - - =OBLONG.=--A rectangle with unequal sides. - - =OVAL.=--A plane figure resembling the longitudinal section of an egg; - or elliptical in shape. - - =OVERALL.=--The entire length. - - =PARALLEL.=--Having the same direction and everywhere equally distant. - - =PATTERN.=--That which is used as a guide or copy in making things. - - _Flat._ One made of paper or other thin material. - - _Solid._ One which reproduces the form and size of the object to be - made. - - =PERIMETER.=--The boundary of a closed plane figure. - - =PERPENDICULAR.=--At an angle of 90°. - - =PERSPECTIVE.=--View; drawing objects as they appear to the eye from - any given distance and situation, real or imaginary. - - =PLAN.=--Plan, horizontal projection and _top view_ have the same - meaning. - - =PLANE FIGURE.=--A part of a plane surface bounded by straight or - curved lines, or by both combined. - -[Illustration: Fig. 39.] - - =POLYGON.=--A plane figure bounded by straight lines called the sides - of the polygon. The least number of sides that can bound a polygon is - three. Polygons bounded by a greater number of sides than four are - denominated only by the number of sides. - - A polygon of five sides is called a _Pentagon_; of six, a _Hexagon_; - of seven, a _Heptagon_; of eight, an _Octagon_; of nine, a _Nonagon_, - etc. - - _Diagonals_ of a polygon are lines joining the vertices of angles not - adjacent. - - The _Perimeter_ of a polygon is its boundary considered as a whole. - - The _Base_ of a polygon is the side upon which the polygon is supposed - to stand. - - The _Altitude_ of a polygon is the perpendicular distance between the - base and a side or angle opposite the base. - -[Illustration: Fig. 40.] - - A _Quadrilateral_ is a polygon having four sides and four angles. - - A _Parallelogram_ is a quadrilateral which has its opposite sides - parallel. - - The side upon which a parallelogram stands and the opposite side are - called respectively its lower and upper bases. - - A _Rectangle_ is a parallelogram having its angles right angles. - - A _Square_ is an equilateral rectangle, fig. 41. - -[Illustration: Fig. 41.] - -[Illustration: Fig. 42.] - -[Illustration: Fig. 43.] - -[Illustration: Fig. 44.] - -[Illustration: Fig. 45.-49.] - - A _Rhomboid_ is an oblique-angled parallelogram. - - A _Rhombus_ is an equilateral rhomboid, fig. 42. - - A _Trapezium_ is a quadrilateral having no two sides parallel, fig. - 43. - - A _Trapezoid_ is a quadrilateral in which two opposite sides are - parallel, and the other two oblique, fig. 44. - - =A POLYHEDRON= is a solid bounded by planes. There are five regular - solids which are shown in figs. 45, 46, 47, 48 and 49. A regular solid - is bounded by similar and regular plane figures. - - Fig. 45.--The _tetrahedron_, bounded by four equilateral triangles. - - Fig. 46.--The _hexahedron_, or cube, bounded by six squares. - - Fig. 47.--The _octahedron_, bounded by eight equilateral triangles. - - Fig. 48.--The _dodecahedron_, bounded by twelve pentagons. - - Fig. 49.--The _icosahedron_, bounded by twenty equilateral triangles. - - =PRISM.=--A solid whose bases or ends are very similar plane figures, - and whose sides are parallelograms; prisms are called triangular, - square, etc., according as the bases are triangles, squares, etc. - - =PRODUCE.=--To continue or extend. - - =PROFILE.=--An outline or contour. - - =PROJECTION.=--The view of an object obtained upon a plane by - projecting lines perpendicular to the plane. - - =QUADRANT.=--The fourth part; a quarter; the quarter of a circle. - - =QUADRISECT.=--To divide into four equal parts. - - =SECTION.=--A projection upon a plane parallel to a cutting plane - which intersects any object. The section generally represents the part - behind the cutting plane, and represents the cut surfaces by diagonal - lines. - - =SECTIONAL.=--Showing the section made by a plane. - - =SHADOW.=--Shade and shadow have about the same meaning. - - =SOLID.=--A solid has three dimensions--length, breadth and thickness. - - =SPHERE.=--A solid bounded by a curved surface every point of which is - equally distant from a point within called the center. - - =SURFACE.=--The boundary of a solid. It has but two dimensions--length - and breadth. Surfaces are plane or curved. - - A _Plane Surface_ is one upon which a straight line can be drawn in - any direction. - - A _Curved Surface_ is one no part of which is plane. - - The surface of the sphere is curved in every direction, while the - curved surfaces of the cylinder and cone are straight in one - direction. - - The surface of a solid is no part of the solid, but is simply the - boundary of the solid. It has two dimensions only, and any number of - surfaces put together will give no thickness. - -[Illustration: Fig. 50.] - -[Illustration: Fig. 51.] - -[Illustration: Fig. 52.] - -[Illustration: Fig. 53.] - -[Illustration: Fig. 54.] - - =SYMMETRY.=--_Design._ A proper adjustment or adaptation of parts to - one another and to the whole. - - =TRISECT.=--To divide into three equal parts. - - =TRIANGLE.=--A triangle is a polygon having three sides and three - angles. _Tri_ is a Latin prefix signifying three; hence a Triangle is - literally a figure containing three angles. - - A _Scalene Triangle_ is one in which no two sides are equal. See fig. - 50. - - An _Isosceles Triangle_ is one in which two of the sides are equal. - See fig. 51. - - An _Equilateral Triangle_ is one in which the three sides are equal. - An _Equiangular Triangle_ is one having its three angles equal. An - _Acute-Angled Triangle_ is one in which each angle is acute. - - A _Right-Angled Triangle_ is one which has one of the angles a right - angle. See fig. 53. - - An _Obtuse-Angled Triangle_ is one having an obtuse angle. Fig. 54. - - Equiangular triangles are also equal sided, and vice versa. - - =VERTICAL.=--Upright or perpendicular. Vertical and perpendicular are - not synonymous terms. - - =VERTEX.=--See Angle, Quadrilateral, Triangle. The vertex of a solid - is the point in which its axis intersects the lateral surface. - - =VIEW.=--See Elevation. Views are called front, top, right or left - side, back, or bottom, according as they are made on the different - planes of projection. They are also sometimes named according to the - part of the object shown, as edge view, end view, or face view. - - =WORKING DRAWING.=--One which gives all the information necessary to - enable the workman to construct the object. - -[Illustration] - - -[Illustration: FREEHAND DRAWING] - -[Illustration: Fig. 55.] - - - - -Free-Hand Drawing. - - -A free-hand drawing is executed with the unaided hand and eye, without -guiding instruments or other artificial help. It is necessary to be -known that all drawing required cannot possibly be done by rule and -compass, but that some portions must be drawn “free-hand,” trusting to -the eye alone. - -Hence, it is important that the student should be able to sketch at -sight from objects he may see, or to draw roughly, with a piece of chalk -or a pencil, pieces of mechanism required to be represented. - -Practice in free-hand should go along with mechanical drawing as -progress is made, and thus cultivating both branches equally. - -“A simple sketch will often,” as has been rather roughly said, “express -more than yards of talk.” - -Even a slight sketch refreshes the memory, and in the case of the -preparation of a complete set of drawings, with a view to the making of -a thoroughly finished mechanical drawing, the proper course to pursue -is, to make a general sketch, letter the various parts for reference, -and then prepare a series of detailed sketches, similarly lettered, and -diffuse with dimensions. - -Everyone, whatever his specialty, feels to-day that the ability to -sketch rapidly and clearly is among the absolute necessities for correct -and prompt transactions of business, in giving and executing orders and -doing business with persons outside his profession. - -Mistakes and misunderstandings may be averted by means of rough sketches -taken at the time and shown for confirmation; this also saves assistants -from getting into trouble, especially if they pin the sketch to the -order, for reference, in case of the arising of any dispute. These are a -few of the advantages of knowing how to sketch quickly and correctly. - -In “free-hand” any sort of pencil is better than none, but there is a -considerable advantage in having a good serviceable article--a pencil -not too soft nor too hard, and one which will retain its point for some -little time. - -Fig. 55 shows the approved position in which the pencil should be held -while sketching. The pencil should be held firmly between the thumb and -first finger of the right hand; press the second finger against the -pencil at the opposite side to the thumb pressure, so that the pencil is -firmly held by the contact of the thumb and two fingers--the third and -fourth fingers just coming into easy reach of the paper surface--the -wrist or ball of the hand resting lightly on the surface of the -work--the arm resting on the desk or drawing-board for steadiness. - -The motion of the pencil is produced from the movement of the fingers -and thumb, principally in the vertical strokes, and the horizontal -strokes are produced by fingers and thumb, combined with a wrist or -elbow motion; the oblique lines and curves are produced with a free -movement, with nothing cramped or confined about the finger joints. - - -POSITION. - -It should be observed that nothing is more prejudicial to good execution -than the habit of leaning over the paper, which ought to be placed on a -surface sufficiently inclined to bring every portion equally under the -eye, thus obviating the necessity of leaning forward. All support to the -figure should be obtained by resting on the left arm, the right being -left free for work. By attention to these rules that awkwardness of -position, so detrimental to a good figure, will be avoided. It is better -to have the light on the left hand, as in this direction the shadow of -the pencil does not interfere with the view of the drawing. - -[Illustration: Fig. 56.] - - -HOW TO CUT A PENCIL. - -Hold the pencil firmly in the left hand, as in the drawing, allowing -about an inch to project beyond the fingers, and turn it gradually as -the knife removes the wood. The knife should be held so that the blade -alone projects beyond the fingers, and the part of it nearest the handle -used for cutting. The pencil should be placed against the inside of the -thumb of the right hand, as in the drawing (fig. 56), and the wood -removed by slight shaving. The lead should not be cut at the same time -as the wood, but rested on the thumb and pared gently afterwards; by -attention to these directions the pencil will be economized. - - -HOW TO DRAW STRAIGHT LINES. - -Before a line is drawn, the point at which it is to commence and the -point where it is to end, should be known; and let it be distinctly -understood that _this judgment of the eye_, and placing of points, -should invariably precede the drawing of every line. - -The first effort should, therefore, be to produce a line of points -exactly parallel with the upper edge of the paper, and at equal -distances from each other. Commence with point _A_ and place the point -_B_ carefully level with it, now place a slip of paper against these -points in the original, mark their distance apart, and see if the same -proportion has been given in your copy; if not, make the necessary -correction. Proceed with the next point, examine it, and so on to the -end of the line. When this is complete, examine each point in -succession, to try if it is at the same distance from the top of the -paper; when this is correct, proceed to draw the first level line. Hold -the pencil as in the drawing, fig. 57, keeping the elbow near the side; -join _A_ to _B_ by one light, steady stroke, produced by a movement of -the wrist, and add stroke upon stroke until the line is of the required -depth. Continue this process to the end of the line of points. Now place -the point _D_ at the right distance below the _A_, proceed with the -points for another line as before, and continue the lines until the -paper is covered. In producing the stroke the pencil should not be -jerked, or any stop be made between the points, but the movement should -be even throughout, and it is much better to produce each line by -several soft strokes, as _the repetition of delicate lines induces -lightness of touch and freedom of hand_; and it is also no small -advantage that lines thus produced are more easily removed by the India -rubber, should they require correction. - - -TO DRAW THE FIRST OBLIQUE LINE. - -Prepare three rows of points down the side of the paper, on the left -hand; examine them to see that they are at equal distances from the side -and from each other; hold the pencil as in the drawing, fig. 58, move -the elbow a little from the side, and join the points _A_ and _B_ with -one light line, produced by a movement of the fingers and thumb, -repeating the strokes until the line is of the requisite depth; proceed -to join _B_ to _C_, taking care previously to bring the hand a little -down the paper, as the line from _A_ to _C_ is too long to be produced -from one position. When the three rows of points are filled, make -another set, examine them and proceed as before. By these means the -paper will be covered with oblique lines, and if the points have been -placed exactly, the sheet will have a neat and regular appearance. - - NOTE.--The drawings of hands are introduced to show the positions for - holding the pencil, and are not intended for copying. - -[Illustration: Fig. 57.] - -[Illustration: Fig. 58.] - -It is a common, and at the same time highly injurious habit, to draw -this line by a movement of the wrist, the fingers remaining rigid. This -may be detected by watching the action of the thumb; if it bends as the -line is produced, all is right; but if it does not the wrist is at work. - - -TO DRAW THE UPRIGHT OR PERPENDICULAR LINE. - -This line demands the greatest attention, and any care bestowed upon it -will be amply repaid in the after studies. - -Commence by placing a line of points down the side of the paper, examine -them very carefully to see that they are all the same distance from its -edge, hold the pencil as in the drawing, fig. 59, move the elbow well -out from the side, and join the points by a movement of the fingers and -thumb. When one line is complete, place the points for the next, and -examine them from the edge of the paper, not from the line just drawn. -Proceed in this manner until the paper is covered. - -There is in most cases a tendency to place the points for this line in a -slightly inclined direction, as in writing, though in some instances the -tendency is the opposite, a thoroughly correct eye in this respect being -a rare gift: and it may be useful to suggest that the paper be so placed -that the line of points to be produced may be exactly in front of the -eye. - - -TO DRAW THE SECOND OBLIQUE LINE. - -Prepare three rows of points down the side of the paper, examine them -for correctness of position, hold the pencil as in the drawing, fig. 60, -remove the elbow as far as possible from the side, and join the points -by a movement of the fingers and thumb, and continue the exercise until -the paper is covered. - -[Illustration: Fig. 59.] - -[Illustration: Fig. 60.] - -It will be noticed that each change in direction of the line to be -drawn, has been accompanied with a corresponding change in the position -of the elbow and wrist. The following simple rule will assist the memory -when placing the hand for any given line; the pencil should be held so -that it may form a T with the line to be drawn: - -[Illustration] - -For the horizontal line, elbow near the side. - -[Illustration] - -For the first oblique, elbow a little removed. - -[Illustration] - -For the perpendicular, elbow more removed. - -[Illustration] - -For the second oblique, elbow most removed. - -[Illustration: Finger and thumb lines. - -Wrist lines. - -Finger and thumb lines. - -Wrist lines.] - -It may also be interesting to notice, with regard to the movements by -which lines are produced, that they are divided into two systems; the -first is that of the wrist, which includes the horizontal, and lines in -nearly the same direction; the second is that of the fingers and thumb, -by which all other lines are formed. The following diagram exhibits the -two systems and their various lines grouped, and it will be observed -that there is a space marked (a) between the two sets, which may be -considered neutral ground. Lines in this direction may be produced by -either movement, as may be most convenient, but it will always be found -that these lines are the most trying to the hand. - - -ON FIGURES FORMED OF STRAIGHT LINES. - -Before commencing this subject, let it be clearly understood that future -success will, in a great measure, depend upon the amount of care -bestowed upon it. The aim should be to obtain absolute accuracy, and for -this end the copies should be tested by the most careful measurements, -and corrected until they are true with the originals, but it should be -distinctly understood that these measurements are only to be made after -the eye and hand have done their best. - - NOTE.--To some it may appear that too much time and care has been - bestowed on mere lines, but let it be understood that a good system of - line drawing is the basis of all education--the slightest outline by a - hand thus trained has a bold, free and masterly character; and with - regard to shading, which is simply an aggregation of good lines, it is - only by such a practiced hand its most charming effects can be - produced. - -Fig. 66: Place the points _A_, _B_. Examine them to see that they are -the same distance apart as in the original, and that they are level; -place the point _C_ exactly under _A_, and make _A C_ equal in distance -to _A B_; now place the point _D_ opposite _C_ and under _B_; try the -distances between each point to see that they are the same; divide each -side by a point half way, and then draw the lines. - -Fig. 67: Repeat the last figure and add the lines _A_ and _B_, taking -great care that the points for them are correctly placed. - -Fig. 68: Commence with the square as before; then join the half-way -points. - -Fig. 69: After the square is drawn, place the points _A_ and _B_ at the -right height above the half-way points, and _C_, _D_ at the proper -distance from the corners, then draw the figure. - -Fig. 70: The greatest care should be taken with the squares for this and -the following figure, as the slightest error in them will destroy the -symmetry of the drawing within; when the square is completed, join the -opposite corners, and place on the crossed lines the points _B_, _C_, -_D_, _E_; examine these to see that they are each at the same distance -from the centre _A_, and that this distance is equal to the space from -_A_ to the sides of the square; when all are proved to be correct, -complete the figure. - -[Illustration: Fig. 66.] - -[Illustration: Fig. 67.] - -[Illustration: Fig. 68.] - -[Illustration: Fig. 69.] - -[Illustration: Fig. 70.] - -[Illustration: Fig. 71.] - -Fig. 71: Repeat the last drawing with, if possible, greater exactness, -and outside the octagon place the points _A_, _B_, _C_, _D_, etc.; -examine each of these points to see that they are all at the same -distance from the centre, and then complete the figure. - - -ON CURVED LINES. - -The right position of the hand for drawing any curved line is that -required for a straight line which would touch the extremities of the -curve. The straight lines given in the exercises are valuable, not only -as a guide to the position of the hand, but as an assistance to the eye -when forming the curves or examining them after they are produced. - -[Illustration] - -The direction given for drawing a straight line was to form it by one -steady movement from point to point, without any jerk or stop by the -way. This instruction requires to be changed for the curve, _which is -better produced by several short strokes_, thus: - -[Illustration] - -or by overlapping lines, any outside bits being cleared away with India -rubber. - -These exercises will test the drawing power and try the patience of the -pupil, but they are worthy of all the care which can be bestowed, which -in future efforts will meet with its full reward. - -Fig. 76: Draw first the square as directed in the previous lesson, join -the points _A_, _B_, _C_ and add the short lines at _E_ and _F_, proceed -with the curve _A B_, drawing it with faint lines at first, and adding -stroke upon stroke until the required depth is obtained; the curve _A C_ -is more difficult to produce, in consequence of the formation of the -hand; it should, therefore, be drawn in shorter pieces, joining them -together afterwards by over strokes. - -[Illustration: Fig. 76.] - -[Illustration: Fig. 77.] - -Fig. 77: Draw the square and straight lines first, then add the curves, -taking care to give the greatest fullness at the right place. - -Fig. 78: Draw the square and straight lines, proceed with the curves, -taking care to make each of the same proportion. - -Figs. 79 and 80: The ovals contained in these figures are simply -foreshortened circles, and as such forms are of frequent occurrence in -sketching from objects, in bridges, wheels, ends of timber, etc., they -should be carefully studied; the greatest difficulty is to turn the -narrow ends, and prevent their looking like corners. For this purpose it -is better to draw the short curves first, thus: - -[Illustration] - -and then join the longer sides to them. - -Fig. 81: If this figure can be drawn correctly, a great success has been -achieved; the circle is a most difficult form to delineate, and without -system could not be accomplished. Draw the square and straight lines -within it with great care, examine each point of the octagon to see that -it is at the same distance from the centre, and then draw the circle. - - -EXAMPLES FOR PRACTICE. - -Several figures, 83 to 96, representing more or less familiar parts of -machines, utilities, etc., are introduced for practice in free-hand, -but-- - -It must be noted that even in free-hand the wise student will -occasionally use the straight edge and compasses, so as to make his -first attempts fairly creditable. Many good draughtsmen have begun by -simply copying such figures and illustrations as are used throughout -this volume and other similar sources; perhaps there is nothing better -for practice or training than the copying and reproducing of samples of -good mechanical drawings, yet it must always be remembered that -advancement in free-hand must be made in the line of less to greater -efforts, and that the why and wherefore will be constantly asked by the -aspiring student; that good and correct drawings are to be aimed for at -all times in every line and dimension--never forgetting the law of -proportion in the smallest outlines of objects to be represented. - -[Illustration: Fig. 78.] - -[Illustration: Fig. 79.] - -[Illustration: Fig. 80.] - -[Illustration: Fig. 81.] - -Fig. 83 is a section, or end view of a bar of angle iron; the student -will find helpful practice in attempting this figure; he may be allowed -to use a straight-edge in drawing the lines, but no measurements; the -work should be tested on completion by a rule, or better by penciling -from the original on tracing paper, and comparing the free-hand with the -copy, when the defective proportions, if any, will be clearly exhibited. - -Fig. 84 is a section of tee iron, and fig. 85 is a section of channel -iron. These three figures on page 75 should be practiced alternately, -although seeming similar in shape. - -Fig. 86 is a side and end view of an angle plate shaded. Fig. 87 is a -wrench shaded. - -Examples of bolt ends are shown in the two next numbers; fig. 88 -exhibits the common square-head bolt, and fig. 89 the hexagon or -six-sided bolt-head; these are also examples of _straight-line shading_. -Fig. 90 is a lathe-dog, and shows an example of _curved shading_; fig. -92 is an engine crank, and an example of _straight and curved shading_; -fig. 91 is a screw clamp. - -Fig. 93 is a section of boiler plates riveted together; a caulking tool -is also shown. - -In the example, fig. 94--a hand-wheel--the principal difficulty, even -for the most advanced student in free-hand, will be in drawing the -circles; a coin, if convenient, can be used to scribe about, in drawing -these; the other parts can afterwards be filled in around the circle. -Fig. 96 is introduced for practice in penciling and shading; the figure -represents a water-wheel on a stone pier. - -The familiar oil can is shown in fig. 95. These all are excellent -objects for practice. - -[Illustration: Fig. 83.--Fig. 84.--Fig. 85.] - -[Illustration: Fig. 86.] - -[Illustration: Fig. 87.] - -[Illustration: Fig. 88.--Fig. 89.--Fig. 90.--Fig. 91.] - -[Illustration: Fig. 92.] - -[Illustration: Fig. 93.] - -[Illustration: Fig. 94.] - -[Illustration: Fig. 95.] - -[Illustration: Fig. 96.] - - -[Illustration: GEOMETRICAL DRAWING] - -[Illustration] - - - - -Geometrical Drawing. - - -Geometry is the science of measurement; it has been known for more than -three thousand years; many lives have been devoted to its development, -and it exists to-day as the foundation of all mathematics. - -Geometrical drawing is the art of representing, to the eye, the problems -“worked out” by geometricians, and the importance of a knowledge of -geometrical drawing is paramount. The student will find that the figures -delineated and explained in the next few pages constantly occur in -mechanical drawing. Says Walter Smith, State Director of Art Education -in Massachusetts, “I have never known a case where a student did not -progress more satisfactorily in his studies after a course of practical -geometry.” - -The elementary conceptions of geometry are few: - - 1.--A point. - 2.--A line. - 3.--A surface. - 4.--A solid, and - 5.--An angle. - -All of which elements are used in mechanical drawings. - -From these, as data, a vast number of mathematical problems have been -deduced; of which a few of the most elementary will be illustrated in -this work; but these few will repay the attention of the student. - -In “freehand” drawing the crayon and pencil are used; in geometrical -drawings the dividers, as shown in illustration, fig. 97, together with -a rule, are all that is necessary to accomplish the work. - -A problem is something to be done, and geometry has been defined as the -science of measurement; the relation between geometry and mechanical -drawing is very close, hence the term “geometrical problem.” - -[Illustration: Fig. 97.] - -[Illustration: Fig. 98.] - -Before proceeding with the examples, a few elementary statements -belonging to the science of geometry are presented; these will be useful -to the student, not only while “doing” the problems, but in many cases -of every-day--future--experience. - -Geometry is one of the oldest and simplest of sciences; it may be -defined as _the science of measurement_; geometry is _the root_ from -which all regular mathematical calculations issue. It has claimed the -best thought of practical men from the times of the Greeks and Romans -two thousand years ago; they derived their knowledge of the science from -the Egyptians, who in turn were indebted to the Chaldeans and Hindoos in -times beyond any authentic history; hence it was under the operations of -the laws explained in geometry, that the pyramids of Egypt and the -temples of Greece were constructed, as well as the engines of war and -appliances of peace of ancient times. - -_A point_ is mere position, and has no magnitude. - -_A line_ is that which has extension in length only. The extremities of -lines are points. - -_A surface_ is that which has extension in length and breadth only. - -_A solid_ is that which has extension in length, breadth and thickness. - -_An angle_ is the difference in the direction of two lines proceeding -from the same point. - -[Illustration] - -Lines, Surfaces, Angles and Solids constitute the different kinds of -quantity called _geometrical_ magnitudes. - -_Parallel lines_ are lines which have the same direction; hence parallel -lines can never meet, however far they may be produced; for two lines -taking the same direction cannot approach or recede from each other. - -[Illustration] - -An _Axiom_ is a self-evident truth, not only too simple to require, _but -too simple to admit of demonstration_. - -A _Proposition_ is something which is either proposed to be done, or to -be demonstrated, and is either a problem or a theorem. - -A _Problem_ is something proposed to be done. - -A _Theorem_ is something proposed to be demonstrated. - -A _Hypothesis_ is a supposition made with a view to draw from it some -consequence which establishes the truth or falsehood of a proposition, -or solves a problem. - -A _Lemma_ is something which is premised, or demonstrated, in order to -render what follows more easy. - -A _Corollary_ is a consequent truth derived immediately from some -preceding truth or demonstration. - -A _Scholium_ is a remark or observation made upon something going before -it. - -A _Postulate_ is a problem, the solution of which is self-evident. - -Let it be granted-- - -I. That a straight line can be drawn from any one point to any other -point; - -II. That a straight line can be produced to any distance, or terminated -at any point; - -III. That the circumference of a circle can be described about any -center, at any distance from that center. - -The common algebraic signs are used in Geometry, and it is necessary -that the student in geometry should understand some of the more simple -operations of algebra. As the terms circle, angle, triangle, hypothesis, -axiom, theorem, corollary and definition are constantly occurring in a -course of geometry, they are abbreviated as shown in the following list: - - Addition is expressed by + - Subtraction is expressed by - - Multiplication is expressed by × - Equality and Equivalency are expressed by = - Greater than, is expressed by > - Less than, is expressed by < - Thus _B_ is greater than _A_, is written _B_ > _A_ - _B_ is less than _A_, is written _B_ < _A_ - A circle is expressed by O - An angle is expressed by L - A right angle is expressed by R. L - Degrees, minutes and seconds are expressed by ° ′ ″ - A triangle is expressed by △ - The term Hypothesis is expressed by (Hy.) - The term Axiom is expressed by (Ax.) - The term Theorem is expressed by (Th.) - The term Corollary is expressed by (Cor.) - The term Definition is expressed by (Def.) - The term Perpendicular is expressed by ⊥ - The difference of two quantities, when it is not - known which is the greater, is expressed by - the symbol ~ - Thus, the difference between _A_ and _B_ is written _A_ ~ _B_ - - -GEOMETRICAL AXIOMS. - -1. _Things which are equal to the same thing are equal to each other._ - -2. _When equals are added to equals the wholes are equal._ - -3. _When equals are taken from equals the remainders are equal._ - -4. _When equals are added to unequals the wholes are unequal._ - -5. _When equals are taken from unequals the remainders are unequal._ - -6. _Things which are double of the same thing, or equal things, are -equal to each other._ - -7. _Things which are halves of the same thing, or of equal things, are -equal to each other._ - -8. _The whole is greater than any of its parts._ - -9. _Every whole is equal to all its parts taken together._ - -10. _Things which coincide, or fill the same space, are identical, or -mutually equal in all their parts._ - -11. _All right angles are equal to one another._ - -12. _A straight line is the shortest distance between two points._ - -13. _Two straight lines cannot enclose a space._ - - -Problems in Geometrical Drawing. - -[Illustration: Fig. 99.] - -EXAMPLE 1.--_To bisect_ (cut in two) _a straight line or an arc of a -circle_, Fig. 99. From the ends of _A B_ as centers, describe arcs -cutting each other at _C_ and _D_, and draw _C D_, which cuts the line -at _E_ or the arc at _F_. - -EX. 2.--_To draw a perpendicular to a straight line, or a radial line to -a circular arc_, Fig. 99. Operate as in the foregoing problem. The line -_C D_ is perpendicular to _A B_; the line _C D_ is also radial to the -arc _A B_. - -[Illustration: Fig. 100.] - -[Illustration: Fig. 101.] - -EX. 3.--_To draw a perpendicular to a straight line, from a given point -in that line_, Fig. 100. With any radius from any given point _A_ in the -line _B C_, cut the line at _B_ and _C_. Next, with a longer radius, -describe arcs from _B_ and _C_, cutting each other at _D_, and draw the -perpendicular _D A_. - -_Second Method_, Fig. 101. From any center _F_ above _B C_, describe a -circle passing through the given point _A_, and cutting the given line -at _D_; draw _D F_, and produce it to cut the circle at _E_; and draw -the perpendicular _A E_. - -[Illustration: Fig. 102.] - -_Third Method_, Fig. 102. From _A_ describe an arc _E C_, and from _E_, -with the same radius, the arc _A C_ cutting the other at _C_; through -_C_ draw a line _E C D_ and set off _C D_ equal to _C E_, and through -_D_ draw the perpendicular _A D_. - -[Illustration: Fig. 103.] - -[Illustration: Fig. 104.] - -EX. 4.--_To draw a perpendicular to a straight line from any point -without it_, Fig. 103. From the point _A_ with a sufficient radius cut -the given line at _F_ and _G_; and from these points describe arcs -cutting at _E_. Draw the perpendicular _A E_. - -If there be no room below the line, the intersection may be taken above -the line; that is to say, between the line and the given point. - -_Second Method_, Fig. 104. From any two points _B C_ at some distance -apart, in the given line, and with the radii _B A_, _C A_, respectively, -describe arcs cutting at _A D_. Draw the perpendicular _A D_. - -[Illustration: Fig. 105.] - -[Illustration: Fig. 106.] - -EX. 5.--_To draw a parallel line through a given point_, Fig. 105. With -a radius equal to the given point _C_ from the given line _A B_, -describe the arc _D_ from _B_, taken considerably distant from _C_. Draw -the parallel through _C_ to touch the arc _D_. - -_Second Method_, Fig. 106. From _A_, the given point, describe the arc -_F D_, cutting the given line at _F_; from _F_, with the same radius, -describe the arc _E A_, and set off _F D_, equal to _E A_. Draw the -parallel through the points _A D_. - -[Illustration: Fig. 107.] - -When a series of parallels are required perpendicular to a base line _A -B_, they may be drawn as in fig. 107 through points in the base line set -off at the required distances apart. This method is convenient also -where a succession of parallels are required to a given line _C D_, for -the perpendicular may be drawn to it, and any number of parallels may be -drawn on the perpendicular. - -[Illustration: Fig. 108.] - -[Illustration: Fig. 109.] - -EX. 6.--_To divide a line into a number of equal parts_, Fig. 108. - -To divide the line _A B_ into, say, five parts. From _A_ and _B_ draw -parallels _A C_, _B D_ on opposite sides; set off any convenient -distance four times (one less than the given number), from _A_ on _A C_, -and on _B_ on _B D_; join the first on _A C_ to the fourth on _B D_, and -so on. The lines so drawn divide _A B_ as required. - -_Second Method_, Fig. 109. Draw the line at _A C_, at an angle from _A_, -set off, say, five equal parts; draw _B_ 5, and draw parallels to it -from the other points of division in _A C_. These parallels divide _A B_ -as required. - -[Illustration: Fig. 110.] - -EX. 7.--_Upon a straight line to draw an angle equal to a given angle_, -Fig. 110. Let _A_ be the given angle and _F G_ the line. With any radius -from the points _A_ and _F_, describe arcs _D E_, _I H_, cutting the -sides of the angle _A_ and the line _F G_. - -Set off the arc _I H_, equal to _D E_ and draw _F H_. The angle _F_ is -equal to _A_ as required. - -[Illustration: Fig. 111.] - -EX. 8.--_To bisect an angle_, Fig. 111. Let _A C B_ be the angle; on the -center _C_ cut the sides at _A B_. On _A_ and _B_ as centers describe -arcs cutting at _D_ dividing the angle into two equal parts. - -[Illustration: Fig. 112.] - -EX. 9.--_To find the center of a circle or of an arc of a circle._ Fig. -112. Draw the chord _A B_, bisect it by the perpendicular _C D_, bounded -both ways by the circle; and bisect _C D_ for the center _G_. - -[Illustration: Fig. 113.] - -[Illustration: Fig. 114.] - -EX. 10.--_Through two given points to describe an arc of a circle with a -given radius_, Fig. 113. On the points _A_ and _B_ as centers, with the -given radius, describe arcs cutting at _C_; and from _C_, with the same -radius, describe an arc _A B_ as required. - -Second, for a circle or an arc, Fig. 114. Select three points _A_, _B_, -_C_ in the circumference, well apart; with the same radius describe arcs -from these three points cutting each other, and draw two lines _D E_, _F -G_, through their intersections according to Fig. 107. The point where -they cut is the center of the circle or arc. - -EX. 11.--_To describe a circle passing through three given points_, Fig. -114. Let _A_, _B_, _C_ be the given points and proceed as in last -problem to find the center _O_, from which the circle may be described. - -This problem is variously useful; in finding the diameter of a large -fly-wheel, or any other object of large diameter when only a part of the -circumference is accessible; in striking out arches when the span and -rise are given, etc. - -[Illustration: Fig. 115.] - -EX. 12.--_To draw a tangent to a circle from a given point in the -circumference_, Fig. 115. From _A_ set off equal segments _A B_, _A D_, -join _B D_ and draw _A E_, parallel to it, for the tangent. - -[Illustration: Fig. 116.] - -EX. 13.--_To draw tangents to a circle from points without it_, Fig. -116. From _A_ with the radius _A C_ describe an arc _B C D_, and from -_C_ with a radius equal to the diameter of the circle, cut the arc at _B -D_, join _B C_, _C D_, cutting the circle at _E F_, and draw _A E_, _A -F_, the tangents. - -[Illustration: Fig. 117.] - -EX. 14.--_Between two inclined lines to draw a series of circles -touching these lines and touching each other_, Fig. 117. Bisect the -inclination of the given lines _A B_, _C D_ by the line _N O_. From a -point _P_ in this line draw the perpendicular _P B_ to the line _A B_, -and on _P_ describe the circle _B D_, touching the lines and cutting the -center lines at _E_. From _E_ draw _E F_ perpendicular to the center -line, cutting _A B_ at _F_, and from _F_ describe an arc _E G_, cutting -_A B_ at _G_. Draw _G H_ parallel to _B P_, giving _H_, the center of -the next circle, to be described with the radius _H E_, and so on for -the next circle, _I N_. - -[Illustration: Fig. 118.] - -[Illustration: Fig. 119.] - -EX. 15.--_To construct a triangle on a given base, the sides being -given._ - -First. An equilateral triangle, Fig. 118. On the ends of a given base _A -B_, with _A B_ as a radius describe arcs cutting at _C_, and draw _A C_, -_C B_. - -Second. Triangle of unequal sides, Fig. 119. On either end of the base -_A D_, with the side _B_ as a radius describe an arc; and with the side -_C_ as a radius, on the other end of the base as a center, describe arcs -cutting the arc at _E_; join _A E_, _D E_. - -This construction may be used for finding the position of a point _C_ or -_E_ at given distances from the ends of a base, not necessarily to form -a triangle. - -[Illustration: Fig. 120.] - -[Illustration: Fig. 121.] - -EX. 16.--_To construct a square rectangle on a given straight line._ - -First. A square, Fig. 120. On the ends _B A_ as centers, with the line -_A B_ as radius, describe arcs cutting at _C_; on _C_ describe arcs -cutting the others at _D E_; and on _D_ and _E_ cut these at _F G_. Draw -_A F_, _B G_ and join the intersections _H I_. - -Second. A rectangle, Fig. 121. On the base _E F_ draw the perpendiculars -_E H_, _F G_, equal to the height of the rectangle, and join _G H_. - -[Illustration: Fig. 122.] - -EX. 17.--_To construct a parallelogram of which the sides and one of the -angles are given_, Fig. 122. Draw the side _D E_ equal to the given -length _A_, and set off the other side _D F_ equal to the other length -_B_, forming the given angle _C_. From _E_ with _D F_ as radius, -describe an arc, and from _F_, with the radius _D E_ cut the arc at _G_. -Draw _F G_, _E G_. Or, the remaining sides may be drawn as parallels to -_D E_, _D F_. - -[Illustration: Fig. 123.] - -EX. 18.--_To describe a circle about a triangle_, Fig. 123. Bisect two -sides _A B_, _A C_ of the triangle at _E F_, and from these points draw -perpendiculars cutting at _K_. On the center _K_, with the radius _K A_ -draw the circle _A B C_. - -[Illustration: Fig. 124.] - -EX. 19.--_To describe a circle about a square, and to inscribe a square -in a circle_, Fig. 124. - -First. To describe the circle. Draw the diagonals _A B_, _C D_ of the -square, cutting at _E_; on the center _E_ with the radius _E A_ describe -the circle. - -Second. To inscribe the square. Draw the two diameters _A B_, _C D_ at -right angles and join the points _A B_, _C D_ to form the square. - -_In the same way a circle may be described about a triangle._ - -[Illustration: Fig. 125.] - -EX. 20.--_To inscribe a circle on a square, and to describe a square -about a circle_, Fig. 125. - -First. To inscribe the circle. Draw the diagonals _A B_, _C D_ of the -square, cutting at _E_; draw the perpendicular _E F_ to one side, and -with the radius _E F_ describe the circle. - -Second. To describe the square. Draw two diameters _A B_, _C D_ at right -angles, and produce them; bisect the angle _D E B_ at the center by the -diameter _F G_, and through _F_ and _G_ draw perpendiculars _A C_, _B -D_, and join the points _A D_ and _B C_ where they cut the diagonals to -complete the square. - -[Illustration: Fig. 126.] - -EX. 21.--_To inscribe a circle in a triangle_, Fig. 126. Bisect two of -the angles _A C_ of the triangle by lines cutting at _D_; from _D_ draw -a perpendicular _D E_ to any side, and with _D E_ as radius describe a -circle. - -[Illustration: Fig. 127.] - -EX. 22.--_To inscribe a pentagon in a circle_, Fig. 127. Draw two -diameters _A C_, _B D_ at right angles cutting at _O_; bisect _A O_ at -_E_, and from _B_ with radius _B E_ cut the circumference at _G H_ and -with the same radius step round the circle to _I_ and _K_; join the -points to form the pentagon. - -[Illustration: Fig. 128.] - -EX. 23.--_To construct a hexagon upon a given straight line_, Fig. 128. -From _A_ and _B_, the ends of the given line, describe arcs cutting at -_G_; from _G_ with the radius _G A_ describe a circle. With the same -radius set off the arcs _A C_, _C F_ and _B D_, _D E_; join the points -so found to form the hexagon. - -[Illustration: Fig. 129.] - -EX. 24.--_To inscribe a hexagon in a circle_, Fig. 129. Draw a diameter -_A C B_; from _A_ and _B_ as centers, with the radius of the circle _A -C_ cut the circumference at _D_, _E_, _F_, _G_, and draw _A D_, _D E_, -etc., to form the hexagon. The points _D E_, etc., may be found by -stepping the radius (with the dividers) six times round the circle. - -[Illustration: Fig. 130.] - -EX. 25.--_To describe an octagon on a given straight line_, Fig. 130. -Produce the given line _A B_ both ways and draw perpendiculars _A E_, _B -F_; bisect the external angles _A_ and _B_ by the lines _A H_, _B C_, -which make equal to _A B_. Draw _C D_ and _H G_ parallel to _A E_ and -equal to _A B_; from the center _G D_, with the radius _A B_, cut the -perpendiculars at _E F_, and draw _E F_ to complete the hexagon. - -[Illustration: Fig. 131.] - -EX. 26.--_To convert a square into an octagon_, Fig. 131.--Draw the -diagonals of the square cutting at _E_; from the corners _A_, _B_, _C_, -_D_, with _A E_ as radius, describe arcs cutting the sides at _G_, _H_, -etc., and join the points so found to complete the octagon. - -[Illustration: Fig. 132.] - -EX. 27.--_To inscribe an octagon in a circle_, Fig. 132. Draw two -diameters _A C_, _B D_, at right angles; bisect the arcs _A B_, _B C_, -at _E_, _F_, etc., to form the octagon. - -[Illustration: Fig. 133.] - -EX. 28.--_To describe an octagon about a circle_, Fig. 133. Describe a -square about the given circle _A B_, draw perpendiculars _H_ and _K_, to -the diagonals, touching the circle to form the octagon. Or, the points -_H_, _K_, etc., may be found by cutting the sides from the corners, by -lines parallel to the diagonals. - -[Illustration: Fig. 134.] - -EX. 29.--_To describe an ellipse when the length and breadth are given_, -Fig. 134. On the center _C_, with _A E_ as radius, cut the axis _A B_ at -_F_ and _G_, the foci, fix a couple of pins into the axis at _F_ and -_G_, and loop on a thread or cord upon them equal in length to the axis -_A B_, so as when stretched to reach the extremity _C_ of the conjugate -axis, as shown in dot-lining. Place a pencil or drawpoint inside the -cord, as at _H_, and guiding the pencil in this way, keeping the cord -equally in tension, carry the pencil round the pins _F_, _G_, and so -describe the ellipse. - - NOTE.--The ellipse is an oval figure, like a circle in perspective. - The line that divides it equally in the direction of its great length - is the _transverse axis_, and the line which divides the opposite way - is the _conjugate axis_. - -_Second Method._ Along the straight edge of a piece of stiff paper mark -off a distance _a c_ equal to _A C_, half the transverse axis; and from -the same point a distance _a b_ equal to _C D_, half the conjugate axis. -Place the slip so as to bring the point _b_ on the line _A B_ of the -transverse axis, and the point _c_ on the line _D E_; and set off on the -drawing the position of the point _a_. Shifting the slip, so that the -point travels on the transverse axis, and the point _c_ on the conjugate -axis, any number of points in the curve may be found, through which the -curve may be traced. See fig. 135. - -[Illustration: Fig. 135.] - - -Trigonometry. - -Trigonometry is that portion of geometry which has for its object the -measurement of triangles. When it treats of plane triangles, it is -called _Plane Trigonometry_; and as the engineer will continually meet -in his studies of higher mathematics _the terms_ used in plane -trigonometry, it is advantageous for him to become familiar with some of -the principles and definitions relating to this branch of mathematics. - -The circumferences of all circles contain the same number of degrees, -but the greater the radius the greater is the absolute measures of a -degree. The circumference of a fly wheel or the circumference of the -earth have the same number of degrees; yet the same number of degrees in -each and every circumference is the measure of precisely the same angle. - -The circumference of a circle is supposed to be divided into 360 degrees -or divisions, and as the total angularity about the center is equal to -four right angles, each right angle contains 90 degrees, or 90°, and -half a right angle contains 45°. Each degree is divided into 60 minutes, -or 60′; and for the sake of still further minuteness of measurement, -each minute is divided into 60″. In a whole circle there are, therefore, -360 × 60 × 60 = 1,296,000 seconds. The annexed diagram, fig. 136, -exemplifies the relative positions of the - - Sine, - Co-sine, - Versed Sine, - - Tangent, - Co-Tangent, - Secant and - Co-secant - -of an angle. - -[Illustration: Fig. 136.] - -These may be defined thus: - - -DEFINITIONS. - -1. The _Complement_ of an arc is 90° minus the arc. - -2. The _Supplement_ of an arc is 180° minus the arc. - -3. The _Sine_ of an angle, or of an arc, is a line drawn from one end of -an arc, perpendicular to a diameter drawn through the other end. - -4. The _Cosine_ of an arc is the perpendicular distance from the center -of the circle to the sine of the arc; or, it is the same in magnitude as -the sine of the complement of the arc. - -5. The _Tangent_ of an arc is a line touching the circle in one -extremity of the arc, and continued from thence, to meet a line drawn -through the center and the other extremity. - -6. The _Cotangent_ of an arc is the tangent of the complement of the -arc. The _Co_ is but a contraction of the word complement. - -7. The _Secant_ of an arc is a line drawn from the center of the circle -to the extremity of the tangent. - -8. The _Cosecant_ of an arc is the secant of the complement. - -9. The _Versed Sine_ of an arc is the distance from the extremity of the -arc to the foot of the sine. - -For the sake of brevity, these technical terms are contracted thus: for -sine _A B_, we write _sin. A B_; for cosine _A B_, we write _cos. A B_; -for tangent _A B_, we write _tan. A B_, etc. - - -[Illustration: INSTRUMENTS AND MATERIALS] - -[Illustration: Fig. 137.] - - - - -Drawing Materials and Instruments. - - -Drawing tools or instruments are contrived solely for mechanical -drawing; aside from this use they are perfectly worthless, hence the -quality of these special utensils is a matter of the first consideration -to the earnest student. - -There are several degrees of excellence to be found in the make-up of -drawing instruments and materials; it may be remarked with truth that -“any kind are good enough, and the best none too good,” i. e., a learner -in this delightful art should not stop at the lack of goodness or the -low grade existing in his “tools,” but rather do the best work possible -with the means at hand. - -However, in order that acceptable work may be accomplished, fairly good -instruments should be procured. The advice of some one experienced in -the use and care of draughting tools should be sought before purchasing. -A drawing board, a single sheet of paper and a pencil is the simplest -“outfit” to be thought of; to this small beginning may be added, soon -afterwards, an inexpensive pair of compasses, a T-square and a couple of -triangles; a vast range of work can be executed with these few tools. - -Nothing else will be needed to do fine work except, perhaps, one or two -pairs of better compasses and a few sweeps or means of drawing irregular -curves; all these had best be purchased separately; for in buying a “box -of instruments,” it may contain some articles which are not desired, or -that are of a wrong size, or even duplicates of those already possessed. - -[Illustration: Fig. 138.] - -An outfit recommended by the author of “Reed’s Hand Book” is as follows. - -Large compasses with movable leg. - -A pair of dividers. - -Bow pencil. - -Bow pen. - -Pencil leg for large compasses. - -Pen leg for large compasses. - -Drawing pen. - -Louis Rouillon, B. S., Instructor of Drawing in Pratt Institute, New -York, recommends the following: - -Compasses, 5¹⁄₂ inches, with needle point; pen pencil and lengthening -bar. - -Drawing pen, 4¹⁄₂ inches. - -T-square, 24-inch blade. - -45-degree triangle, 9 inches. - -30 and 60 degree triangle, 9 inches. - -1 Scroll. - -Dixon’s V. H. pencil. - -12-inch boxwood scale, flat, graduated ¹⁄₁₆ inch the entire length. - -Bottle of liquid India ink, four thumb-tacks, pen and ink eraser. - -20 sheets drawing paper, 11 × 15 inches, and a drawing-board about 16 × -23 inches will also be necessary; students can usually make the board -themselves for less money than it can be bought. - - NOTE.--The purchasing of drawing tools is one of the most difficult - points to settle that can present itself to a person about to buy a - drawing outfit for the first time. Nothing can be so productive of - distress to a person drawing as to have his tools getting out of - order, joints one day too tight, next day too slack, points getting - blunt or perhaps turning up altogether; if needle points, then the - needles slip up, and drawing spoiled; in fact, the purchaser can be - annoyed in numberless different ways.--W. H. THORN. - -[Illustration: Fig. 139.] - -[Illustration: Fig. 140.] - - -THE DRAWING BOARD. - -A drawing-board should be made of well seasoned pine of a convenient -size, say 23 × 16, which will take half a sheet of imperial paper, -leaving ¹⁄₂-inch margin all around. - -The working surface of the board--or its front side--should be perfectly -smooth, but instead of being flat it should have a very slight camber, -or rounding, breadthways, this latter feature in its construction being -to prevent the possibility of a sheet of paper when stretched on its -surface having any vacuity beneath it. - -The _four edges_ of the board need not form an exact rectangle, as much -valuable time is often wasted in the attempts to produce such a board; -but it will answer every purpose of the draughtsman so long as the -adjacent edges at the lower left-hand corner of it are at right angles, -or square to each other. - -An English authority recommends the use of two drawing-boards, 42 inches -long and 30 inches wide, made of plain stuff, without cleets, 1¹⁄₄ -inches thick--seasoned--with edges perfectly straight and at right -angles to each other. _With two boards, one may be used for sketching -and drawing details and the other for the finished drawing._ - -The board should be ³⁄₄ inch in thickness, and fitted at the back, at -right angles to its longest side, with a couple of hardwood battens, -about 2 inches wide and ³⁄₄ inch thick; the use of these battens being -to keep the board from casting or winding and to allow of its expansion -or contraction through changes of temperature. This latter purpose, -however, is only effected by attaching the battens to the back of the -board in the following manner: ... At the middle of the length of each -batten--which should be one inch less than the width of the board--a -stout, well-fitted wood screw is firmly inserted into it, and made to -penetrate the board for about ¹⁄₂ inch, the head of the screw being made -flush with the surface of the batten; on either side of the central -screw, two others, about 3¹⁄₂ inches apart, are passed through oblong -holes in the battens, and screwed into the body of the board until their -heads are flush with the central one; fitted in this way the board -itself can expand or contract lengthwise or crosswise, while its surface -is prevented from warping or bending. - -[Illustration: Fig. 141.] - -A further improvement in such a drawing board as above shown is made by -cutting lengthwise along its ends a narrow groove and inserting an ebony -or hardwood strip; this is cut or sawn apart at about every inch to -admit of contraction; this strip serves as a guide to the stock of the -drawing square, allowing an easy sliding movement. - -To produce really good work in the shape of a mechanical drawing, one -perfect straight edge only is required on a drawing board, and that the -left one, which is always known as the _working-edge_; but for the -convenience of being able to draw a long line across the board at right -angles to its lower edge, this edge is made truly square with that on -the left side of the board. - -The details for building these drawing boards are given, because they -are easy to be made by one who understands the use of a few wood-working -tools; while the boards themselves are difficult of transportation--in -case of the change of residence of their owners--quite unlike the -instruments which are to accompany them. - -Fig. 141 represents the board which has been described in the text, with -provisions for the contraction and expansion; the very dark lines are -intended to represent the ebony insertions--as described. Fig. 140 -represents a plain pine board with dovetailed battens. - -[Illustration: Fig. 142.] - -Fig. 142 represents the common means used to attach or secure slightly -or temporarily the drawing paper to the drawing board; these are called -thumb-tacks, and are usually forced through the paper into the wood by -the hand, whence they are easily detached. These are made to have as -slight a projection as may be, so as not to interfere with the free -movement of the tee-square. - -For mechanical drawing the invariable practice is to secure the paper on -which the drawing is to be made to the drawing board by pinning it; this -is effected by various kinds of _drawing pins_ or _thumb-tacks_. - -The best kind for this purpose have a head as thin as possible without -cutting at its edges, slightly concave on the under side next the paper, -and only so much convex on its upper side as will give it sufficient -thickness to enable the pin to be secured to it; better use four or more -small pins along the edge of a sheet of paper, than use one clumsy, -badly made pin at each end. - -[Illustration: Fig. 143.] - -[Illustration: Fig. 144.] - -Fig. 143 and fig. 144 represent a pair of plain trestles or horses in -common use for supporting large size drawing boards. This pattern is -found frequently in the laying-out shop. Fig. 145 and fig. 146 represent -_adjustable_ horses or trestles--these are designed, primarily, for -office use. As will be seen by viewing the illustration, the upper part -is supported by two hard-wood sliding pieces; these are provided with -strong pins and numerous holes, and pass through the frame of the -trestle, as shown, so that the upper portions can be arranged at any -angle convenient to the draughtsman, as he lies over his work or stands -by it. - -[Illustration: Fig. 145.] - -[Illustration: Fig. 146.] - -Fig. 137 is introduced to exhibit the paper attached to the drawing -board with the thumb-tacks, and with the T-square and set-squares -arranged to commence work; the paper should not extend to the edges of -the board; three, four or more tacks may be used on each edge of the -sheet of paper, instead of two, as shown in illustration. - -[Illustration: Fig. 147.] - -A most convenient--and except for its extreme lightness, which is not -good in a drawing stand--a most admirable device is shown in fig. 138. -The drawing table is simply a drawing board with folding legs; these are -made from hard-wood, while the top is made of soft, seasoned pine, with -square corners; while the device is strong and well braced, it can be -folded and easily carried about--all as shown in the illustration. - -[Illustration: Fig. 148.] - - -THE TEE-SQUARE. - -This is an instrument in the form of a letter T, as shown in the figures -149 and 150; the two parts are known as _stock_ and _blade_; the -horizontal part of the letter (T) is the stock, and the vertical part -the blade--hence the name, T-square; to form the square, the two parts -are joined together in such a way as to make them exactly at right -angles to each other; the stock, which is applied to the working edge of -the drawing board, being about one-third the length of the blade, and -about three times its thickness. - -[Illustration: Figs. 149 and 150.] - -To be perfect in construction, a tee-square should be as light as is -consistent with its necessary strength and stiffness of parts; it should -be made of suitable material easily manufactured, put together, and -repaired, and withal as truly correct as is possible to be made. Such a -square is represented in fig. 148; it has a taper blade, which is -generally about double the width where secured to the stock as it is at -the end. - -[Illustration: Figs. 151 and 152.] - -The manner in which the stock is united to the blade determines its -adaptability or otherwise to the use made of it; in some the stock is -rectangular in section, and the blade mortised into it; in others the -blade is dovetailed and let into the stock for the whole of its -thickness. - - -ADJUSTABLE BLADED SQUARE. - -In cases where many parallel lines have to be drawn, of lengths beyond -the capabilities of ordinary set-squares, and in directions other than -square with or parallel to the working edge of the drawing board it is -convenient to have for use an _adjustable_ bladed tee-square, or one -whose blade can be set at any desired angle. The blade of such a square -should be tapered as in illustration, but shaped at its wide end as -shown, and having a stock wide enough to allow for the surface required -in the washers of the fittings necessary to make the blade adjustable. -These fittings, though requiring to be well made and neatly finished, -are not expensive or difficult to make, as they consist merely of two -washers, a square-necked bolt, and a fly nut. - -The tee-square, as shown, has four parts: 1, blade; 2, fixed head; 3, -shifting head; 4, swivel. The head is held firmly by the left hand to -the left edge of the drawing board, and the blade serves as a -straight-edge for horizontal lines that may extend the whole length of -the paper. It can be used for either horizontal, or, by reversing to the -bottom of the board, for vertical lines; and, by turning it over, so -that the shifting head is against the edge of the drawing board instead -of the fixed head, lines at different angles may be drawn. The length of -the blade should be the length of the drawing board; if it is shorter, -inconvenience will be experienced when lines the whole length of the -board are wanted. - - -TRIANGLES, OR SET-SQUARES. - -Set-squares are invariably used in connection with the tee-square, as -shown in fig. 148. The illustrations below show several patterns of the -device; by these, vertical lines, triangles, squares and hexagonal, -octagonal and twelve-sided figures, diagonal section lines, etc., can be -easily drawn. For ordinary purposes, a triangle or set-square with -angles of 45° may be 4 inches long and the other 8 inches in length, but -a six-inch set-square having angles 90°, 45° and 45°, and an eight-inch -one having angles of 90°, 60° and 30°, will be found sufficient for all -purposes; there are other triangles used specially for making letters. - -[Illustration: Fig. 153.--Fig. 154.--Fig. 155.--Fig. 156.] - -In practice the triangles or set-squares are slid along the edge of the -blade, and need not be any thicker than it. - - -PARALLEL RULE. - -[Illustration: Fig. 157.] - -This instrument is used to mark lines which are neither horizontal nor -vertical (usually these are drawn by the square and set-square), and -which are parallel to one another; by adjusting the edge of the parallel -ruler to a line, it can be extended or opened out (or vice-versa -closed), and the line or lines drawn will be parallel or equally distant -from the base or first line it was set by. See fig. 157. - -Fig. 158 is a parallel ruler, constructed with two rollers fixed on a -rod so that they move the same distance, carrying the ruler parallel to -the starting line. - -[Illustration: Fig. 158.] - - NOTE.--It has been said that “a workman may be known by his tools,” - but the statement must be taken with a good deal of allowance. Some - workmen may possess a very fine set of tools and never use them, - because they have not the ability or inclination to learn how; - especially is this the case with drawing instruments. - - If all the fine sets of drawing instruments that are owned by workmen - were put to frequent use the owners would find a marked improvement in - their abilities in other lines as well as drawing; for it is a - noticeable fact that when a person’s mind has been trained in a - business that requires close calculation and a knowledge of materials, - he is capable of showing good qualifications in other lines, and the - more skilled he is in one the easier can he acquire skillfulness in - another, if he applies the same amount of energy, thought and interest - as he did to acquire skill in the first. - - -SECTION LINER. - -Fig. 159 shows an improved section liner which can be adjusted to any -angle, and will space the parallel lines at any desired regular -distance. - -[Illustration: Fig. 159.] - - -IRREGULAR CURVE OR SCROLL. - -Irregular curves, or, as they are sometimes termed, sweeps, represented -by figs. 160-166, are used for curves that cannot be put in by the other -instruments. They are very useful when elliptical or parabolic curves -are desired, in preference to circles or arcs of a circle. They are much -used in design and architectural drawing. They are made of thin hardwood -or rubber, and sometimes of horn. - -[Illustration: Figs. 160-166.] - -[Illustration: Fig. 167.] - -[Illustration: Fig. 168.] - -Curves are irregular lines; a circle is a regular line. If a curve is to -be passed through a number of predetermined points it should be first -sketched in lightly, free-hand; a section of the scroll is then applied -to the curve so as to embrace as many points as possible; only the -central points of those thus embraced should be inked in; this process -is continued until the desired curve is completed. - -Curves are made of various material, pearwood, cardboard, xylonite, hard -rubber, and a strip of soft lead is sometimes used, which may be easily -adjusted to the curve required. - -The curves generally used in mechanical drawing are shown on previous -page. - -Fig. 208, page 134, is a logarithmic _spiral_ curve. It is -mathematically constructed and contains every curve within the limit of -its size. - - -ELLIPSES. - -An ellipse is a geometrical figure, and can be drawn as described in -geometrical problem 29, page 96; many drawing offices keep sets of hard -rubber ellipses, to economize time constructing them. - -[Illustration: Fig. 169.] - -[Illustration: Fig. 170.] - - -DRAWING PENCILS. - -These are instruments for marking, drawing or writing, formed of -graphite, colored chalk or materials of similar properties, and having a -tapering end, inclosed, generally, in a cylinder of softwood. Fig. 167 -represents a ruling pencil; its point is a parallelogram or of a wedge -shape. In ruling, the length view rests against the square; its shape -gives considerable strength to the lead and allows the making of a very -fine line. Fig. 168 differs in the point of the pencil shown, as may be -observed in the illustration. - -A pencil that is hard is best for mechanical drawing; one that will -retain a good point for some considerable time. Pencil lines should be -made as light as possible; the presence of lead on the surface of the -paper tends to prevent the ink passing to the paper, and in rubbing out -pencil lines the ink is reduced in blackness, and the surface of paper -is roughened, which is a disadvantage. As little erasing or rubbing out -as possible should be done. - - -DIVIDERS AND COMPASSES. - -These instruments, while they appear alike, have a separate use: the -dividers are used to space off distances and dimensions; especially are -they necessary in reading drawings made to scale. _Compasses_ are used -for describing circles, curves, etc., _dividers_ are used for marking -out spaces. - -Two forms of the dividers are shown in figs. 169 and 170; the simplest, -plainest form is shown in fig. 169; these are used for rough spacings; -fig. 170 represents a pair of dividers fitted with an adjustable screw -controlled by a steel spring in one leg; by this a very exact -measurement can be made. Fig. 170 is intended to exhibit what is called -a “hair-spring divider.” - - -PROPORTIONAL DIVIDERS. - -These dividers differ from the ordinary ones shown in figs. 169 and 170 -in that they are provided with four steel points, one pair of which -being set to the full dimension will be reproduced by the other pair, -but in a smaller, or reduced size. - -Fig. 171 are “bisecting” dividers, being proportional dividers, which, -when open, one end measures double the distance of the other. - -Fig. 172 are proportional dividers; the points at one end are capable of -being changed, to measure practically any desired proportion at the -other end, by altering the position of the pivot where the legs cross -one another. The lower connecting link is a micrometer adjustment, for -minute measurements. - -Fig. 173 are proportional dividers which are marked for the proportions -of lines and radii of circles, being provided with a rack movement for -adjustment. - -Fig. 174 represents three-leg dividers, used for taking the position of -three points; this instrument is very useful in finding the position of -a point in a figure. - -[Illustration: Fig. 171.--Fig. 172.--Fig. 173.--Fig. 174.] - -[Illustration: Fig. 175.--Fig. 176.--Fig. 177.--Fig. 178.--Fig. 179.] - - -COMPASSES. - -Compasses consist of two pointed legs; they are instruments for -describing circles or for--sometimes--measuring figures, in absence of -dividers. Fig. 175 represents compasses fitted as dividers. - -Compasses should have jointed legs, which will allow the points to be -placed at right angles to the paper, whatever the size of the circle to -be drawn. Compasses should not be used for circles which are too large -to allow the points to be thus placed; a lengthening bar is generally -provided, which greatly increases the diameters of circles which may be -drawn by this attachment; it is shown in fig. 176. - -One leg of the compasses is usually provided with a socket to which are -fitted three points: a divider point, fig. 179; a pencil point, fig. -177; and a point, fig. 178, carrying a special pen for the inking of -circles. Each of these points is generally provided with a joint, so -that it may be placed at right angles to the paper. - -The other leg should be jointed; it is often provided with a socket -which receives two points, one a divider point, and the other carrying a -needle point. Such an instrument may be used as dividers for spacing, or -as compasses for penciling or inking circles. - -The joint at the head of the compasses (see fig. 175) is the most -important feature. It should hold the legs firmly in any position, so -that in going over a circle several times only one line will result. It -should allow the legs to move smoothly and evenly, and should be capable -of adjustment. - -[Illustration: Fig. 180.--Fig. 181.--Fig. 182.--Fig. 183.--Fig. 184.] - -As shown in fig. 174, one leg has a hinge or joint, and a needle point, -which can be regulated by a thumb screw; the other leg has a socket or -recess into which interchangeable parts can be inserted. The four -figures to the right of the compasses show the parts which are provided -with shanks or insertion pieces. Fig. 180 and fig. 181 represent -compasses specially used for making small circles, and work too minute -for the larger instruments described above. - -To do work of this nature easily a pair of spring dividers are -frequently used. This instrument has one point attached to a spring, -which is regulated by a screw, so that very slight changes in the space -may be made with ease. - -Compasses specially used for putting in fine circles and dimensions are -called “bows.” When a pen point it is a “bow pen,” with a pencil point a -“bow pencil,” and if with needle point a “bow dividers.” Fig. 180 is a -“bow dividers”, this fitted with screw for fine adjustment in one leg, -fig. 181, is called a “hair-spring bow dividers”; for small details, -bows with steel spring legs without any joint are used; these are called -“steel-spring bows.” - -[Illustration: Fig. 185.--Fig. 186.--Fig. 187.] - - -SPRING BOWS. - -These were originally developed from the common form of compasses, with -a single spring leg; later, the demand for smaller sizes made changes -necessary, and spring bows are now made symmetrical, both sides of the -bow being made to “spring.” - -Fig. 182 are spring dividers. - -Fig. 183 is a spring pencil. - -Fig. 184 is a spring pen. - -In these figures it will be seen that the two threads, a right and a -left, are moved with one central thumbscrew; in the figs. 185 to 187 a -single screw is used. - -In choosing spring bows, care must be exercised to select a sufficiently -strong, stiff spring, as the relation between spring pressure and -thumbscrew is important. - - -BEAM COMPASSES AND TRAMMELS. - -In fig. 188 is shown a set of beam compasses, together with a portion of -the wooden rod or beam on which they are used. - -[Illustration: Fig. 188.] - -The latter, as will be seen by the section drawn to one side, _A_, is in -the shape of a T. This form has considerable strength and rigidity. Beam -compasses are provided with extra points for pencil or ink work, as -shown. - -While the general adjustment is effected by means of the clamp against -the wood, minute variations are made by the screw, _B_, shifting one of -the points, as shown in the figure. - -This instrument is quite delicate, and, when in good order, is very -accurate. It should be used only for fine work on paper, and never for -scribing on metal. - -[Illustration: Fig. 189.] - -A coarser instrument, and one especially designed for use upon metal, is -shown in fig. 189, and is called a trammel. There are various forms of -this instrument, all being the same in principle. The engraving shows a -form in common use. A heavier stick is used with it than with the beam -compasses, and no other adjustment is provided than that which is -afforded by clamping against the stick. - -In the illustration, a carrier at the side is shown, in which a pencil -may be placed. Some trammels are arranged in such a manner, that either -of the points may be detached and a pencil substituted. - -A trammel, by careful arrangement, can be made to describe very accurate -curves, and hence can be used in place of the beam compasses in many -instances. For all coarse work it is to be preferred to beam compasses. -It is useful for all short sweeps upon sheets of metal, but for curves -of a very long radius a strip of sheet iron or a piece of wire will be -found of a more practical service than even this tool. - -The length of rods for both beam compasses and tramels, up to certain -limits, is determined by the nature of the work to be done. The extreme -length is determined by the strength and rigidity of the rod itself. It -is usually convenient to have two rods for each instrument, one about -1¹⁄₂ or 2 feet in length and the other considerably longer--as long as -the strength of the material will admit. - - -DRAWING SCALES. - -Scales are proportioned rules or mathematical instruments of wood, -metal, etc., on which are marked lines and figures for the purpose of -measuring sizes and distances. It is usual to make scales in the -proportion of parts of an inch equalling a foot; the most generally -adopted scale for machine drawing is one and a half inches, equalling -one foot; that is, twelve-eighths of an inch (each eighth of an inch -representing one inch); there is no fixed rule in the choice of a scale, -as they are varied according to the coarseness or fineness of the parts -of the machine to be drawn and the space or surface of paper to be -utilized. - -When objects are of moderate proportions they may be represented full -size; but when large, the drawings must be smaller. Standard scales for -mechanical drawings are ¹⁄₂, ¹⁄₄, ¹⁄₈ and ¹⁄₁₆ full size. These scales -are often written 6″ = 1 ft.; 3″ = 1 ft.; 1¹⁄₂″ = 1 ft., and ³⁄₄″ = 1 -ft. - -[Illustration: Fig. 190.] - -[Illustration: Fig. 191.] - -Instead of selecting one of the scales named or one found upon the -ordinary scales used by draughtsmen, drawings may be made to any scale -whatever. Thus, if any object is to be represented in a certain space, a -scale should be constructed which will cause the whole of the object to -be shown. - -Drawing to Scale.--The meaning of this is, that the drawing when done -bears a definite proportion to the full size of the particular part, or, -in other words, is precisely the same as it would appear if viewed -through a diminishing glass. - -The two-foot rule shown in fig. 192 is the most useful instrument for -the comparison of linear dimensions--it can be used as a scale of -one-twelfth, or 1 inch equal to a foot, 12 inches = 12 feet, it being -divided into portions or spaces, each of which is subdivided into -halves, quarters, eighths and sixteenths; frequently in the latter class -of two-foot rules there are graduations of scales, and it is then also -called a draughting scale. - -Fig. 190 represents a flat scale, graded so that one inch represents a -foot--¹⁄₁₂th size--etc., as shown. - -Fig. 191 represents a triangular scale (broken). The triangular scale -should read on its different edges as follows: Three inches and 1¹⁄₂″ to -one foot, 1″ and ¹⁄₂″ to one foot, ³⁄₄″ and ³⁄₈″ to one foot, ¹⁄₄″ and -¹⁄₈″ to one foot, ³⁄₁₆″ and ³⁄₃₂″ to one foot, and one edge read -sixteenths the whole 12″ of its length. - -Fig. 190 shows such a scale broken. An explanation of the 1″ and ¹⁄₂″ -side will suffice for all. Where it is used as a scale of 1″ to one -foot, each large space, as from 0 to 12 or 0 to 1, represents a foot, -and is a foot at that scale. There being 12″ in one foot, the twelve -long divisions at the left represent inches; each inch is divided into -two equal parts, so from 0 to one division at the left of 9 is 9¹⁄₂″ and -so on. The 1″ and ¹⁄₂″ scales being at opposite ends of the same edge, -it is obvious that one foot on the 1″ scale is equal to two feet on the -¹⁄₂″ scale, and conversely, one foot on the ¹⁄₂″ scale is equal to six -inches on the 1″ scale; and 1″ being equal to one foot, the total feet -in length of scale will be 12; at ¹⁄₂″ to 1 foot the total feet will be -24. - -In working to regular scales, such as ¹⁄₂, ¹⁄₈, or ¹⁄₁₆ size, a good -plan is to use a common rule, instead of a graduated scale. There is -nothing more convenient for a mechanical draughtsman than to be able to -readily resolve dimensions into various scales, and the use of a common -rule for fractional scales trains the mind, so that computations come -naturally, and after a time almost without effort. - -The protractor shown in fig. 193 is an instrument for laying down and -measuring angles on paper; it is used in connecting with a scale to -define the inclination of one line to another. - -Protractors have the degrees of a half circle marked upon them; as the -whole circle contains 360 degrees, half of it will contain 180, -one-quarter 90, etc. Hence, protractors showing 180° exhibit all that is -needed. To protract means to extend, so this instrument is also useful -in “extending” the lines of inclination at the circle. - -[Illustration: Fig. 192.] - -[Illustration: Fig. 193.] - - -DRAWING-PENS. - -A special pen called a drawing-pen, and also special ink, are required -to ink a drawing; figs. 194 and 195 represent two sizes of -drawing-pens--one being best adapted for fine work, and the other for -coarse or heavy line work. The points, as will be observed in the -illustration, are made of two steel blades which open and close as -required for thickness of lines by a regulating screw. - -[Illustration: Fig. 194.] - -[Illustration: Fig. 195.] - -A good drawing pen should be made of properly tempered steel, neither -too soft nor hardened to brittleness. The nibs should be accurately set, -both of the same length, and both equally firm when in contact with the -drawing paper. The points should be so shaped that they are fine enough -to admit of absolute control of the contact of the pen in starting and -ending lines, but otherwise as broad and rounded as possible, in order -to hold a convenient quantity of ink without dropping it. The lower -(under) blade should be sufficiently firm to prevent the closing of the -blades of the pen, when using the pen against a straightedge. - -[Illustration: Fig. 196.] - -[Illustration: Fig. 197.] - -The spring of the pen, which separates the two blades, should be strong -enough to hold the upper blade in its position, but not so strong that -it would interfere with easy adjustment by the thumbscrew. The thread of -the thumbscrew must be deeply and evenly cut so as not to strip. - -An important requisite after the pencil lines have been put in is ink, -with which to line the drawing. This should be of the best that can be -procured. The pen is filled by dropping the ink between the blades, or -nibs, while held in a nearly vertical position, as shown in fig. 196. - -Liquid India ink can be procured in bottles with glass tube feeders, -which are very good, and keep the hands and fingers free of the ink. -Fig. 197 is a sectional view of such a bottle and “filler,” or feeder. -This generally answers all requirements, but the dry ink of good -quality, in sticks or bars, cannot be surpassed, although it requires -skill for its preparation. Fig. 198 represents a sloping dish or “tile” -for mixing, which should be done with little pressure, in clean, -filtered or distilled water, care being taken to keep the liquid free of -dust, which obstructs the free flow of the ink in the pens. - -[Illustration: Fig. 198.] - -The bars of India ink are shown, as they are imported, in figs. 199 to -202. - -Pure India or China ink is only made in those countries, because the -special wood from which it is prepared is found only in those regions. -So-called India inks, made of lampblack and animal glue, are only -imitations; therefore India ink should be purchased from a reliable -importing house--shape is little guarantee of quality. - -[Illustration: Fig. 199.--Fig. 200.--Fig. 201.--Fig. 202.] - -[Illustration: Fig. 203.] - -Soft gray vulcanized rubber (fig. 203) should be used for cleaning -drawing paper; for erasing any portion of a line in pencil, a piece of -prepared white vulcanized rubber is the best, small in size and of -rectangular shape (see fig. 205). - -[Illustration: Fig. 204.] - -[Illustration: Fig. 205.] - -[Illustration: Fig. 206.] - -[Illustration: Fig. 207.] - -An ink eraser is made of a composition of rubber and ground glass, and -it should be used as sparingly as possible on drawings, as it roughens -the paper and removes the gloss from its surface (see fig. 204). Steel -ink erasers are useful in removing defects, overrun lines, joint of -lines if swollen, etc.; they have a fine point and can be used to -advantage with a little practice; they are used with a scratching, not a -cutting, motion (see figs. 206, 207). - - -DRAWING PAPER. - -The first thing to be considered in selecting drawing paper is the kind -most suitable for the proposed plan. Paper may be purchased in sheets 22 -× 30 inches, that make four exercise sheets 11 × 15 inches; this may be -of several grades and tints. - -The qualities that constitute good paper are strength, uniformity of -thickness and surface, neither repelling nor absorbing liquids, -admitting of considerable erasing without destroying the surface, not -becoming brittle nor discolored by reasonable exposure or age, and not -buckling when stretched, or when ink or color is applied. - -The sizes and names of paper made in sheets is as follows: - - Cap 13×17 ins. - Demy 15×20 „ - Medium 17×22 „ - Royal 19×24 „ - Super Royal 19×27 „ - Imperial 22×30 „ - Atlas 26×34 „ - Double Elephant 27×40 „ - Antiquarian 30×53 „ - -For large drawings paper is made in rolls. “Detail paper” is especially -made for marking out new designs; it is made in rolls 36, 42 and 54 -inches wide; it has excellent erasing qualities and takes ink and color -with facility. - -When working by artificial light it is desirable that the paper be of a -light-brown color, which is less trying to the eyes than a pure white. - -If it is a shop drawing or sketch not to be preserved, use detail paper, -which is the most economical and will stand a great deal of handling -without becoming soiled. If it is a detailed plan, finished drawing or a -picture, use the best white drawing paper to be obtained, so that your -drawings can be preserved indefinitely without danger of fading, which -is due either to the paper being poorly made and discoloring with age, -or being of poor fiber and absorbing the ink or color, and the drawing -consequently losing its brightness. - -After deciding on the size of paper most suitable for the work, then -carefully select the paper embracing the most qualities of value for the -proposed drawing. - -[Illustration: Fig. 208.] - - -[Illustration: MECHANICAL DRAWING] - -[Illustration: Fig. 209.] - - - - -Mechanical Drawing. - - -In distinction to “free-hand,” mechanical drawing is executed almost -entirely by the use of the instruments previously described; hence its -other term, instrumental drawing. To define it particularly it may be -said that,-- - -Mechanical drawing is the correct reproduction of any figure or part of -a machine, whether of full size or reduced in the proportion of one part -to another; it also comprises the art of delineating the interior parts -which are hidden from view in solid bodies. - -A mechanical drawing is the vehicle for conveying the ideas of the -designer to those who are to embody them in wood and metal, and the -considerations which should govern its production are those which affect -its clearness and legibility or those which facilitate reference to it. - -Drawings consist of plans, elevations and sections; plans being views on -the top of the object in a horizontal plane; elevations, views on the -sides of the object in vertical planes; and sections, views taken on -bisecting planes, at any angle through an object. - -Drawings in true elevation or in section are based upon flat planes, and -given dimensions parallel to the planes in which the views are taken. - -Two elevations taken at right angles to each other fix all points, and -give all dimensions of parts that have their axis parallel to the planes -on which the views are taken; but when a machine is complex, or when -several parts lie in the same plane, three and sometimes four views are -required to display all the parts in a comprehensive manner. - -A man must have either a natural talent for hand-drawing or years of -experience, before he can produce a sketch and “dimension” it, fit to -work from; hence the elementary character of the examples given for -practice. A “pretty” drawing is not expected from a beginner; it should -always be borne in mind, that correctness of dimensions and general -clearness, rather than elaborate finish, are what will save the battle -in the days of competition. - -Mechanical drawings should be made with reference to all the processes -that are required in the construction of the work, and the drawings -should be responsible, not only for dimensions, but for adaptation to -fitting, forging, pattern-making, moulding, and so on. - -Every part laid down should have something to govern it that may be -termed a “base”--some position which, if understood, will suggest size, -shape and relation to other parts. Searching after a base for each and -every part and detail, the draughtsman should proceed upon a regular -system, continually maintaining a test of what is done. - -A mechanical drawing consists chiefly of three views: - - 1. The plan or top view. - 2. The side elevation. - 3. The end elevation. - -In addition to the above, drawings are used to show interior portions of -the figure; these are termed “sections,” and they may be taken where any -plane crosses another. - - NOTE.--The word _elevation_, as applied to mechanical drawings, means - simply a view; hence a side elevation is a side view, or an end - elevation is an end view. - - The word _plan_ is employed in place of the word top; hence a plan - view is a top view or a view looking down upon the top of the piece. - - A _general_ view means a view showing the machine put together or - assembled, while a detail drawing is one containing a detail, as a - part of the machine or a single piece disconnected from the other - parts of the whole machine. - - - - -Penciling. - - -[Illustration: Fig. 211.] - - It is very nearly true, as has been said, that every mechanical - drawing is, in the first instance, penciled; if this is so, then more - work is done with the pencil than with the pen; therefore the first - attention of the student of mechanical drawing should be directed to - the following instructions for penciling a drawing. - - With all necessary materials in hand, and in good order for the - beginning of a drawing, the first thing to do is to pin the paper on - the board quite square. - - To do this effectively, lay the paper flat and put on the T-square - with its head at the left side of the board; slide the square up - nearly to the top, and arrange the paper level with the blade; with - the right hand hold the paper still and move the square down a little; - now, pin the top of the paper with thumb-tacks. - -Next, pressing the square lightly to the paper, slide it down to the -bottom and pin that part of the paper to the board. The paper must not -project outside or over the edges of the board, and the pins or -tack-heads should be forced down flush with the paper, so as not to -interfere with the free movement of the tee-square up and down the board -as occasion may require. - -The accuracy of the work depending upon their condition, it is first -needful to see that the pencil and pencil compasses are properly -sharpened. Reference is made to valuable directions contained on pages -beginning with 55, under the heading of “Free-hand Drawing,” to which -may be added that-- - -All lines should be drawn with the pencil slightly inclined in the -direction in which it is moved. - -Any and all lines not needed in the finished drawing should be erased at -one time after the final lines have been determined, for the surface of -the paper is soiled very quickly when worked upon after erasures have -been made. - -The working lines and other lines that are to be removed should be -erased when the drawing is ready to finish and before its outlines have -been strengthened, in order that the final lines may be left in perfect -condition. - -To show where the lines meet or terminate it is needful that all pencil -lines pass the actual ending place, making a distinct intersection. This -does not apply to “inking in” the lines, but rather to prevent the -over-drawing of the ink lines, because the edge of the rule and the pen -itself obstruct or partly cover the view of the line, it is very liable -to pass over or beyond the required point in inking the lines, which -must not occur. - -In the preliminary operation of producing a regular mechanical or -instrumental drawing, it is necessary to make a “sketch,” in pencil, of -the object to be represented. The AMERICAN MACHINIST has given in a few -words the order to be followed, in effecting the best results; we quote, -as follows: - -“In making a free-hand sketch of an object from the model it is well to -observe the following order: Look the model over carefully and determine -the number of views necessary to illustrate it fully, drawing the same, -free-hand, in their proper relation to each other, on sketching paper. -Look the sketch over carefully to see that nothing has been omitted, and -put on dimension lines, after which scale the model carefully and put on -dimensions. Do not put in the dimensions at the same time the dimension -lines are drawn; have all the dimension lines in place before attempting -to insert dimensions. - -“Follow the same order in making the drawing with instruments as was -used in making the sketch; that is, draw the views in their proper -relation to each other, put in dimension lines, then dimensions, and -lastly notes and title. If section drawing is made, do not draw section -lines in pencil.” - - -POINTS TO BE OBSERVED IN SKETCHING. - -1. Especial attention is to be paid to outlines--edges of plane surfaces -are lines; when a line is made it represents the edge or outside of -something. - -2. Learn to be accurate before being rapid. - -3. A sketch should be intelligible to any one, even if they are -unacquainted with drawing. - -4. Horizontal and vertical lines and a few curves will enable one to -make almost any simple sketch. - -5. It must be also remembered in making drawings from actual measurement -that the instruments are not in the first place employed; the rough -sketch is first made and then it is converted into a drawing. The -draughtsman makes a rough sketch entirely by the hand and eye, measures -the various parts, and jots down the measurements in his sketch; after -this he reduces the whole to the desired scale, and proceeds to make his -mechanical drawing. - -6. Let the sketch book be the constant companion of the student; it may -be advantageously filled with outlines of machine or other work suitable -for preservation, to be made into finished drawings, or for reference. -Sketches are often valuable for reference as aids in originating new -designs. - -7. A sketch, when possible, should have all the dimensions written upon -it, but-- - -8. Sketches in shop practice should not take the place of -working-drawings; the latter have a check upon them in being drawn to a -scale--hence the figures written upon them and dimensions by scale must -agree. - -9. Place title and date on each sketch--no matter how seemingly -unimportant--for future reference. - -10. Practice sketching at every favorable opportunity. There is no -necessity for detail at first--simply the outlines of the article and -its parts. - -11. Sketch-books, with paper bound in cloth covers, are utilized for -bold, off-hand sketches by experienced draughtsmen, but a single sheet -of paper, used on both sides, is not unworthy of service in an -emergency--or even the blank side of a letter may be available. -Sketching-blocks, or paper “pads,” 4 × 6, or more, in size, and -containing 48 sheets, are sold by stationers, and are found to be most -convenient to have in hand and for practical use. Portfolio-envelopes, -made of extra length paper (manila) are useful in filing away sketches -and drawings. The size 10¹⁄₂ × 15 is used for United States Patent -Office drawings. - -The function of the pencil--in mechanical drawing--is to make a path for -the pen to follow. If it were possible to make a drawing with all its -lines ending at the proper place, at the first time, there would be no -necessity for using the pencil. One is obliged, however, so to use the -pencil that all lines pass beyond the actual ending place, thus making a -distinct point for the drawing pen to stop at. - -The pencil should be pressed to the paper just enough to make a clean, -fine line, and no more; once over the path is sufficient, if the line is -visible and true. - -To sharpen drawing pencils, 1, use a fine file, after taking off enough -of the wood with a knife; 2, make a conical point for the free-hand -drawing pencils and a chisel point for ruling and marking distances. - -Pencil compasses are instruments where one leg is provided with a pencil -point. Fig. 209 shows the mode of manipulation of those shown in fig. -180 and fig. 181. - -The pencil compass is held by the projection above the joint between the -thumb and first finger, which enables it to be rotated by a movement of -the finger without causing any undue pressure on the points. Should much -pressure be applied, there is a tendency to force the point or center -through the paper, making an ugly center mark; at the same time the -pressure tends to break off the pencil point. - -These few illustrations from fig. 212 to fig. 219 are made designedly -simple, so that they may be utilized in “free-hand” work, for which they -are good practice, as well as serving for examples in mechanical -drawing. - -[Illustration: Figs. 212.] - -Figs. 212, showing a spool or bobbin, exhibit three views, viz.: front -elevation or plan, and section; both are drawn with simple lines, the -end elevation by circles. - -[Illustration: Fig. 213.--Fig. 214.] - -Figs. 213 and 214 are two side views of a hexagon head bolt. Figs. 216 -and 217 are a square head bolt--two side views and end view. Figs. 218 -and 219 are a front and edge view of a forked or double joint. - -[Illustration: Figs. 215.] - -Figs. 215 show three views of a file handle; the front view and section -are practice for compound curves and curved lines meeting straight ones; -all these are capable of being produced by instruments. - -Moreover, many of the views and illustrations used to instruct and -explain machine tools and other devices in other parts of the volume, -are drawn so that they may be used also as examples in advanced -instrumental practice. This is a “hint” to the diligent and painstaking -student worthy of remembering. - -[Illustration: Fig. 216.--Fig. 217.--Fig. 218.--Fig. 219.] - -The designing and drawing of arcs and whole circles occupy a large -proportion of space in nearly all mechanical drawings. The making of a -complete circle is a matter of no great difficulty, but the beginning -and termination of parts of circles require both judgment and -considerable practice. - -To aid the student these two illustrations of circles are introduced. To -draw fig. 220 with a pencil, using the upper edge of the blade of the -T-square as a guide, draw a center line, _A B_, mark on it a distance of -4 inches, space this into half inches, using the dividers and making the -points with it; then with the pencil compasses or bow pencil, which must -be held as shown on page 209, and rotated from left to right, or -clockwise, draw a series of circles through these points, tangent to one -another or all touching at _A_, care being taken that the pencil lines -exactly meet at _A_, and also cut the divided points as shown in the -illustration. For fig. 221 divide the center line as before, and draw -the semi-circles on it _A B_, _B C_, meeting at _B_, and _C D_, _D E_, -etc. - -[Illustration: Fig. 220.] - -[Illustration: Fig. 221.] - -Now from center _B_ draw circles _A C_, _C E_, meeting in _C_, and so on -with the circles, arcs or segments; success in drawing this figure -depends on the correct spacing of the center line in the first instance -into equal parts. - -The T-square should be used for drawing horizontal lines only. Its head -should always be placed upon the left edge of the board. Vertical lines -should be drawn by the use of a triangle placed upon the T-square and -not by means of the T-square only; because the edges of a board are -seldom at right angles to each other, and the blade of the T-square is -often not at right angles to the head, so that lines at right angles to -each other will not result from the use of the T-square upon all edges -of the board. Only the upper edge of the T-square should be used, as the -edges are often not quite straight or parallel. - -The 45° triangle has two angles of 45° and one of 90°. The 30° and 60° -triangle has an angle of 30°, one of 60°, and one of 90°. By placing -these triangles upon the T-square, lines at any of these angles with a -vertical or horizontal line may be drawn. - -Drawings finished in ink are much more effective and desirable than -pencil drawings; but as a good inked drawing cannot be made except upon -an accurate pencil drawing, students should begin with the pencil, and -should not use ink until they are able to produce satisfactory results -in pencil. - - - - -Projection. - - - The word projection means to throw forward, and in ordinary machine - drawing it is the projecting or throwing forward of one view from - another view. - - In drawings the lines in one view or plan may be availed of to find - those of others of the same object, and also to find their shape or - curvature as they would appear in the other representations; this is - called projection-drawing. - -Fig. 222 is the illustration as shown in fig. 212 on page 143, with the -addition of dotted projection line, which illustrates the method of -throwing forward the section and the end view of the object; these two -views are procured from the plan or first figure, as shown in fig. 222. - -Fig. 224 represents the square bolt and nut shown in fig. 216, and the -mode of projecting is similarly shown by dotted lines. - -Fig. 223 shows file handle shown in fig. 215 and the mode of projecting. - -[Illustration: Fig. 222.] - -[Illustration: Fig. 223.] - -[Illustration: Fig. 224.] - -The principles upon which “projection” in drawing is based, are -illustrated in the following examples and text: As a real object can be -scaled with a foot rule, so a drawing must permit of scaling and -measuring. This measuring may take place as with the real object in full -size or the drawing may, for the sake of convenience, be reproduced and -measured in a reduced scale, as half, or in still smaller sizes. -Sometimes it may prove convenient to enlarge the drawing to twice the -natural size of the object, as a means of making it stand out more -clearly than the real size would accomplish. - -For practical purposes, it is productive of economy of time to mark the -dimensions of height and width or depth on the drawing in figures, to -avoid the scaling. This marking of the dimensions is best done at the -time of making the drawing, while the conception of the object is clear. - -To convey a correct impression of the object, all lines that are marked -to be of equal length should appear equally long on the drawing and be -capable of being scaled to such equal length; for this, it must be -assumed that the eye of the observer is equally distant from all points -of a plane through the nearest point, or one of the axes of the object, -and that the lines of sight are all parallel to each other and square to -this plane. - -In fig. 225 these lines of sight are seen as directed toward one side of -a cube or block; it will be readily understood, however, that in this -way nothing is visible and accessible for scaling and dimensioning -except this front face of the block, thus, a determination of the -dimensions of only height and width would be possible, while the -dimension of depth is entirely undetermined. - -[Illustration: Fig. 225.] - -It is thus necessary to get a view of the block from another side; the -direction in which it will be most instructive to obtain additional -views is in the direction of the breadth and of the length and square to -the lines of sight of the first view. - -Fig. 226 shows how the lines of sight would strike the object in the -three directions. If these lines should be rays of light, some of them -would pass by the body until they squarely strike the large plane -surfaces, _I_, _II_, _III_; naturally the rays of light on the faces of -the object will be retained, and cannot strike the plane surfaces, thus -leaving dark shadows of exactly the same outlines as the block; these -would be exact drawings of the faces, and if by some means they can be -fixed and retained on the plane they can be completely measured and -dimensioned. - -This throwing forward of the outline of the object in different views on -the planes is called projection of the object, and furnishes a highly -important means of fixing the outlines and dimensions in the three main -directions of height, width and depth; evidently, the light rays passing -by the front face may not all reach the plane of projection, but they -may be retained by protruding parts of the object behind the front face. -These protruding parts naturally would also be projected on the plane in -the same manner as the main body of the block. - -These projections of the protruding part are plainly visible in plane -_II_ and plane _III_, while the part would not be drawn in outline in -plane _I_. It may be imagined, however, that the greater thickness of -the body in the direction of the protruding part would intensify also -the shadow, thus outlining the face of the protruding part in plane _I_. - -It is apparent that these three projections are all needed, but as -drawing is all done in one single plane, the three projections will for -the sake of convenience have to be brought into a single plane. This can -be realized if plane _II_ is swung around axis _O Z_ and plane _III_ -around axis _O Y_, until all three surfaces are in one single plane, -which would then appear as shown in fig. 227. - -It is also possible to assume transparent planes in front of the body -and extend the parallel lines of sight forward instead of backward. Thus -an outline picture will be created on each of the three planes _I_, -_II_, _III_ in fig. 228, in a manner similar to fig. 226. For drawing -purposes, all three views again have to be brought into a single plane, -which is done by swinging _II_ around _O Z_, and _III_ around _O Y_ in -the same manner as fig. 227 was evolved from fig. 226. - -It will be noted that in fig. 229 plane _III_ is now above _I_ and plane -_II_ on the left-hand side of _I_, while in fig. 227 they were below and -at the right-hand side of plane _I_. As the swinging of the top and side -planes takes place around the edges of the front plane _I_ two systems -may thus be distinguished, according to the position of plane _I_ in -regard to the object. Fig. 226 thus represents the system of backward -projection, while fig. 228 represents the system of forward projection. - -Either system can have, however, the plane _II_ at the right- or -left-hand side edge, while plane _III_ may be attached to the top or -bottom edge of plane _I_; it is readily understood that a number of -combinations are possible for each system, as it is not necessary to -adhere absolutely to one rule. The system of forward projection is the -one generally practiced and further examples are all executed by this -system, meaning that the planes are always between the observer and the -object. - -[Illustration: Fig. 226.] - -[Illustration: Fig. 227.] - -[Illustration: Fig. 228.] - -For the clearness of the drawing, it is desirable to have all corners, -edges and outlines appear in such solid lines as they appear to the eye. -If, therefore, certain sharply defined outlines occur on one side more -than on the opposite one, it is most desirable to take the view against -the side that has the most definitely marked outlines. - -If the opposite side should show a number of wholly different features, -it may prove even desirable to show this side also, thus gaining four -views instead of the usual three, and so obtain a more complete -understanding of the shape of the object, besides giving increased -facilities for dimensioning each part of the body. - -This additional view may be taken from the sides, as well as up or down, -thus making a maximum of five views possible by which the outside of the -object may be delineated. - -Fig. 230 shows why it may be desirable to take five views of a block -that has a receding space of different outlines in each of the side and -top and bottom faces. - -It is not necessary, however, to resort to five views in such a case as -is represented in fig. 230, as the only differing feature, the circular -space in _III_ bottom, might be shown in _III_ top by dotted lines, and -the difference of _II_ right might be shown in _II_ left, also by dotted -lines. It is desirable to show four or five views only where great -complication and consequent lack of clearness through numerous dotted -lines would result from having a less number of views. - -[Illustration: Fig. 229.] - -So far the projections and views have only represented the outlines of -the object; it may often be desirable, however, to show central holes or -other perforations or variations of sections; in this case it is -possible to imagine the object cut in slices, by planes, through certain -well defined axes, or other lines of distinctive importance, and then -take a view of this sectional plane with its newly created intersections -or sharply marked outlines; thus, a section may often take the place of -the third, fourth or fifth view to great advantage. - -[Illustration: Fig. 230.] - -It is not always possible to get a view against a face or side of the -object, but, with irregularly shaped bodies or under special conditions, -it may be necessary to take a view of corners, sloping planes, curved or -irregularly shaped surfaces. - -Fig. 231 shows a hexagonal nut in the three normal projections; from the -top view it is readily seen that the front view shows the side of the -hexagon in a contracted scale, and that therefore the scaling and -dimensioning for the horizontal direction have all to be done in the top -or plan view; the front and side views convey, however, the dimension of -height correctly, and these are therefore the right places for scaling -and dimensioning in the vertical direction. - -Fig. 231 shows how a cylindrical outline appears in the three views; the -hole in the nut presents itself as circular in the top view, while it -appears in the front and side views as a rectangle. For simple objects, -it is unnecessary to show the edges of the planes, and the three views -are grouped, as regards distances and positions, as most convenient for -the execution of the drawing. - -Where sloping surfaces are of irregular form, it may be necessary to -employ help lines for their full determination in the three views. Fig. -232 shows how the surface that is produced by a slanting cut through a -cylinder would appear in the three views. The help lines are placed in -the top view in eight equal divisions around the circumference of the -cylinder. These division lines are shown by dotted lines on the cylinder -in the front and side views. Their intersections, with the sloping cut -in the front view, furnish also the height of the corresponding points -for the side view. - -[Illustration: Fig. 231.] - -By progressive determination of points, lines and surfaces, even the -most complicated bodies can be completely represented for their -reproduction in any application to mechanical or industrial purposes. - -[Illustration: Fig. 232.] - -The spur wheel shown on page 163 is an example of projection drawing. - -The wheel is illustrated in three views: fig. 235 is a side view, or -elevation; fig. 234 is a front elevation; fig. 233 is a section view on -_C D_. The section is projected from the front elevation by drawing -parallel lines from the points in front elevation where the lines are -intersected by the center line _C D_, cutting the plane of view and -showing the interior shape at _C D_. - -The side elevation, fig. 235, is projected from the front elevation, -fig. 234, by drawing parallel lines from the edges in the front view -across its face. - -In actual drawing practice the figures should be made about three times -larger than the example, and as follows: - -For the front elevation draw the center lines _A B_, _C D_, and from -their point of intersection as a center, with the compasses draw the -inner circle or hole, also the pitch line _E E_. With the dividers space -this line into nineteen equal divisions; each point on this line will be -the center of a tooth, and the distance from one point to the next one -is the pitch of the tooth. - -Now, with the dividers mark off the thickness of tooth at each side of -these points on pitch line; with the compasses draw the outer circle for -points of teeth, the inner circle for the root of teeth, and circles for -thickness of rim and hub; also circles representing the fillets at rim -and hub. For clearness and to prevent confusion these lines are shown on -one-half the wheel only, terminating in center line _C D_; all the lines -of the front elevation are now complete excepting the teeth. - -In the drawing office it is unusual to delineate all the teeth in a gear -wheel, the lines without the teeth as now completed being deemed -sufficient, giving all particulars; however, to prevent the error of -mistaking the circles it is well to represent one or two teeth on the -drawing. - -[Illustration: Fig. 233.--Fig. 234.--Fig. 235.] - -In the example, proceed and complete the entire wheel, taking on the -compasses a radius equal to the pitch of the tooth; set them on the -pitch line at the point _G_, already spaced for thickness of tooth, draw -line _H_ from pitch line to root of tooth; proceed similarly all around -the circle, completing one side. Next, reverse the operation and draw -the corresponding side of the root of tooth. Now take a radius equal to -half the space between teeth and the thickness of tooth on pitch line, -and, with center in pitch line, as shown at _I_, draw the outside or -addendum of tooth, _J_; it will be apparent that the reverse addendum of -the tooth next adjoining can be formed at the same time, with one -setting of the compasses; finish all the teeth similarly; the front -elevation will thus be completed. - -Fig. 233 is an excellent example of sectional drawing, to be executed as -follows: - -First draw the center line _L M_ of fig. 233, lay off at each side of it -half the breadth of face and of the hub; now project, from center line -_C D_, in the front elevation, fig. 234, with the T-square the upper and -lower teeth, the fillets, the chamfers, the hub and center hole, also -dot in the pitch line _F F_, take the radii and draw the fillets and -chamfers; draw section lines and the view will appear as shown in fig. -233. - -Fig. 235 is a side elevation of fig. 234, and is a fine sample of -projection, to be executed as follows: - -Proceed similarly as for fig. 233, projecting the lines from the outside -edges of the front elevation, instead of from the center line _C D_ as -in last figure, and the end elevation will be as shown in fig. 235. - -The student will be assisted in understanding this working drawing by -consulting the pages under the heading of “Gearing.” - - -[Illustration: LETTERING, INKING - -SECTION LINING] - -[Illustration: Fig. 236.] - - - - -“Inking In” Drawings. - - -When a drawing is completely finished in penciling, it should next be -“inked in” for preservation. - -Care should be used that the pen may be perfectly clean; the pen should -be held nearly vertical, leaning just enough to prevent it from catching -on the paper; the pen should be held between the thumb and first and -second fingers, the knuckles being bent, so that it may be at right -angles with the length of the hand. - -The ink should be rubbed up fresh whenever it is about to be used, for -it is better to waste a little time in preparing ink slowly than to be -at a continual trouble with pens, which will occur if the ink is ground -too rapidly or on a rough surface. - -To test ink, a few lines can be drawn on the margin of a sheet, noting -the shade, how the ink flows from the pen, and whether the lines are -sharp. After the lines have dried, cross them with a wet brush; if they -wash readily, the ink is too soft; if they resist the water for a time -and then wash tardily, the ink is good. - -Care must be exercised not to overload the pen with ink, and, like the -pencil, the pen should always be moved from left to right and from the -bottom to the top of the board. When inking, both “nibs” of the pen -point must rest evenly on the paper and the pen be pressed only lightly -against the T-square. Never ink any portion of a drawing until the -penciling is complete. - -In inking long, fine lines it is well to go over each line twice, -without moving the T-square, trying not to widen the line on the second -passage; also see that the pen contains ink enough to finish a line, as -it is difficult to continue with the same width of line after -re-filling. - -To produce finished drawings, it is necessary that no portion should be -erased, otherwise the color applied will be unequal in tone; thus, when -highly finished mechanical drawings are required, it is usual to draw an -original and to copy it. Where sufficient time cannot be given to draw -and copy, a very good way is to take the surface off the paper with fine -sand-paper before commencing the drawing; if this be done, the color -will flow equally over any erasure it may be necessary to make -afterwards. - -The rules of procedure in drawing the lines in “inking” are, 1, ink in -the small circles and curves; 2, ink in the larger circles and curves; -3, then all the horizontal lines, beginning at the top of the drawing -and working downward; 4, next ink in all the vertical lines, commencing -at the left and moving back to the right; 5, draw in the oblique lines; -6, all the center lines and dimension and reference lines. The figuring -and lettering should be always done with India ink, thoroughly black; -the last lines to be drawn are the section lines. The reason why -irregular curves and arcs of circles are inked in first is, that it is -easier to draw a straight line up to a curve than to take a curve up to -a straight line. - -In practice, the flat side of the drawing-pen is laid against the -tee-square or ruler; the taper of the blade of the pen is sufficient to -throw the point enough away from the edge to prevent blotting; the pen -is drawn from left to right and from the bottom to the top of the board. - -This is shown in fig. 237, intended to represent “short work” with the -drawing-pen. The wrist is shown resting upon the blade of the square. - -[Illustration: Fig. 237.] - -[Illustration: Fig. 238.] - -In a similar figure, 238, the position of the hand holding the pen -indicates the best relative posture for inking long lines. In one of -these illustrations the work is executed principally by the wrist--in -the other by the arms and fingers working together. - -The pen should be held with even pressure against the straight-edge or -curve. If the pressure varies, the blades will spring and the width of -the line will change. The blades should be of such length that both will -bear equally upon the paper when the pen is inclined slightly, so as to -bring the inner blade near the straight-edge; the angle of the pen -should not be changed while drawing any line. - -When the inking is finished, the whole drawing may be cleaned by rubbing -it with bread which is not greasy or so fresh as to stick to the paper. -If the paper is much soiled it may be necessary to use an eraser. A soft -pencil eraser should be used and great care taken that the ink lines are -not lightened and broken by it. - -To avoid the necessity of using an eraser upon a finished drawing, -instruments and paper must be kept free from dust and dirt. The -triangles and T-square should be cleaned often, by rubbing them -vigorously upon rough, clean paper. - -Pounce is a powder used to prevent blotting in rewriting over erasures; -it is held in a bag or small box with a perforated lid for convenience -in sprinkling on paper; when used it should be distributed evenly with a -piece of chamois, and the surplus or loose particles removed before -applying the ink. - -A drawing is made to be read, and the skill in inking, as in “free-hand” -and in penciling, does not consist so much in the fineness of the lines -as in their clearness. - - - - -Lettering Drawings. - - -Lettering is an important part of making drawings, the object aimed at -being to identify any portion by reference letter or letters; thus in -fig. 239 the line _A C_ describes the line extending from - -[Illustration: Fig. 239.] - -Any information which cannot be expressed in the drawing is always -expressed by lettering, and it is desirable to confine the lettering of -drawings to one or two standard alphabets that are plain and distinct, -and the principles of which are easily acquired. These conditions are -fulfilled in the Gothic fonts shown on page 173. - -Both letters and figures must be carefully made and of uniform -proportion; it is well to “lay out” these by regular measurement before -permanently inking them. Letters should not be less than one-eighth of -an inch in height and penciled carefully before inking. - -On page 173 are printed two forms of numerals and letters of the -alphabet; it is recommended that these be used both for practice in -free-hand and for regular office work. - -For easy reference, letters should not be crowded nor allowed to -interfere with one another; they should be drawn neatly, avoiding all -lines of the drawing; plain letters are always used on mechanical -drawings, whether for title, scale, reference, etc. - -Arrow-heads, figures and letters should be in black, and made with a -writing pen. A pen with a ball point is preferable, giving an equal -thickness of line, no matter in which direction the stroke is made. - -[Illustration: Fig. 240.] - -Neat, well-lettered drawings go far towards establishing a high standing -for the aspiring draughtsman. All lettering should be done free-hand, -first with the pencil, sharpened to a fine round point, and afterwards -written in ink. For this purpose common writing pens are best to be -used; fig. 240 represents the several numbers of the approved Gillott’s -pens adapted to this purpose. - -In lettering, it is well, for a guide for size and location, to draw, -with a round-pointed pencil, two horizontal lines just the height the -letters are to be; the letters are also best made with careful use of -the instruments, rather than free-hand. - -In order to letter systematically, it is a good plan to start with the -middle letter of the inscription and work in both directions; making too -prominent letters should be avoided, plain and distinct letters being -most desirable. - -Finally, with an ordinary writing pen, trace over the penciling in ink; -the pencil guide lines being erased after the letters are inked in -completes the operation. - -[Illustration] - -An important matter in connection with lettering a drawing is the -location of the letters; these should be so placed as not to interfere -with the lines of the drawing and should clearly point out the part -intended to be described. When single letters are used, they should be -inked in before the shade or section lines are drawn. - -Fig. 241 is an example introduced to show the method of lettering a -descriptive mechanical drawing. - -The figure shows a “blow-off valve” of approved design drawn in section; -the letters designate the several parts; the reading to accompany the -lettering is as follows: - - =A=--Inlet communicating with annular passage (_C_) to admit steam, - which blows off scale and sediment from seat (_E_) before disc (_L_) - comes in contact with same. - - =B=--Plug to permit passage of rod to clean out blow-off pipe. - - =C=--Annular steam passage around casing (_D_) and communicating with - inlet (_A_). - - =D=--Removable bronze casing, in which plug (_L_) fits snugly. - - =E=--Removable bronze seat ring, which also holds casing (_D_) in - place. - - =J=--Slot in casing (_D_) arranged to discharge sheet of steam from - _C_, which, blowing across seat (_E_), cleans off scale and sediment - before contact with disc (_L_). - - =K=--Non-rotating washer to prevent loosening of locknut (_H_) when - opening valve. - - =L=--Reversible disc, having two Babbitt-metal seating faces (_F_) - (_F_). - -[Illustration: Fig. 241.] - - - - -Dimensioning Drawings. - - -To “dimension” working drawings is to place measurements upon the parts -represented, to enable the workman to proceed without measuring the -drawing itself. - -These dimensions should be placed so as not to interfere with nor crowd -the lines of the drawing, nor yet interfere with one another. - -Arrow-heads are used at the extreme points of measurement, the figures -are generally inserted midway between the arrows; a dot and dash line -reaches from the figure to the arrow-heads, as shown below. - -[Illustration] - -When the dimension is short these lines are omitted and the dimension is -placed outside the drawing, thus - -[Illustration] - -and connected by a curved line; at other times it is found needful to -place arrow-heads outside the drawing and the measurement inside. - -[Illustration] - -When the dimension is long and narrow it is usual to carry the -dimensions under the drawing by dotted and dash lines, as shown below. - -[Illustration] - -Arrow-heads and figures should be drawn free-hand with a common writing -pen. - -Usually dimensions are given in inches, up to 24 inches, as it is found -less confusing; for instance, if written 1′ 1″ it may be mistaken for -11″; if written 13″ no mistake could be made. - -Again, 1′ 0″ may be mistaken for 10″; if written 12″ it would not; in -addition to being more distinct, it occupies less space on the drawing. -In large measurements there is more room for the figures, and, -therefore, they can be spaced further apart--in feet and inches. - -All figures should be made of a fairly large size. Vertical dimensions -should read from the right hand, thus, as shown: - -[Illustration] - -Measurements of importance, such as the diameter of a circle, the pitch -or distance apart of rivets and bolts, etc., should be marked in figures -on the drawing. When rough or unfinished work is mixed with machined or -finished portions, it is usual to mark F, or “fin.,” after the latter -dimension. - -In practice, at times, instead of dimensions reference letters are used, -thus: - -[Illustration: - - D = diam. of shaft, 2¹⁄₂ inches. - L = length of bearing, 3³⁄₄ inches. - T = thickness of collar, ⁷⁄₈ inch. - d = diam. of collar, 3¹⁄₂ inches.] - -Generally it is preferable to give the diameters of turned and bored -work on a section, instead of an end drawn separately; confusion is -sometimes caused by a number of radial dimensions. - -Fig. 242 and fig. 243 are introduced to show the principal measurements -required in practical work, and the usual way in which such dimensions -are marked when ordering parts of machinery. - -Fig. 242 is a pedestal, or metal frame; three views are shown, the -center figure being an elevation, the lower figure is the plan of the -base, the upper figure is a view of the top, on which is bolted the -bearing block, it being on the outside of the center figure. The -essential measurements are marked by letters. _H_ is the vertical height -from base to the seat of bearing block: _L_ being the length, and _W_ -the width of the base; _P_ is the length between checks, and _B_ the -width of seat for bearing block; _C_ is the distance from center to -center of the holding down bolt holes, and _T_ is the depth of the holes -in the base; _K_ is the distance from center to center of the bolt holes -in the top for bearing block. - -Fig. 243 is a hanger, or metal bracket, and shows the center figure or -elevation, the plan of the top and the plan of the seat for bearing -block, which is bolted on the interior of the center figure. _H_ is the -vertical distance from the top to the seat for bearing block; the other -measurements required are marked by letters similar to figure 242. - -Now, one of the important matters in connection with dimensioning a -drawing is the location of the figures. One rule, whose utility cannot -be gainsaid, is that they should be so located that they can be altered -or erased without damage to the lines of the drawing, as changes may be -necessitated either by original errors in writing down the figures or by -changes in the design being found desirable during the construction of -the machine. - -[Illustration: Fig. 243.--Fig. 244.] - - - - -Shading Drawings. - - -To produce an effect, drawings are shaded; that is, shadow lines about -twice the width of the regular line are drawn according to a recognized -rule, which always represents the same peculiarity of form in the same -way. - -In working drawings light lines only are permitted; shade lines are -wider than the working lines, and in reading scale measurements the -extra thickness of line would make a difference. - -Instead of representing the shadow as it is really cast by the object, -the edges which cast the shadow are determined, and all the views are -treated as if the light came from behind and from the left, downwards, -at an angle of 45° to the horizontal line, as shown by the arrows in -figs. 245 to 248. - -[Illustration: Fig. 245.--Fig. 246.--Fig. 247.--Fig. 248.] - -The lower and right-hand outlines of projecting parts will cast shadows, -and the student should make them of extra width. - -Fig. 249: In shading curves, divide the center line as before (see page -146) and describe circles from center, _D_; these lines are not to be -shaded in penciling, but when inking. The figures represent two rings, -_A_ and _C_, and spaces, _B_ and _D_. The outside of a surface is shaded -according to circle 1, and the inside surface according to circle 2. -This will give the desired shading, but it makes a drawing incorrect, -and therefore shading is not used in working drawings. - -This shading is accomplished by inking the circle first with regular -width of line; then with the same radius remove the point of compasses -from the _true center_, placing it outside, according to the desired -position of the shaded line, and describe an arc of a circle. - -In fig. 250 divide center line as before; with the 45° triangle or set -square, draw through the center the diagonals shown by dotted lines; and -through the points _A_, _B_, _C_, etc., draw the perpendiculars, cutting -the diagonals; from the points of intersection draw the horizontal -lines, completing the squares. - -[Illustration: Fig. 249.] - -[Illustration: Fig. 250.] - -Now, take a radius of half an inch, and in the corner of each square -draw with the bow-pencil a circular arc meeting the pencil lines -exactly; with the bow-pen _ink in the arcs_ first, then ink carefully -the lines joining with the arcs; all lines must be of the same width; -put in the shade on arcs and lines as in fig. 249; and, finally, erase -the pencil lines at corners, etc. - - - - -Section-Lining. - - -Cross-hatching has been defined in the “preliminary definitions” to -drawing; this term represents the practice of drawing diagonal lines -representing the interior of an object, shown as a piece cut in half or -when a piece is broken away. This is done to make more of the parts -show, or to exhibit more clearly the nature of the materials; hence -section lining and cross-hatching tell the same thing, _i. e._, the -drawing of diagonal lines, usually at an angle of 45°, to show that the -object is broken away and the interior designed to be represented. - -[Illustration: Cast iron. - -Fig. 251.] - -[Illustration: Wrought iron. - -Fig. 252.] - -[Illustration: Steel. - -Fig. 253.] - -[Illustration: Composition. - -Fig. 254.] - -[Illustration: Vulcanite. - -Fig. 255.] - -[Illustration: Wood. - -Fig. 256.] - -[Illustration: Leather. - -Fig. 257.] - -[Illustration: Brick. - -Fig. 258.] - -[Illustration: Fig. 259.] - -Figs. 251 to 258, inclusive, show the section lining and cross-hatching -by which it is customary to represent the various materials entering -into a construction. - -In fig. 259 is outlined a representation of a section of a cog-wheel; -section 1 being the wood cogs; 2, the iron wheel, and 3 the wedges at -the root of the gear. It would be impossible to convey the same ideas by -ordinary plan or elevation drawing; all the objects on the same page are -more clearly represented by the use of section lines or cross-hatching. - -Sectioning is executed by drawing a series of parallel lines about ³⁄₃₂ -inches apart. Lay the 45° triangle on the upper edge of the T-square and -draw the top-most line of the sectioning. Then slide the triangle along -the T-square for each successive line. The sectioning should be inked in -without previous penciling and the lines should be finer than the lines -of the general drawing. - -Various devices are in use for mechanically equalizing the distances in -section lining, but the trained eye is the most practical method. When -two abutting pieces are sectioned, the section lining on one piece -slants in an opposite direction to that on the other. - -To draw an object to be sectioned on both sides of its center line, only -one side is sectioned, while the other side is drawn in full. - -Sections are necessary in nearly all machine drawings; they are usually -taken horizontally or vertically, but they may be taken in any -direction; the position of a section should be shown by a line upon the -object; this line is called the cutting plane. - -In fig. 261 is shown the hub of a wheel, it is also a sample of work for -practice. - -[Illustration: Fig. 260.] - -[Illustration: Fig. 261.] - -Fig. 260 shows the mode of representing two different materials in one -plane, or a section may be represented by the darker portion, and the -lighter shaded portion being a surface resting on the section. - -Fig. 261 shows the section of a shaft surrounded by the surface of a -wheel. - - -TINTS AND COLORS. - -For special purposes of illustration drawings are made which must be -tinted. In such cases the paper must be expanded and stretched evenly -all over its surface; otherwise when the moist tint is applied the paper -will wrinkle and get out of shape; to do this cut the paper at least -half an inch less in size than the drawing board; lay the paper face -down, turn up a margin or edge of about three-fourths of an inch all -round, then dampen the paper with a sponge and clean water; allow it to -soak for a few minutes until it is evenly dampened or moistened all -over, turn the paper upside down (face up). - -Apply strong paste to the under side of the margin all round; rub down, -on the drawing-board, working from the center of the board outwards so -as to exclude the air and prevent creases or furrows. The board is then -inclined and left to dry slowly; make sure that the paper is all well -pasted and every part of the edges attached to the board. - -If tracings are required to be tinted or shaded, the color may be -applied before the tracing is cut off, or what is more usual, the color -may be applied on the back of the tracing; then there is no liability to -wash out the lines. - -Mechanical drawings are seldom tinted, but are mainly produced in India -ink. Where, however, a fine effect is desired, working drawings are -colored, so as to show at a glance the material of which the different -parts are to be made. - -The colors required are few but should be of the best quality. Besides -India ink the following water-colors are generally used: - -1, Neutral-tint. 2, Prussian Blue. 3, Chrome Yellow. 4, Gamboge. 5, Raw -Sienna. 6, Carmine. 7, Vermillion. 8, Venetian Red. 9, Sepia. 10, -Indigo. These come in hard cakes. - -Certain colors and tints represent different metals and materials as -follows: - -Wrought Iron--Prussian Blue. - -Steel--Carmine and Prussian Blue, mixed to give a purple shade. - -Steel Casting--Same as the above darkened by Venetian Red. - -Cast-Iron--Neutral Tint made of India Ink, indigo, mixed with a little -carmine. - -Brass--Gamboge or Chrome Yellow. - -Babbitt--Emerald Green; sometimes light mixture of India Ink. - -Copper--Purple Lake. - -It is sometimes found necessary to prepare a highly finished and shaded -drawing of the work in hand. Such elaborations, in fact, are much -admired by the uninitiated, although the complete shading of the drawing -is no criterion as to the scientific value of the machine. An -illustration of this is told in the note. - - NOTE.--A consulting engineer had to lay before a board of directors - plans of horizontal engines for their consideration. One of these - drawings was of a very superior machine, but being only depicted - lineally was at once rejected by them, for a highly finished - representation of a very inferior apparatus. The engineer, wishing to - induce the board to decide for the best, suggested that the matter - should be postponed to a future day, and in the meantime had the - drawing of the superior machine highly colored and finished. At the - next meeting the directors unanimously decided that this was the very - one which they preferred and had chosen. - - - - -Reproducing Drawings. - - -When once finished, one or more copies of drawings are frequently -required; these are produced, 1, by blue printing, as described before; -2, by tracing. A tracing is a mechanical copy of a design or drawing, -made by reproducing its lines as seen through a transparent medium--as -tracing-cloth or tracing-paper. - -Tracing-cloth is a thin linen fabric, coated with size; this is called -_tracing-lines_; _tracing-paper_ is so prepared as to be transparent, so -that it will receive marks either in pencil or with pen and ink. - -Tracing-cloth must be fastened to the board, over the drawing, by pins -or other tacks; moisture or dampness should be carefully avoided and the -drawing done on the smooth side of the cloth. - -When tracing cloth will not take ink readily a small quantity of pounce -may be applied to the surface of the cloth and distributed evenly with a -piece of cotton waste, chamois, or similar material, but the pounce -should be thoroughly removed--by washing--before applying the ink. - -In making tracings the same order is followed as described under the -section “Inking”--to repeat: 1, ink in the small circles and curves; 2, -ink in the larger circles and curves; 3, then all the horizontal lines, -beginning at the top of the drawing and working downward; 4, next ink in -all the vertical lines, commencing at the left and moving back to the -right; 5, draw in the oblique lines; 6, all the center lines red -(carmine), and dimension and reference lines in blue (Prussian blue) or -_vice versa_. The figuring and lettering should always be done with -India ink, thoroughly black. - - -BLUE PRINTING. - -Copies of drawings or parts representing details and measurements are -frequently needed for the office, pattern shop, machine and blacksmith -shop, etc. These copies are best made by printing on sensitized or -specially prepared paper, from tracings drawn on transparent cloth or -paper, as hereabove described. The original design may be guarded with -the utmost care for long preservation, but the blue prints, so called, -are for ready reference and use without much regard to the length of -time they are to be in existence. - -The usual practice is to carefully trace from the drawing on transparent -cloth or paper an exact reproduction of it, filling in all detail -lettering and sizes or figured dimensions. - -This tracing is fixed in a frame similar to a picture frame, with the -side on which the drawing is made next to the glass: 1, place the -sensitized side of the paper (which has been prepared previously) -against the back of the tracing; 2, fix soft padding against the back of -the paper and fasten it up so that both paper and tracing are compressed -firmly against the glass, permitting no creases or air spaces between -them. - -[Illustration: Fig. 262.] - -This should be done in a darkened room; 3, expose for three to six -minutes, according to the intensity of the sun; 4, take the sensitized -paper out of the frame and quickly wash well in clean running cool -water, and the drawing will appear in white lines on blue ground; 5, -hang the print up by one edge so that the water will run off and the -print will soon dry and be ready for use. - - -TEST-PIECES. - -To make good blueprints, being guided only by the appearance of the -exposed edge of sensitized paper, requires considerable experience. Very -often, especially on a cloudy day, the edge looks just about right, but -when taken out of the frame and given a rinsing, it is only to find that -the print looks pale because it should have been allowed to remain -exposed for a longer period. - -Now simply take a small test-piece of the same paper (say about 4 inches -square) and a piece of tracing cloth with several lines on its surface -and lay these small pieces out at the same time the real print is being -exposed, and cover these samples with a piece of glass about 4 inches -square. As a general rule, we can find a place on top of the frame for -the testing-piece, and by having a small dish of water at hand for -testing the print by tearing off a small bit and washing same to note -its appearance, the novice can get just as good results as the -experienced hand without danger of failure. - - -BLACK PROCESS COPYING. - -This is accomplished by specially sensitized paper by which a fac-simile -of the original drawing can be made; that is, black lines upon white -ground. It also avoids the objection to the blue print paper of shaded -drawings which show light and shade reversed. - -The prints made by the process are said to be absolutely permanent and -can be altered, added to or colored the same as original drawings. - -The sensitized paper is sold ready for use, but it can be prepared by -dissolving two ounces of citrate of iron and ammonium in eight ounces of -soft water; keep in a dark bottle, also, one and one-third ounces of red -prussiate of potash in eight ounces of water; keep in another dark -bottle; when about to use mix an equal quantity of each in a cup and -apply in a dark room with a soft brush or sponge to one side of white -rag paper, similar to envelope paper, let it dry and put away in a dark -place until required for use. - - -[Illustration: DRAWING OFFICE RULES] - - - - -Drawing Office Rules.[1] - - - [1] NOTE.--A. W. Robinson, M.E., Montreal, must have all credit for - these admirable rules and regulations. They bring into a single focus - the whole science and art of mechanical drawing. - - There are drawing offices where from ten to nearly one hundred people - are busily employed in making new plans and sketches by the hundreds, - and where thousands of completed drawings are filed for reference or - for changes, as these are needed in the shop management. - - To be introduced for the first time into such a company is a trial for - the “new man” both of nerve and manners, and a test as well of skill; - nothing helps more at such a time than an acquaintance with the rules - and routine of the office, for the old saying holds good in a drawing - office, of “doing in Rome as the Romans do.” The author of this book - has felt this strangeness in a new position and so adds the following - model-rules for the guidance of the student when first entering a - regular position in an office where many are employed and where - success depends upon a systematic ordering of the work in hand. - - -SIZE OF DRAWINGS. - -1. The standard size shall be 23 inches by 36 inches, subdivided into -half, quarter and eighth sheets. - -2. Full-size drawings shall be reserved, as far as possible, for general -views and parts not capable of being shown on smaller sheets. - -3. All shop detail shall, as far as possible, be shown on quarter and -eighth sheets. - - -CHARACTER OF DRAWINGS. - -4. Detail drawings shall, as far as possible, classify the different -kinds of works, such as castings, forgings, shafts, levers, piping, etc. -Different kinds of work shall not be shown on the same detail drawing. - -5. All shop drawings liable to repetition shall be traced and -blue-printed. All temporary details, requiring only one copy, may be -made on sketch sheets and press copied. - -6. A shop drawing is to be considered as an order or instruction to the -shop, and not merely as a statement or illustration. For this purpose it -must convey clearly and distinctly all the information necessary to make -the article. - -7. Every dimension necessary to the execution of the work is to be -clearly stated by figures on the drawing, so that no measurements need -to be taken in the shop by scale. All measurements to be given with -reference to the base or starting point from which the work should be -laid out, and also with reference to center lines. - -8. All figured dimensions on drawings to be plain, round vertical -figures, not less than one-eighth inch high, and formed by a line of -uniform width and sufficiently heavy to insure printing well. No thin, -sloping, or doubtful figures, or diagonal-barred fractions will be -tolerated. All figured dimensions below two feet to be expressed in -inches. - -9. All center lines to be alternate dot and dash in fine black line. All -dimension lines to be double dot and dash, with a central space for the -figure, and of such strength as to show on blue-print more faintly than -lines of drawing. Lines of drawing to be bold and clearly defined in -proportion to the scale, and may be shade-lined by making the right-hand -and bottom lines heavier. No ornamental shading or other “frills” -allowed on shop drawings. - -10. Every drawing, whether whole or half-sheet, shall have the title, -date, scale and number of the sheet stamped in lower right-hand corner, -and the quarter and eighth sheets printed on top. - -11. The name of the drawing, as given in the title, is invariably to -consist of two divisions in one line separated by a hyphen. The first -division is to state the general name of the thing or machine, and the -second name is to clearly designate the part or parts represented (or if -a general view should so state). The wording of titles should be -submitted to the chief engineer or head draughtsman for approval. - -12. Each drawing shall bear the name of the draughtsman and examiner, -the surname being used without initials. - -13. Drawings of piping details shall be made in diagram form, using -standard symbols. - -14. All detail parts for standard or repetition work shall be shown -unassembled as far as possible. - - -DRAWING SYMBOLS. - -15. Detail shop drawings should state: - -(a) The pattern number of every casting in plain figures of larger size -than the dimension figures. - -(b) The material of which the parts are made, using symbols as follows: -C.I.--Cast iron. W.I.--Wrought iron. M.S.--Machinery steel. -H.S.--Hammered steel. Bs.--Brass. Bbt.--Babbitt. Bz.--Bronze. -C.R.S.--Cold rolled steel. - -Other materials write full name. - -(c) Finished surfaces will be indicated by “f” written on the line or -surface to be finished. When not so marked it is understood that the -part is to be left black or rough. In cases where finish might be -presumed but not required, follow the figured dimensions by the word -“cast,” if a casting, and “rough,” if a forging. - - -STANDARDS. - -16. The following standards shall be strictly adhered to as given in the -tables noted: - -(1.) Table of standard diameters of shafting and key seats. - -(2.) Table of standard stock sizes of rounds. - -(3.) Table of standard stock sizes of flat steel. - -(4.) Table of standard clearance fits. - -(5.) Table of standard symbols for notation of riveting. - -(6.) Table of standard symbols for pipe fittings. - -Also such other standards as may be adopted from time to time. - - -NUMBERING OF DRAWINGS. - -17. Drawers and filing cases shall be numbered consecutively. Drawers -shall contain 100 sheets each, and filing cases 200 sheets each, and to -be fully indexed. Drawings shall be numbered by a number indicating both -drawer number and serial number in the drawer--thus, 7,604 is the fourth -sheet in drawer 76, etc. - -18. Drawing numbers shall be checked off the index as required, and the -index posted up in uniform handwriting by the clerk. - -19. Standard size drawings shall be kept in drawers and quarter and -eighth sheets in filing cases. All drawings shall be indexed by an index -sheet kept in each drawer or case. - - -CHECKING. - -20. All drawings must be approved before being traced. When tracing is -completed it will be given immediately to the chief draughtsman, who -will have a preliminary print made and carefully checked, before being -used. - - -PATTERNS. - -21. All patterns shall bear the number of the drawing on which they are -first detailed, followed by a serial letter, according to the number of -patterns on the drawing. - -22. Standard patterns used repeatedly and liable to be ordered from in -repairs must not be changed. Other patterns may only be changed when -absolutely necessary and by order. When so changed they will bear the -original number and letter, followed by A for the first change, B for -the second change, and so on thus: 4860 AB is the second change in -pattern 4860 A. - - -SKETCH BOOKS. - -23. Each draughtsman will be supplied with a sketch book by the company, -in which he shall make all his notes, calculations and data referring to -his work, and under no circumstances shall notes of value be made on -loose sheets. Each entry should invariably be commenced with the subject -and date, and full notes made of data on which the calculations were -based, and the results obtained clearly stated. These books are to -remain the property of the company. - - -IN GENERAL. - -24. Changes in drawings, sketches or order lists issued to the shop -shall only be made when authorized by the chief engineer, or, in his -absence, by the chief draughtsman, and when so authorized shall be made -by the order clerk. - -25. The names of all similar parts in order lists and drawings are to be -uniform. - -26. Tracings must be kept in safe, for blue-printing purposes only. -Office copies of blue-prints must be used for references. - -27. No drawing, print or photograph shall be taken from office without -permission. - - -NUMBERING WORKING DRAWINGS. - -There are a great many different systems used in indexing drawings, most -of which have some good points, but very few are sufficiently elastic to -cover a wide field. A plan based upon the decimal system of notation is -very simple, and, as there is no practical limit to the number of -subdivisions, it can be expanded indefinitely. Following are the main -outline features of the system as adapted to the needs of drawing -offices belonging to large works. - -The main division numbers, 000, 100, 200, 300, etc., are used -respectively for all plans and general sheets referring to the division -concerned. 100 includes general plans covering more than one department, -and all small-scale plans with cross references to departments covered. - -The class or tens divisions contain general drawings of the -subdivisions, the subclasses or units divisions being limited to details -only. Further subdivisions would probably be necessary in some cases. A -card index with cross references and written by someone who knew what to -do is an essential part of the system. - - -[Illustration: GEARING AND DESIGN] - -[Illustration: Fig. 263.] - - - - -Gearing. - - - Under this heading the author has grouped some information relating to - a subject of wide interest and one sure to interest a student of - mechanical drawing. - - The diagrams are intended for exercises in drawing, _i. e._, to be - redrawn as parts of practice; the text is to be studied not only for - the good to be gained from the study of gearing, but as an example of - the way in which written or printed descriptions are necessary to - explain a subject illustrated by drawings. - -_A gear_ is primarily a toothed wheel; gearing is a train of toothed -wheels for transmitting motions; there are two chief sorts of toothed -gearing, viz., spur gearing and bevel gearing. - -_A spur wheel_ has teeth around the edge pointing to the center; -commencing at the center, a spur wheel may be said to consist of a hole, -square, octagonal or round, for its axle or shaft; a hub; the web, body -or arms; a rim, and the teeth; see fig. 263. - -_A spur wheel_ has teeth on its circumference which run parallel to its -shaft; wheels as shown in fig. 271 are termed _helical wheels_; these -are similar to spur wheels except their teeth are arranged upon -different angles to the shaft. - -_A bevel_ is a slant or inclination of a surface from a right line, -hence a bevel wheel is one whose teeth stand beveling or at an oblique -angle to the shaft, or towards the center; see fig. 267. - -_Miter wheels_ are bevel wheels of the same size, working at right -angles with one another; see fig. 268. - -_The diameter of both spur and bevel wheels_ is measured and calculated -neither from the outside nor from the bottom of the teeth, but on the -pitch circle. When we speak of the diameter of a spur or bevel wheel, we -mean the diameter of the pitch circle, without any reference to the form -of tooth. - -_The addendum circle_ of a toothed wheel is as shown in illustration, -fig. 264; _addendum_ means “something added,” and, as shown in the -figure, it is the part added beyond the pitch “line” or circle. - -[Illustration: Fig. 264.] - -_The pitch line_ is the most important one in gearing; the “pitch line” -or “pitch circle” is supposed to be the working circle. This is shown in -P--P in fig. 274. - -_The periphery_ of a wheel is the extreme circumference, as N in fig. -274. - -All parts of gear-wheels consist of portions, to which have been given -generally accepted names. Fig. 264 shows the “addendum circle” and the -“pitch line” as marked. The teeth and rim are shown in white, and the -other portions are indicated by the names. - -_The circular pitch line_, as opposed to the diametral pitch, is the -same as the pitch circle. It is a line which bisects all the teeth of a -toothed wheel. - -_The rolling circle_ is the same as the circular pitch line. - -_Diametral_ means pertaining to a diameter or the length of a diameter; -hence a diametral pitch is a system of measures or enumeration based -upon the diameter instead of the circular pitch line; it is used very -generally in spacing for fine tooth gear. Wheels of this description -usually have their teeth cut in a gear-cutting machine, _i. e._, medium -and fine tooth gears. - -[Illustration: Fig. 265.] - -_A cog wheel_ is the general name for any wheel which has a number of -cogs placed around its circumference. - -When the teeth of a wheel are made of the same material and formed of -the same piece as the body of the wheel, they are called _teeth_; when -they are made of wood or some other material and fixed to the -circumference of the wheel, they are called _cogs_; see fig. 265. - -_A pinion_ is a small wheel. When two toothed wheels act upon one -another, the smaller is generally called the pinion. The terms _trundle_ -and _lantern_ are applied to small wheels having cylindrical bars -instead of teeth. The teeth in pinions are sometimes termed _leaves_; in -a trundle, _staves_. See fig. 273. - -The wheel which acts is called a _leader_ or _driver_; and the wheel -which is acted upon by the former is called a _follower_ or the -_driven_. When a screw or _worm_ revolves in the teeth of a wheel, the -latter is termed a _worm wheel_ or _worm gear_; see fig. 270. When a -pinion acts with a rack having teeth, we speak of _rack_ and pinion. -When the teeth are on the inside of the rim, and not on the periphery, -the wheel is termed an _internal gear_; see fig. 272. - -Two wheels acting upon one another in the same plane are called _spur -gear_; the teeth are parallel with the axis. When wheels act at an -angle, they are called _bevel gear_. - -_Friction gear-wheels_ are those which communicate motion one to the -other by the simple contact of their surfaces. - -In frictional gearing the wheels are toothless and one wheel drives the -other by means of the friction between the two surfaces which are -pressed together. - -Grooved friction wheels are used to give greater cohesion than can be -obtained by the plain surface. - -Fig. 263 shows a pair of spur-wheels in gear. The dotted circles which -meet are the rolling circles, called the “pitch line” or “pitch circle.” - -[Illustration: Fig. 266.] - -A spur mortise wheel is similarly shown in fig. 266; it is very like in -appearance to a spur wheel; it differs essentially in that the teeth are -separate cogs, fixed in singly to the rim; see also fig. 265, page 201. - - NOTE.--The teeth of spur wheels cast from a pattern must of necessity - be larger at one side than at the other, because the teeth must have - taper to permit the extraction of the pattern from the mould; - therefore, in fixing wheels to gear, the large side of one should meet - the smaller side of the other; should the two large sides come - together the teeth will meet only at the large side, and the teeth - will probably break away from the excessive strain on that point. - -_Skew gearing_ are bevel wheels working out of center; the teeth do not -form radial lines from the wheel center. - -Fig. 267 shows a pair of bevel wheels in gear as described on page 199. -A bevel mortise wheel, _i. e._, one having cogs inserted in its rim -instead of teeth. - -[Illustration: Fig. 267.] - -A bevel wheel and pinion must be made to suit one another by both having -teeth forming together an angle of 90°, therefore they are pairs, or -proportioned in the number of teeth one to the other. Any other -proportion used would not exactly gear and would be termed a “bastard” -gear. - -[Illustration: Fig. 268.] - -Fig. 268 represents a pair of miter wheels in gear; it will be noted -that the shafts, when connected, will be at right angles to each other, -the wheels being in all particulars of the same dimensions; the figure -answers the purpose of a much longer description, if given in words. - -A miter-wheel can easily be known by putting a square upon the face of -the teeth, which are always at an angle of 45° with one another, -irrespective of size. - -_A miter-wheel_ is a particular kind of bevel-wheel, the bevel being -limited to an angle of 45° in each wheel. - -The curve of the teeth in bevel-gears, when correctly formed, changes -constantly from one end of the tooth to the other, therefore bevel-gears -whose teeth are produced with a forced cutter are not theoretically -correct. - -Fig. 269 represents a rack and pinion: the teeth in this form of gear -are shaped similarly to those in the spur wheel, shown on page 198, with -the difference that the teeth of one are on a circle and on the rack are -made on a straight line. - -[Illustration: Fig. 269.] - -A flange or addition to the end of a tooth and the rim connecting them -together is used to strengthen the teeth. This extends from the root to -pitch line when the wheel and pinion are both flanged: if only one is -flanged it extends from the root to the addendum. - -[Illustration: Fig. 270.] - -Fig. 270 illustrates a worm and a worm wheel, sometimes called screw -gears. This is a slow but powerful method of transmitting power, one -revolution of the worm only moving the wheel the distance of one tooth -and space. - -_A worm gear_ is a spur wheel with teeth at an angle to the axis, so as -to work with a worm which is a _screw_, or has teeth shaped in the form -of a spiral wound round its circumference; the screw or worm is called -an endless screw, because it never comes to a stopping place in the -circumference of the wheel. - -[Illustration: Fig. 271.] - -Fig. 271 represents a gear with helical teeth. It is similar to a spur -wheel, and is used in place of same in heavy and slow moving machinery, -the formation of teeth preventing--in large measure--the jar or -concussion noticeable in common spur gears. - -In recent years the speed at which gearing is run has been greatly -increased. A striking instance is that of a pair of _cast-iron_ helical -wheels, 6 ft. 3 in. diameter, 12 in. wide, making 220 revolutions per -minute, the speed of the pitch line being 4,319 feet per minute; these -wheels are running continuously and with little noise. There is also a -_cut_ gear in a mill in Massachusetts, 30 feet in diameter, and the -speed of pitch line is 4,670 feet per minute. - -[Illustration: Fig. 272.] - -_An internal or annular gear wheel_ is one in which the faces of the -teeth are within and the flank without the pitch circle, hence the -pinion operates within the wheel. See fig. 272. - -In internal geared wheels there is almost an entire absence of friction -and consequent wear of the teeth, as compared to ordinary spur gearing. - -Fig. 273 shows a _crown-wheel_ which has pin teeth which are fixed by -one end only, on its side face and gear into a trundle wheel. - -[Illustration: Fig. 273.] - -_A trundle wheel_ has no teeth, properly speaking. Instead of teeth, it -has pins as shown on illustration, fig. 273, arranged like the rungs of -a ladder between two walls. See page 201. - -_Trains of Gears._--When two wheels mesh--that is, engage with each -other--as in fig. 263, one axle revolves in the opposite direction to -the other; but when internal gears mesh as shown in fig. 272, the shafts -revolve in the same direction; three or more gears running together are -often called _a train of gears_. - -Maximum speed of gears under favorable conditions for safety is -comparatively-- - - Ordinary cast-iron wheels, 1,800 feet per minute. - Helical cast-iron wheels, 2,400 feet per minute. - Mortise wood cog wheels, 2,400 feet per minute. - Ordinary cast-steel wheels, 2,600 feet per minute. - Helical cast-steel wheels, 3,000 feet per minute. - Cast-iron machine cut wheels, 3,000 feet per minute. - -It is not, however, advisable to run gears at their maximum speeds, as -great noise and vibration are caused. - - - - -Designing Gears. - - - This section is introduced into the work for a double purpose; 1, as - an exercise in drawing; 2, as a study in accurate measurements. It is - a sample of the work that the advanced student in mechanical drawing - will be confronted with as he puts in practice the theory of the art - of drawing. - - Some sample rules are given in the following pages to aid in - calculations relating to gears, and still others are given under the - section “Useful Rules and Tables” at the end of the volume; these are - to be carefully studied. - -To accurately divide the pitch circle of a gear wheel by hand requires -both patience and skill. On the accuracy of spacing lies the essential -requisite of a good gear wheel. - -The drawing in plate, fig. 274, illustrates a pair of spur wheels, shown -in gear, the office instructions for which being: - -“Required, _a detail plan_ of a pair of spur wheels; dimensions: wheel, -76 teeth, 3¹⁄₂ inches pitch, 7-inch eye, 6 arms; pinion, 19 teeth; -scale, 1¹⁄₂ inches = 1 foot.” - -The drawing, as illustrated, is the result of the above instructions, -all pencil lines being removed, and this result is worked out as -follows: - -76 teeth × 3¹⁄₂ inches, pitch = 266 inches in circum. = 7 ft. 0¹¹⁄₁₆ in. -diam. = 3 ft. 6¹¹⁄₃₂ in. radius; with this measurement as represented on -scale, draw line _P P_ on drawing. This is called the pitch line. - -Draw next diameter line, produce or extend this diameter line for -pinion, and with radius of 10¹⁹⁄₃₂ (19 teeth × 3¹⁄₂) from pitch line of -wheel, draw pitch line of pinion. - -Take any point in this pitch line of wheel, mark off 3¹⁄₂ inches as -represented on scale, mark this around the pitch line, it will be the -center of each of the 76 teeth; then the breadth of thickness of each -tooth (= pitch × 0.475) must be marked from these centers, then mark -from _P L_, length of tooth to point (= pitch × 0.35) and _P L_ to root -(= pitch × 0.4), draw circles for outside of teeth _N_ and root of tooth -_O_; now with compass set to the pitch (3¹⁄₂) of the wheel, draw the -outer portion from pitch line of tooth. - -The radius will center in the pitch line of next tooth where thickness -of tooth has been marked; after finishing outer portion of both sides of -teeth, set the compass from _center_ of tooth with radius to the -thickness marked on pitch line and draw the portion of tooth from pitch -line to root. - -Now mark off with dividers and draw thickness of rim (= pitch × 0.5), -divide this line into six parts, draw radii for centers of arms; draw -the bore hole 7″ and the thickness of metal for hub same as pitch. - -On radii lines of arms, draw the breadth of arm at rim (= pitch and -thickness of tooth), increase in breadth approaching the center (1″ per -foot), draw the thickness of feather of arm (= pitch × 0.35); draw web -on inside of rim (= pitch × 0.375); fill in arcs for the joining of arms -in rim and hub (radii = pitch × 0.8) and feather to rim and hub (radii = -pitch × 0.37). - -Proceed in similar manner, completing the teeth of pinion, and when -pencil lines are all in, ink the drawing, erasing all needless lines. - -_P P_ shows the pitch line; _B_, thickness of tooth; _c_, breadth of -space; _A_, the pitch; _E_, clearance at root; _N_, the addendum of -tooth; _O_, the root of tooth; _H_, length of tooth from pitch line to -point; _I_, length of tooth pitch line to root; _G_, whole length of -tooth; _F_, thickness of rim; _J_, web or feather on rim; _K_, breadth -of arm; _L_, thickness of feather; _M_, hub, or thickness round the eye. - - NOTE.--It must be remembered that no fixed standard has ever been - agreed upon for these proportions, and workshops differ considerably - in practice. - -[Illustration: Fig. 274.] - -The number of teeth, their proportions, pitch and diameter of pitch -circle are frequently determined on the “Manchester” principle. This -system originated in Manchester (Eng.), and is now generally used in the -United States for determining diameters and number of teeth, which, of -course, regulate speeds. The principle is not applicable to large -wheels, but is limited in its application to small wheels, or wheels -having “fine pitch,” as will be seen in the following explanation, which -is introduced as very useful and indispensable knowledge for the -acquisition of the student in mechanical drawing. - -[Illustration: Fig. 275.] - -The “pitch” of teeth has already been stated to be the distance from -center of one tooth to the center of another on the “pitch line,” -measured on the chord of the arc. In determining the number of teeth or -pitch of wheels on this principle, the pitch is reckoned on the -_diameter_ of the wheel, _in place of the circumference_, and -distinguished as wheels of “4 pitch,” “6 pitch,” “8 pitch,” etc. In -other words, this means that there are four, six, or eight teeth in the -circumference of the wheel for every inch of diameter. - -In designing gears to transmit power the stress on a tooth is -calculated; it determines the breadth or width and also the thickness of -the tooth on pitch line; the space between the teeth is in proportion to -the thickness of tooth, and the thickness of both combined (one tooth -and one space), measured on the pitch line or circle, is the pitch of -the wheel. - -From the pitch all the proportions and measurements for the sizes and -strength of the parts of the wheel are taken _by rule_, and a -symmetrical form is produced. - -In machine drawing the practice is to represent wheels by circles only; -the teeth are never shown except on enlarged details and then only in -very rare instances; the circles drawn are always the _pitch lines_ or -the rolling points of contact of the wheels. - -The addendum circle is seldom if ever used in practical drawing. Should -it be necessary to show it in an exceptional case, the circle would be -represented by “dotted” line. - -The shape of tooth and mode of constructing it, as practiced in drawing -offices, differs from the true theoretical curve of the tooth, although -very minutely. - -In all calculations for the speed of toothed gears the estimates are -based upon the pitch line, the latter standing in the same place as the -circumference of a pulley. - -To find the _diameter of a gear-wheel_ multiply the number of teeth by -the pitch, divide by 3.1416. - -To find the _pitch of a gear-wheel_ multiply the diameter by 3.1416 and -divide by the number of teeth. - -To find the _number of teeth in a gear-wheel_ multiply the diameter by -3.1416 and divide by the pitch. - -The _breadth of wheels_, where practicable, should be at least three -times the pitch. - -[Illustration: Fig. 276.] - -Fig. 276 shows a scale for proportions of teeth; it is divided into -tenths and used thus: - -Say wheel is 2″ pitch, then from pitch circle to addendum will be 3¹⁄₂ -tenths, and from pitch circle to root of tooth will be 4 tenths measured -at the 2″ line on scale, and so on. - -The decimal proportions already given in example, page 210, are adopted -in many workshops. Many others use the proportions approved of by Sir -William Fairbairn, which are: - -Table of proportion of gears: - - Depth of tooth above pitch line .35 of the pitch. - Depth of tooth below pitch line .40 of the pitch. - Working depth of tooth .70 of the pitch. - Total depth of tooth .75 of the pitch. - Clearance at root .05 of the pitch. - Thickness of tooth .45 of the pitch. - Width of space .55 of the pitch. - -The diameter of a wheel or pinion is invariably the diameter measured on -pitch circle, except it is specially described otherwise, thus the -diameter “over all,” etc. - -The shape of the curved face of the teeth of gears extending from the -root to the addendum is the curve conforming to the passage of the teeth -described on its fellow entering and leaving, as they rotate or roll -together on their pitch circles. - -The curve of teeth outside the pitch circle is called “the face,” and -the curve from pitch circle to root is called “the flank.” - -The difference between the width of a space and the thickness of a tooth -is called clearance or side clearance. - -The play or movement permitted by clearance is called the backlash; -clearance is necessary to prevent the teeth of one wheel becoming locked -in the spaces of the other. - -Wheels are in gear or geared together when their pitch lines engage, _i. -e._, when the pitch circles meet. - -Wheels to be geared together must have their teeth spaced the same -distance apart, or in other words, of the same pitch. - -The teeth of spur wheels are arranged on its periphery parallel to the -wheel axis, or shaft on which it is hung. - -The teeth of a bevel wheel or bevel gears are always arranged at an -angle to the shaft. - -When the _teeth_ of bevel gears form an angle of 45° they are called -miter wheels. - -Miter wheels to gear must be of equal sizes. - -A crown wheel is a disc that has teeth which are on its side face; that -is, teeth on a flat circular surface all parallel to the axis of the -wheel. - -A rack has teeth on a flat surface or plane all parallel to one another. - -A gear cut by machine is called a _cut gear_. It has teeth with less -clearance than cast wheels, which are not so true or perfect, and -therefore require more clearance. - -A worm with even a light load is liable to heat and cut if run at over -300 feet of rubbing surface travel. The wheel teeth will keep cool, as -they form part of a large radiating surface; the worm itself is so small -that its heat is dissipated slowly. - -A worm throws a severe end thrust or strain on its shaft. - -_Steel Gears._--There is great economy in the use of cast-steel over -cast-iron in gears; the average life of the former is nearly twice as -great as of cast-iron gears. And, apart from their longer life and -efficiency, there is less danger of breaking. - -The most accurate teeth, strongest and most uniform in wearing, are to -be found in steel gears cut from solid stock, or made by cutters of -proper shape. - -Fig. 275 shows an elevation and a vertical section of a spur wheel. From -these views the various parts in spur gears can be better understood, as -they are represented here in combination, and the wheel in its entirety. - -_AA_ is the horizontal center line, _BB_, _BB_ the vertical center -lines, _II_ and _II_ the pitch lines, _N_ thickness of tooth, _O_ space -of tooth, _D_ total depth of tooth, _C_ breadth of face, _F_ diameter on -pitch line, _P_ diameter over all, _G_ diameter of hub, _E_ diameter of -hole, _H_ depth of hole, _L_ thickness of rim, _M_ thickness of web. - -Much has been and still is being written on gearing. No general rule is -followed by the writers; the elementary principles given will enable the -student to master spur gearing, and bevel and combinations of many kinds -of wheels will afterwards be found easier to delineate than the numerous -lines seem to indicate. - - -[Illustration: WORKING DRAWINGS] - -[Illustration: Fig. 277.] - - - - -Working Drawings. - - - From the “plans” made in the office are produced “working - drawings”--which represent in detail the work to be done to exact - measurement and of material, as indicated, by the pattern-maker, the - foundry, the forge, the shop, and finally, by the erector of the - completed mechanism. - - How to satisfactorily fulfill the directions contained in these - drawings, representing only a part of the work, so that it will fit, - with needed accuracy, to all other parts of the design, is the task - before each separate worker. - -It is by means of this division of the process of manufacture through -these drawings, that scores and hundreds of men can be employed at the -same time upon a single engine or machine; thus, while handwork has been -superseded by machines in many quarters, the art of drawing has not been -narrowed nor diminished, for no drawings or designs have yet been made -by machinery, nor are they likely to be. - -It is thus that a good designer and draughtsman “projects” or extends -himself, to the advantage of many fellow workers. - -The drawing, fig. 277, shows a simple form of pillar crane: it consists -of an upright cast-iron pillar, which is bolted on a cap stone, under -which is the foundation plate not shown in the drawing; the boom is of -rolled steel, supported by steel tie rods, and provided with rollers at -the base; the hoisting gear is shown in broken lines and circles; all as -seen in the drawing. - -[Illustration: Fig. 278.--Fig. 279.] - -[Illustration: Fig. 280.] - -Figs. 278, 279 and 280 show a drawing of a “hydraulic beam bending -machine” in three views; fig. 280 is a plan, fig. 278 is an end -elevation, and fig. 279 a side elevation, and a portion of the latter in -section shows the interior construction. - - NOTE.--These three views are a practical illustration of drawings for - a machine of the following dimensions: this machine has a bed 3 × 5 - feet in area, with 27 holes in each side for the bending pins. The - frame and cylinders are made of cast iron, the rams of machinery - steel, and the slides for holding the bending blocks, of steel - casting. The distance between the bending blocks is 17 inches. The - cylinders are copper lined, 8 inches diameter, and the rams have a - 6-inch stroke. The rams, which are independent and single acting, are - returned by counterweights placed as shown under the table. The - cylinders can be operated independently from either side of the - machine by an arrangement of valves and levers. The machine complete - weighs about 7,500 lbs. - -[Illustration: Fig. 281.--Fig. 282.--Fig. 283.] - -The drawing, page 222, shows three views of a power punching press. - -Fig. 282 is a side elevation. - -Fig. 283 a front elevation. - -Fig. 281 a vertical sectional view; from these views the proportion, -general arrangement and disposition of the automatic devices can be -easily understood; it may be well to call particular attention to the -automatic clutch on the top shaft and the tripping device. - -[Illustration: Fig. 284.] - -This drawing, fig. 284, shows a side elevation in section of a -self-adjusting piston-rod packing. - -_A_ is the gland, _B_ is the piston rod, _C_ is a brass sleeve which -contains the packing _D_, _E_ is the cylinder cover, _F_ is a coil -spring. It will be seen that the spring _F_ abuts on a bushing in the -bottom of the stuffing box and is prevented from scoring the piston rod -by stepping over the ends of the bushing and follower. All as shown in -the drawing. - -The drawing, fig. 285, shows a sectional view of a large pulley fixed on -a “quill,” or hollow shaft: the driving shaft passes through the hollow -shaft and is attached to the friction clutch shown at the right-hand -end; this friction clutch drives the hollow shaft and pulley. - -[Illustration: Fig. 285.] - - Fig. 286 shows the mechanism, called the link-motion, employed to - reverse an engine, or to enable it to be run in either direction. Many - forms of link-motion have been devised, but the Stephenson form, as - shown in the figure, is, however, the one in almost universal use. - -[Illustration: Fig. 286. - -This drawing shows shading and the mode of figuring the parts for -identification.] - -[Illustration: Fig. 287.--Fig. 288.--Fig. 289.] - -Figs. 287 to 289 represent a bumping-post for the end of railway tracks, -reproduced on an enlarged scale from the columns of the _Engineering -News_. - -In addition to the lettering and dimensions, admirably shown in the -drawings, the following description is appended to show how printed text -and mechanical drawings mutually aid in practical--or commercial--usage. - -The unique feature of the arrangement shown, is that the center line of -the post does not coincide with the track, thus adapting itself to the -nature of the blows of a car-bumper, as received in the single-post -style of the mechanism. - - -BUMPING POST FOR RAILWAY TRACKS. - -The post is a 15-in. steel I-beam, resting on a base plate ³⁄₄-in. -thick, and supported by anchor rods 1³⁄₄ ins. diameter, with upset ends -held by nuts on a heavy forging bolted to the top of the post. These -rods extend forward and outward to clear the rails, and then pass -vertically through a 4 × 6-in. angle iron crosstie, and an ordinary -wooden tie, extending down to an anchor block or deadman buried in the -ground 6¹⁄₂ ft. below the top of the rail. - -Vertical braces or spreaders are fitted between the anchor timber and a -longitudinal timber under the ties, so as to prevent the loosening of -the anchor rods when the post is struck. The rods are held in position -against the rails by steel forgings bolted to the rail with 1-in. turned -bolts. An oak striking block, 12 × 12 ins., 3 ft. long, is bolted -between angle iron brackets on the face of the post. - -[Illustration: Fig. 290. - -Front View. - -Fig. 291. - -Side View. - -Scale, 3 in. = 1 ft.] - - - - -To Read Working Drawings. - - - One of the advantages resulting from a knowledge of practical - draughting is, that it enables a mechanic to _read_ a drawing when - given him as a guide for his work. It is getting every day more - general among draughtsmen to figure exactly and minutely every part of - their drawings which are made to a scale. - -Drawings are almost always made “finished size,” that is, the dimensions -are for the work when it is completed. Consequently all the figures -written on the different parts indicate the exact size of the work when -finished, without any regard to the size of the drawing itself, which -may be made to any reduced and convenient scale. - -Even in full size drawings this system of figuring is not objectionable. -It is a system which should be followed whenever a drawing is made “to -work to,” for it allows the workman to comprehend at a glance the size -of his work and the pieces he has to get made. Figuring makes a drawing -comprehensible even to those who cannot make drawings. - -A working drawing should be made, primarily, as plain as possible by the -draughtsman; second, the workman should patiently and carefully study -it, so that it is thoroughly understood. - -In studying a drawing, the object it is intended to represent should be -made as familiar as possible to the mind of the student, so that he may -fill out in imagination the parts designedly left incomplete--as in a -gear wheel where only two or three teeth are drawn in, that he may see, -mentally, the whole. - -The following is a description of reading drawings when dimensions are -not figured. Here we have a piece of machinery represented by fig. 290, -and the information we have is that it is to scale, three inches = one -foot. Now, with scale and dividers, we can arrive at its actual -dimensions. - -Measurements should be first taken _with the dividers from the drawing_, -and then the dividers applied to the scale to which the drawing is made; -this scale is always marked on the working drawing; if the dividers are -set to the length of the base of the example, fig. 290, they will -measure, on an ordinary two-foot rule, three and three-fourths inches, -_but if applied to the three-inch scale they will read_ one foot three -inches, the actual length of the part; the “reading” is from the scale; -thus, in both figures the drawings are “three-inch scale.” - -Now, 3 inches is one-fourth of a foot, hence 3³⁄₄ × 4 = 1 ft. 3 in., the -full size, and so on for all parts of the drawing. - -Fig. 291 shows _a side view_ of the “steady rest,” illustrated in front -elevation, fig. 290; from the scale as before we get the sizes; the two -views combined give length, breadth and thickness of the parts. - -In some figures it is necessary to show end views, also section views, -to enable all measurements to be read from the drawing. - - -[Illustration: PATENT OFFICE DRAWINGS] - - - - -Patent Office Drawing Rules. - - -U. S. PATENT OFFICE RULES. - -AS APPLIED TO PREPARATION OF DRAWINGS. - -Each applicant for a patent is required by law to furnish a drawing of -his invention whenever the nature of the case admits of it. The drawing -must be signed by the inventor or the name of the inventor may be signed -on the drawing by his attorney-in-fact, and in either case must be -attested by two witnesses. The drawing must show every feature of the -invention covered by the claims. - -When the invention consists of an improvement on an old machine, the -drawing must exhibit, in one or more views, the invention proper, -disconnected from the old structure, and also, in another view, so much -only of the old structure as will clearly show the connection of the -invention with the old machine. - -Several editions of the patent-drawings are printed, the smallest of -which is about 3 × 4³⁄₄ inches, so that the drawing must be so made that -it will stand a reduction of about one-fourth. This work is done by the -photo-lithographic process, and therefore the character of the original -drawing must be brought as nearly as possible to a uniform standard of -excellence suited to the requirements of the process. - - NOTE.--These rules will be found most useful to many readers of this - work--hence their introduction at this point. Nearly 50,000 patents - are “applied for” in the United States every year. - -The following rules are given by the Patent Office for guidance: - -1. Drawings must be made upon pure white paper of a thickness -corresponding to three-sheet Bristol board. The surface of the paper -must be calendered and smooth. India ink alone must be used, so as to -secure perfectly black and solid lines. - -2. The size of a sheet on which a drawing is made must be exactly 10 × -15 inches. One inch from its edges a single marginal line is to be -drawn, leaving the “sight” precisely 8 × 13 inches. Within this margin -all work and signatures must be included. One of the shorter sides of -the sheet is regarded as its top, and measuring downwardly from the -marginal line, a space of not less than 1¹⁄₄ inches is to be left blank -for the heading of title, name, number and date. - -3. All drawings must be made with the pen only. Every line and letter, -signature included, must be absolutely black. This direction applies to -all lines, however fine, to shading, and to lines representing cut -surfaces in sectional views. All lines must be clean, sharp, and solid, -and they must not be too fine or crowded. Surface shading, when used, -should be open. Sectional shading should be made by oblique parallel -lines about ¹⁄₂₀ of an inch apart. Solid black should not be used for -sectional or surface shading. - -4. Drawing must be made of the fewest lines possible, consistent with -cleanness. The plane upon which a sectional view is taken should be -indicated by a broken or dotted line. Heavy lines on the shade side of -objects should be used, except where they tend to thicken the work and -obscure letters of reference. The light is always supposed to come from -the upper left hand corner at an angle of 45 degrees. - -5. The scale to which a drawing is made should be large enough to show -the mechanism without crowding. The number of sheets used must never be -more than is absolutely necessary. - -[Illustration: Fig. 292.] - -6. The different views should be consecutively numbered. Letters and -figures of reference must be carefully formed. They should, if possible, -measure at least one-eighth of an inch in height. - -If the same part of an invention appears in more than one view of the -drawing it must always be represented by the same character. - -7. The signature of the inventor is to be placed in the lower right-hand -corner of each sheet, and those of the witnesses at the lower left-hand -corner. - -The title should be written with pencil on the back of the sheet. - -Drawings should be rolled for transmission, never folded. - -On page 235, fig. 292 exhibits a reproduction of a patent office -drawing, used in connection with specification papers in an application -for a United States patent. - - -ENGLISH PRACTICE. - -The rules for patent drawings in England are practically the same as in -the United States; the paper sizes are, however, different. They must be -on sheets of one of the two following sizes (the smaller being -preferable), 13 inches at the sides by 8 inches at the top and bottom, -or 13 inches at the sides by 16 inches at the top and bottom, including -margin, which must be one-half an inch wide. - -If there are more figures than can be shown on one of the smaller-sized -sheets, two or more of these sheets should be used in preference to -employing the large size. When an exceptionally large drawing is -required, it should be “_continued_” on subsequent sheets. There is no -limit to the number of sheets that may be sent in. - - -[Illustration: PRACTICAL POINTS] - -[Illustration: Fig. 293. See page 244.] - - - - -Useful Hints and “Points.” - - - Many of these “points” are repetitions, with but little variation from - the way they have been previously stated; they are thus repeated to - emphasize their practical worth. - -A good draughtsman leaves his work in such a state that any competent -person can without difficulty ink in what he has drawn. - -The criterion of a good set of drawings is that with a properly prepared -specification they are complete in themselves and require no -explanation. - -A “break” in a figure or object in a drawing is shown in rough irregular -lines, as in fig. 134, on page 131; this is useful when the paper is not -large enough to show the whole. - -Never use a sloping line in writing fractions on a drawing. The -objection arises from the fact that such a dimension as 1³⁄₁₆, if -written with the inclined line, unless very distinctly executed, may be -read as ¹³⁄₁₆. - -In inking do not draw the lines further than you wish them to go, but in -penciling it is well to extend the lines, free up. - -Never use a scale for a ruler. - -Do not overload the pen with ink. - -Having filled the pen, nearly close the nibs and try the width of the -line on a piece of paper or the margin of the drawing. - -Never refill or lay the pen aside without first cleaning it. - -The application of the science of geometry to the drawing-board is -absolutely necessary to success, for the reason that the whole fabric of -mechanical drawing rests on the principles of geometry, which is well -termed the science of measurements. - -Section lines should be the last inked and always without previous -penciling. - -Center lines are necessary in working drawings. - -In choosing T-squares, care should be exercised to see that the head -slides up and down the _left_-hand side of the board easily, and that -when pressed against the board with the left hand there is no “slogging” -of the blade up or down, or in other words, that the head is bearing -firmly for its whole length against the board. - -The best place for the title of a drawing is said to be the upper -left-hand corner; this facilitates the filing of the sheet. - -Never use a soft pencil except for finishing in shadow lines. - -The rubber should always be kept clean. - -Great care should be taken to keep drawing boards out of the way of heat -or damp, as these cause the wood to warp. - -Circles and curves are to be “inked in” before straight lines. First ink -the smallest and afterwards the larger curves. - -Do not press heavily on the pencil so as to cut the paper, but draw -lightly, so that the mark can be erased and leave no trace, especially -if the drawing is to be inked. - -The draughtsman should commence his work at the top of the paper, -keeping the lower part covered over until he needs to use it. - -Shade lines should be avoided in all working drawings, as their use -interferes with accurate measurements. - -To make ink stick to the tracing cloth, with a woolen cloth rub some -powdered chalk or pounce over the surface on which the ink lines are to -be drawn, then wipe the surface clean and use a good quality of ink. - -For striking small circles a small bow pen should be used. - -To fix lead pencil marks on sketches so that they cannot be readily -erased, sponge them with milk carefully skimmed, then lay blotting paper -over them and iron with a hot flat-iron. - -To have the ink preserve its fluidity and to keep out all dirt and dust, -keep the cover on the ink slab; the mistake is often made of putting too -liberal a supply of water in ink well, which causes a waste of both time -and ink; no more should be prepared than to meet immediate requirements. - -Always draw on the right side of the sheet, which can be found by -holding the sheet up to the light and looking across its surface with -the eye nearly in the same plane as the paper; note which side is the -smoothest and has the least number of blemishes on it; this is the right -side to draw on. - -As to sharpening pencils, it is always best to cut a chisel point on the -pencil used for drawing, and put a circular point on the pencils in the -bow pencil and pencil leg. The chisel point makes a finer line and lasts -much longer than a round point. - -The varnish used in many large drawing-rooms is simply white shellac -dissolved in alcohol; it requires a little experience to mix these to a -proper consistency, but this is soon acquired. - -Never sharpen your pencil over the drawing. - -A center line of a drawing is the line upon which the figure is to be -constructed; the center line is the first line to be drawn. - -The T-square belongs to the left side of the drawing-board, and is -operated by the left hand. The right hand should be kept free for the -purpose of picking up pencil, pen and bows, adjusting and marking off. -The left hand controls the T-square and the triangle that slides along -the upper edge of the square; the right hand is for the instruments. - -The advantage of a paper rule or scale is that the paper will expand and -contract under varying degrees of atmospheric moisture the same as the -drawing does. - -Avoid rubbing out and constantly cleaning the drawing with India rubber; -if wrong lines are made or it is desired to make alterations, the part -to be changed should be rubbed out and completely re-drawn. - -When using the bows see to it that the steel-pointed leg that is put -down first on the paper, to secure a center for a curve or a circle, is -a trifle longer than the pencil or pen leg. - -To clearly indicate the position of a center which is to be used again, -lightly pencil a small circle about it; never put the point of a pencil -in the center hole to enlarge or blacken it; the prick point made by the -dividers and needle points should be no more than can be just seen, -hence the circle to be made as advised above. - -[Illustration: Fig. 294.] - -Be particular in having the legs of the dividers exactly the same -length, and sharp, so that in pricking off distances, and dimensions, -and centers, the indent or hole made in the paper is as small as -possible. - -The term “plane” means a perfectly flat surface; that is, something -which has length and breadth but no thickness. - -The best way to indicate on the drawing the surfaces which are to be -finished is to write on the lines which represent the finished surfaces -“finished,” tool-finish, or “faced,” according to the degree of finish -required. The single letter _f_ is frequently used. - -Avoid fingering the drawing sheet as much as possible; in pointing to -any part of the drawing use a pencil and not the finger. - -Remember that a drawing is made to be read. - -The skill in inking does not depend on the fineness of the line, but on -its clearness. - -A soft pencil should never be used on a mechanical drawing unless in -rare cases when it is used for pencil shading; the hardness or softness -of pencils is denoted by letters. - -Never ink any portion of a drawing until the penciling is entirely -finished. - -Stretching or pasting the paper to the board is very seldom resorted to, -for the reason that the mechanical drawings are _to scale_ and the paper -is natural when pinned to the board and more correct than if under a -strain. Mechanical drawings are always required in practice _right -away_, and time would be wasted and lost in damping and pasting and -drying again. - -A working drawing, whether made to a scale or not, must have all the -dimensions plainly written upon it, for a workman should never be -compelled to measure a drawing. - -In marking off distances, centers, etc., a fine needle point is useful; -the hole should not be punctured through the paper, merely a prick -point, so that it will leave an impression, which will not be -obliterated by the use of rubber; drawing-pens are often equipped with -such a needle point in the end of the handle, that is visible only when -the pen is unscrewed from the handle; but in the absence of one of this -kind the point of the divider leg will be of use. - -Mechanical construction drawings represent a large amount of mental and -manual work, as well as a considerable cost in money; hence, they are of -value quite as much as property which has been acquired by the -expenditure of either labor or capital. It is wise to keep copies of -original designs and sketches, as well as data and formulæ, for record -and comparison. - -The best system for keeping drawings is to make them of certain standard -sizes, and to keep them flat, unrolled, in drawers, numbered, lettered -and labeled. - -In an office where space is limited and drawings have to be rolled it is -well to use a number of pasteboard cases about three feet long and three -inches in diameter. These are shown in fig. 294. - -A puncture can be made near the top and, when a new drawing or -blue-print is inserted in this cylindrical case, a cardboard tag can be -looped through the puncture. This label will give the title and number -of drawings in that case. - -A manuscript book methodically and neatly kept should tell immediately -the number of the drawing and the case. - -Fig. 293 is good for practice in line drawing and also as an optical -illusion. “You look and are deceived. At first glance you say, ‘Of -course, those two lines are curved.’ You are mistaken. They are exactly -parallel. In order to prove this hold them up edgewise to the eye. It -is, of course, the subsidiary lines which lead the vision astray. It is -a case of first impressions being quite wrong.” - - -[Illustration: LINEAR PERSPECTIVE] - -[Illustration: Fig. 295.] - - - - -Linear Perspective. - - -It should be mentioned that this subject is outside the limits of -mechanical drawing, which only deals with objects that can be measured, -projected or dimensioned to an accurate scale. - -But, in rounding out the more formal subjects it is well to look a -little outside the rigid lines of mechanics into the methods of nature, -for no system of teaching drawing is complete that does not include some -explanations for sketching from nature--the objects being always around -the student, the eye always clear to see and the hand only needing the -training to make permanent the impressions received. - -The word perspective means to _see through_; the word perspective being -derived from the Latin word _perspicere_, to look through, hence, -perspective is a science which teaches us to _see_ correctly and enables -us to represent the _appearance_ of anything we may wish to draw; care -should be taken in perspective drawing, to select objects interesting in -themselves, and the best specimens of their class, so as to cultivate -taste, while they at the same time afford useful and instructive drawing -lessons. - -The meaning of the term linear perspective is a line view; the previous -examples have been composed of surfaces placed fronting the eye; -perspective is the science which treats of the changes of form produced -by viewing them in various oblique positions. - -The slightest alteration of _position_ will change the _appearance_ of -an object; this can be easily shown--for illustration take a coin, the -actual shape of which is a perfect round; or, strictly speaking, a -circle. If we take the coin between the thumb and the first finger, -holding it in an upright position, and exactly facing the eyes, as in -Fig. 296, it appears of its true form, viz., a circle. If we alter its -position, balancing it upon the thumb, in a level position, with its -edge directly opposite the eye, as in Fig. 297, its appearance is -changed, and what we know to be really a circle, appears to us as a -straight line. - -[Illustration: Fig. 296.--Fig. 297.--Fig. 298.] - -Now, still balancing the coin upon the thumb, but changing its position -with regard to the eye, by holding it a little lower than in the last -position, that is slightly beneath the level of the eye, as in fig. 298, -we see both the edge and the surface, the coin now appearing neither a -circle nor a straight line, but a curved figure of an elliptical form. -Thus the same coin held in three different positions has assumed three -different shapes. - -Let us take two coins of the same size, holding (in the position shown -at fig. 296) one in each hand. Now, closing one eye, (which will make -the experiment more clear), hold one coin out at arm’s length, and the -other at about the distance of a foot from the eye. On comparing them, -we find that the coin which is further from the eye appears less than -the nearer one. We know that the coins are really equal in size, yet one -appears smaller than the other. - -We thus see that when we change the position of an object, we have as a -consequence a change of appearance; also that the change of appearance -may affect both the shape and the size of the object. - -These diversities of appearance may be remarked in everything around us. -We can observe them in the street by looking at a building from -different points of view, or by comparing the apparent sizes of the -street lamps; in the railway station, by watching the arriving or -departing train; and at sea, by noticing the vessels as they approach, -or as they retire, ultimately vanishing from our sight in that line -where the sea and sky appear to meet. - -All these interesting variations of appearance are in strict accordance -with the laws of =GEOMETRY= and =OPTICS=. The former subject has been -enlarged upon beginning with page 81 of this work, where a line, a -point, an angle, etc., are defined; other terms are explained at page 41 -and the following pages; to these we add a few definitions essential to -the subject. - -=A PLANE= is a surface which is perfectly even and flat; to use a -familiar illustration, a plane is like the surface of a sheet of plate -glass; recollect particularly, that a surface which is at all curved, is -not a plane. - -The =GROUND-PLANE= is the plane on which we stand; the _base-line_ is an -imaginary line passing through the middle of the feet as we stand square -and erect; and the _vertical plane_ is supposed to stand on the -base-line and perpendicular to it. - -[Illustration: Fig. 299.--See page 255.] - -Planes are parallel to each other when they are throughout their entire -surfaces the same distance apart. - -[Illustration: Fig. 300.] - -=THE PERSPECTIVE PLANE= is an upright square of glass, usually framed -like a picture, with a base, so that it can stand up alone. This is -placed between the eye of the spectator and the subject to be drawn, and -as the drawing is sometimes made directly upon it, it is sometimes -called the _Picture_ or the _Plane of the Picture_. - -[Illustration: Fig. 301.] - -=HORIZONTAL= means perfectly level, like the surface of still water. We -must be careful to understand perfectly the difference between the terms -“level” and “even” or “flat.” A surface may be even or flat, without -being level. Thus the wall is even and flat, but it is upright, not -level; level means a fixed, constant position. - -In fig. 301 a house is shown in perspective in which the line _H L_ is -the line of the horizon and _V P_ is the,-- - -=VANISHING POINT.=--The vanishing point is familiarly represented by the -rails on a trolley track on a straight road, which seem to approach each -other in the distance, as shown in fig. 302 at _V P_. - -[Illustration: Fig. 302.] - -_All parallel lines seen in perspective appear to meet in the same -vanishing point._ - -The value of the vanishing point may be seen in the view of a wooden -house, fig. 303, where it (_V P_) gives direction to the retiring lines -of the roof, side planks and door. - -[Illustration: Fig. 303.] - -=POINT OF SIGHT.=--This is that point in the eye where the lines or rays -from the object cross each other, as shown at _P_ in fig. 305, also in -fig. 299 at _S_. - -=VERTICAL= means perfectly upright. If we attach a piece of thread to a -weight, a small piece of lead for example, and hold the thread with the -lead hanging downwards, the thread will fall in an upright or vertical -position. - -=PARALLEL= lines are said to be parallel to each other when they are -throughout their whole lengths the same distance apart. - -=PERPENDICULAR.= When one straight line, meeting another, makes the -angles at the point of contact equal, each of the angles is called a -right angle, and the lines are said to be perpendicular to each other. -Remember especially that perpendicular and vertical have not the same -meaning. Vertical means an unvarying upright position. Perpendicular -means that one line or plane meets another line or plane at right -angles. - -The fig. 295 on page 246 is a study in perspective, showing a water -reflection. As rays from every visible part of the object are reflected, -all following the same law, the reflection will appear to the eye -_inverted_, and of the _same size as the object_. The arch itself forms -the upper half of a hollow cylinder, and the reflection forms the lower -half. The reflection shows much more of the interior of the arch than -can be seen directly. The leaning tree, the boy fishing, and the -receding banks, all are seen in accordance with the laws of reflection -and perspective. - -=THE HORIZONTAL LINE, THE POINT OF SIGHT AND THE VANISHING POINTS= are -the principal items. These should be studied in every room and during -every walk, and the more pleasing accidents of form stored in the mind -or committed to paper for future use. - -[Illustration: Fig. 304.] - -=OPTICS=, the science of sight, gives us the following laws: - -1. That we see by the agency of light. - -2. That light passes from objects to our eyes. - -3. That light travels in straight lines, which are called Visual Rays. - -The human eye may be briefly described as a chamber of a spherical or -globular form, with a circular opening in front. This circular opening -is called the pupil, and through it the visual rays pass to the interior -of the eye. The visual rays, passing from space in all directions -through the small pupil, are received upon what may be called the -interior wall of the globular chamber forming the eye (see fig. 305). -This interior wall is called the retina, and upon it the impressions of -external objects are received, just as they are received upon a screen -in a dark chamber. These impressions are conveyed by the optic nerve -from the retina to the brain. - -In front of the pupil is a segment of a small sphere, composed of the -cornea and the aqueous humor, both of which are transparent, and from -their shape and density have a convergent effect upon the rays passing -through them. - -Behind the pupil is the transparent crystalline lens, which, from its -shape and its elasticity, is a powerful agent in aiding the convergence -of the rays, and in bringing objects at various distances to a clear -focus upon the retina. - -[Illustration: Fig. 305.] - -The pupil has the power of contraction and dilation, which is influenced -by the quantity of light entering the eye, but when it is dilated to the -utmost its size is very small in comparison with the great chamber -forming the body of the eye. - -In fig. 305 we have a rough sectional diagram of the eye and an object -in front of it. This object, an arrow, is seen by means of the visual -rays proceeding from it, the principal two of which are shown. The -visual ray from _A_ passes through the pupil and is received upon the -retina at _a_. In the same way the visual ray from _B_ passes through -the pupil and is received upon the retina at _b_. It will thus be seen -that the impressions or images received upon the retina are inverted; -but, by long reason and experience, the mind has acquired the habit of -determining the real positions of objects, and does not, though the -image is so received, imagine them to be upside down. - -It will also be observed, in the same way, that that portion of an -object which is upon the right will be pictured upon the retina upon the -left, and _vice versa_, but the mind, for the reasons before stated, -never imagines the object to be reversed. This fact is another proof -that, as mentioned at the commencement of our study, to see accurately -is a matter of education and practice. - -And first of Optics; it was asserted, page 252, that we see by the -agency of light which passes from objects to our eyes in straight lines -which are called Visual Rays. - -We see by the agency of light, as all objects, except such as may be -styled self-luminous, when placed in a dark chamber are not perceivable -by us, except by touch, smell or hearing; we cannot _see_ them; they are -invisible. But when, by removing a shutter or igniting a flame, we -introduce something to the chamber which was not present when the -chamber was dark, we become at once conscious of the appearance of the -object, we perceive it by the sense of sight. - -This something which must always be present to enable us to see, is -called Light; all objects are made visible to the sense of seeing by its -agency. - -Without light, natural or artificial, it would be impossible to -distinguish one object from another. - -[Illustration: Fig. 306.] - -That the Visual Rays pass from objects in straight lines to the eye may -be proved by the following experiment (see fig. 306):--Pierce two -screens with a large pin, and place them so that the holes are in a -straight line with a flame, as the light of a candle or lamp. On fixing -the eye to one of these holes we are able to see the flame; but if we -slightly move the flame, one of the screens, or the eye, the flame is no -longer visible. To be visible, the flame, the holes in the screens, and -the eye must all be in the same straight line. See fig. 306. - -In fig. 299 the picture plane is represented by the rectangle _W X Y Z_. -Although the picture plane is here shown as a rectangle, it may be of -any shape or of any size. - -The observer is at _S_, looking through the picture plane at the cross -_R C O H_. The observer is standing upon a horizontal surface, which is -called the ground plane. If we are in a room, the window may be called a -picture plane and the floor a ground plane. - -The picture plane rests, as it were, upon the ground plane, in a line -which passes from _Y_ to _Z_. The two planes meet or intersect in this -line, which is called the ground line. The ground line is sometimes -called the picture line, or the measuring line. - -The visual rays, by means of which the observer sees the cross, will, in -their course from it to the eye, pass through the picture plane. These -visual rays will intersect the picture plane in a number of points, and -if we mark the true positions of these points the result will be a -perspective image of the cross. - -The rays are shown passing from the cross to the eye of the observer, -and meeting the picture plane in points _r_, _c_, _o_, _h_; _r_ being -joined to _o_, and _c_ to _h_, we have the perspective image of the -cross as it would appear to the observer at _S_. Of course an infinite -number of rays proceed from the cross to the eye of the observer; but it -is quite evident that we need only consider those proceeding from the -extremities of the object. - - -Scale or Approximate Perspective. - - Real, or true perspective, represents the object exactly as it is seen - in nature, where the parts that are far away from the eye of the - observer appear smaller than those nearby. Occasions arise, however, - in practical life, with its numerous phases of industrial - requirements, where the convenience of showing the complete form of - the object in a single view might preferably be coupled with the - convenience of scale dimensions. - -This has led to a modified perspective, that sacrifices some of the -accuracy in the appearance of the object to gain the advantage of scale -dimensions; this form of perspective may be distinguished by the -name--_approximate or scale perspective_--which does not represent the -object exactly as seen in nature, but where those parts that are afar -off are shown of the same size as those that are near by, and where the -lines that run out into space are parallel to each other and do not -converge into a vanishing point. - -To represent an object in perspective, the horizon and the point of -vision will have to appear in the drawing as the fundamental starting -points. - -Three dimensions are distinguished for the fixing of an object in space -from a certain reference point. They are height, breadth and thickness, -and are in their direction square to each other. The height is the -fundamental direction, being derived from the direction of gravity, that -invariably extends to the center of the earth. - -All directions in the perspective determination of an object are -parallel to these. - -Vertical lines and planes point toward the center of the earth, while -horizontal planes, including the directions of breadth and thickness, -are square to the vertical direction. In this, the principal visual ray -extends in the direction of thickness. - -For a clear understanding of perspective, it must be firmly fixed in -mind, that for each prominent point of the object behind the picture -plane, a corresponding point lies in the picture plane, in that position -where a straight line or ray of sight that is going from the eye to the -point of the object, cuts through the picture plane. - -Suppose we could replace these rays of sight by thin, visible threads of -wire that would go through little holes in the picture plane, we could -then walk around this bundle of rays, and by looking at it from three -different directions, we would get three different views of it. We may -look upon it from the top, from the side or from the end, where the -bundle of rays all concentrate in the eye of the observer. - -[Illustration: Fig. 307.] - -Figs. 307 and 308 show, in two cases, how these three views would -appear. The end views are those where the perspective picture appears on -the plane, while the top and side views only show where the rays -intersect the picture plane. The top view shows how far, for example, -point _A_ is distant from a vertical line _O Z_, while the side view -shows how far point _A_ is below horizontal line _O X_, which is at the -same height above the ground as the eye of the observer, _O_. Thus, all -points of the cube can be located on the picture plane, and the outlines -of the cube reproduced in perspective. - -[Illustration: Fig. 308.] - -Modified arrangements are shown in figs. 309 and 310 for parallel and -angular perspective. - -[Illustration: Fig. 309.] - -The views are so arranged in relation to each other that the picture -plane in the top view is parallel to the horizon and the ground-line, -which latter is the intersection of the picture plane with the level -ground of the end or perspective view. At the same time the eye of the -observer is in one and the same vertical line for both views, two -vanishing points may be found in the horizon outside of the principal -visual ray. To find the position of these two vanishing points in the -picture plane, the modified top view, fig. 310, is used. - -[Illustration: Fig. 310.] - -As all lines that end in a vanishing point must be parallel in reality, -this parallelism may be seen in the top view and lines through the eye -of the observer, parallel to the directions of the main lines of the -object, will cut the picture plane at the vanishing points. - -Through these two vanishing points the directions of two sets of lines -are found, the starting points of which are determined from the plane of -measurement. The third set of lines, being vertical, also appears -vertical and parallel in the picture. - -The position of each vertical line is found in the top view, where the -light rays from the observing eye to the ends of the vertical lines -intersect with the picture plane. Projecting these points down upon the -rays to the vanishing points produces the vertical lines in the picture. - -For example, in fig. 311, the purpose of perspective is entirely -defeated by placing the eye of the observer directly in front of the -object and arriving at the view taken in mechanical drawing which needs -supplementary views for complete comprehension of the form of the -object. - -[Illustration: Fig. 311.] - -In fig. 312 the eye of the observer is first placed directly opposite -the object, then it sees the object to the left but a short distance -away, while in the third figure the observer is farther away from the -object. In each case the picture plane and plane of measurement is at -the front face of the cube. - -[Illustration: Fig. 312.] - -[Illustration: Fig. 312 (second part).] - -For such simple objects, it is not necessary to draw the top view at -all. The only reminder of the top view is the eye or point of vision, -the picture plane that falls together for the sake of convenience with -the horizon of the end view and the ray that determines the measurement -point _M_, which is, in this suppressed reproduction, absolutely -necessary, in order to find the apparent position of the real corners -behind the picture plane. - -So far, only square or sharp-cornered objects have been represented in -perspective. - -It is evident, however, that round objects can also be shown in linear -perspective, placing reference lines on the object and representing -these as if they were real lines. A cylinder is thus shown in fig. 313 -of which the end planes will appear very distinctly in sharp outlines. - -[Illustration: Fig. 313.] - -Vertically, only the outlines of the cylinder, as contrasted against -space, will appear as distinct outlines, while the reference lines will -not appear and are therefore shown only as dotted lines. - -Fig. 314 shows the approximate or scale perspective with all the axes -drawn and the corresponding angles and scales marked. The outlines of -the object running in these directions appear all parallel to the axes. - -The approximate or scale perspective completely avoids all the -difficulties of choosing a point of sight, of having several views, -vanishing points and measurement points, and thus offers a -representative view, with a great saving of time and labor. Particularly -for mechanical purposes, where an artistic impression is not called for, -it presents a distinct advantage over the true or real perspective. - -[Illustration: Fig. 314.] - - -[Illustration: TABLES - -AND - -INDEX] - -[Illustration: MARINERS’ COMPASS.] - - - - -Useful Tables for Draughtsmen. - - -TABLE OF DECIMAL EQUIVALENTS. - -8ths, 16ths, 32ds and 64ths of an Inch. - - 8ths. - - ¹⁄₈ = .125 - ¹⁄₄ = .250 - ³⁄₈ = .375 - ¹⁄₂ = .500 - ⁵⁄₈ = .625 - ³⁄₄ = .750 - ⁷⁄₈ = .875 - - 16ths. - - ¹⁄₁₆ = .0625 - ³⁄₁₆ = .1875 - ⁵⁄₁₆ = .3125 - ⁷⁄₁₆ = .4375 - ⁹⁄₁₆ = .5625 - ¹¹⁄₁₆ = .6875 - ¹³⁄₁₆ = .8125 - ¹⁵⁄₁₆ = .9375 - - 32nds. - - ¹⁄₃₂ = .03125 - ³⁄₃₂ = .09375 - ⁵⁄₃₂ = .15625 - ⁷⁄₃₂ = .21875 - ⁹⁄₃₂ = .28125 - ¹¹⁄₃₂ = .34375 - ¹³⁄₃₂ = .40625 - ¹⁵⁄₃₂ = .46875 - ¹⁷⁄₃₂ = .53125 - ¹⁹⁄₃₂ = .59375 - ²¹⁄₃₂ = .65625 - ²³⁄₃₂ = .71875 - ²⁵⁄₃₂ = .78125 - ²⁷⁄₃₂ = .84375 - ²⁹⁄₃₂ = .90625 - ³¹⁄₃₂ = .96875 - - 64ths. - - ¹⁄₆₄ = .015625 - ³⁄₆₄ = .046875 - ⁵⁄₆₄ = .078125 - ⁷⁄₆₄ = .109375 - ⁹⁄₆₄ = .140625 - ¹¹⁄₆₄ = .171875 - ¹³⁄₆₄ = .203125 - ¹⁵⁄₆₄ = .234375 - ¹⁷⁄₆₄ = .265625 - ¹⁹⁄₆₄ = .296875 - ²¹⁄₆₄ = .328125 - ²³⁄₆₄ = .359375 - ²⁵⁄₆₄ = .390625 - ²⁷⁄₆₄ = .421875 - ²⁹⁄₆₄ = .453125 - ³¹⁄₆₄ = .484375 - ³³⁄₆₄ = .515625 - ³⁵⁄₆₄ = .546875 - ³⁷⁄₆₄ = .578125 - ³⁹⁄₆₄ = .609375 - ⁴¹⁄₆₄ = .640625 - ⁴³⁄₆₄ = .671875 - ⁴⁵⁄₆₄ = .703125 - ⁴⁷⁄₆₄ = .734375 - ⁴⁹⁄₆₄ = .765625 - ⁵¹⁄₆₄ = .796875 - ⁵³⁄₆₄ = .828125 - ⁵⁵⁄₆₄ = .859375 - ⁵⁷⁄₆₄ = .890625 - ⁵⁹⁄₆₄ = .921875 - ⁶¹⁄₆₄ = .953125 - ⁶³⁄₆₄ = .984375 - - -TABLE OF DECIMAL EQUIVALENTS - -Of Millimeters and Fractions of Millimeters. - - _mm._ _Inches._ - - ¹⁄₅₀ = .00079 - ²⁄₅₀ = .00157 - ³⁄₅₀ = .00236 - ⁴⁄₅₀ = .00315 - ⁵⁄₅₀ = .00394 - ⁶⁄₅₀ = .00472 - ⁷⁄₅₀ = .00551 - ⁸⁄₅₀ = .00630 - ⁹⁄₅₀ = .00709 - ¹⁰⁄₅₀ = .00787 - ¹¹⁄₅₀ = .00866 - ¹²⁄₅₀ = .00945 - ¹³⁄₅₀ = .01024 - ¹⁴⁄₅₀ = .01102 - ¹⁵⁄₅₀ = .01181 - ¹⁶⁄₅₀ = .01260 - ¹⁷⁄₅₀ = .01339 - ¹⁸⁄₅₀ = .01417 - ¹⁹⁄₅₀ = .01496 - ²⁰⁄₅₀ = .01575 - ²¹⁄₅₀ = .01654 - ²²⁄₅₀ = .01732 - ²³⁄₅₀ = .01811 - ²⁴⁄₅₀ = .01890 - ²⁵⁄₅₀ = .01969 - ²⁶⁄₅₀ = .02047 - ²⁷⁄₅₀ = .02126 - ²⁸⁄₅₀ = .02205 - ²⁹⁄₅₀ = .02283 - ³⁰⁄₅₀ = .02362 - ³¹⁄₅₀ = .02441 - ³²⁄₅₀ = .02520 - ³³⁄₅₀ = .02598 - ³⁴⁄₅₀ = .02677 - ³⁵⁄₅₀ = .02756 - ³⁶⁄₅₀ = .02835 - ³⁷⁄₅₀ = .02913 - ³⁸⁄₅₀ = .02992 - ³⁹⁄₅₀ = .03071 - ⁴⁰⁄₅₀ = .03150 - ⁴¹⁄₅₀ = .03228 - ⁴²⁄₅₀ = .03307 - ⁴³⁄₅₀ = .03386 - ⁴⁴⁄₅₀ = .03465 - ⁴⁵⁄₅₀ = .03543 - ⁴⁶⁄₅₀ = .03622 - ⁴⁷⁄₅₀ = .03701 - ⁴⁸⁄₅₀ = .03780 - ⁴⁹⁄₅₀ = .03858 - 1 = .03937 - 2 = .07874 - 3 = .11811 - 4 = .15748 - 5 = .19685 - 6 = .23622 - 7 = .27559 - 8 = .31496 - 9 = .35433 - 10 = .39370 - 11 = .43307 - 12 = .47244 - 13 = .51181 - 14 = .55118 - 15 = .59055 - 16 = .62992 - 17 = .66929 - 18 = .70866 - 19 = .74803 - 20 = .78740 - 21 = .82677 - 22 = .86614 - 23 = .90551 - 24 = .94488 - 25 = .98425 - 26 = 1.02362 - - 10 mm. = 1 Centimeter = 0.3937 inches. - 10 cm. = 1 Decimeter = 3.937 „ - 10 dm. = 1 Meter = 39.37 „ - 25.4 mm. = 1 English Inch. - - -RULES RELATIVE TO THE CIRCLE. - -The circle contains a greater area than any other plane figure bounded -by an equal perimeter or outline. - -TO FIND CIRCUMFERENCE-- - - Multiply diameter by 3.1416. - Or divide diameter by 0.3183. - -TO FIND DIAMETER-- - - Multiply circumference by 0.3183. - Or divide circumference by 3.1416. - -TO FIND RADIUS-- - - Multiply circumference by 0.15915. - Or divide circumference by 6.28318. - -TO FIND SIDE OF AN INSCRIBED SQUARE-- - - Multiply diameter by 0.7071. - Or multiply circumference by 0.2251. - Or divide circumference by 4.4428. - -TO FIND SIDE OF AN EQUAL SQUARE-- - - Multiply diameter by 0.8862. - Or divide diameter by 1.1284. - Or multiply circumference by 0.2821. - Or divide circumference by 3.545. - -SQUARE-- - - A side multiplied by 1.4142 equals diameter of its circumscribing - circle. - A side multiplied by 4.443 equals circumference of its circumscribing - circle. - A side multiplied by 1.128 equals diameter } - A side multiplied by 3.545 equals circumference } of an equal circle. - A side multiplied by 1.273 equals circle inches } - -TO FIND THE AREA OF A CIRCLE-- - - Multiply circumference by one-quarter of the diameter. - Or multiply the square of diameter by 0.7854. - Or multiply the square of circumference by .07958. - Or multiply the square of ¹⁄₂ diameter by 3.1416. - - Contents of cylinder = area of end × length. Contents of wedge = area - of base × ¹⁄₂ altitude. Surface of cylinder = area of both ends × - length × circumference. Surface of sphere = diameter squared × 3.1416, - or = diameter × circumference. Contents of sphere = diameter cubed × - .5236. Contents of pyramid or cone, right or oblique, regular or - irregular = area of base × ¹⁄₃ altitude. Area of triangle = base × ¹⁄₂ - altitude. Area of parallelogram = base × altitude. Area of trapezoid = - altitude × ¹⁄₂ the sum of parallel sides. - - -ROMAN TABLE. - - I. denotes One. - II. „ Two. - III. „ Three. - IV. „ Four. - V. „ Five. - VI. „ Six. - VII. „ Seven. - VIII. „ Eight. - IX. „ Nine. - X. „ Ten. - XI. „ Eleven. - XII. „ Twelve. - XIII. „ Thirteen. - XIV. „ Fourteen. - XV. „ Fifteen. - XVI. „ Sixteen. - XVII. „ Seventeen. - XVIII. „ Eighteen. - XIX. „ Nineteen. - XX. „ Twenty. - XXX. „ Thirty. - XL. „ Forty. - L. „ Fifty. - LX. „ Sixty. - LXX. „ Seventy. - LXXX. „ Eighty. - XC. „ Ninety. - C. „ One hundred. - D. „ Five hundred. - M. „ One thousand. - X̅. „ Ten thousand. - M̅. „ One million. - - -SOLID MEASURE, OR CUBIC MEASURE. - -This is used in measuring bodies, or things having length, breadth and -height or depth. - -TABLE. - - 1728 cubic inches (cu. in.) make 1 cubic foot (cu. ft.). - 27 cubic feet, „ 1 cubic yard (cu. yd.). - 128 cubic feet, „ 1 cord (C.). - - -CIRCULAR MEASURE. - - 60 seconds (″) make 1 minute (′). - 60 minutes „ 1 degree (°). - 360 degrees „ 1 circum. (C.). - -The circumference of every circle whatever, is supposed to be divided -into 360 equal parts, called _degrees_. - -A degree is ¹⁄₃₆₀ of the circumference of any circle, small or large. - -A quadrant is a fourth of a circumference, or an arc of 90 degrees. - -A degree is divided into 60 parts called minutes, expressed by the sign -(′), and each minute is divided into 60 seconds, expressed by (″); so -that the circumference of any circle contains 21,600 minutes, or -1,296,000 seconds. - - -LONG MEASURE--MEASURES OF LENGTH. - - 12 inches = 1 foot. - 3 feet = 1 yard. - 5¹⁄₂ yards = 1 rod. - 40 rods = 1 furlong. - 8 furlongs = 1 common mile. - 3 miles = 1 league. - -The mile (5,280 feet) of the above table is the legal mile of the United -States and England, and is called the statute mile. - - -Tables of Diameters, Circumferences and Areas of Circles. - - -----+-----------+---------- - Diam.| Area. | Circum. - -----+-----------+---------- - 0.0 | | - .1 | .007854| .31416 - .2 | .031416| .62832 - .3 | .070686| .94248 - .4 | .12566 | 1.2566 - | | - .5 | .19735 | 1.5708 - .6 | .28274 | 1.8850 - .7 | .38485 | 2.1991 - .8 | .50266 | 2.5133 - .9 | .63617 | 2.8274 - | | - 1.0 | .7854 | 3.1416 - .1 | .9503 | 3.4558 - .2 | 1.1310 | 3.7699 - .3 | 1.3273 | 4.0841 - .4 | 1.5394 | 4.3982 - | | - .5 | 1.7671 | 4.7124 - .6 | 2.0106 | 5.0265 - .7 | 2.2698 | 5.3407 - .8 | 2.5447 | 5.6549 - .9 | 2.8353 | 5.9690 - | | - 2.0 | 3.1416 | 6.2832 - .1 | 3.4636 | 6.5973 - .2 | 3.8013 | 6.9115 - .3 | 4.1548 | 7.2257 - .4 | 4.5239 | 7.5398 - | | - .5 | 4.9087 | 7.8540 - .6 | 5.3093 | 8.1681 - .7 | 5.7256 | 8.4823 - .8 | 6.1575 | 8.7965 - .9 | 6.6052 | 9.1106 - | | - 3.0 | 7.0686 | 9.4248 - .1 | 7.5477 | 9.7389 - .2 | 8.0425 | 10.0531 - .3 | 8.5530 | 10.3673 - .4 | 9.0792 | 10.6814 - | | - .5 | 9.6211 | 10.9956 - .6 | 10.1788 | 11.3097 - .7 | 10.7521 | 11.6239 - .8 | 11.3411 | 11.9381 - .9 | 11.9456 | 12.2522 - | | - 4.0 | 12.5664 | 12.5664 - .1 | 13.2025 | 12.8805 - .2 | 13.8544 | 13.1947 - .3 | 14.5220 | 13.5088 - .4 | 15.2053 | 13.8230 - | | - .5 | 15.9043 | 14.1372 - .6 | 16.6190 | 14.4513 - .7 | 17.3494 | 14.7655 - .8 | 18.0956 | 15.0796 - .9 | 18.8574 | 15.3938 - | | - 5.0 | 19.6350 | 15.7080 - .1 | 20.4282 | 16.0221 - .2 | 21.2372 | 16.3363 - .3 | 22.0618 | 16.6504 - .4 | 22.9022 | 16.9646 - | | - .5 | 23.7583 | 17.2788 - .6 | 24.6301 | 17.5929 - .7 | 25.5176 | 17.9071 - .8 | 26.4208 | 18.2212 - .9 | 27.3397 | 18.5354 - | | - 6.0 | 28.2743 | 18.8496 - .1 | 29.2247 | 19.1637 - .2 | 30.1907 | 19.4779 - .3 | 31.1725 | 19.7920 - .4 | 32.1699 | 20.1062 - | | - .5 | 33.1831 | 20.4204 - .6 | 34.2119 | 20.7345 - .7 | 35.2565 | 21.0487 - .8 | 36.3168 | 21.3628 - .9 | 37.3928 | 21.6770 - | | - 7.0 | 38.4845 | 21.9911 - .1 | 39.5919 | 22.3053 - .2 | 40.7150 | 22.6195 - .3 | 41.8539 | 22.9336 - .4 | 43.0084 | 23.2478 - | | - .5 | 44.1786 | 23.5619 - .6 | 45.3646 | 23.8761 - .7 | 46.5663 | 24.1903 - .8 | 47.7836 | 24.5044 - .9 | 49.0167 | 24.8186 - | | - 8.0 | 50.2655 | 25.1327 - .1 | 51.5300 | 25.4469 - .2 | 52.8102 | 25.7611 - .3 | 54.1061 | 26.0752 - .4 | 55.4177 | 26.3894 - | | - .5 | 56.7450 | 26.7035 - .6 | 58.0880 | 27.0177 - .7 | 59.4468 | 27.3319 - .8 | 60.8212 | 27.6460 - .9 | 62.2114 | 27.9602 - | | - 9.0 | 63.6173 | 28.2743 - .1 | 65.0388 | 28.5885 - .2 | 66.4761 | 28.9027 - .3 | 67.9291 | 29.2168 - .4 | 69.3978 | 29.5310 - | | - .5 | 70.8822 | 29.8451 - .6 | 72.3823 | 30.1593 - .7 | 73.8981 | 30.4734 - .8 | 75.4296 | 30.7876 - .9 | 76.9769 | 31.1018 - | | - 10.0 | 78.5398 | 31.4159 - .1 | 80.1185 | 31.7301 - .2 | 81.7128 | 32.0442 - .3 | 83.3229 | 32.3584 - .4 | 84.9487 | 32.6726 - | | - .5 | 86.5901 | 32.9887 - .6 | 88.2473 | 33.3009 - .7 | 89.9202 | 33.6150 - .8 | 91.6088 | 33.9292 - .9 | 93.3132 | 34.2434 - | | - 11.0 | 95.0332 | 34.5575 - .1 | 96.7689 | 34.8717 - .2 | 98.5203 | 35.1858 - .3 | 100.2875 | 35.5000 - .4 | 102.0703 | 35.8142 - | | - .5 | 103.8689 | 36.1283 - .6 | 105.6832 | 36.4425 - .7 | 107.5132 | 36.7566 - .8 | 109.3588 | 37.0708 - .9 | 111.2202 | 37.3850 - | | - 12.0 | 113.0973 | 37.6991 - .1 | 114.9901 | 38.0133 - .2 | 116.8987 | 38.3274 - .3 | 118.8229 | 38.6416 - .4 | 120.7628 | 38.9557 - | | - .5 | 122.7185 | 39.2099 - .6 | 124.6898 | 39.6841 - .7 | 126.6769 | 39.8982 - .8 | 128.6796 | 40.2121 - .9 | 130.6981 | 40.5265 - | | - 13.0 | 132.7323 | 40.8407 - .1 | 134.7822 | 41.1549 - .2 | 136.8478 | 41.4690 - .3 | 138.9291 | 41.7832 - .4 | 141.0261 | 42.0973 - | | - .5 | 143.1388 | 42.4116 - .6 | 145.2672 | 42.7257 - .7 | 147.4114 | 43.0398 - .8 | 149.5712 | 43.3540 - .9 | 151.7468 | 43.6681 - | | - 14.0 | 153.9380 | 43.9823 - .1 | 156.1450 | 44.2965 - .2 | 158.3677 | 44.6106 - .3 | 160.6061 | 44.9248 - .4 | 162.8602 | 45.2389 - | | - .5 | 165.1300 | 45.5531 - .6 | 167.4155 | 45.8673 - .7 | 169.7167 | 46.1814 - .8 | 172.0336 | 46.4956 - .9 | 174.3662 | 46.8097 - | | - 15.0 | 176.7146 | 47.1239 - .1 | 179.0786 | 47.4380 - .2 | 181.4584 | 47.7522 - .3 | 183.8539 | 48.0664 - .4 | 186.2650 | 48.3805 - | | - .5 | 188.6919 | 48.6947 - .6 | 191.1345 | 49.0088 - .7 | 193.5928 | 49.3230 - .8 | 196.0668 | 49.6372 - .9 | 198.5565 | 49.9513 - | | - 16.0 | 201.0619 | 50.2655 - .1 | 203.5831 | 50.5796 - .2 | 206.1199 | 50.8938 - .3 | 208.6724 | 51.2080 - .4 | 211.2407 | 51.5221 - | | - .5 | 213.8246 | 51.8363 - .6 | 216.4243 | 52.1504 - .7 | 219.0397 | 52.4646 - .8 | 221.6708 | 52.7788 - .9 | 224.3176 | 53.0929 - | | - 17.0 | 226.9801 | 53.4071 - .1 | 229.6583 | 53.7212 - .2 | 232.3522 | 54.0354 - .3 | 235.0618 | 54.3496 - .4 | 237.7871 | 54.6637 - | | - .5 | 240.5282 | 54.9779 - .6 | 243.2849 | 55.2920 - .7 | 246.0574 | 55.6062 - .8 | 248.8456 | 55.9203 - .9 | 251.6494 | 56.2345 - | | - 18.0 | 254.4690 | 56.5486 - .1 | 257.3043 | 56.8628 - .2 | 260.1553 | 57.1770 - .3 | 263.0220 | 57.4911 - .4 | 265.9044 | 57.8053 - | | - .5 | 268.8025 | 58.1195 - .6 | 271.7164 | 58.4336 - .7 | 274.6459 | 58.7478 - .8 | 277.5911 | 59.0619 - .9 | 280.5521 | 59.3761 - | | - 19.0 | 283.5287 | 59.6903 - .1 | 286.5211 | 60.0044 - .2 | 289.5292 | 60.3186 - .3 | 292.5530 | 60.6327 - .4 | 295.5925 | 60.9469 - | | - .5 | 298.6477 | 61.2611 - .6 | 301.7186 | 61.5752 - .7 | 304.8052 | 61.8894 - .8 | 307.9075 | 62.2035 - .9 | 311.0255 | 62.5177 - | | - 20.0 | 314.1593 | 62.8319 - .1 | 317.3087 | 63.1460 - .2 | 320.4739 | 63.4602 - .3 | 323.6547 | 63.7743 - .4 | 326.8513 | 64.0885 - | | - .5 | 330.0636 | 64.4026 - .6 | 333.2916 | 64.7168 - .7 | 336.5353 | 65.0310 - .8 | 339.7947 | 65.3451 - .9 | 343.0698 | 65.6593 - | | - 21.0 | 346.3606 | 65.9734 - .1 | 349.6671 | 66.2876 - .2 | 352.9894 | 66.6018 - .3 | 356.3273 | 66.9159 - .4 | 359.6809 | 67.2301 - | | - .5 | 363.0503 | 67.5442 - .6 | 266.4354 | 67.8584 - .7 | 369.8361 | 68.1726 - .8 | 373.2526 | 68.4867 - .9 | 376.6848 | 68.8009 - | | - 22.0 | 380.1327 | 69.1150 - .1 | 383.5963 | 69.4292 - .2 | 387.0756 | 69.7434 - .3 | 390.5707 | 70.0575 - .4 | 394.0814 | 70.3717 - | | - .5 | 397.6078 | 70.6858 - .6 | 401.1500 | 71.0000 - .7 | 404.7078 | 71.3142 - .8 | 408.2814 | 71.6283 - .9 | 411.8707 | 71.9425 - | | - 23.0 | 415.4756 | 72.2566 - .1 | 419.0993 | 72.5708 - .2 | 422.7327 | 72.8849 - .3 | 426.3848 | 73.1991 - .4 | 430.0526 | 73.5133 - | | - .5 | 433.7361 | 73.8274 - .6 | 437.4354 | 74.1416 - .7 | 441.1503 | 74.4557 - .8 | 444.8809 | 74.7699 - .9 | 448.6273 | 75.0841 - | | - 24.0 | 452.3893 | 75.3982 - .1 | 456.1671 | 75.7424 - .2 | 459.9606 | 76.0265 - .3 | 463.7693 | 76.3407 - .4 | 467.5947 | 76.6549 - | | - .5 | 471.4352 | 76.9690 - .6 | 475.2916 | 77.2832 - .7 | 479.1636 | 77.5973 - .8 | 483.0513 | 77.9115 - .9 | 486.9547 | 78.2257 - | | - 25.0 | 490.8739 | 78.5398 - .1 | 494.8087 | 78.8540 - .2 | 498.7592 | 79.1681 - .3 | 502.7255 | 79.4823 - .4 | 506.7075 | 79.7965 - | | - .5 | 510.7052 | 80.1106 - .6 | 514.7185 | 80.4248 - .7 | 518.7476 | 80.7389 - .8 | 522.7924 | 81.0531 - .9 | 526.8529 | 81.3672 - | | - 26.0 | 530.9292 | 81.6814 - .1 | 535.0211 | 81.9956 - .2 | 539.1287 | 82.3097 - .3 | 543.2521 | 82.6239 - .4 | 547.3911 | 82.9380 - | | - .5 | 551.5459 | 83.2522 - .6 | 555.7163 | 83.5664 - .7 | 559.9025 | 83.8805 - .8 | 564.1044 | 84.1947 - .9 | 568.3220 | 84.5088 - | | - 27.0 | 572.5553 | 84.8230 - .1 | 576.8043 | 85.1372 - .2 | 581.0690 | 85.4513 - .3 | 585.3494 | 85.7655 - .4 | 589.6455 | 86.0796 - | | - .5 | 593.9574 | 86.3938 - .6 | 598.2849 | 86.7080 - .7 | 602.6282 | 87.0221 - .8 | 606.9871 | 87.3363 - .9 | 611.3618 | 87.6504 - | | - 28.0 | 615.7522 | 87.9646 - .1 | 620.1582 | 88.2788 - .2 | 624.5800 | 88.5929 - .3 | 629.0175 | 88.9071 - .4 | 633.4707 | 89.2212 - | | - .5 | 637.9397 | 89.5354 - .6 | 642.4243 | 89.8495 - .7 | 646.9246 | 90.1637 - .8 | 651.4407 | 90.4779 - .9 | 655.9724 | 90.7920 - | | - 29.0 | 660.5199 | 91.1063 - .1 | 665.0830 | 91.4203 - .2 | 669.6619 | 91.7345 - .3 | 674.2565 | 92.0487 - .4 | 678.8668 | 92.3628 - | | - .5 | 683.4928 | 92.6770 - .6 | 688.1345 | 92.9911 - .7 | 692.7919 | 93.3053 - .8 | 697.4650 | 93.6195 - .9 | 702.1538 | 93.9336 - | | - 30.0 | 706.8583 | 94.2478 - .1 | 711.5786 | 94.5610 - .2 | 716.3145 | 94.8761 - .3 | 721.0662 | 95.1903 - .4 | 725.8336 | 95.5044 - | | - .5 | 730.6167 | 95.8186 - .6 | 735.4154 | 96.1327 - .7 | 740.2299 | 96.4469 - .8 | 745.0601 | 96.7611 - .9 | 749.9060 | 97.0752 - | | - 31.0 | 754.7676 | 97.3894 - .1 | 759.6450 | 97.7035 - .2 | 761.5380 | 98.0177 - .3 | 769.4467 | 98.3319 - .4 | 774.3712 | 98.6460 - | | - .5 | 779.3113 | 98.9602 - .6 | 784.2672 | 99.2743 - .7 | 789.2388 | 99.5885 - .8 | 794.2260 | 99.9026 - .9 | 799.2290 |100.2168 - | | - 32.0 | 804.2477 |100.5310 - .1 | 809.2821 |100.8451 - .2 | 814.3322 |101.1593 - .3 | 819.3980 |101.4734 - .4 | 824.4796 |101.7876 - | | - .5 | 829.5768 |102.1018 - .6 | 834.6898 |102.4159 - .7 | 839.8185 |102.7301 - .8 | 844.9628 |103.0442 - .9 | 850.1229 |103.3584 - | | - 33.0 | 855.2986 |103.6726 - .1 | 860.4902 |103.9867 - .2 | 865.6973 |104.3009 - .3 | 870.9202 |104.6150 - .4 | 876.1588 |104.9292 - | | - .5 | 881.4131 |105.2434 - .6 | 886.6831 |105.5575 - .7 | 891.9688 |105.8717 - .8 | 897.2703 |106.1858 - .9 | 902.5874 |106.5000 - | | - 34.0 | 907.9203 |106.8142 - .1 | 913.2688 |107.1283 - .2 | 918.6331 |107.4425 - .3 | 924.0131 |107.7566 - .4 | 929.4088 |108.0708 - | | - .5 | 934.8202 |108.3849 - .6 | 940.2473 |108.6991 - .7 | 945.6901 |109.0133 - .8 | 951.1486 |109.3274 - .9 | 956.6228 |109.6416 - | | - 35.0 | 962.1128 |109.9557 - .1 | 967.6184 |110.2699 - .2 | 973.1397 |110.5841 - .3 | 978.6768 |110.8982 - .4 | 984.2296 |111.2124 - | | - .5 | 989.7980 |111.5265 - .6 | 995.3822 |111.8407 - .7 |1000.9821 |112.1549 - .8 |1006.5977 |112.4690 - .9 |1012.2290 |112.7832 - | | - 36.0 |1017.8760 |113.0973 - .1 |1023.5387 |113.4115 - .2 |1029.2172 |113.7257 - .3 |1034.9113 |114.0398 - .4 |1040.6212 |114.3540 - | | - .5 |1046.3467 |114.6681 - .6 |1052.0880 |114.9823 - .7 |1057.8449 |115.2965 - .8 |1063.6176 |115.6106 - .9 |1069.4060 |115.9248 - | | - 37.0 |1075.2101 |116.2389 - .1 |1081.0299 |116.5531 - .2 |1086.8654 |116.8672 - .3 |1093.7166 |117.1814 - .4 |1098.5835 |117.4956 - | | - .5 |1104.4662 |117.8097 - .6 |1110.3645 |118.1239 - .7 |1116.2786 |118.4380 - .8 |1122.2083 |118.7522 - .9 |1128.1538 |119.0664 - | | - 38.0 |1134.1149 |119.3805 - .1 |1140.0918 |119.6947 - .2 |1146.0844 |120.0088 - .3 |1152.0927 |120.3230 - .4 |1158.1167 |120.6372 - | | - .5 |1164.1564 |120.9513 - .6 |1170.2118 |121.2655 - .7 |1176.2830 |121.5796 - .8 |1182.3698 |121.8938 - .9 |1188.4724 |122.2080 - | | - 39.0 |1194.5906 |122.5221 - .1 |1200.7246 |122.8363 - .2 |1206.8742 |123.1504 - .3 |1213.0396 |123.4646 - .4 |1219.2207 |123.7788 - | | - .5 |1225.4175 |124.0929 - .6 |1231.6300 |124.4071 - .7 |1237.8582 |124.7212 - .8 |1244.1021 |125.0354 - .9 |1250.3617 |125.3495 - | | - 40.0 |1256.6371 |125.6637 - .1 |1262.9281 |125.9779 - .2 |1269.2348 |126.2920 - .3 |1275.5573 |126.6062 - .4 |1281.8955 |126.9203 - | | - .5 |1288.2493 |127.2345 - .6 |1294.3189 |127.5487 - .7 |1301.0042 |127.8628 - .8 |1307.4052 |128.1770 - .9 |1313.8219 |128.4911 - | | - 41.0 |1320.2543 |128.8053 - .1 |1326.7024 |129.1195 - .2 |1333.1663 |129.4336 - .3 |1339.6458 |129.7478 - .4 |1346.1410 |130.0619 - | | - .5 |1352.6520 |130.3761 - .6 |1359.1786 |130.6903 - .7 |1365.7210 |131.0044 - .8 |1372.2791 |131.3186 - .9 |1378.8529 |131.6227 - | | - 42.0 |1385.4424 |131.9469 - .1 |1392.0476 |132.2611 - .2 |1398.6685 |132.5752 - .3 |1405.3051 |132.8894 - .4 |1411.9574 |133.2035 - | | - .5 |1418.6254 |133.5177 - .6 |1425.3092 |133.8318 - .7 |1432.0086 |134.1460 - .8 |1438.7238 |134.4602 - .9 |1445.4546 |134.7743 - | | - 43.0 |1452.2012 |135.0885 - .1 |1458.9635 |135.4026 - .2 |1465.7415 |135.7168 - .3 |1472.5352 |136.0310 - .4 |1479.3446 |136.3451 - | | - .5 |1486.1697 |136.6593 - .6 |1493.0105 |136.9734 - .7 |1499.8670 |137.2876 - .8 |1506.7393 |137.6018 - .9 |1513.6272 |137.9159 - | | - 44.0 |1520.5308 |138.2301 - .1 |1527.4502 |138.5442 - .2 |1534.3853 |138.8584 - .3 |1541.3360 |139.1726 - .4 |1548.3025 |139.4867 - | | - .5 |1555.2847 |139.8009 - .6 |1562.2826 |140.1153 - .7 |1569.2063 |140.4292 - .8 |1576.3255 |140.7434 - .9 |1583.3706 |141.0575 - | | - 45.0 |1590.4313 |141.3717 - .1 |1597.5077 |141.6858 - .2 |1604.5999 |142.0000 - .3 |1611.7077 |142.3142 - .4 |1618.8313 |142.6283 - | | - .5 |1625.9705 |142.9425 - .6 |1633.1255 |143.2566 - .7 |1640.2962 |143.5708 - .8 |1647.4826 |143.8849 - .9 |1654.6847 |144.1991 - | | - 46.0 |1661.9025 |144.5133 - .1 |1669.1360 |144.8274 - .2 |1676.3853 |145.1416 - .3 |1683.6502 |145.4557 - .4 |1690.9308 |145.7699 - | | - .5 |1698.2272 |146.0841 - .6 |1705.5392 |146.3982 - .7 |1712.8670 |146.7124 - .8 |1720.2105 |147.0265 - .9 |1727.5697 |147.3407 - | | - 47.0 |1734.9445 |147.6550 - .1 |1742.3351 |147.9690 - .2 |1749.7414 |148.2832 - .3 |1757.1635 |148.5973 - .4 |1764.6012 |148.9115 - | | - .5 |1772.0546 |149.2257 - .6 |1779.5237 |149.5398 - .7 |1787.0086 |149.8540 - .8 |1794.5091 |150.1681 - .9 |1802.0254 |150.4823 - | | - 48.0 |1809.5574 |150.7964 - .1 |1817.1050 |151.1105 - .2 |1824.6684 |151.4248 - .3 |1832.2475 |151.7389 - .4 |1839.8423 |152.0531 - | | - .5 |1847.4528 |152.3672 - .6 |1855.0790 |152.6814 - .7 |1862.7210 |152.9956 - .8 |1870.3786 |153.3097 - .9 |1878.0519 |153.6239 - | | - 49.0 |1885.7409 |153.9380 - .1 |1893.4457 |154.2522 - .2 |1901.1662 |154.5664 - .3 |1908.9024 |154.8805 - .4 |1916.6543 |155.1947 - | | - .5 |1924.4218 |155.5088 - .6 |1932.2051 |155.8230 - .7 |1940.0042 |156.1372 - .8 |1947.8189 |156.4513 - .9 |1955.6493 |156.7655 - | | - 50.0 |1963.4954 |157.0796 - .1 |1971.3572 |157.3938 - .2 |1979.2348 |157.7080 - .3 |1987.1280 |158.0221 - .4 |1995.0370 |158.3363 - | | - .5 |2002.9617 |158.6504 - .6 |2010.9020 |158.9646 - .7 |2018.8581 |159.2787 - .8 |2026.8299 |159.5929 - .9 |2034.8174 |159.9071 - | | - 51.0 |2042.6206 |160.2212 - .1 |2050.8895 |160.5354 - .2 |2058.8742 |160.8495 - .3 |2066.9245 |161.1637 - .4 |2074.9905 |161.4779 - | | - .5 |2083.0723 |161.7920 - .6 |2091.1697 |162.1062 - .7 |2099.2829 |162.4203 - .8 |2107.4118 |162.7345 - .9 |2115.5563 |163.0487 - | | - 52.0 |2123.7166 |163.3628 - .1 |2131.8926 |163.6770 - .2 |2140.0843 |163.9911 - .3 |2148.2917 |164.3053 - .4 |2156.5149 |164.6195 - | | - .5 |2164.7537 |164.9336 - .6 |2173.0082 |165.2479 - .7 |2181.2785 |165.5619 - .8 |2189.5644 |165.8761 - .9 |2197.8661 |166.1903 - | | - 53.0 |2206.1834 |166.5044 - .1 |2214.5165 |166.8186 - .2 |2222.8653 |167.1327 - .3 |2231.2298 |167.4469 - .4 |2239.6100 |167.7610 - | | - .5 |2248.0059 |168.0752 - .6 |2256.4175 |168.3894 - .7 |2264.8448 |168.7035 - .8 |2273.2879 |169.0177 - .9 |2281.7466 |169.3318 - | | - 54.0 |2290.2210 |169.6460 - .1 |2298.7112 |169.9602 - .2 |2307.2171 |170.2743 - .3 |2315.7386 |170.5885 - .4 |2324.2759 |170.9026 - | | - .5 |2332.8289 |171.2168 - .6 |2341.3976 |171.5310 - .7 |2349.9820 |171.8451 - .8 |2358.5821 |172.1593 - .9 |2367.1979 |172.4735 - | | - 55.0 |2375.8294 |172.7876 - .1 |2384.4767 |173.1017 - .2 |2393.1396 |173.4159 - .3 |2401.8183 |173.7301 - .4 |2410.5126 |174.0442 - | | - .5 |2419.2227 |174.3584 - .6 |2427.9485 |174.6726 - .7 |2436.6899 |174.9867 - .8 |2145.4471 |175.3009 - .9 |2454.2200 |175.6150 - -----+-----------+-------- - - -UNITED STATES STANDARD SIZES OF WROUGHT IRON WELDED PIPE. - - ------+-------+------+------+--------+--------+--------+--------+ - | | | | | | | | - | | | | | | Length | Length | - | | | | | | of pipe| of pipe| - | | | | | | per | per | - Inside| Actual| |Actual| | | square | square | - diam-|outside| |Inside|External|Internal| foot of| foot of| - eter | Diam- |Thick-| Diam-| circum-| circum-| outside| inside | - nom. | eter. | ness.| eter.|ference.|ference.|surface.|surface.| - ------+-------+------+------+--------+--------+--------+--------+ - ¹⁄₈ | .405| .068 | 0.269| 1.272 | 0.848 | 9.440 | 14.15 | - ¹⁄₄ | .54 | .088 | 0.364| 1.696 | 1.144 | 7.075 | 10.50 | - ³⁄₈ | .675| .091 | 0.493| 2.121 | 1.552 | 5.657 | 7.67 | - ¹⁄₂ | .840| .109 | 0.622| 2.652 | 1.957 | 4.502 | 6.13 | - ³⁄₄ | 1.050| .113 | 0.824| 3.299 | 2.589 | 3.637 | 4.635 | - 1 | 1.315| .134 | 1.047| 4.134 | 3.292 | 2.903 | 3.679 | - 1¹⁄₄ | 1.660| .140 | 1.38 | 5.215 | 4.335 | 2.301 | 2.768 | - 1¹⁄₂ | 1.90 | .145 | 1.61 | 5.969 | 5.061 | 2.010 | 2.371 | - 2 | 2.375| .154 | 2.067| 7.461 | 6.494 | 1.611 | 1.848 | - 2¹⁄₂ | 2.875| .204 | 2.467| 9.032 | 7.754 | 1.328 | 1.547 | - 3 | 3.50 | .217 | 3.066| 10.996 | 9.636 | 1.091 | 1.245 | - 3¹⁄₂ | 4.0 | .226 | 3.548| 12.566 | 11.146 | .955 | 1.077 | - 4 | 4.50 | .237 | 4.026| 14.137 | 12.648 | .849 | 0.949 | - 4¹⁄₂ | 5.0 | .247 | 4.506| 15.708 | 14.153 | .765 | 0.848 | - 5 | 5.563| .259 | 5.045| 17.475 | 15.849 | .629 | 0.757 | - 6 | 6.625| .280 | 5.065| 20.813 | 19.054 | .577 | 0.630 | - 7 | 7.625| .301 | 7.023| 23.954 | 22.063 | .505 | 0.544 | - 8 | 8.625| .322 | 7.981| 27.096 | 25.076 | .444 | 0.478 | - 9 | 9.688| .344 | 9.00 | 30.433 | 28.277 | .394 | 0.425 | - 10 | 10.750| .366 |10.018| 33.772 | 31.475 | .355 | 0.381 | - ------+-------+------+------+--------+--------+--------+--------+ - - ------+------+-------+-------+-------+-------+------ - | | | | | | - | | | Length| | | - | | |of pipe| | | - | | | con- | Weight|No. of | - Inside| | Actual|taining| per |threads|Length - diam-| Ex- | in- | one | foot | per | per- - eter |ternal| ternal| cubic | of |Inch of| fect - nom. | area.| area. | foot. |length.| screw.|screw. - ------+------+-------+-------+-------+-------+------ - ¹⁄₈ | .129| .0572|2500. | .243| 27 | 0.19 - ¹⁄₄ | .229| .1041|1385. | .422| 18 | 0.29 - ³⁄₈ | .358| .1916| 751.5 | .561| 18 | 0.30 - ¹⁄₂ | .554| .3048| 472.4 | .845| 14 | 0.39 - ³⁄₄ | .866| .5333| 270.0 | 1.126| 14 | 0.40 - 1 | 1.357| .8627| 166.9 | 1.670| 11¹⁄₂ | 0.51 - 1¹⁄₄ | 2.164| 1.496 | 96.25| 2.258| 11¹⁄₂ | 0.54 - 1¹⁄₂ | 2.835| 2.038 | 70.65| 2.694| 11¹⁄₂ | 0.55 - 2 | 4.430| 3.355 | 42.36| 3.667| 11¹⁄₂ | 0.58 - 2¹⁄₂ | 6.491| 4.783 | 30.11| 5.773| 8 | 0.89 - 3 | 9.621| 7.388 | 19.40| 7.547| 8 | 0.95 - 3¹⁄₂ |12.566| 9.837 | 14.56| 9.055| 8 | 1.00 - 4 |15.901|12.730 | 11.31| 10.728| 8 | 1.05 - 4¹⁄₂ |19.635|15.939 | 9.03| 12.492| 8 | 1.10 - 5 |24.299|19.990 | 7.20| 14.564| 8 | 1.16 - 6 |34.471|28.889 | 4.98| 18.767| 8 | 1.26 - 7 |45.663|38.737 | 3.72| 23.410| 8 | 1.36 - 8 |58.426|50.039 | 2.88| 28.348| 8 | 1.46 - 9 |73.715|63.633 | 2.26| 34.677| 8 | 1.57 - 10 |90.762|78.838 | 1.80| 40.641| 8 | 1.68 - ------+------+-------+-------+-------+-------+------ - -Thread taper three-fourths inch to one foot. - -All pipe below 1¹⁄₂ inches is butt-welded, and proved to 300 pounds per -square inch; 1¹⁄₂ inch and above is lap-welded and proved to 500 pounds -per square inch. - - -STANDARDS FOR WIRE GAUGES IN USE IN THE UNITED STATES. - -Dimensions of Sizes in Decimal Parts of an Inch. - - ------+--------+--------+---------+------+------+----------+------ - | | | Washburn| | | | - Number|American|Birming-| & Moen | Impe-| | U. S. |Number - of |or Brown| ham, or|Mfg. Co.,| rial |Stubs’| Stand. | of - Wire | & | Stubs’ | Worces- | Wire | Steel| for | Wire - Gauge.| Sharpe.| Wire. | ter, Ms.|Gauge.| Wire.| Plate. |Gauge. - ------+--------+--------+---------+------+------+----------+------ - 000000| .... | .... | .... | .464 | .... |.46875 |000000 - 00000| .... | .... | .... | .432 | .... |.4375 | 00000 - 0000| .46 | .454 | .3938 | .400 | .... |.40625 | 0000 - 000| .40964 | .425 | .3625 | .372 | .... |.375 | 000 - 00| .3648 | .38 | .3310 | .348 | .... |.34375 | 00 - 0| .32486 | .34 | .3065 | .324 | .... |.3125 | 0 - 1| .2893 | .3 | .2830 | .300 | .227 |.28125 | 1 - 2| .25763 | .284 | .2625 | .276 | .219 |.265625 | 2 - 3| .22942 | .259 | .2437 | .252 | .212 |.25 | 3 - 4| .20431 | .238 | .2253 | .232 | .207 |.234375 | 4 - 5| .18194 | .22 | .2070 | .212 | .204 |.21875 | 5 - 6| .16202 | .203 | .1920 | .192 | .201 |.203125 | 6 - 7| .14428 | .18 | .1770 | .176 | .199 |.1875 | 7 - 8| .12849 | .165 | .1620 | .160 | .197 |.171875 | 8 - 9| .11443 | .148 | .1483 | .144 | .194 |.15625 | 9 - 10| .10189 | .134 | .1350 | .128 | .191 |.140625 | 10 - 11| .090742| .12 | .1205 | .116 | .188 |.125 | 11 - 12| .080808| .109 | .1055 | .104 | .185 |.109375 | 12 - 13| .071961| .095 | .0915 | .092 | .182 |.09375 | 13 - 14| .064084| .083 | .0800 | .080 | .180 |.078125 | 14 - 15| .057068| .072 | .0720 | .072 | .178 |.0703125 | 15 - 16| .05082 | .065 | .0625 | .064 | .175 |.0625 | 16 - 17| .045257| .058 | .0540 | .056 | .172 |.05625 | 17 - 18| .040303| .049 | .0475 | .048 | .168 |.05 | 18 - 19| .03589 | .042 | .0410 | .040 | .164 |.04375 | 19 - 20| .031961| .035 | .0348 | .036 | .161 |.0375 | 20 - 21| .028462| .032 | .03175 | .032 | .157 |.034375 | 21 - 22| .025347| .028 | .0286 | .028 | .155 |.03125 | 22 - 23| .022571| .025 | .0258 | .024 | .153 |.028125 | 23 - 24| .0201 | .022 | .0230 | .022 | .151 |.025 | 24 - 25| .0179 | .02 | .0204 | .020 | .148 |.021875 | 25 - 26| .01594 | .018 | .0181 | .018 | .146 |.01875 | 26 - 27| .014195| .016 | .0173 | .0164| .143 |.0171875 | 27 - 28| .012641| .014 | .0162 | .0149| .139 |.015625 | 28 - 29| .011257| .013 | .0150 | .0136| .134 |.0140625 | 29 - 30| .010025| .012 | .0140 | .0124| .127 |.0125 | 30 - 31| .008928| .01 | .0132 | .0116| .120 |.0109375 | 31 - 32| .00795 | .009 | .0128 | .0108| .115 |.01015625 | 32 - 33| .00708 | .008 | .0118 | .0100| .112 |.009375 | 33 - 34| .006304| .007 | .0104 | .0092| .110 |.00859375 | 34 - 35| .005614| .005 | .0095 | .0084| .108 |.0078125 | 35 - 36| .005 | .004 | .0090 | .0076| .106 |.00703125 | 36 - 37| .004453| .... | .... | .0068| .103 |.006640625| 37 - 38| .003965| .... | .... | .0060| .101 |.00625 | 38 - 39| .003531| .... | .... | .0052| .099 | .... | 39 - 40| .003144| .... | .... | .0048| .097 | .... | 40 - ------+--------+--------+---------+------+------+----------+------ - - -UNITED STATES STANDARD SIZES OF BOLTS. - - +------+--------+--------+--------+ - | |DISTANCE| | | - | | ACROSS |DISTANCE| | - | | FLATS, | ACROSS | THICK- | - | DIAM-| SQUARE |CORNERS,| NESS | - | ETER.|AND HEX.| HEX. |OF HEAD.| - +------+--------+--------+--------+ - | ¹⁄₄″| ¹⁄₂″ | ⁹⁄₁₆″ | ¹⁄₄″ | - | ³⁄₈ | ¹¹⁄₁₆ | ²⁵⁄₃₂ | ¹¹⁄₃₂ | - | ¹⁄₂ | ⁷⁄₈ | 1 | ⁷⁄₁₆ | - | ⁵⁄₈ | 1¹⁄₁₆ | 1⁷⁄₃₂ | ¹⁷⁄₃₂ | - | ³⁄₄ | 1¹⁄₄ | 1⁷⁄₁₆ | ⁵⁄₈ | - | ⁷⁄₈ | 1⁷⁄₁₆ | 1²¹⁄₃₂ | ²¹⁄₃₂ | - | 1 | 1⁵⁄₈ | 1⁷⁄₈ | ¹³⁄₁₆ | - | 1¹⁄₈ | 1¹³⁄₁₆ | 2³⁄₃₂ | ²⁹⁄₃₂ | - | 1¹⁄₄ | 2 | 2⁵⁄₁₆ | 1 | - | 1³⁄₈ | 2³⁄₁₆ | 2¹⁄₂ | 1³⁄₃₂ | - | 1¹⁄₂ | 2³⁄₈ | 2³⁄₄ | 1³⁄₁₆ | - | 1⁵⁄₈ | 2⁹⁄₁₆ | 2¹⁵⁄₁₆ | 1⁹⁄₃₂ | - | 1³⁄₄ | 2³⁄₄ | 3³⁄₁₆ | 1³⁄₈ | - | 1⁷⁄₈ | 2¹⁵⁄₁₆ | 3¹³⁄₃₂ | 1¹⁵⁄₃₂ | - | 2 | 3¹⁄₈ | 3⁵⁄₈ | 1⁹⁄₁₆ | - | 2¹⁄₄ | 3¹⁄₂ | 4¹⁄₁₆ | 1³⁄₄ | - | 2¹⁄₂ | 3⁷⁄₈ | 4¹⁄₂ | 1¹⁵⁄₁₆ | - | 2³⁄₄ | 4¹⁄₄ | 4⁷⁄₈ | 2¹⁄₈ | - | 3 | 4⁵⁄₈ | 5¹¹⁄₃₂ | 2⁵⁄₁₆ | - +------+--------+--------+--------+ - - -UNITED STATES STANDARD SCREW THREAD GAUGE. - -[Illustration: Fig. 316.] - - - - -Some Things Personal. - - -In the preparation of this work the idea of self-help has never been -forgotten; nothing has been held back or omitted which would, in the -author’s opinion, tend to advance the student in the draughtsman’s art. - -The volume contains the experience in practical drawing, as related to -engineering and mechanics, of over one hundred years, i. e., the -author’s, Mr. Perrott’s and Mr. Lucas’ experiences added together exceed -that period. - -Hence, the work should be really helpful; it has been aimed also to be -entertaining and with easy tasks; all the illustrations of the book are -recommended as models for practice--they have been selected with that -view. - -Moreover, as between author and publishers, the latter have agreed to -issue the work in the most thorough style possible--as to paper, -printing and binding--and to sell it at a very generously low price, -considering all things. - -With the closing words “hail and farewell,” the author bids adieu (God -be with you) to the reader and the student. - -[Illustration: BEACON LIGHT.] - - - - - Self - Hawkins’ Mechanical Drawing. - Help - -Index. - - - PAGE - - =Acute Angle=, def. 42 - - =Acute-Angled Triangle=, def. 50 - - =Addendum Circle=, illus. 200 - - =Alphabets=, Gothic, desc. 171 - - =Altitude=, def. 41 - - =Altitude of a Polygon=, def. 47 - - =American Machinist=, Quotation from 141 - - =Angle=, def. 41, 83 - To bisect an 89 - To draw an 89 - - =Angle-Iron=, illus. 75 - - =Angle-Plate=, illus. 75 - - =Angular Perspective Drawing=, illus. 260, 265 - - =Apex=, def. 42 - - =Apex of an Angle=, def. 42 - - =Arc=, def. 42 - Complement of an 99 - Cosecant of an 99 - Cotangent of an 99 - Sine of an 99 - Supplement of an 99 - Tangent of an 99 - Versed Sine of an 99 - - =Arc of Circle=, To find the center of an 90 - - =Arcs=, Drawing 146 - Illus. 146 - - =Arrow-Heads=, how made 172, 176 - - =Axiom=, def. 83 - - =Axis=, Conjugate, def. 96 - Of a Figure, def. 43 - Of a Solid, def. 42 - Transverse, def. 96 - - - =Backlash in Gearing= 215 - - =Backward Projection= desc. 152, illus. 152-3 - - =Base=, def. 43 - Of a Polygon, def. 47 - - =Beacon Light=, illus. 282 - - =Beam Compasses= desc. 124, illus. 124 - - =Bending Machine=, Hydraulic desc. 221, illus. 220-1 - - =Benjamin, Prof. Chas.=, “How and What to Study” 23 - - =Bevel-Gear=, desc. 201 - - =Bevel Mortise Wheel=, desc. 203 - - =Bevel Wheel= desc. 199, illus. 203 - - =Bisect=, def. 43 - - =Bisector=, def. 43 - - =Blackboard=, illus. and desc. 27 - - =Blackboard Drawing= 28 - - =Blue Printing= desc. 186, illus. 187 - Test pieces 188 - - =Blue Prints=, Office rules for 196 - - =Black Process Copying= 188 - - =Boiler-Plate=, Riveted, illus. 76 - - =Bolt=, Square-head, illus. 76 - - =Bolt and Nut=, Square-head desc. 144, illus. 145 - Hexagon desc. 144, illus. 76, 144 - - =Bow-Dividers= desc. 123, illus. 122 - - =Bow-Pencil=, desc. 123 - - =Bows=, Use of 242 - - =Brick=, Section Lining, illus. 182 - - =Broken Lines=, How to Draw 33 - Def. 46 - - =Bumping Post=, illus. 226 - - - =Calipers=, illus. vi - - =Cast-Iron=, Section-Lining, illus. 182 - - =Caulking Tool=, illus. 76 - - =Center Line=, def. 46 - - =Center Lines=, In Drawings 240-1 - In Shop Drawings 192 - - =Chalk-Crayon= desc. 27, illus. 27, 37 - - =Chalk-Work= 27 - Instruments for Drawing, illus. 29 - - =Channel-Iron=, illus. 75 - - =Checking Drawings= 194 - - =Circle=, def. 43 - To describe about a square 93 - about a triangle 93 - through two points 90 - through three points 90 - To find the center 90 - To inscribe in a square 94 - in a triangle 94 - - =Circles=, Drawing desc. 146, illus. 146 - - =Circumference of a Circle= 43 - - =Circumscribe=, def. 43 - - =Circular Pitch Line=, illus. 200 - - =Classifying Drawings=, Office Rules for 191 - - =Clearance in Wheel Teeth= 215 - - =Cog-Wheel= def. 201, illus. 201 - Section Lining, illus. 183 - - =Coin=, Perspective View of a, illus. 248 - - =Color and Tints= 185 - - =Compass=, Mariners’, illus. 268 - - =Compasses=, Beam, desc. and illus. 124 - Desc. and illus. 121 - For holding chalk, illus. 29 - How to hold, illus. 136 - - =Complement of an Arc= 99 - - =Composition=, Section-Lining, illus. 182 - - =Concave=, def. 44 - - =Cone=, def. 44 - - =Conjugate Axis=, def. 96 - - =Construction=, def. 44 - Line, def. 46 - - =Contents=, Table of 24 - - =Contour=, def. 44 - - =Convergence=, def. 44 - - =Convex=, def. 44 - - =Copying Drawings=, Black Process 188 - Blue Printing 186 - Tracing 184 - - =Copyright= viii - - =Corner=, def. 44 - - =Corollary=, def. 83 - - =Co-secant of an Arc=, def. 99 - - =Co-sine of an Arc=, def. 99 - - =Co-tangent of an Arc=, def. 99 - - =Crane=, desc. 219 - Working drawing of, illus. 218 - - =Cross-Hatches=, def. 44 - - =Cross-hatching Drawings= 182 - - =Crown Wheel=, illus. 208 - - =Curve=, def. 44 - - =Curved Line=, def. 46 - - =Curved Lines=, Drawing, desc. 69 - Drawing Figures of 70, 72, 73 - How to Draw 35 - Illus. 70 - - =Curved Surface=, def. 49 - - =Curve or Scroll=, illus. 117 - - =Cut Gears=, desc. 215 - - =Cylinder=, def. 44 - - =Cylindrical=, def. 44 - - =Cylindrical Projection=, illus. 161 - - - =Dash Line=, def. 46 - - =Decimal Equivalents=, Table of 269 - - =Dedication= ix - - =Definitions=, Preliminary 41 - - =Degree=, def. 41, 44 - - =Describe=, def. 44 - - =Design=, def. 44 - Symmetry in, def. 50 - - =Designing Gears= 209 - - =Detail Drawings=, Office Rules for 191 - - =Develop=, def. 45 - - =Diagonal=, def. 45 - - =Diagonals of a Polygon=, def. 47 - - =Diameter=, def. 45 - - =Diameter=, Of a Circle 43 - - =Diameters of Wheels=, How to Measure 199 - - =Diametral Pitch=, desc. 200 - Of Gears 212 - - =Dimensioning Drawings= desc. 176, illus. 177, 179 - - =Dimension Line=, def. 46 - - =Dimensions on Drawings=, Office Rules for 192 - - =Dividers and Compasses=, desc. 119 - - =Dividers=, illus. 82, 119 - Bisecting, desc. 119 - Bow, illus. 122 - “Point” 242 - Proportional desc. 119, illus. 120 - Spring Bow, desc. 124 - - =Dodecahedron=, def. 48 - - =Dot-and-Dash Line=, def. 46 - - =Dotted Line=, def. 46 - - =Draughtsmen=, Useful Tables for 269-280 - - =Drawing=, Blackboard 28 - Free hand 55 - Linear Perspective 247 - Parallel Perspective 260-1 - Projection, desc. 148 - Scale Perspective 257 - Spur Gear, illus. 163, 211 - Spur Wheel, desc. 162 - Straight Line Figures desc. 65, illus. 66-7-8 - Symbols 193 - The Pitch Line 209 - To Scale, desc. 126 - Working, def. 51 - - =Drawing-Board= desc. 107, illus. 102, 106, 108 - Expansion and Contraction of 108 - How to Construct 107 - Trestles, illus. 110, 111 - - =Drawing Instruments=, desc. 103 - How to Select 104 - Illus. 105 - Outfit Recommended 104 - - =Drawing Materials=, desc. 103 - - =Drawing Office Rules= 191 - - =Drawing Paper=, desc. 132 - Erasing Lines 140 - Fixing of, illus. 102 - Fixing on the Board 139 - Pasting 184 - “Points” 243 - Patent Office Sizes 234 - Sizes of 133 - - =Drawing Pencils=, desc. and illus. 118 - How to Use 56 - - =Drawing Pen=, Filling with ink, illus. 130 - - =Drawing Pens=, desc. and illus. 129 - - =Drawing-Pins=, desc. 110 - - =Drawing Scales=, desc. 126 - - =Drawings=, Cleaning, desc. 169 - Color and Finish, note 185 - Dimensioning, desc. 176 - Inking in, desc. 167 - Lettering, desc. 171 - Patent Office Rules for 233 - Reproducing 186 - Section-Lining, desc. 182 - Shading desc. 180, illus. 180, 225 - Size of, Office Rules 191 - Tint and Color, desc. 184 - - =Drawing-Table=, desc. 111 - Folding Legs, illus. 104 - - - =Edge=, def. 45 - - =Elevation=, def. 45 - - =Elevation and Section=, Spur Wheel desc. 216, illus. 212 - - =Ellipse, An=, desc. 96, 118 - To describe when length and breadth are given 96 - - =Envelopes=, Portfolio, desc. 142 - - =Equiangular Triangle=, def. 45 - - =Equilateral Triangle=, def. 50 - - =Erasing=, “Points” on 242 - - =Eye=, illus. and desc. 253 - Effect of Light on 253 - - - =Face=, def. 45 - - =Faced Surfaces=, Points, etc. 242 - - =Figures=, Drawing Straight-line 65 - Numerals, examples of 173 - Straight-line, illus. 66 - - =File Handle=, desc. and illus. 144 - - =Finger and Thumb Lines=, illus. 64 - - =Finished Surfaces=, How to indicate 193 - “Points” 242 - - =Finishing=, def. 45 - - =Flanged-tooth Wheel=, desc. 205 - - =Flat Pattern=, def. 47 - - =Foreshortening=, def. 45 - - =Forward Projection= desc. 151, 152, illus. 155 - - =Free-hand=, def. 45 - - =Free-hand Drawing= 55, 78 - First Lesson in 30 - Penciling, illus. 54 - - =Free-hand Illustration=, a Water Wheel 78 - - =Friction-Clutch and Pulley=, desc. 224 - - =Friction Gear-Wheels=, desc. 201 - - =Full Line=, def. 46 - - - =Gear=, def. 199 - - =Gearing=, desc. 199 - Drawing desc. 162, illus. 163 - - =Gearing and Design= 197 - - =Gears=, A Train of, desc. 208 - Speed of 208 - - =Gear-Wheels=, Spur, illus. 198 - - =Generated=, def. 45 - - =Geometric=, def. 45 - - =Geometrical Axioms= 84 - - =Geometrical Drawing=, desc. 81 - Problems in 86 - - =Geometrical Signs= 84 - - =Geometry=, defs. 249 - Elements of 81 - - =Gothic Letters=, illus. 173 - - =Grooved Friction Wheels=, desc. 201 - - =Ground-Plane= def. 249, 255, illus. 249 - - - =Half-Tint=, def. 45 - - =Hand-Wheel=, illus. 77 - - =Hanger=, illus. 179 - - =Helical Wheel= desc. 199, illus. 207 - - =Hemisphere=, def. 45 - - =Heptagon=, def. 47 - - =Hexagon=, def. 47 - To Construct a 95 - - =Hexagon-Head Bolt=, illus. 144 - - =Hexahedron=, def. 48 - - =Horizontal=, def. 45, 250 - - =Horizontal Line=, in Drawing, illus. 250 - How to Draw 32 - In Perspective 252 - - =Hypothesis=, def. 83 - - - =Icosahedron=, def. 48 - - =India Ink=, desc. and illus. 131 - Dish or Tile, illus. 131 - - =India Rubber Eraser=, illus. 132 - Use of 240 - - =Ink=, Preparing, for Drawings, desc. 167 - Test for Good, desc. 167 - - =Ink Eraser=, Steel, desc. and illus. 132 - - =Inking=, illus. 166, 169 - Long Lines, illus. 170 - Rules of Procedure 168 - Short Work, illus. 169 - - “=Inking in=” Drawings, desc. 167 - “Points,” etc. 239, 243 - Patent Office Drawings 234 - - =Inscribe=, def. 45 - - =Instrumental=, def. 45 - - =Instruments= for Chalk-work, illus. 29 - - =Internal Gear=, desc. 201 - - =Internal-Gear Wheel=, illus. 207 - - =Introduction= 15 - - =Isosceles Triangle=, def. 50 - - - =Lantern-Wheel=, desc. 201 - - =Lathe-Dog=, illus. 76 - - =Lemma=, def. 83 - - =Lettering=, Blow-Off Valve, illus. 175 - Drawings, desc. 171 - - =Light=, and Sense of Seeing 254 - Experiment with, illus. 255 - Laws of 252 - - =Line=, def. 82 - To Divide into Equal Parts 89 - - =Linear Perspective Drawing= 247 - - =Lines=, def. 45 - Note 65 - Parallel, def. 83 - - =Link Motion=, Stephenson’s, illus. 225 - - =Longitudinal=, def. 46 - - =Lucas, Theo.=, Acknowledgement 23, 266 - - - =Marking Measurements= on Drawings 243 - - =Mechanical Drawing= 137, 247 - Elevation 138 - Examples 143 - Procedure 141 - - =Minerva=, Free-hand Sketch of iii - - =Miter-Wheel= desc. 199, 204, illus. 204 - - =Model=, def. 46 - - - =Nonagon=, def. 47 - - =Numbering= Drawings, Office Rules for 194-5 - - =Numerals=, illus. 173 - - - =Oblique=, def. 46 - - =Oblique Lines= How to Draw, 32, 58, illus. 59, 63 - - =Oblong=, def. 46 - - =Obtuse Angle=, def. 42 - - =Obtuse-Angled Triangle=, def. 50 - - =Octahedron=, def. 48 - - =Octagon=, def. 47 - To Describe on a given straight line 95 - To Inscribe in a circle 96 - - =Oil Can=, illus. 77 - - =Optic Nerve=, illus. 253 - - =Optics=, in Drawing, def. 252 - - =Outline Picture=, desc. 152 - - =Oval=, def. 46 - How to Draw an 71 - - =Overall=, def. 46 - - - =Paper=, Fastening Drawing on 184 - Rule or Scale 242 - Sensitized, desc. 188 - The Right Side of 241 - - =Parallel=, def. 46 - - =Parallel Lines=, def. 83, 252 - To Draw 88 - In Perspective 251 - - =Parallel Perspective Drawing=, illus. 260 to 263 - - =Parallel Rule= desc. 115, illus. 114-5 - - =Parallelogram=, def. 47 - To Construct 93 - - =Pasting Drawing Paper=, “Points” 243 - - =Patent Office Drawings= 235, illus. 236 - Rules of Great Britain 236 - Rules of U. S. 233 - - =Patterns=, def. 46 - Numbering, from drawings 194 - - =Pedestal=, illus. 178-9 - - =Pen=, Hand holding, illus. 166, 169, 170 - - =Pencil=, Function of a 142 - How to Cut, illus. 57 - How to Hold, illus. 54, 59, 60 - How to Use 56 - - =Pencil-Compasses=, desc. 143 - How to Hold 143 - - =Penciling=, desc. and illus. 139 - “Points” 143, 240 - - =Pencil Lines=, How to Make 119 - - =Pencils=, Sharpening Points of 143, 241 - - =Penknife=, illus. 52 - - =Pens=, Drawing, illus. 129 - Lettering and Figuring, illus. 172 - - =Pentagon=, def. 47 - To Inscribe in a circle 94 - - =Perimeter=, def. 47 - Of a Polygon, def. 47 - - =Periphery of a Wheel=, def. 200 - - =Perpendicular=, def. 47, 252 - - =Perpendicular Lines=, How to Draw 31, 61, 62 - - =Perrott, Geo.=, Acknowledgement 23, 266 - - =Personal=, Note from the Author 266 - - =Perspective Drawing= def. 47, illus. 250 - Definitions of Terms used in 249 - Geometrical Terms used in 248 - Of a Bridge, illus. 249 - Scale or Approximate 257 - Vanishing Point, illus. 251 - Water Reflection, illus. 246 - - =Perspective=, Linear, def. 247 - - =Perspective Plane=, def. 250 - - =Picture Plane=, desc. 255 - In Drawing, illus. 249, 250 - In Perspective, illus. 258 - - =Pinion-Wheel=, desc. 201 - - =Piston Rod=, desc. 223 - - =Pitch Circle= in Gearing 200 - - =Pitch Line=, Drawing the 209 - In Gearing 200 - - =Plan=, def. 47 - - =Plan of the Work= 21 - - =Plane=, in Perspective, def. 249 - - =Plane of the Picture=, def. 250 - - =Plane Surface=, def. 49 - - =Point=, def. 82 - - =Point of Sight=, in Perspective Drawing 251 - - =Points=, relating to Chalk Drawings 36 - To be observed in Sketching 141-2 - Useful Hints and 170, 239 - - =Polygon=, def. 47 - - =Polyhedron=, def. 48 - - =Postulate=, def. 83 - - =Pounce=, How to Use, desc. 170, 186 - - =Preface= 13 - - =Preparatory Practice= in Drawing 30 - - =Prism=, def. 49 - - =Problems=, def. 83 - Geometrical 86 - - =Produce=, def. 49 - - =Profile=, def. 49 - - =Projection=, Backward, illus. 153-4 - Def. 49 - Cylindrical Outline, desc. 159 - Cylindrical Surface, illus. 161 - Forward desc. 152, illus. 155 - Hexagon Nut, illus. 160 - Marking Dimensions 150 - Lines of Sight 150 - Scaling and Measuring in 149 - Sight-Lines, illus. 151 - Sloping Surface, desc. 159 - Spur-Wheel, desc. 162 - - =Projection Drawing=, desc. 148 - Illus. 149, 151, 153, 154, 155, 157, 158, 160, 161, 163 - Principles of 148 - Spur-Wheel, illus. 163, 212 - - =Proportional Dividers=, illus. 120 - - =Proportions= of Teeth of Wheels 210 - - =Proposition=, def. 83 - - =Protractor=, desc. 128 - - =Pulley and Friction Clutch=, illus. 224 - - =Punching Press= illus. 222, desc. 223 - - - =Quadrant=, def. 49 - - =Quadrilateral=, def. 47 - - =Quadrisect=, def. 49 - - - =Rack and Pinion=, desc. 205 - - =Radius of a Circle= 43 - - =Reading Working Drawings=, desc. 229 - - =Rectangle=, def. 48 - To Construct a 92 - - =Reproducing Drawings=, desc. 186 - - =Reverse Curve=, def. 44 - - =Rhomboid=, def. 48 - - =Rhombus=, def. 48 - - =Right-Angle=, def. 42 - Triangle, def. 50 - - =Robinson, A. W., M. E.=, note, Office Rules 191 - - =Rolling-Circle= in Gear-Wheel 200 - - =Rule=, Area of a Circle, To find the 271 - Circumference of a Circle, To find the 271 - Cylinder, To find the contents of a 271 - Diameter of a Circle, To find the 271 - Illus. 82 - Inscribed Square, To find side of an 271 - Parallelogram, To find the area of a 271 - Pyramid or Cone, To find the contents of a 271 - Radius of a circle, To find the 271 - Sphere, To find the contents of a 271 - Square, To find the side of an equal 271 - Trapezoid, To find the area of a 271 - Triangle, To find the area of a 271 - Two-foot desc. 127, 128, illus. 128 - Wedge, To find the contents of a 271 - - =Rules= for Drawing Office 191 - Wheels, To find proportions of 209 - - - =Sand Paper=, Removing surface of paper 168 - - =Scale=, Drawing to, desc. 126 - - =Scale Drawings=, To read 230 - - =Scale=, Flat, illus. 127 - Triangular, illus. 127 - - =Scalene Triangle=, def. 50 - - =Scale or Approximate Perspective= 257 - - =Scale Rule=, for Proportions of Teeth 214 - - =Scales=, Drawing, desc. 126 - - =Scholium=, def. 83 - - =Screw-Thread=, U. S. Standard Gauge 280 - - =Scroll or Curve=, desc. 117 - Universal Curve, illus. 134 - - =Secant of an Arc= 99 - - =Section=, def. 49 - Drawing illus. 163, desc. 160 - - =Sectional=, def. 49 - - =Section-Liner=, illus. 116 - - =Section Lines= in Drawings 240 - - =Section-Lining=, Cast-iron, etc., illus. 182 - Cog-Wheel, illus. 183 - - =Section-Lining Drawings= desc. 182, illus. 183 - Wheel Hub, illus. 184 - - =Selecting Drawing Instruments= 103, 240 - - =Semi-Circle=, def. 43 - - =Sensitized Paper=, desc. 188 - - =Set-Square=, illus. 102, 114 - - =Shading Curves=, illus. 181 - - =Shading Drawings= 180 - - =Shadow=, def. 49 - - =Shadow Line=, def. 46 - - =Shadow Lines=, “Points” 241 - - =Sharpening Pencils= 57 - - =Shop Drawings=, Office Rules 192 - - =Sight=, Point of, in Perspective Drawing 252 - Sense of 254 - - =Sight Lines= in Projection, illus. 157 - - =Sine= of an Arc 99 - - =Skew-Gearing=, desc. 203 - - =Sketch Books=, desc. 142 - Office Practice 194 - - =Sketches=, details 142 - - =Sketch=, Free-hand, Advantage of 55 - - =Sketching= desc. 141, note 17 - Points to be observed in 141-2 - - =Sloping Surface= in Projection desc. 159, illus. 166 - - =Solid=, def. 49, 83 - - =Solid Pattern=, def. 47 - - =Speed of Gears= 208 - - =Sphere=, def. 49 - - =Spiral Curve=, def. 44 - - =Spring Bows= illus. 122, 123, desc. 124 - - =Spur-Gear= desc. 201, illus. 211 - - =Spur Mortise-Wheel=, illus. 202 - - =Spur-Wheel=, desc. 199, 216 - How to Draw 209 - Illus. 198 - Projection desc. 162, illus. 163 - Teeth of cast-iron, desc. 202 - - =Square=, def. 48 - To Convert into an Octagon 95 - To Describe about a Circle 94 - To Inscribe in a Circle 93 - - =Standards=, for U. S. 278 - Office Rules for 193 - - =Steady Rest=, Scale, illus. 228 - - =Straight Line=, def. 46 - To Draw a Perpendicular to a 86 - Drawing Figures of 66-7-8 - How to Draw a 57, 59 - To Bisect a 86 - - =Steel=, Section-Lining, illus. 182 - - =Steel Gears=, Economy of 216 - - =Surface=, def. 49, 82 - - =Symbols=, representing Materials 193 - - - =Table= of Areas of Circles 273-277 - Bolts, Standard Sizes of 280 - Circular Measure 272 - Circumferences of Circles 273-277 - Contents 24 - Decimal Equivalents 269 - Diameters of Circles 273-277 - Land Measure 272 - Metric Equivalents 270 - Pipe, Standard Sizes of Welded 279 - Roman Figures 272 - Solid Measure 272 - Wire Gauges 278 - Useful for Draughtsmen 266-280 - - =Tangent= of an Arc, def. 99 - To Draw a, to a Circle 91 - - =Tee-Iron=, illus. 75 - - =Tee-Square=, Adjustable, desc. and illus. 113 - Desc. 112 - How to Use, illus. 111, 147 - Illus. 102, 112 - Points about 242 - - =Teeth in Bevel Gears=, desc. 204 - - =Terms and Definitions=, Preliminary 41 - - =Test-Pieces=, Use of, in Blue Prints 188 - - =Tetrahedron=, def. 48 - - =Theorem=, def. 83 - - =Thumb-Tack= illus. 109, desc. 110 - - =Tints and Colors=, desc. 184 - - =Title and Date=, on Sketches 142 - - =Title, Date, Scale, etc.=, Office Rules for 192 - - =Title Page= vii - - =Tracing-Cloth=, desc. 186 - - =Tracings=, Office Rules for Keeping 195 - Tinted and Shaded, desc. 184 - - =Trammels=, desc. and illus. 125 - - =Trapezium=, def. 48 - - =Trapezoid=, def. 48 - - =Trestles=, Drawing-Board, desc. and illus. 110 - - =Triangle=, def. 50 - To Describe a Circle about a 93 - - =Triangle, or Set-Square=, desc. and illus. 114 - How to Use desc. 147, illus. 166 - - =Triangles=, illus. 102 - To Construct 92 - - =Trigonometry=, principles and def. 98 - - =Trisect=, def. 50 - - =Trundle Wheel= desc. 201, illus. 208 - - =Two-foot Rule=, illus. 128 - - - =U. S. Standards=, Pipes 279 - - =Upright Lines=, How to Draw 61 - - =Useful Tables for Draughtsmen= 269-280 - - - =Valve=, Blow-off desc. 174, illus. 175 - - =Valve Gear=, or Link Motion, illus. 225 - - =Vanishing Point=, def. and illus. 251 - In Drawing, illus. 250 - In Perspective Drawing 252 - - =Varnish=, Shellac, for Drawings 241 - - =Versed Sine= of an Arc 99 - - =Vertex=, def. 51 - Of an Angle, def. 42 - - =Vertical=, def. 51, 251 - - =Vertical Lines=, in Perspective 257 - - =Vertical-Plane=, def. 249 - - =View=, def. 51 - - =Visual Ray= in Perspective Drawing 256 - - =Visual Rays of Light= desc. 252, illus. 255 - - =Vulcanite=, Section-Lining, illus. 182 - - - =Water Reflection= in Perspective Drawing, illus. 246 - - =Wheel=, Bevel, A, illus. 203 - Crown, illus. 208 - Helical, A, illus. 207 - Hub, Section-Lining, illus. 184 - Internal Gear, desc. 201, 207 - Miter, A, illus. 204 - Proportions of the Teeth of a 210 - Spur illus. 198, desc. 199, 201 - Worm, desc. 201 - - =Wood=, Section-Lining, illus. 182 - - =Working Drawing=, Bending Machine 220-1 - Def. 51 - Illus. 218, 220, 221, 222, 223, 224, 225, 226, 228 - Bumping Post, illus. 226 - Points, etc. 243 - Power Punching Press 222 - Numbering a, Office Rules for 195 - To Read a 229 - - =Working Drawings=, desc. 219, 227 - - =Worm-Gear=, desc. 206 - - =Worm-Wheel= desc. 201, illus. 206 - Speed of a 216 - - =Wrench=, illus. 75 - - =Wrist-Lines=, illus. 64 - - =Wrought-Iron=, Section-Lining, illus. 182 - - - - -Transcriber’s Notes - - - Inconsistent and unusual spelling, capitalisation and hyphenation have - been retained, unless mentioned below. The discrepancies between the - General List of Contents and the structure of the text have not been - corrected. - - Depending on the hard- and software used to read this text and their - settings, not all elements may display as intended. Where scales are - given for illustrations, these can, of course, not be relied upon for - measurements. - - The numbering of illustrations in the book is inconsistent. - Illustrations that have no numbered captions may or may not be - counted; this has not been standardised, except as listed below. - Missing illustration numbers have been added only where they are - referenced in the text. The illustrations referred to in the text as - Figs. 242 and 243 are captioned Fig. 243 and 244. - - Page 65-69 and Figs. 66-71: the text uses upper case reference - letters, the illustrations lower case, which has not been - standardised. - - Page 84, table of symbols: the source document uses letters (O and L) - rather than symbols for circle and angle. - - Page 94, Bisect two of the angles _A C_ of the triangle: presumably - the unmarked point A is in the lower left-hand corner of the - illustration. - - Page 113, The tee-square, as shown, ...: there is no illustration with - the relevant reference numbers; probably the reference is to Fig. - 151/152. - - Page 239, rough irregular lines, as in fig. 134, on page 131: there is - no Fig. 134 on page 131; page 134 nor page 131 show the rough - irregular lines. Several other illustrations in the book do show such - lines. - - Index: the order of entries is as printed in the source document. - - - Changes made: - - Illustrations and Notes have been moved out of text paragraphs. - - Some obvious minor punctuation and typographical errors have been - corrected silently. - - In some tables the ditto symbol („) has been replaced with the dittoed - text; some tables have been split or re-arranged. The use of × and x - in multiplications and dimensions has been standardised to ×. - - Page 48: Figure numbers 45-49 added to illustration. - - Page 89: and on _B_ on _B I_ changed to and on _B_ on _B D_. - - Page 177: D = diam. of collar changed to d = diam. of collar. - - Page 193: Bbt.--Babbit. changed to Bbt.--Babbitt. - - Page 203: as described on page 191 changed to as described on page - 199. - - Page 214: 4 tenths measured at the 2′ line changed to 4 tenths - measured at the 2″ line. - -*** END OF THE PROJECT GUTENBERG EBOOK SELF-HELP MECHANICAL -DRAWING *** - -Updated editions will replace the previous one--the old editions will -be renamed. - -Creating the works from print editions not protected by U.S. copyright -law means that no one owns a United States copyright in these works, -so the Foundation (and you!) can copy and distribute it in the -United States without permission and without paying copyright -royalties. 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