LIBRARY OF CONGRESS. 
 
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 UNITED STATES OF AMERICA. 
 
NATIONAL DRAWING COURSE. 
 
 TEXT-BOOKS. 
 
 Free-Hand Drawing. 
 
 Mechanical Drawing. 
 
 Color Study. 
 
 Light and Shade. (/n preparation.) 
 
 Historic Ornament and Design. (/n preparation.) 
 
 TEACHERS' MANUALS. 
 Outline of Drawing Lessons for Prinnary Grades. 
 Outline of Drawing Lessons for Gramnnar Grades. 
 
 DRAWING CARDS. 
 National Drawing Cards for Primary Grades. 
 
 DRAWING BOOKS. 
 One book each for the 4th, 5th, 6th, 7th, and 8th 
 years of school. 
 
 SPECIAL MATERIAL FOR THE NATIONAL 
 DRAWING COURSE. 
 The Cross Transparent Drawing Slate.^ 
 The Cross Pencil for use with the slate. 
 The National Drawing Models. 
 The National Model Support or Desk Easel. 
 

 Mechanical Drawing 
 
 A MANUAL FOR 
 
 TEACHERS AND STUDENTS 
 
 BY 
 
 ANSON K. CROSS 
 
 Instructor in the Massachusetts Normal Art School, and in the School of Drawing and 
 Painting, Museum of Fine Arts, Boston. Author of '''' Free-H and Drawing, 
 Light and Shade, and Free-Hand Perspective,'''' and a Series of '~ " 
 Text and Drawing Books for the Public Schools. 
 
 '^^^\v^ 
 
 SEP 95 IMS 
 
 B05T0N, U.S.A. 
 
 GINN & COMPANY, PUBLISHERS 
 
 1895 
 
 Of ^ 
 
 l^ 
 
< '^';>^ 
 
 Copyright, 1895 
 By ANSON K. CROSS 
 
 ALL RIGHTS RESERVED 
 
PREFACE. 
 
 The following notes are intended for students, and for 
 teachers of elementary work, particularly for public school 
 teachers. There are many books on the subjects of projec- 
 tion and working drawings, but none which present the prin- 
 ciples in ways suited to the needs of the large number of 
 teachers who are required to give instruction in these subjects 
 in the public schools. Most of these teachers have had little 
 instruction in the subjects and frequently do not understand 
 problems which they are expected to explain. This occurs 
 because too difficult work is often planned for their grades, and 
 also because the instruction which many public school teachers 
 have received has been so advanced and theoretical that the 
 simple principles which alone are necessary for elementary 
 work have been lost in the attempt to understand descriptive 
 geometry and the drawings of machine and other details, whose 
 nature and use are often not known. 
 
 To understand descriptive geometry certain qualities of mind 
 are absolutely necessary, and many find it impossible to com- 
 prehend even the simpler problems of this subject. The 
 draughtsman or drawing teacher of advanced work who is 
 without practical knowledge of the subject of descriptive 
 geometry is very poorly equipped for his duties ; but this 
 knowledge is not necessary for the public school teacher, who 
 will find it best to treat the subject of working drawings in a 
 much simpler way. 
 
 This book presents principles and not a graded course of 
 lessons. It covers more than many teachers may require, 
 
IV PREFACE. 
 
 though special students or classes of the high school may study 
 work as advanced as any of that presented. Teachers of draw- 
 ing in the high school should understand all the problems of 
 the book. 
 
 It is hoped that the book may assist teachers of both gram- 
 mar and high schools so to understand the subject that they 
 may give to classes instruction suited to their capacity and 
 needs. 
 
 Anson K. Cross. 
 
CONTENTS. 
 
 PAGE 
 
 CHAPTER I. — Materials and their Uses i 
 
 Paper . . . . . . i 
 
 Drawing Boards . . .2 
 
 Pencils , , , , 3 
 
 T-Square = . . 4 
 
 Triangles ....♦...,,.. 4 
 
 Compasses 5 
 
 Dividers 7 
 
 Needle , . 7 
 
 Scales 8 
 
 French Curves 9 
 
 Penciling . 10 
 
 CHAPTER II. — Geometrical Problems 12 
 
 CHAPTER HI. — Working Drawings 28 
 
 Nature and Use of Working Drawings 28 
 
 Study of Principles 32 
 
 Making Working Drawings 35 
 
 Views of a Circular Plinth 36 
 
 Views of an Hexagonal Plinth ^7 
 
 Views of a Plinth and Disc • 39 
 
 Views of a Box and Pyramid 40 
 
 Dimensioning • . . 43 
 
 Lettering 45 
 
 CHAPTER IV. — Developments 47 
 
 The Cube 47 
 
 Prisms 49 
 
 The Cylinder 49 
 
 The Cone 50 
 
 Pyramids • . . . .51 
 
 The Sphere t;i 
 
VI CONTENTS. 
 
 PAGE 
 
 CHAPTER v. — Shadow Lines ........ 54 
 
 CHAPTER VI. — Inking 58 
 
 India Ink . . . ' 58 
 
 Inking a Drawing 59 
 
 Blue Prints 61 
 
 Erasing and Cleaning 61 
 
 Sharpening the Pen 62 
 
 Stretching Paper 62 
 
 CHAPTER VII. — Machine Sketching and Drawling . . .64 
 
 CHAPTER VIII. — Orthographic Projection 67 
 
 Projection Principles 70 
 
 Axes of Projection 71 
 
 Views of a Point 71 
 
 Position of a Point 72 
 
 Views of a Straight Line 74 
 
 Views of a Plane Surface . 75 
 
 Views of a Solid 76 
 
 Views of a Rectangular Card 77 
 
 Views of a Pyramid 78 
 
 True Length and Position of a Straight Line ... 81 
 
 Statements of Principles 83 
 
 Projection Problems . ^ 86 
 
 CHAPTER IX. — Sections 89 
 
 Sections of the Sphere 90 
 
 Sections of the Cube 90 
 
 Sections of the Cylinder 92 
 
 Sections of the Pyramid 93 
 
 Sections of the Cone 94 
 
 CHAPTER X. — Intersections 96 
 
 Intersections of a Line and a Plane Surface ... 97 
 Intersections of a Line and a Curved Surface . . -99 
 
 Intersections of Solids 102 
 
 CHAPTER XI. — Arrangement and Names of Views . . . 104 
 
 CHAPTER XII. — Plates and Explanations .... 109 
 
 DEFINITIONS ... 184 
 
MECHANICAL DRAWING, 
 
 CHAPTER I. 
 MATERIALS AND THEIR USES. 
 
 1. Good work cannot be done without good instruments. The 
 best work cannot be done without steel T-squares and triangles, 
 steel-edged drawing boards, and drawing instruments of the best 
 make. Students of art and technical schools should provide them- 
 selves with the best instruments. 
 
 In the public schools, no more can be done than to give a little 
 knowledge of the principles of instrumental drawing, which will be 
 valuabk to all. To do the best work technically is impossible, 
 because the pupils are too young, because they do not have the time 
 necessary for practice, and because they do not have the materials 
 necessary to produce the best work. Any one of these reasons is 
 sufficient to prevent the making of first-class drawings, and together 
 they make it impossible for us to expect that perfect results can be 
 obtained. 
 
 Although pupils of the public schools labor under the above-men- 
 tioned disadvantages, it will not do for the teacher to expect bad 
 work, or to be satisfied with work which at first glance is seen to be 
 inaccurate ; for even with the imperfect materials provided it is pos- 
 sible, by careful work, to obtain drawings which are neat and suffi- 
 ciently accurate for the requirements of simple working drawings. 
 
 2. Paper. — The paper must be tough and should have a surface 
 which is not easily changed or roughened by erasing lines drawn 
 upon it. This is most important when drawings are to be inked. 
 For free-hand sketching a soft paper is best ; but for all mechanical 
 work, the paper should be hard and strong. 
 
 For pencil drawings a paper which is not smoothly calendered 
 is best, because the pencil marks more readily upon an unpolished 
 paper, and because its surface will not show erasures as quickly as 
 
2 MECHANICAL DRAWING. 
 
 that of a smooth paper. For public school use, several kinds of 
 cheap paper, which are good enough for the work, may be obtained 
 both in sheets, in block form, and also made up in blank books. 
 
 Whatman's paper is the best for drawings which are to be inked. 
 There are two grades, hot and cold pressed, suitable for this use ; 
 the cold-pressed having the rougher surface. If the paper is not 
 to be stretched, the cold-pressed is preferable, as its surface show^s 
 erasures less than that of the hot-pressed. The side from which the 
 water-marked name is read is the right side, but there is little differ- 
 ence between the two sides of hot and cold pressed papers. Stretch- 
 ing the paper is unnecessary except when colors are to be applied by 
 the brush, or when the most perfect inked drawing is desired. 
 
 The use, in the public schools below the high, of sheets of blank 
 paper instead of blank or other drawing books has its advantages 
 and its disadvantages. If a drawing upon a sheet of paper is spoiled, 
 it may be thrown away and another begun upon a new sheet ; but 
 this fact tends to careless work. To the teacher loose sheets of 
 paper are a source of great care, even if kept in portfolios or large 
 envelopes and retained by the pupils, for the drawings must be 
 examined, and it is not easy to keep them arranged in the order in 
 which they are made. 
 
 Blank books cost little more than paper ; and their use tends to 
 neatness and care on the part of the pupils, each of whom is inter- 
 ested to produce the best book. Drawings in the books are always 
 arranged in order and ready for examination. The chief objection 
 to their use is that they cannot be handled as a block for free-hand 
 purposes, or be used with a T-square for instrumental work. This 
 difficulty is avoided by fastening the book, by means of two rubber 
 bands, to a drawing board made to receive it. When fastened to 
 this board the book may be used for instrumental drawing as advan- 
 tageously as paper upon a board. Used in this way a book is prefer- 
 able to sheets of paper ; therefore a board should be provided 
 whenever books are used, whether the books are blank books or 
 those of any system. 
 
 3. Drawing Boards. — Drawing boards should be made of clear 
 white pine, and should not be painted or varnished; they should 
 have cleats upon the back, so that upon the whole working surface 
 
MATERIALS AND THEIR USES. 
 
 of the board the grain of the wood runs in one direction ; for 
 when paper is stretched upon cleats which are fastened at the ends 
 of the board, it is often spoiled by the swelling of the board, which 
 moves back and forth upon the cleats. 
 
 The cleats should not be glued or otherwise firmly secured to the 
 board, since the board must change its width with the weather, and 
 if the cleats are firmly secured to it, the board will split or warp. 
 The cleats should be fastened by means of screws and washers, the 
 screws being placed in slots, which allow the board to move. This 
 construction is not necessary for small boards required in the gram- 
 mar schools. 
 
 The best board for use with a drawing book is one provided with 
 a groove to receive the head of the T-square, which should be thick 
 enough to remain in the groove when the book is between the square 
 and the board. 
 
 4. Pencils. — To do good instrumental work two grades of 
 pencils should be used, a hard one for the fine working lines and a 
 softer one for the result lines. The hard pencil 
 should be sharpened to the wedge-shaped point 
 illustrated, which is as thin as possible one way, 
 and about one-half the width of the lead the other. 
 
 The softer pencil should have a point of the 
 same shape but thicker, so as to give the width 
 required for the result line. 
 
 To sharpen the pencil a knife should be used 
 to give the wood the flattened form required in 
 the lead. The lead should then be worked down 
 by the use of a fine file. A substitute for the 
 file, which may be used in the public schools, 
 is given by gluing a straight-edged piece of sand- 
 paper to a strip of wood. A point with which good work can be 
 done cannot be obtained by the use of the knife alone. 
 
 A pencil with a round point should be used for all free-hand 
 lettering and figuring, and for other work of a free-hand nature, such 
 as the drawing of irregular curves. 
 
 The hard pencil should be used very lightly, as much pressure 
 will indent the paper so that its marks cannot be removed. 
 
4 MECHANICAL DRAWING. 
 
 All lines should be drawn with the pencil slightly inclined in the 
 direction in which it is moved. The pencil should not be moved in 
 the opposite direction, as it will then act as a plough to tear the sur- 
 face of the paper. 
 
 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 Avorking 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, and may not 
 require retouching on account of use of the eraser. 
 
 5. T-Square. — A T-square is an instrument used in connection 
 with a drawing board, for drawing straight lines ; it consists of two 
 parts, a blade and a head, which are secured at right angles to each 
 other. The blade of the T-square should be placed upon the head, 
 which should never be cut to receive it, and should be secured to it 
 by means of screws. This construction allows the parts to be sepa- 
 rated for straightening, and the triangles to be moved across the head, 
 which is often desirable, but cannot be done when the blade is set 
 into the head. 
 
 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 the board are seldom exactly 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 the 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. 
 
 6. Triangles. — The usual forms are illustrated. The 45° tri- 
 angle 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 mav be drawn. 
 
MATERIALS AND THEIR USES. 
 
 Lines parallel to any given line, AB, may be drawn by first plac- 
 ing together two triangles (C and D), so that a side of one (C) coin- 
 cides with AB, and then sliding triangle C upon triangle D, being 
 careful not to allow D to move. 
 
 Any parallel lines are most conveniently drawn by sliding one 
 triangle upon the other, or upon the T-square. 
 
 Lines perpendicular to AB may be drawn by placing either tri- 
 angle upon the other, so that its hypotenuse coincides with AB, 
 and then revolving the triangle through an angle of 90°, into the 
 position illustrated by the dot and dash lines. 
 
 Lines at 30°, 45°, and 60° with AB, may be drawn by placing one 
 triangle upon the other, so that an edge coincides with AB, and then 
 reversing the triangle, as will be shown by experiments. 
 
 When paper is to be cut by the use of a knife and straight edge, 
 the T-square or triangle should not serve as the straight edge, for 
 its edges would soon be nicked and spoiled by the knife. 
 
 7. Compasses. — Compasses suitable for any use 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. 
 
 The joint at the head of the compasses 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 
 
6 MECHANICAL DRAWING. 
 
 of adjustment, as the parts soon wear, and the joint then becomes 
 too loose. The joint should never be so tight that much pressure is 
 required to move the legs. 
 
 One leg of compasses is usually provided with a socket to which 
 are fitted three points ; a divider point, a pencil point, and a point 
 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, 
 and will be all that is needed for public school work. The pen point 
 is not necessary for grammar school use, as inking should not be 
 attempted before the high school. 
 
 The cheaper grades of compasses are provided with sharp, pointed, 
 solid legs, and are very objectionable, because these points work into 
 the paper as the compasses revolve and make large holes which spoil 
 the drawing. Even for public school work, compasses should have 
 a needle point with a shoulder which prevents the making of holes 
 in the paper. It may not be possible to provide for the 
 grammar schools compasses with jointed legs ; but the 
 needle point with the shoulder must be insisted upon, if 
 even fair work is desired. 
 
 The compasses should be held very lightly between the 
 thumb and forefinger, and should be inclined slightly in the 
 direction in which the line is drawn. No more pressure should 
 be applied than is necessary to obtain the line. In inking, little more 
 than the weight of the compasses is needed ; but in penciling more 
 pressure will be required for result lines. Two grades of pencils are 
 more necessary for use in the compasses than for straight line work ; 
 they should be sharpened as shown on page 3, and so that the wide 
 side of the point is at right angles to a line extending from the 
 pencil point to the needle point. As this is difficult to do, pupils 
 below the high school may use a sharp conical point. 
 
MATERIALS AND THEIR USES. 7 
 
 The legs of the compasses should be moved, to give, any desired 
 radius, by taking hold of each leg. The needle point may be placed 
 on one point of the drawing while this is done ; but if the strain due 
 to changing the radius is brought upon the needle point, it will tear 
 the paper and spoil the drawing. 
 
 8. Dividers. — The compasses are changed into dividers as 
 already explained. To set off equal spaces on any line, hold the 
 dividers lightly between the thumb and forefinger and place one 
 divider point at any desired point in the line to be spaced ; then 
 revolve the dividers about this point as centre until the other divider 
 point comes to the line, when the dividers are to be revolved about 
 it as about the first point. Revolve the divider points first on one 
 side and then on the other of the line to be spaced, and never lift 
 both points from the paper at the same time. Continue thus until 
 the line is spaced. 
 
 To divide a line of given length into any number of equal parts, 
 by use of the dividers, it will be necessary to use so little pressure 
 that visible punctures are not made until the correct space has been 
 obtained; if this is not done in two or three trials a second line 
 should be drawn and the spacing continued upon it. If one line is 
 gone over many times, the divider points will fall into the punctures 
 previously made, and accurate work will not be obtained. When 
 the correct space is found the points must be marked in the line so 
 that they can be readily seen. To do this the line should be gone 
 over several times, each time more pressure being applied to the 
 dividers, until the punctures become visible. If the pressure re- 
 quired to place the points is applied at once, equal spaces will not 
 be given, as the dividers will spring and move while marking the 
 points. 
 
 To do work of this nature easily, a pair of spring dividers should 
 be 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. 
 
 9, Needle.. — The needle may be used by the draughtsman and 
 by advanced pupils with great advantage as regards both accuracy 
 and speed. It should not, however, be used by young pupils, or 
 below the high school. 
 
8 MECHANICAL DRAJVIXG. 
 
 The needle may be used to set off distances from the scale and 
 to mark the intersections of lines ; it may also be used when lines 
 are to be drawn through two points, to hold the triangle so that only 
 one point requires the attention of the draughtsman. This is done 
 by placing the needle in one point, holding the triangle against it, 
 and revolving the triangle until it comes to the second point. 
 
 Points should never be marked by holes in or through the paper, 
 but by the smallest punctures which can be seen. These are much 
 more definite than pencil marks, and have the advantage of locating 
 points so that they are not lost by erasures. When a point is marked 
 in this way a small pencil circle should be drawn about it free-hand, 
 in order that its position may be readily seen. 
 
 The needle point may be placed in a handle of soft wood, and 
 should project just far enough to be used, but no farther, as acci- 
 dents will happen if it is not carefully handled. The best place for 
 the needle point is at the unsharpened end of a pencil. A double- 
 €nded pencil holder for round leads may have the pencil in one end 
 and the needle in the other. This holder should be about 4+" long, 
 so that it may be reversed readily. 
 
 10. Scales. — When objects are small they may be represented 
 full size ; but when large, the drawings must be smaller. Common 
 scales for mechanical drawings are \. \, -J, and ^j. full size. These 
 scales are often written 6"=i ft.; 3"=! ft.; ii-"=i ft. and f "= i ft. 
 
 Large objects are often drawn to very small scales ; in maps an 
 inch often represents many miles. 
 
 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 
 drawing to fill the space in the best way. 
 
 To determine the scale by which the drawing of any object may 
 be made of any desired size, divide the length of the object by the 
 length of the drawing desired. Thus, suppose an object 21" long is 
 to be represented in a space which will allow the drawing to be i o^" 
 long. The drawing must be half size, and may be made by measur- 
 ing the lines of the object and making those of the drawing half 
 as long. If the drawing can be but 7" long, it will be one-third 
 
MATERIALS AND THEIR USES. 9 
 
 size. To make the scale for this drawing, draw a Hne 4'' long and 
 divide it into twelve equal parts, which represent inches. Divide 
 one of these spaces into eight equal parts to represent eighths of 
 inches. By means of this scale, the drawing may be readily made 
 by taking from it the dimensions of the different parts. 
 
 If views are to be made of a mallet, the length of the front 
 view to be 6", while the actual length of the object is 15", a scale 
 from which the sizes of the different parts can be taken may be 
 made by drawing a line six inches long and dividing it into fifteen 
 equal parts, each part representing one inch. One space may be 
 divided into eight equal parts, and by means of this any part of an 
 inch may be obtained. 
 
 The triangular scale (architects') has upon it the following scales : 
 ^V 8"? Tfi-j i? f> i» 4? ^j ^2" ^^<i 3 inches. These are not provided and 
 are not necessary in the grammar schools, w^here a cheap foot rule 
 having three or four scales ranging from :|- to f full size will generally 
 be all that is required. If a drawing cannot be made full or half 
 size, or by one of these scales, a scale can be constructed by the 
 pupils, as explained. 
 
 When views of objects are to be made, they should be of such 
 size and so arranged as to produce a pleasing effect upon the sheet 
 or the page, and when none of the scales provided will give the best 
 size to the drawing, a special scale should be constructed. 
 
 II. Irregular or French Curves. — These curves are not re- 
 quired below the advanced work of the high school. They are used 
 for curves which cannot be drawn with the compasses. 
 To draw an ellipse or other curved line by their use, 
 care must be taken to have the French curve exactly 
 cover as many points in the line as possible, and then 
 only the central points should be connected. If the 
 line is drawn to all the points covered by the French 
 curve, it will generally fail to continue smoothly 
 through the points in the line beyond the curve. 
 
 When French curves are to be used the lines 
 should be very lightly sketched free-hand through the 
 determined points, before the curve is applied to the 
 drawing to obtain the final lines. 
 
lO MECHANICAL DRAWING. 
 
 12. Penciling. — Penciling is generally done to prepare for ink- 
 ing or for the making of tracings on tracing cloth, but sometimes 
 practical shop drawings are finished in pencil lines. In this case 
 the drawing is generally little more than a rough diagram, hastily 
 made, and not intended for continued use. 
 
 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, pupils should begin with 
 the pencil, and should not be allowed to use ink until they can pro- 
 duce satisfactory results in pencil. 
 
 The aim in pencil work should be to approach as nearly as pos- 
 sible the accuracy and decision which are only attainable in an ink 
 drawing. Care in selecting and using pencils, paper, and instru- 
 ments will produce pencil drawings which have firm, even, black 
 lines, that are very effective, if they are not quite so strong as those 
 made with ink. 
 
 It is customary to represent all visible edges by full lines, and all 
 invisible edges by dotted lines. These dotted lines should be as 
 strong as the full lines ; they should be regular and composed not of 
 dots, but of very short dashes, whose lengths are uniform and greater 
 than the spaces between them. Dotted lines are often very poorly 
 drawn, and spoil the effect of drawings otherwise good. The wadth 
 of the line and the length of the dash depend upon the size of 
 the drawing and its purpose. For the drawings which pupils should 
 make, the line given may serve as a copy. 
 
 Care must be taken to have all lines stop just where they are 
 intended to end. If lines do not quite meet, or if they pass by each 
 other, as illustrated, when this is not intended, the 
 drawing is spoiled. 
 
 The perfect union of lines which should be tangent 
 to each other is also necessary to good work. Straight 
 lines should run into arcs, and arcs into each other, without the 
 slightest suggestion of a break, or of two lines instead of one. 
 
 Centre lines are necessary in working drawings. In the study of 
 projection, it is not necessary to show them unless dimensions are 
 
MATERIALS AND THEIR USES. II 
 
 to be given. They are represented by dot and dash lines, as illus- 
 trated, and extend some slight distance outside the drawing, to show 
 
 that they are centre lines. Dimensions are placed as explained in 
 Art. 44. 
 
 Different materials may be conventionally shown in pencil draw- 
 ings by using different kinds of section lines. The materials most 
 commonly used are often represented as illustrated in Art. 76. 
 
 An accurate and neat drawing is very pleasing and effective ; 
 every draughtsman and advanced student should be able to produce 
 such a drawing with ease. Neatness and accuracy, the results of 
 care and long practice, are essential to good work ; and pupils should 
 understand that without the exercise of much patience, work of real 
 value cannot be obtained. 
 
CHAPTER 11. 
 
 GEOMETRICAL PROBLEMS. 
 
 The following problems are those most likely to prove valuable 
 to grammar and high school pupils. They are given in order that 
 pupils may be able to apply them in the study of design and working 
 drawings, and that they may learn to make accurate drawings. 
 They are not given with the intention of teaching geometry. 
 
 The illustrations represent the working lines by light lines, the 
 given lines by lines of medium strength, and the result lines by heavy 
 lines. In finishing pencil drawings the best effects are obtained by 
 using dotted lines for the working lines ; but if time necessary to 
 obtain perfectly regular lines cannot be given, it is better to use fine, 
 light lines. When drawings are inked the working lines may be red, 
 the given lines blue, and the result lines black. 
 
 In working all problems, to obtain accurate results, arcs de- 
 scribed as working lines should be of large radii. 
 
 Problem i . — To bisect a straight line AB, or 
 an arc of a circle ACB. 
 
 With A and B as centres and any radius 
 greater than half AB, describe arcs which inter- 
 sect at I and 2, Join i and 2 by a straight 
 line, and 1-2 is perpendicular to AB^ and 
 bisects it in j, and the arc in C 
 
 Problem 2. — To erect a perpendicular to a given 
 lifie at a given point A in the line. 
 
 With A as centre and any radius, set off equal 
 distances A i and A 2 from A. From points i and 2 
 as centres, and with a radius greater than half 7-2, 
 describe arcs which intersect in j and 4. Join j and 
 ^, and j-4. is the required perpendicular. 
 
 1^ 
 
GEOMETRICAL PROBLEMS. 
 
 13 
 
 %? 
 
 Problem 3. — To draw a perpeitdicular to a given 
 line BC^from a point A oictside the line. 
 
 With A as centre and any radius, intersect the 
 given line in points i and 2. With points i and 2 
 as centres and any radius describe arcs intersecting in 
 J. Join A and j ; ^ j is the required perpendicular. 
 
 Problem 4. — To erect a perpendicular to a given line AB, from 
 Q, point B, at or near its end. 
 
 With B as centre and any radius, draw an arc 
 "t of a circle 1-2-j, With i as centre and the same 
 
 radius, cut this arc in 2, and with 2 as centre and 
 the same radius, describe the arc j-4. With j as 
 centre and the same radius, intersect j-4 in 4, 
 Join 4 B ; this is the required perpendicular. 
 
 Problem 5. — To draw a line^ EF^ pa^^allel to a given line^ ABy 
 and the distance CD from it. 
 
 From any two points, A and ^, in the line 
 as centres, with radius CZ>, describe arcs E 
 and F, Erect perpendiculars at A and B, 
 which intersect the arcs in E and F, Join E 
 and F\ this is the required parallel. In prac- 
 tice, it is not necessary to draw the perpen- 
 diculars. 
 
 Problem 6. — To construct an equilateral tri- 
 angle on a give?t base, A B. 
 
 With A and B as centres and A B '^s radius, 
 describe arcs which intersect at i. Join i A and 
 I B. 
 
 Problem 7. — 2b construct a square on a 
 giveii base, AB. 
 
 Draw A I perpendicular to AB and equal 
 to it. (Problem 4.) With B and, 7 as centres 
 and radius AB, describe arcs intersecting in 2, 
 Join 1-2 and 2 B, 
 
14 
 
 MECHANICAL DRAWING. 
 
 Problem 8. 
 
 To co?istnict a recta?tgle of given sides, AB and 
 CD, 
 
 At A, by Problem 4, erect a perpendicular 
 A I equal to CD. With i as centre and radius 
 AB, describe an arc, and intersect this arc in 2, 
 by one described from B with CD as radius. 
 Join 1-2 and 2 B. 
 
 Problem 9. — To inscribe a square within a 
 given circle. 
 
 Draw AB, a diameter of the circle, which 
 wdll be a diagonal of the square. By Problem 
 I bisect AB, and continue the bisector to inter- 
 sect the circle in i and 2. Join A i, i B, B 2, 
 and 2 A. 
 
 Note. — A square may be constructed upon a given diagonal by draw- 
 ing a circle through its ends, A and B, and proceeding as explained above. 
 
 Problem 10. — To bisect a given aiigle, CAB. 
 
 With A as centre and any radius, describe an 
 arc intersecting AC and AB in points i and 2. 
 With points i and 2 as centres and any radius, 
 describe arcs which intersect in j. Join A and j. 
 
 Problem 11. — To trisect a right angle, CAB. 
 
 With A as centre and any radius, describe an arc 
 intersecting AC in i and AB in 2. With i and 
 2 as centres and the same radius intersect the arc 
 in J and 4. Join A 4 and Aj. 
 
 Problem 12. — To co7istruct at E, in the line E D, 
 an angle equal to a gii^en angle, ABC. 
 
 With B as centre and any radius, intersect 
 AB and BC in i and 2. With E as centre and 
 same radius describe an arc intersecting E D 
 in 4. With 4 as centre and 1-2 as radius, inter- 
 sect the arc in j. Join E and j. 
 
GEOMETRICAL PROBLEMS. 
 
 15 
 
 Problem 13. — To co7istruct an angle of go", dd", 4^, 30"", ^5°, 
 or any other given magniticde. 
 
 There are 360° in the entire circumference 
 of a circle. Draw a diameter 1-2, and there 
 are 180° on either side of this line. Draw a 
 second diameter 3-4, at right angles to 1-2, 
 and the circle is divided into four equal angles of 
 90°. Trisect one of these by Problem 11, and 
 angles of 30°, 2c6, and of 60°, 20^ are ob- 
 tained. Bisect one of these angles of 30°, and 
 angles of 15°, 2 c y, and "/ c 6 are obtained. Bisect an angle of 90° 
 and angles of 45° are obtained. Trisect, by spacing, an angle of 15° 
 and angles of 5° are given. Divide one of these into five equal 
 parts and degrees are given. In this way any angle may be obtained. 
 
 Problem 14. — To divide a given line, A B, into any number of 
 equal parts, as five. 
 
 Draw AC at any angle to AB. On AC set 
 off any distance five times. Join B and the 5th 
 point, and through the other points draw parallels 
 to B ^ (Art. 6), which will divide AB as required. 
 
 Problem 15. — To construct a triangle, hav- 
 ing given its sides, AB, C, and £>. 
 
 With A as centre and radius C, describe an 
 arc. With B as centre and radius D, intersect 
 this arc in i. Join i A and i B. 
 
 Problem 16. — To inscribe a regular hexag07i withiii a given circle. 
 Draw the diameter AB of the circle, which 
 will be a long diagonal of the hexagon. With 
 point A as centre and the radius of the circle 
 as radius, intersect the circle by arcs i and 2. 
 With B as centre and same radius, intersect the 
 circle by arcs 3 and 4. Join A-i, i-j, J-B, 
 B-4, 4-2, and 2-A. 
 Note. — By joining every odier point an equilateral triangle will be obtained. 
 
i6 
 
 MECHANICAL DRAWING. 
 
 Problem 17, — To coiistruct a regular hexago?t 
 iipoji a given base AJB. 
 
 With A and B as centres and AB as radius, 
 describe arcs which intersect at i. With i as 
 centre and same radius, describe a circle, and in 
 its circumference place points 2, j, 4, and 5, which 
 are equidistant and the distance AB apart. Join 
 A-2, 2-3, 3-4, 4-3, and 3B, 
 
 Note. — The ladius of any circle applied, as a chord, six times to the 
 circumference divides it into six equal parts. 
 
 Problem 18. — To consh^iict a regular octagon 
 within a give7i squa7X ABCD. 
 
 Draw the diagonals of the square, which 
 intersect at its centre, z, and with points A, By 
 C, and T> as centres and a radius, A r, equal to 
 half the diagonal, describe arcs intersecting the 
 sides of the square in points 2, j, 8, 9, 6, 7, ^, 
 and 3. Join 4-g, 2-7, 8-3, and 6-3. 
 
 Problem 19. — To inscribe, a regular pentagon 
 within a given circle C. 
 
 Draw the diameter 1-2 and a radius 3-4 per- 
 pendicular to it. Bisect 2-3 in 5, and with 3 as 
 centre and radius 3-4, intersect 2-1 in 6. With 
 4 as centre and radius 4-6 intersect the circle in 
 7. Set off the distance 4-7 from y \.o 8, 8 to g, 
 g to 10. Join ^-7, 7-6", 8-g, g-io, and 10-4. 
 
 Problem 20.. — To draw a tangent at any 
 poi?it A, in a give?i circiunference. 
 
 Draw a radius, cA, and erect a perpen- 
 dicular, 3-4, to the radius at A, by Problem 
 2. This is the required tangent. 
 
GEOMETRICAL PROBLEMS. 
 
 17 
 
 Problem 21 . — 7^ inscribe a circle ivithin a given 
 tria?tgle ABC, 
 
 By Problem 10, bisect any two angles of the 
 triangle, as CAB and ABC The bisecting lines 
 intersect at z, the centre of the required circle. A 
 perpendicular, as z-2, from i to any side of the tri- 
 angle, is the radius of the required circle. 
 
 Problem 22. — To draw an arc of a 
 given radius EF^ tangent to given straight 
 lines AB and CD. 
 
 Draw parallels to AB and CD by Prob- 
 lem 5, at the distance EFixovd them. The 
 parallels intersect at z, the centre of the 
 required arc. The arc is tangent to AB 
 and CD^ at points j and 2, in perpendiculars 
 to these lines from i. 
 
 Problem 23. — To draw an arc of a given radius DE^ tangent to a 
 straight line AB and a circle C 
 
 Draw by Problem 5 a parallel Bj to AB^ 
 the distance DE from it. With i, the centre 
 of circle C, as centre and a radius 1-2, equal 
 to the radius of the given circle plus DE, 
 that of the required arc, describe an arc to 
 intersect the parallel Bj in j, the centre of 
 the required arc. A perpendicular from j to 
 AB gives 5, the point of contact of the arc 
 and AB, and a straight line from J to i gives 4, the point of contact 
 of the arc and the circle. 
 
 Problem 24. — To draw an arc of a given 
 radius CD tangent to tu'o circles A and B. 
 
 With I, the centre of the circle A, as 
 centre and radius 1-2, equal to the radius 
 of A plus CD, describe an arc 2-^. Inter- 
 sect this arc in 5 by one described from j, 
 the centre of the circle />, with a radius j-^, 
 equal to the radius of B plus CD, Point 5 
 
i8 
 
 MECHANICAL DRAWING. 
 
 is the centre of the required arc, and lines from 5 to z and from 
 J to J givx 6 and 7, the points of contact of the arc and circles A 
 and B, 
 
 Problem 25. — 
 
 7^ di^aw^ wit /lift a giveii equilateral triaiigle ABC, 
 three equal circles, each ta?igent to two 
 others a?id to one side of the ti^iangle. 
 
 Bisect the angles of the triangle and 
 obtain i. Bisect the angle i AB and 
 obtain 2, the centre of one circle. With 
 I as centre and radius 1-2 draw an arc 
 to obtain centres j and 4. With centres 
 2, J, and 4 and radius 2-5, describe 
 the required circles. . 
 
 Problem 26. — To draw arcs of circles taiigent at points C and B, 
 to two parallel straight lines AB and CD and passing thi'ough a?iy 
 
 point as E, in the line CB. 
 
 At C and B erect perpendiculars. 
 Bisect BE and CE, continuing the lines 
 of bisection until they meet the perpen- 
 diculars from B and C in points 2 2iY\d i, 
 which are the centres of the required arcs. 
 With I as centre and z C as radius draw 
 the arc CE, and with 2 as centre and 2 B 2iS radius draw the arc BE, 
 
 Problem 27. — To circumscril?e a circle ahont a 
 give?! triangle ABC. 
 
 By Problem i bisect any two sides as AB 
 and BC by perpendiculars meeting at z. With 
 I as centre and radius i A describe the circle. 
 
 NoTi-: I. — If any three points as A, B, C not in the same straight 
 line are given, a circle may be passed through them by connecting the 
 points and proceeding as above. 
 
 NoTK 2. — The centre of any circle may be found by assuming any 
 three points in it and proceeding as above. 
 
GEOMETRICAL PROBLEMS. 
 
 19 
 
 Problem 28. — To construct a regular octagon on a given side AB, 
 
 At A and B erect perpendiculars by 
 Problem 4. Continue AB and bisect the 
 outer right angles, making the bisecting 
 lines A i and B 2 equal to AB. Draw 1-2 
 cutting the perpendiculars in j and 4. 
 
 From J and 4 set off the distance 3-4.^ 
 giving 5 and 6. Draw through 5 and 6 a 
 straight line, and set off from 5 and 6 dis- 
 tances 5-7 and 5-^, (5-p and d-zo equal 
 to 3-1 or J A. Join the points z-7, ^-8^ 8-g^ 9-10^ and zo-2. 
 
 Problem 29. — 7^ inscribe^ within a given equilateral tiHangle 
 ABC, three equal circles, each tangent to two others a?id to two sides of 
 the triangle. 
 
 Bisect the angles A, B, and C by lines 
 meeting at i ; bisect the right angle C 2 B 
 to obtain J, the centre of one of the required 
 circles. With i as centre and i-j as radius, 
 describe a circle giving points 4 and 5. 
 J, 4, and 5 are the centres of the required 
 circles. The shortest distance, as j-(5, from 
 a centre to a side of the triangle, is the 
 radius of the required circles. 
 
 Problem 30. — Within a given square, 
 ABCD, to dra7v four equal circles, each 
 touching two others and tivo sides of the 
 square. 
 
 Draw the diagonals of the square to 
 obtain its centre, and then lines parallel 
 to the sides through the centre to inter- 
 sect the sides in points i, 2, j, and 4. 
 Join i-j, j-2, 2-4, and 4-1, obtaining 
 points 5, 6, 7, and 8, the centres of the 
 required circles. The length, 7-g, of a radius is given by drawing /-(?. 
 
 
 
 
 
 
 
 J) 
 
 
 ./ 
 
 r 
 
 
 y\ 
 
 ^1 
 
 A 
 
 
 ^'^3 
 
 
 "X 
 
 >C^ 
 
 
 ■r 
 
 ji 
 
 ' ^ 
 
 y 
 
 V 
 
 y. 
 
 
 A 
 
 
 i 
 
 1 ' 
 
 ) 
 
 
 
 \ 
 
 
 
20 
 
 MECHAXICAL DRA JVIXG. 
 
 Problem 31. — To divide a straight line, 
 CD^ into the sa7ne proportio?ial parts as a 
 given divided line, AB. 
 
 Draw CD parallel to AB by Problem 5. 
 Draw lines through AC and BD to meet in 
 7. From points a, h, and c in AB, draw to 
 I lines, which will divide CD as required. 
 
 Problem 32. — Within a square, ABCD, 
 to draw four equal se?ni-eire/es, each toueh- 
 ing one side, and forjning by their diaineters a 
 square. 
 
 Draw the diagonals and the diameters 
 of the square and join points 1-2, 2-j,j-4, 
 and 4-1, obtaining 5, 6, 7, and 8, Join 5-^, 
 6-7, 7-8, and <?-5, obtaining g, 10, 11, and 
 12, which are the centres of the required 
 arcs, of which g-2 is the length of a radius. 
 
 Problem 33. — To eon struct an equilat- 
 eral triang/e, its altitude, AB, being given. 
 
 At A and ^ erect perpendiculars to AB 
 by Problem 4. With A as centre and any 
 radius describe a semi-circle, which inter- 
 sects the perpendicular in i and 2, and from 
 / and 2 set off the radius of the semi-circle 
 in J and 4. Draw A j and A 4 to intersect, 
 in J and 6, the perpendicular erected at B, 
 
 Problem 34. — To circumscribe an equi- 
 lateral triangle about a given circle C, 
 
 Draw a diameter, 7-2, of the circle. With 
 2 as centre and 2-j, the radius of C, as 
 radius, describe a circle intersecting 1-2 
 produced in 4 and the circle in 5 and 6. 
 With I, 5, and 6 as centres, and radius 7-5, 
 ^ describe arcs which intersect at 7 and 8. 
 Join 4~y, y-8, and 8-4. 
 
GEOMETRICAL PROBLEMS. 
 
 21 
 
 Problem 35. — To circumscribe a square 
 about a given circle. 
 
 Draw the diameter 1-2^ and a diameter 
 j-4, at right angles to 1-2, With z, 2, j, 
 and 4 as centres and radius 7-5, describe 
 arcs which intersect at 6, 7, (?, and p. Join 
 p-d, 6-7, 7-8, and (S'-p. 
 
 Problem 36. — To circumscribe . a regular hexagon about a give7i 
 
 circle C. 
 
 Draw a diameter, z-2, and with the radius 
 of the circle set off, from i and 2, points j, 4^. 
 5, and 6 ; through these points draw radii, ex- 
 tending them beyond the circle. Bisect the 
 sector z ^ J by <r 7, which intersects the circle 
 at 8. At 8, by Problem 2, erect a perpendicular 
 and obtain g and 14. With <: as centre and c g 
 as radius, describe a circle to intersect the 
 radii extended in 10. 11, 12^ and zj, the vertices 
 
 of the required hexagon. Join p-zo, lo-ii, 11-12, 12-ij, and 13-14. 
 
 Problem 37. — To draw circles tangeiit to each other and to two 
 
 lines ^ AC and BC, not parallel, the radius., DE. of one ci^^cle being 
 
 given. 
 
 Bisect the angle ACB by C i\ Draw 
 parallel to AC, and distant from it by the 
 radius of the given circle, a line which, in- 
 tersecting C z at 2, gives the centre of 
 the first circle, which intersects C z in j- 
 At J erect a perpendicular to C z and 
 obtain 4, and bisect the angle j-4-C 
 obtaining 5, the centre of the second 
 
 circle, whose radius is the distance 5-j. In this way any number of 
 
 circles may be drawn. 
 
 Problem 38. — To construct a regular polygon of any nufnbcr of 
 sides {271 this case sevoi), upon a given base, AB, 
 
22 
 
 MECHANICAL DRAWING. 
 
 Extend BA^ and with A as centre and 
 AB as radius describe a semi-circle. Divide 
 the semi-circle into as many equal parts as 
 there are sides in the required polygon, and 
 draw from A to the second point, a- second 
 side of the polygon. Bisect A 2 and AB ; 
 the bisectors meet at 7, the centre of the 
 required polygon. With 7 as centre and 
 7 ^ as radius, describe a circle and upon it 
 set off the distance AB^ beginning at 2, and 
 
 obtain points 8, g, 10^ and zz, the vertices of the required polygon. 
 
 Join 2-8, 8-g, g-io, lO-ii, and 11 B. 
 
 Problem 39. — To inscf-ibe a regular polygon of any 7iii7nber of 
 sides {in this case five) witJmi a given circle. 
 
 Draw a diameter AB and divide it 
 into as many equal parts as the poly- 
 gon is to have sides. With A and B as 
 centres and radius AB^ describe arcs 
 intersecting in 5. From 5 draw a straight 
 line through the second point, 2, to inter- 
 sect the circle in d. ^ d is a side of the 
 required polygon. Beginning at 6. set 
 off ^ d upon the circle to obtain points 7, <?, g. Join A d, 6-j 
 7-8, 8-g, and g A. 
 
 Note. — This method is approximate only. 
 
 Problem 40. — Withi?i a given circle to d?^aw any niwiber of 
 equal se77ii-circles ta7ige7it to the circle and fo7'77ii7ig by their dia7neters a 
 
 regular polygon. 
 
 Draw radii i~2 and i-j at right angles 
 to each other. Beginning at 2, divide the circle 
 into twice as many equal parts as the number 
 of semi-circles required, and draw diameters to 
 these points. Join 2 and j ; the intersection g 
 of 2-j with the diameter ^-7 is a vertex of the 
 polygon whose sides are the diameters of the 
 required semi-circles. Describe a circle through 
 
GEOMETRICAL PROBLEMS, 
 
 23 
 
 g with I as centre, and join its points of intersection, 9, zo, and 
 zz, obtaining the diameters which contain points 72, zj, and 14^ 
 the centres of the required semi-circles, of which 12-2 is . the 
 length of a radius. 
 
 Problem 41. — Within a giveii circle A to draw any member of 
 
 equal circles tangent to each other and to the circle A. 
 
 Divide the circumference of the cir- 
 cle into as many equal parts as inscribed 
 circles are required, and draw radii 
 to these points of division. Bisect the 
 angle z ^ 5, and at 5 draw a tangent 
 which intersects the bisector at 6. Bisect 
 the angle 5-6-^ and obtain 7, the centre 
 of one of the required circles. With c as 
 centre and <: 7 as radius, describe a circle 
 
 to cut the radii previously drawn, in points 8^ p, zo, and 11^ which 
 
 are centres of the required circles. The radius of each circle is the 
 
 distance 7-5. 
 
 -iX 
 
 Problem 42. — About a given circle A ^ to draw any nwnber of 
 equal circles tangent to each other and to the give?! circle. 
 
 Divide the circumference of the circle 
 into twice as many equal parts as the number 
 of circles required, and draw radii extend- 
 ing them beyond the points of division. At 
 any point, as 2, draw a tangent to the circle 
 intersecting a radius, extended, at 11. Bisect 
 the angle 2-11-12^ and obtain zj, the centre 
 of one of the required circles. With c as 
 centre and c-ij as radius, describe a circle 
 to intersect every other radius in z^, zj, zd, and z/. These points 
 are the centres of the required circles of which the radius is the 
 distance 13-2, 
 
 Problem 43. — To draw an- ellipse^ its axes AB and CD being 
 give7i. 
 
24 
 
 MECHAXICAL DRAWIXG. 
 
 
 
 / 
 
 P 
 
 
 
 'i 
 
 
 VA 
 
 ^ 
 
 >< 
 
 )/z 
 
 / 
 
 ''^- 
 
 \ 
 
 ^ 
 
 ■^\ 
 
 
 /7 
 
 '^v^^ 
 
 
 js 
 
 \ 
 
 \^ 
 
 / 
 
 \^\a. 
 
 I 
 
 j\ 
 
 z\ 
 
 ^ 
 
 4. 
 
 J^ 
 
 >, 
 
 Ai 
 
 
 i*. 
 
 \ 
 
 L> 
 
 <i 
 
 
 Problem 44. 
 
 axes are given. 
 
 Upon the axes draw circles and divide 
 both circles by diameters i-i ; 2-2, etc., 
 into any number of parts, equal or un- 
 equal. From the points in the large circle 
 draw parallels to the short axis, and from 
 those in the small circle draw parallels to 
 the long axis. The intersections of lines 
 from points of the same number are points 
 in the ellipse. 
 
 To draw^ by means of a travifnel^ an ellipse^ whose 
 
 rvA/— 
 
 Set off AB^ half the long axis, on the 
 edge of a straight piece of paper from F 
 to S^ and AC^ half the short axis, from P 
 to Z. Move this paper trammel so that 
 xS is upon the short axis and L upon the 
 long axis, and P will always be a point in 
 the ellipse. Its position may be marked 
 by a sharp lead pencil. The best way is 
 to use a needle point, and make very fine 
 punctures in a straight line drawn near the edge of the trammel. 
 The needle may then be placed through L and on the long axis, and 
 the paper revolved about L till the point S is over the short axis, 
 when the needle may be used to mark, through P^ a fine puncture 
 in the paper. 
 
 To enable the axes to be seen, the edges of the paper may be 
 notched as indicated. 
 
 Problem 45. 
 
 proximate ellipse. 
 c 
 
 To draia upon give?i axes, AP and CD, an ap- 
 
 With the centre, d, of the ellipse as 
 centre and half the short axis, d C, as 
 radius describe an arc C i. Draw CP and 
 from C set off C2 equal to P i, or the 
 difference between half the short and half 
 the long axis. Bisect P 2 and continue 
 the bisecting line to intersect AP in j, 
 
GEOMETRICAL PROBLEMS. 
 
 25 
 
 and CD in 4. With j as centre and radius j B describe an arc 
 B 5, and with 4 as centre and 4-^ as radius, describe an arc which 
 will pass through C and complete one quarter of the curve. From 
 d, set off d-7 equal to 6-j, and 6-8 equal to 6-4^ and draw lines 
 joining ^-7, 8-"/^ and (^-j, and corresponding to ^-5. With 
 points J, ^, 7, and 8^ as centres, complete the ellipse as ex- 
 plained above. 
 
 Problem 46. 
 
 To draw a7i equable spiral. 
 
 Draw AB and upon it place any two points, 
 I and 2, as centres. With i as centre, and 
 radius z-2, draw a semi-circle. With 2 as 
 centre and radius 2-j, draw a semi-circle ; 
 continue the process, using i and 2 as centres, 
 and the distance to the end of the diameter of 
 the semi-circle last drawn as radius. 
 
 Problem 47. — To draw a variable spiral^ its greatest^ diameter^ 
 
 AB, being given. 
 
 Divide AB into eight equal parts by 
 points a, b, c, etc. On d e, the fifth space, 
 as diameter, draw a circle which is called 
 the eye of the spiral. Inscribe a square in 
 the circle and draw its diameters, as shown 
 in the enlarged drawing of the eye. Divide 
 these diameters into six equal parts and 
 number the points as shown. With i as 
 centre and i-ij as radius, draw the arc 
 13-14. With 2 as centre and 2-14 as 
 radius, draw the arc 14-15. ^^'ith j as 
 centre and 3-1S as radius, draw the arc 
 
 15-16 ; and so proceed, using all the points up to and including 12, 
 
 as centres, and limiting each arc by a line drawn from its centre 
 
 through the centre of the next arc. 
 
26 
 
 MECHANICAL DRAWING. 
 
 Problem 48. — To draw the mvohite of a ci?rie. 
 
 From any point, as i, in the circum- 
 ference set off any number of equidistant 
 points, 2, J, 4, and J. Draw tangents to 
 the circle at these points, and on the tan- 
 gent at 2 set off the distance 2-1^ giving 6. 
 On the tangent at j, set off the length of 
 the arc j-z, giving 8. At 4 set off the 
 length of the arc 4-1, giving 11. At 5 set 
 off the length of the arc 5-7, giving zj. 
 A curve through z, d, 8, 11^ and zj is 
 the involute of the circle. 
 
 Note. — The distance between points i, 2, 3, etc. must be such that 
 the difference in leno^th between the chord and the arc is slis^ht. 
 
 Problem 49. — To draw the cycloid curve traced by a poi7it in the 
 circiimfo'ejice of a given circle^ Ay which rolls on the li?ie CD, 
 
 Divide the circumference into 
 any number of equal parts, as 
 twelve, and set off on CD the 
 equidistant points in which the 
 points marked in the circle will 
 come to the straight line as the 
 circle rolls upon CD. When the 
 circle is tangent at 2' its centre 
 will be in a perpendicular erected 
 at 2' and in a parallel to CD through o, the centre of A. In the 
 same way its centre will be found when it is tangent at 3', 4', 5', etc. 
 As the circle rolls, point i describes the cycloid. In any position 
 of the circle point i will be found by setting off as many equal 
 spaces on the circle from points 2', 3 ', 4', etc., as the circle has rolled 
 over, starting from /'. Thus, one space from 2' to ^ ; two spaces 
 from j' to by and so on, for the complete curve. Points <?, b, c, etc., 
 may also be found by noting the level of the points d, 5, 4, etc., in 
 the given circle A. Thus, a horizontal line from 6 will give a ; one 
 from 5 will give by and so on. In either of these ways, the complete 
 
GEOMETRICAL PROBLEMS. 
 
 27 
 
 curve, of which one-half only is shown, may be obtained. The 
 length of the arc between the points- in the circle of this and Prob. 
 48 may be computed, or a result practically correct obtained by 
 dividing the circle into many parts. 
 
 Problem 50. — To construct the epicycloid curve traced by a point 
 in the circumference of a circle, A, which rolls on the outside of a circle B. 
 
 Divide the circumference of the 
 circle A into any number of equal 
 parts, as twelve, and set off from 7 
 on B the points 6\ 5', /, j', 2', i\ 
 in which the points d, 5, 4, j, 2, i 
 oi A will coincide with B ?iS> A rolls. 
 Describe an arc of A when it is tan- 
 gent, at each of the points 6\ 5', 4', 
 j', 2' ; the centres will be in an arc 
 concentric with B, passing through 
 0, and in radii of B, extended 
 through the points 6', 5', ^', j\ and 
 2'. When tangent at 2' the marking 
 point I is at a, and is obtained by 
 setting off from 2' one of the equal 
 spaces into which it was divided, or 
 by drawing an arc through 6, with D as centre, to intersect at a the 
 circle A, when tangent at 2'. When tangent at j' the marking point 
 I will be at b, and will be obtained by setting off two equal spaces 
 from j' to b, or by an arc through 5. In this way all the points, c, 
 d, e, of the curve may be found ; also the other half of the curve. 
 
 Problem 51. — To construct the hypocycloid traced by a point in 
 circle C, which rolls inside the circle B. 
 
 The process is the same as for the outer epicycloid explained in 
 Problem 50. 
 
 Note. — When the diameter of the rolling circle is equal to the radius of 
 the circle within which it rolls, the hypocycloid becomes a straight line and 
 is a diameter of B. 
 
CHAPTER III. 
 
 WORKING DRAWINGS. 
 
 13. Nature and use of Working Drawings. — The workman 
 who builds a house, engine, or any other object, obtains the neces- 
 sary information concerning the size, shape, kind of material, and 
 amount of finish, etc., of all the different parts, from different projec- 
 tions or views of the object to be made. These views are called 
 working drawings. 
 
 14. Free-hand drawings, that is, perspective views, give the appear- 
 ance of objects, but they represent gnly part of the surface of an 
 object, and do not give the actual sizes or shapes of any of its parts ; 
 therefore such drawings are not suitable for working drawings. 
 
 A perspective drawing may be used as a working drawing when 
 the object illustrated is very simple, or of a form commonly used, and 
 the dimensions of the parts are placed upon the drawing ; but when 
 such a drawing is used the knowledge and experience of the work- 
 man supply the information which the perspective drawing does not 
 give. 
 
 Generally, perspective views of machinery and architecture are 
 made simply as illustrations of objects which either have not been 
 made, or which are not placed so they can be photographed. The 
 best and cheapest perspective possible to obtain is given by a photo- 
 graph. 
 
 15. A working drawing must give the actual shapes of the parts 
 it represents. It may be the full size of the object or smaller, but it 
 must be drawn to some determined scale and show the true propor- 
 tions and relations of all the parts. In projection the actual appear- 
 ance of any object is given upon a plane at right angles to the direc- 
 tion in which the object is seen. It follows that, as far as possible, 
 working drawings should be made upon planes which are parallel to 
 the principal faces of the object to be represented. 
 
WORKING DRAWINGS, 
 
 29 
 
 16. The nature and relations of the different views usually made 
 as working drawings will be shown by the following experiments : 
 Place a cube, shown 
 
 by Fig. I, at the level 
 
 of the eye so that only 
 
 one face is visible, the 
 
 corners of this face 
 
 being equally distant 
 
 from the eye. Hold a 
 
 glass slate vertically in 
 
 front of the cube and 
 
 parallel to the visible face, that is, at right angles to the direction in 
 
 which the cube is seen. Trace the appearance of the cube upon the 
 
 slate. The tracing will be a square, for the apparent shape of any 
 
 surface at right angles to the direction in which it is seen agrees with 
 
 its real shape. 
 
 If the slate is now held horizontally above the top of the 
 cube and the eye is placed above the object, so that the top face 
 appears its real shape, a tracing of the cube from this position will 
 be a square, as in the first case. The first tracing gives the real 
 shape of the front face of the cube, and the second tracing the real 
 shape of the top face of the cube. The tracing on the slate when 
 vertical shows that the height of the object is equal to its width, 
 while that upon the slate when horizontal shows that the distance 
 from front to back is equal to the width. The two tracings together 
 express the fact that the three dimensions of the cube are equal. 
 This will be shown whatever the distances of the slate and the eye 
 from the cube ; but to have the two tracings of the same size, the 
 eye and slate must have the same relative positions with reference 
 to the cube when making both. 
 
 17. The tracings are smaller than the cube because the visual 
 rays converge toward the eye, and the slate is nearer the eye than is 
 the cube. If the eye is placed farther away from the object, and the 
 position of the slate remains the same, the tracing will be larger. If, 
 as the tracing is made, the eye is moved so as to see every visible 
 point of the object by means of a visual ray perpendicular to the 
 slate, the tracing will be the full size of the object. 
 
OQ MECHANICAL DRAWING. 
 
 18. If instead of one slate which is held in the hand to receive 
 the different views of the cube, two slates are used which are sup- 
 ported at right angles to each other, as in Fig. 2, and if instead of 
 tracing the appearance when the eye is in one position, the eye is 
 moved with the pencil point so that every point of the object is seen 
 in a perpendicular to the slate, the tracing upon the vertical slate 
 will give the real dimensions of the front face of the cube, and that 
 upon the horizontal slate will give the real dimensions of the top 
 face of the cube. The two tracings will give all the actual dimen- 
 sions of the object, and the dimensions of one tracing may be com- 
 pared with those of the other. These tracings illustrate the nature 
 of constructive drawings, and the slates in their different positions 
 illustrate the imaginary planes upon which these drawings are sup- 
 posed to be made. 
 
 19. These tracings or drawings are called "projections" or 
 "views," and when dimensioned they are called working drawings. 
 The tracing upon the vertical slate is called the vertical projection, 
 or the front view; that upon the horizontal slate is called the 
 horizontal projection, or the top view. Working drawings are 
 generally called views, which will be the term used throughout this 
 book. 
 
 The pupil who has used the transparent drawing slate for free- 
 hand purposes will have no difficulty in making and understanding 
 these experiments ; he will readily see that the tracings made upon 
 the vertical and horizontal slates give the three actual dimensions of 
 the cube seen through them. 
 
 20. These drawings might be used to form the cube represented, 
 by so shaping any material that, when it is looked at horizontally, 
 it will cover the tracing upon the vertical slate, and when it is looked 
 at from above, it will cover that upon the horizontal slate. 
 
 Objects are sometimes shaped by cutting them until they exactly 
 cover the drawing which represents them : this is the case particu- 
 larly with irregularly curved parts. Such parts are, however, gener- 
 ally obtained by the use of templets, which are forms of thin wood 
 or metal made from the drawings, and applied to the object to show 
 when the desired form has been obtained. In this way the frames 
 of boats are shaped; also other irregularly curved objects. 
 
WORKING DRAWINGS, 
 
 31 
 
 21. Objects are seldom shaped by placing them upon the draw- 
 ings, since the drawings are often less than the full size of the object. 
 Even when this is not the case it is difficult to obtain accurate work 
 in this way. 
 
 22. Drawings which are intended for shop use always give in 
 figures the dimensions of each part ; also directions as to the material 
 of which it is to be made, and the amount of finish which it is to 
 receive. The machinist readily works to hundredths of inches ; 
 accuracy such as this, or even far less, would be impossible by such 
 a clumsy method as placing the object upon the drawing. 
 
 23. The dimensioning of the drawings is to the workman of first 
 importance, as he is not allowed to apply a scale to obtain a dimen- 
 sion omitted from the drawing ; but to the student it is not at first 
 important. His attention should be given to the study of the princi- 
 ples necessary to enable him to make drawings. When able to do 
 this he should study dimensioning, which is advisable in the seventh 
 and eighth grades. 
 
 24. Drawings are made upon paper, which cannot be used as 
 was the glass slate to trace the appearance of an object ; but as 
 practical working drawings are not made by tracing, and since the 
 chief use of a working drawing is to represent what exists only in 
 the brain of the engineer, architect, or designer, working drawings 
 would be of little service if they could be obtained only by tracing or 
 by drawing from objects actually existing. 
 
 Drawings represent so exactly the conceptions of the designer 
 that the greatest mechanical constructions are made in small sections 
 in the workshop, and are then taken away and put together, every 
 screw and bolt hole of every part being properly placed, so that the 
 ^' Ferris Wheel," or the engine of the steamship ^' Paris," or even the 
 entire boat with all her fittings, can be put together the first time as 
 surely as can the parts of a simple box. 
 
 The following paragraph from the Century for July, 1894, gives 
 an idea of the importance of the draughtsman's duties. 
 
 ^' For the hull alone of the battleship Indiana, 25 principal plans must 
 be made, and fully 400 separate drawings must be prepared, and duplicated 
 by photographs. This of itself is enough work to keep a force of expert 
 draftsmen busy continuously for eight months. For the engines more than 
 
32 
 
 MECHANICAL DRAWING, 
 
 250 separate drawings are required, and these in all their intricate details 
 would take a force of 50 men nearly a year to complete, if engaged continu- 
 ously at the task. Not only must every rivet and every joint be marked out 
 and noted, but there must be the most complicated computation of strains 
 and weights." 
 
 25. Study of Principles. — In order to understand readily the 
 principles of the subject of working drawings, two slates should be 
 hinged together so that they can be placed at right angles to each 
 other, and can be revolved into the same plane, as shown by Figs. 2, 
 3, and 4. 
 
 Fig. 2 represents a cube and the vertical and horizontal glass 
 slates, which represent the imaginary planes upon which the front 
 
 rM 
 
 TRONT PLAflf 
 
 Fig. 2. 
 
 Fig. 3. 
 
 Fig. 4. 
 
 and top views are supposed to be made ; Fig. 3 represents the slates 
 after the horizontal one has been revolved to coincide with the plane 
 of the vertical slate ; Fig. 4 is a drawing which gives the real shapes 
 of the two slates and of the views upon them. 
 
 26. It will be seen that after the slates have been placed as illus- 
 trated by Figs. 3 and 4, the views will be in line with each other, the 
 top view being above the front view, and the tw^o views of any point 
 being in the same vertical line. 
 
 The drawings also show a vertical line 1-2, and its views upon the 
 two slates. 
 
WORKING DRAWINGS. 
 
 zz 
 
 27. It often happens that these two views are not enough to de- 
 scribe an object. Thus, if the cube is bisected to form two triangu- 
 lar blocks, the views of either triangular block upon the vertical and 
 horizontal slates will be the same as the views of the entire cube. 
 
 Fig. 5 represents a triangular block formed by bisecting the cube, 
 illustrated in Fig. 2, by a plane passing through the front upper and 
 
 Fig. 5. 
 
 Fig. 6. 
 
 
 
 TOP 
 
 J 
 2. 
 
 
 3 
 
 
 
 
 
 
 
 
 LEFT 
 
 A 
 
 
 FRONT 
 
 Jl 
 
 
 3 
 
 
 RiCrtr 
 
 
 ■ 
 
 
 
 
 X/ 
 
 Fig. 7. 
 
 back lower edges ; in Fig. 5 these edges are 2-j and 1-4 respec- 
 tively. It also represents side slates attached to the vertical and 
 horizontal slates, and at right angles to both. If a tracing of the 
 block is made upon one or both of these side planes, or slates, as 
 
34 
 
 MECHANICAL DRAWING. 
 
 upon the front and top slates, the information not given by the first 
 two views will be supplied. 
 
 Figs. 6 and 7 represent the slates when the side slates and the top 
 slate have been revolved to coincide with the plane of the front slate. 
 
 28. The views upon the side slates are at the same level as that 
 upon the front slate, and when the slates are as shown in Figs. 6 
 and 7, the right and left side views of any point are in a horizontal 
 line through the front view of the point. 
 
 29. A fifth slate may be placed at the back, and thus a view of 
 the back may be obtained. The surface upon which the object rests 
 may be that of a sixth slate, and thus a view^ of the bottom may be 
 traced. All these, together with additional views, are sometimes 
 necessary. This arrangement of slates and object really amounts to . 
 placing about the object a glass box and tracing upon the sides of 
 the box, by means of perpendiculars to the sides, the different 
 appearances of the object. 
 
 30. These perpendiculars, or visual rays, which produce the dif- 
 ferent views are called projecting lines. When the planes are 
 revolved to coincide with the plane of the front plane, as shown by 
 Figs. 6 and 7, these projecting lines are represented by the vertical 
 and horizontal lines which contain the different views of the various 
 points, and are the projecting, or working, lines of the drawings. 
 
 31. The figures show that the different views are arraiiged with 
 reference to the fro7it view, so that the line of each view nearest the front 
 view represents the front face of the object. 
 
 By means of such glass slates hinged together, these simple prin- 
 ciples may be so illustrated that pupils in the upper grammar grades 
 can understand the nature of the drawings made for the different 
 views of the various objects chosen for study, the reason for arrang- 
 ing working drawings as explained ; also the fact that the front and 
 top views of the same point are in the same vertical line, and the 
 front and side views in the same horizontal line. 
 
 32. Some teachers prefer to think of the object as revolved so 
 that its different surfaces are, one after another, placed parallel to 
 one plane (or glass slate) upon which the drawings are made in the 
 same manner as a drawing made upon any one of the slates at right 
 angles to each other. There is no objection to this method ; indeed 
 
WORKING DRAWINGS. 
 
 35 
 
 in the practical work of making drawings from the object it is really 
 always done. The slates at right angles to each other are used 
 simply to illustrate the principles governing the making and arrange- 
 ment of the views, and in the work of the draughting office, as well 
 as that of the elementary school-room, no further use is made of the 
 planes ; for shop and elementary drawings are not arranged, with 
 reference to the edges of the planes, as are drawings in which pro- 
 jection methods are used. 
 
 33. Teachers who prefer to consider the object as seen from one 
 direction and turned to present the surfaces to be represented, must 
 explain the arrangement of views desired, and also the fact that a 
 point of the top view is in a vertical line from the same point in the 
 front view, and that a point in a side view is in a horizontal line 
 through the front view of the same point. Pupils may be told these 
 facts and be taught to make their drawings in accordance ; but the 
 reasons for these facts are most easily understood when they are 
 clearly illustrated by use of the slates. 
 
 To present the principles as explained above will take little 
 longer than to state only the facts regarding the desired positions of 
 the views ; when the principles are not explained, these positions are 
 very likely to be forgotten. 
 
 34. After the pupils understand the arrangement and nature of 
 the views, mention of the slates, or planes of the drawing, is unnec- 
 essary, and the work is practically the drawing of different views of 
 an object, which is so turned that these views may be seen and 
 drawn in their proper relations to the front view. 
 
 If the subject is presented to pupils too young to understand 
 what has been explained, the work must be simply dictation or copy- 
 ing. Much of the time now spent on working drawings in the lower 
 grades of the pubHc schools is wasted, for the pupils can do no more 
 than copy ; instead of copying, they might spend their time profitably 
 upon free-hand drawing, and, when old enough to reason, take up the 
 study of working drawings, and thus not be obliged to copy. 
 
 35. Making Working Drawings. — When given any object for 
 which drawings are to be made, the first question is, what position of 
 the object shall be represented, and the next, what views will best 
 show its construction and require the least time to make. 
 
36 
 
 MECHANICAL DRAWING. 
 
 The front view of an object should represent its front surface, 
 when it is in the position in which it is intended to be used. If the 
 object has no portion which is the *' front surface,'' or which is more 
 important than the others, and has no special position, its position is 
 immaterial. 
 
 Practical drawings give no more views than are needed to show 
 all the construction ; but while studying the subject, many views of 
 simple objects may be made. 
 
 36. When possible, objects for study should be placed so that 
 one principal surface will appear of its real shape, in either the front, 
 the top, or the side view, and the pupils should begin with this vie\^, 
 or with the one concerning which they know the most. To work 
 advantageously upon subjects at all complicated, it is necessary to 
 begin a second view as soon as the first view represents all the parts 
 which exhibit their real shapes in it. The parts which show their 
 real shapes in the second view can be drawn after the second view 
 has given all that the first view represents, and then the parts whose 
 real shapes are seen in the second view may be projected to the first 
 view. Circular parts should always be represented first in the view 
 in which they appear as circles. By drawing vertical projecting lines 
 from the front to the top or bottom views, or horizontals from the 
 front to the side views, the different views will be made to agree with 
 and to complete one another. In this way several different views can, 
 and should be carried along together. In order to make them accu- 
 rately and rapidly, the dimensions of circles, or of any parts which 
 have the same size in two or more views, should be taken in the com- 
 passes or dividers and set off at once in all the views. This method 
 is much more accurate and rapid than working by projecting lines, 
 which often are not drawn quite parallel ; this is the draughtsman's 
 method, and though it may not be possible for young pupils to work 
 in this way, it should be explained, and older pupils should use it. 
 
 37. Views of a Circular Plinth. — Fig. 8 is a perspective of a 
 
 circular plinth of which working drawings are to be 
 ^^ ^L^ made. 
 
 If the plinth is a circular box it will be placed 
 
 horizontally ; if it is a clock it will be placed with its 
 
 Fig. 8. circles vertical. In general the object may occupy 
 
WORKING DRAWINGS. 
 
 Z1 
 
 Fig. 9. 
 
 any position, but it is better to have it placed in 
 the position in which it is intended to be used. 
 Suppose the circles to be vertical and the object 
 to be seen from the front, so that the front surface 
 appears of its real shape. The circle F^ Fig. 9, 
 will then be the front view of the object. This 
 circle should be drawn before any of the lines of 
 the top view are drawn. 
 
 To see what the top view will represent, hold 
 the plinth so that its front face appears a circle, as in F^ Fig. 9, 
 and then turn the top of the plinth toward the eye until the circles 
 appear as straight lines ; that is, as nearly straight as it is possible 
 for them to appear when held in the hand. In this position the 
 curved surface is seen, and the outline of the object is very nearly a 
 rectangle. The circles cannot appear as straight lines at the same 
 time, but the block can be held so that the circles, one at a time, 
 will appear straight ; and, since the circles are perpendicular to the 
 top plane, this is the appearance which both must present in the top 
 view, 7J Fig. 9. The length of the rectangle of the top view must be 
 the same as the diameter of the circle F^ and its width must be the 
 same as the thickness of the plinth. The horizontal lines which 
 represent the two circles must then be as far apart as the plinth is 
 thick, and the short vertical lines must, if extended, be tangent to 
 the circle of the front view ; they should be placed by drawing tan- 
 gents (projecting lines) to the front view, or by setting off, with the 
 compasses, the required distance from either side of the centre line, 
 as explained in Art. 36. 
 
 38. Views of an Hexagonal Plinth. — Fig. 10 is a perspective 
 of an hexagonal plinth, or box, of which working drawings are to be 
 made from the object. 
 
 The plinth being horizontal when in use, the top 
 hexagon will appear its real shape in the top view. 
 This view should be drawn first, the plinth being 
 placed so that if any vertical faces are more important 
 than others they may be seen in the front and side 
 views of the object. This may cause the hexagon of the top view to 
 have a long diagonal horizontal or vertical. 
 
 Fig. 
 
38 
 
 MECHANICAL DRAWING. 
 
 / 
 
 / 
 
 \ 
 
 
 
 
 \ 
 
 1 
 
 A^ 
 
 
 t 
 
 
 
 
 
 
 
 
 
 r 
 
 
 \A.S. 
 
 
 
 Fi 
 
 ^. II. 
 
 
 
 Suppose the hexagon T^ Fig. ii, to be the top view of the box, 
 
 and the front and right side views 
 to be required. To see what the 
 front view will represent, the box 
 must be looked at horizontally 
 when placed as shown in the top 
 view ; or, if we move the object to 
 present the different views, the box 
 should be held so that its hexagonal 
 faces are vertical, each has two 
 edges horizontal, and so that only the hexagonal face, originally the 
 top of the object, is visible. When in this position the lower horizon- 
 tal edge of the visible hexagon represents the face of the plinth, 
 originally its front vertical face. The object is now to be revolved 
 so that this lower edge is brought upward toward the eye until the 
 hexagons appear, as nearly as possible, straight lines. This position 
 gives the appearance which the front view must represent; in this 
 view three vertical faces are visible as surfaces, and the outline of 
 the view is a rectangle. 
 
 Knowing the appearance which the front viev/ must represent, it 
 may be drawn in its proper position below the top view. The verti- 
 cal lines of the front view are equal in length to the thickness of the 
 plinth, and must be vertically under the corners of the hexagon which 
 in the top view represent these lines. The horizontal lines which 
 represent the hexagons are thus as long ^s a long diagonal of the 
 hexagon. See F^ Fig. ii. 
 
 To obtain the right side view, the object should be held to appear 
 as shown in the front view, and then be revolved about a vertical 
 axis, the right vertical edge coming toward the eye, until the object 
 is seen at right angles to the direction in which the front view was 
 visible. In this position the two visible vertical faces will appear of 
 equal widths, the vertical surface originally at the front of the object 
 will appear a vertical line at the left of the object, and the vertical 
 surface originally at the back, a vertical line at the right of the object. 
 The distance between these surfaces is the length of a short diagonal 
 of the hexagon. These facts having been noted, the right side view 
 may be drawn in its proper position at the right of and on the same 
 
WORKING DRAWINGS. 
 
 39 
 
 level as the front view, its width being the distance 1-2 of the top 
 view. See R.S.^ Fig. 11. 
 
 39. Views of a Plinth and Disc. — Fig. 12 
 represents an equilateral triangular plinth support- 
 ing, by means of two wires, a vertical circular disc, 
 which appears a circle in the front view. 
 
 To represent this object only the front and side 
 views are necessary ; but the top view is added. 
 
 It is natural to draw first the triangular plinth, 
 because it supports the disc. The disc appears a 
 circle in the front view, and the plinth appears a 
 triangle in the side view, so it is evident that the 
 two views should be carried along together as ex- 
 plained in Art. 36. 
 
 The side view S^ Fig. 13, in which the plinth appears a triangle, 
 
 should first be drawn ; but this 
 view cannot be completed to rep- 
 resent the circle until the front 
 view of the circle is drawn ; for 
 th^ wires ^ " long, which support 
 the circle, cause its lowest point 
 to come below the ends of these 
 wires an unknown distance. 
 
 Having completed the triangle 
 of the side view, the front view of 
 the plinth should be drawn. To 
 discover its appearance, hold the 
 plinth in the position represented 
 by the side view, and then revolve 
 the object on a vertical axis so 
 that the right hand points of the 
 plinth move toward the eye until 
 the vertical triangles appear as 
 nearly as possible vertical lines. 
 One sloping face is then visible ; the horizontal base appears a hori- 
 zontal line whose length is equal to the thickness of the plinth ; and 
 the triangles, one at a time, appear vertical lines, whose length is 
 
 Fig. 13. 
 
40 
 
 MECHANICAL DRAWING. 
 
 equal to the distance 1-2 of S. The front view of the plinth may 
 now be drawn at the right of and on the same level with the side 
 view. In this view the wires which support the disc are seen of their 
 real lengths ; they should be represented when the front view of the 
 plinth is completed. 
 
 To obtain the circle which represents the disc, describe arcs, whose 
 radius is that of the circle, from the upper ends of the wires. These 
 arcs intersect at 5, the centre of the circle, from which the circle 
 may be described. In the side view the circle appears a vertical 
 line ; this view may now be completed by limiting this line by 
 a horizontal projecting line tangent to the circle at its highest 
 point. 
 
 The top view of the plinth alone might have been given before 
 this, but as there are no parts which must be represented first in the 
 top view, the time of drawing it is not important. To see the appear- 
 ance that the top view represents, hold the object as it appears in 
 the front view, and then revolve the top toward the eye until the 
 circular disc appears a horizontal line. In this position the 
 length of the rectangle which represents the plinth will be equal 
 to 3-4, or the length of the base of the object as seen in the 
 side view. 
 
 The appearance having been discovered, the top view may be 
 drawn above and in line with the front view, by means of verti- 
 cal projecting lines through the points of the front view. See T^ 
 
 fig. 13- 
 
 Views of a Box and Pyramid. — Fig. 14 is a perspective 
 of a tin box with its cover opened back 
 and resting on the surface that supports 
 the box. A square pyramid is centrally 
 placed on the top of the box, the edges 
 of its base being at 45° to those of the 
 top of the box. The view which shows 
 the front of the box, we will call the front 
 view. This and the top and side views 
 are required. The box is 6" long, 4" 
 wide, and 3" deep. The base of the pyra- 
 mid is 3^" square ; its axis is 6" long. 
 
 40. 
 
WORKING DRAWINGS. 
 
 41 
 
 In the front view the box is represented by a horizontal rectangle 
 3" high and 6" long. This may be drawn first, and then the top and 
 the end views of the box. 
 The top view of the box is 
 a horizontal rectangle 4" 
 wide and 6" long ; the side 
 view is a horizontal rec- 
 tangle 3" high and 4" long, 
 and is on the same level 
 as the front view, while 
 the top view is directly 
 over the front view. In 
 the side view the cover is 
 seen extending obliquely 
 from the top of the box to 
 the level of the bottom. 
 To draw the side view of 
 the cover, an arc of a cir- 
 cle whose radius is 4" ^^' '^" 
 should be described from the edge, where the cover swings (point 3) 
 to intersect the line of the table in 2. The cover must be 
 represented by a line drawn from 2 to 3. The top view of the cover 
 may now be drawn ; its width in this, view is the distance 1-2 of the 
 side view. 
 
 In the top view the square base of the pyramid is seen of its real 
 shape ; this should be the first view of the pyramid drawn. To ob- 
 tain the square, draw the diagonals of the top view of the box; these 
 intersect at its centre. With this point as centre describe a circle 
 3^" in diameter. Tangent to this circle and at 45° with the edges 
 of the box, draw the sides of the square which represents the base of 
 the pyramid. After this is drawn, the top view may be completed 
 by drawing the diagonals of the square to represent the lateral edges 
 of the pyramid. 
 
 In the front and side views the base of the pyramid coincides 
 with the top of the box. In the front view the axis of the pyramid 
 is a vertical line just under the centre of the square, — the point 
 which represents the axis in the top view. 
 
42 MECHANICAL DRAWING. 
 
 In the side view the axis is a vertical line midway between the 
 front and back of the box, as is shown by the top view. In the front 
 view the width of the base is given by projecting from a and b in the 
 top view ; and in the side view the width of the base must be equal 
 to c-d of the top view. From c and d in the side view, and a and b in 
 the front view, the contour lateral edges extend to the top of the axis. 
 In both front and side views, the nearest lateral edge of the pyramid 
 is a vertical line ; this line represents also the axis of the pyramid. 
 
 41. It will be difficult to hold the models so that they have the 
 proper relations and present to the eye the appearances of the differ- 
 ent views of the complete group. The box and the pyramid may be 
 held separately and turned to give the appearances of the different 
 views of the single objects, as has been explained. It may happen 
 that views of a group are desired when the relations are not so easy 
 to see as in this group ; in 'such cases it will be much better to 
 arrange the group and to look at it in the directions of the different 
 views required. 
 
 42. The making of simple working drawings in the manner ex- 
 plained, requires but slight knowledge of the principles of projection. 
 The essential points are that the front and top views of the same 
 point shall be in the same vertical line, and that the front and side 
 views of any point shall be in one horizontal line. When the drawings 
 are made from the object, they, are simply representations of the ap- 
 pearances which it presents when seen from different directions, per- 
 spective effects of vanishing not being given. Pupils will readily 
 learn to look at an object from different directions so as to realize 
 the appearances which its different views will present ; they will 
 also learn to revolve the object to present these different appear- 
 ances. Both methods should be explained, and pupils will then have 
 little difficulty in using either. 
 
 43. The simplest way to illustrate the manner in which an object 
 should be turned to present the appearances w^hich the different 
 views are to represent, is to fold about the front, top, bottom, left, 
 and right sides of a cube, squares of paper which form the develop- 
 ment of these faces, and write front, top, bottom, left, and right 
 sides respectively upon these different surfaces. If the cube is held 
 so that the front view is visible, and the other marked sides occupy 
 
WORKING DRAWINGS. 43 
 
 their proper positions, and if the squares upon them are then re- 
 volved about the edges of the front face into the plane of this face, 
 the manner in which the cube must be turned to present the different 
 appearances will be understood at once, and also the relations of the 
 different views to the front view. This experiment should be made 
 by all the pupils. 
 
 DIMENSIONING. 
 
 44. To dimension a drawing is to place upon it all measurements 
 of the object represented, so that the workman may construct the ob- 
 ject from the given measurements. 
 
 In order to be easily read the dimensions should be so placed as 
 not to crowd or interfere with each other or with the lines of the 
 drawing, and the figures must be neatly made ; poor figures will spoil 
 the appearance of the best drawing. 
 
 Dimensions should be put down in inches and fractions of inches 
 up to two feet, thus : 16 J". 
 
 Distances greater than two feet should be given in feet and 
 inches, thus : 3' yf". 
 
 Dimensions should be placed upon dimension lines which, by 
 means of arrow-heads at each ,end, indicate the position of the 
 dimension. These lines should not be continuous, a space being 
 left to receive the figures, which should be symmetrically placed 
 upon the line, thus: 1 U-i-'^ ^ 
 
 The line may be omitted •* ^ 
 
 when the space between the arrow-heads is short ; and when there 
 is not room for both arrow-heads and dimension, the arrow-heads 
 may be turned in the direction of the measurement and , ^„, 
 placed outside the line, thus: ^^ I 
 
 When there is not room even for the dimension, arrow-heads may 
 be used either outside or inside the dimension lines, and the 
 dimension placed where there is room for it, thus : LI 
 
 Arrow-heads and figures should be drawn free-hand; (^^., 
 when the drawing is inked they should be drawn with a com- 
 mon writing pen and with black ink, while the dimension 
 lines should be red. The line separating the figures indicating the 
 fraction must be parallel to the dimension line. 
 
44 
 
 MECHANICAL DRAWING. 
 
 Vertical dimensions should be placed so as to read right- 
 
 I handed, thus : 
 The dimensions may be placed upon the drawing when 
 I there is room ; but when the space is small it is better to 
 carry the dimension outside the drawing by means of dot and 
 
 •^1^ dash lines, thus : 
 CO 
 
 ] 
 
 J^ 
 
 K '3g'' — ^ 
 
 The space between the different views is often the best 
 position for many of the dimensions. 
 
 When an object is divided into different parts and the lengths 
 are given in detail, an over-all dimension should be given. 
 
 •Dimensions should not be placed upon centre lines. 
 
 Distances between centres of all parts, such as rods, bolts, or 
 any evenly spaced parts, should be given ; and when the parts are 
 arranged in a circle, the diameter of the circle passing through their 
 centres should be given. 
 
 The diameter of a circle and the radius of an arc should be 
 given. The centre of an arc, when not otherwise shown, should be 
 marked by a small circle placed about it, and used instead of an 
 arrowhead. The dimension line should begin at this circle. 
 
 Dimensions should be clearly given in some one view, and not 
 repeated in other views ; they should seldom be placed between a 
 full and a dotted line, or between dotted lines when they can be 
 placed in a view where the part is represented by full lines. 
 
 45. Simple objects, circular in section, are often shown by only 
 one view. The fact that they are circular is indicated by D. or Dia.^ 
 understood to mean diameter, placed after the dimension. 
 
 46. When several pieces are alike, only one is drawn, and the 
 number to be made is expressed by lettering. 
 
 47. When parts are to be fitted together, it is customary to write 
 whether the fit is to be "tight'' or "loose." 
 
 48. Surfaces which are to be finished by turning or planing are 
 often indicated by placing upon a line drawn parallel to their outline, 
 the letter y, which means finish. 
 
WORKING DRAWINGS. 
 
 45 
 
 49. Any special information which cannot be expressed by draw- 
 ing is always expressed by lettering. 
 
 The name of the object represented, the scale of the drawing, its 
 number and date of completion, should be placed upon the drawing. 
 
 Simple letters should be used for all this work ; they should be 
 small and not more prominent than the drawing ; pupils will find 
 block letters made of a single line the best to use. 
 
 The draughtsman will find Soennecken's system of round writing 
 neat and practical.^ 
 
 The following are styles of lettering suitable for use : 
 
 ROUND WRITING. 
 
 12 £3 l\T6tZS 90 
 
 COPYKUIHTED BY KEUp-PEL & ESSER, 1877. 
 
 1 " This is not a system of lettering, but a scientilically evolved system of 
 writing which has the effect of lettering without requiring the same amount of 
 skill and consumption of time.". 
 
46 MECHANICAL DRAWING. 
 
 GEOMETRIC LETTERS. 
 
 ABCnE'FGHI 
 
 JKLMNDPgR 
 
 STUVWXYZ 
 
 aticdefglii 
 jklmnDpqr 
 
 S t U V "TO" X 7 z 
 
 1234567330 
 
 GOTHIC LETTERS. 
 
 ABCDEFGHI 
 JKLMNOPQR 
 S T U V W X Y Z 
 
 abcdefghi 
 j k I m n o p q r 
 stuvwxyz 
 
 1234597890 
 
CHAPTER IV. 
 
 DEVELOPMENTS. 
 
 50. The development of any object gives the real shapes of all 
 the surfaces of the object. It is obtained by unrolling or unfolding 
 the surface and placing it upon a single plane surface. Objects are 
 developable or non-developable, according as their surfaces may or 
 may not be laid out on a plane surface. 
 
 The cube, cone, and cylinder are types of developable objects ; 
 the sphere is a type of an undevelopable object. 
 
 The Cube. — The development of the cube. Fig. 16, may be 
 obtained as follows : Place any one of its faces, 
 as A^ upon paper, and with a sharp pencil trace 
 its outline. Then tip the cube over, revolving 
 it upon the edge i of the face A, until a second 
 face, B, rests on the paper ; then trace face B. 
 In the same way trace faces C and D, after 
 revolving upon edges 2 and 3. 
 
 The development of 
 these four faces is a rec- 
 tangle whose width is 
 equal to a side of the 
 cube, and whose length 
 is equal to four times the 
 side of the cube. After 
 the four faces A^ B^ C 
 and D have been placed 
 upon the sheet of paper 
 and drawn, revolve the 
 i^it'- ^7- cube so as to bring the 
 
 faces E and i^to the paper, and trace them ; these faces should be 
 so placed that one edge of each coincides with an edge of the face 
 A, B, C, or B, 
 
 
 
 F 
 
 
 
 
 
 6 
 
 
 
 A 
 
 
 £ 
 
 C 
 
 J) 
 
 
 1 
 
 Z 
 s 
 
 3 
 
 H 
 
 
 
 £ 
 
 ■ 
 
 
48 MECHANICAL DRAWING. 
 
 The real shape of each face is thus placed upon paper in what is 
 the most convenient way, supposing the cube is to be constructed of 
 tin, paper, or any other thin material capable of being bent ; for if 
 the material is cut to the given outline and then bent at the lines 
 7, 2, J, 4, 5, and 6, only the other edges of the object require 
 fastening by glue or solder. See Fig. 17. 
 
 The development of an object is thus a pattern. All objects of 
 sheet metal, from the simplest tin dish to the most complicated sheet- 
 iron work in pipes, ventilators, and boats are obtained by means of 
 such patterns. 
 
 52. When the form developed is to be constructed of paper, the 
 outside edges of the development should be provided with projecting 
 pieces called laps^ by which the different parts maybe held together. 
 These laps are shown in some of the developments given in the 
 plates of this book. Pupils making the drawings should construct 
 the forms by developing, cutting, folding, and gluing the parts 
 together ; unless the objects are made in this way, many of the pupils 
 will not understand the subject of developments. 
 
 The paper to be used for this work should be as stiff as a good 
 quality of drawing paper. A medium weight manilla is satisfactory. 
 All the edges to be folded may be cut partly through on one side by 
 a knife, to cause them to fold sharply and neatly. The laps should 
 be placed inside the object and fastened by glue or thick mucilage. 
 
 Manual and mental training of the greatest value are given by the 
 making of objects which have been drawn and developed. This 
 work has the advantage of being equally adapted for boys and girls, 
 and of requiring no special or expensive materials or tools. 
 
 53. When these experiments have been made and understood, 
 pupils will be able to apply the principles to the development of 
 solids, represented by different views, without using the object ; or, 
 if the object is used to obtain the views, it will not be necessary to 
 place it upon the paper and trace its different surfaces. 
 
 This method of tracing is, at best, an inaccurate one ; its chief 
 value is to illustrate principles so that pupils may understand that 
 the development of any object gives the actual size and shape of 
 every one of its plane surfaces, and the actual dimensions of all of 
 its curved surfaces which are developable. After these points are 
 
DE VEL OP ME NTS. 
 
 49 
 
 understood, pupils will be able to make accurate developments by 
 working from the different views of an object. 
 
 54. Prisms. The development of any prism will be obtained 
 by placing its different faces upon the paper, and tracing their real 
 forms upon it. The order in which these surfaces are drawn is of 
 little consequence. The most natural way is to revolve the object 
 in one direction, tracing the different faces, until all the lateral faces 
 have been drawn. The bases may then be drawn so that a side of 
 each coincides with an equal side of any one of the faces already 
 developed. 
 
 55. The Cylinder. The development of the curved surface of 
 
 a cyHnder will be obtained 
 by rolling the cylinder along, 
 and tracing as it moves, 
 until its entire curved sur- 
 face has passed over the 
 paper. 
 
 A right cylinder will thus 
 produce a rectangle whose 
 width is equal to the length 
 of the cylinder, and whose 
 length is equal to the cir- 
 cumference of a base of the 
 cylinder. 
 
 In exact work the cir- 
 cumference of the circular 
 base should be calculated, but in practice it may be obtained by 
 dividing the circle into such a number of equal parts that the differ- 
 ence in the distances between two points in the circle measured in a 
 straight line (the chord) and measured- on the circle (the arc) is very 
 slight. 
 
 56. By using the dividers and spacing the circle accurately, the 
 study of working drawings may be carried on without the introduc- 
 tion of mathematics ; if the points in the circle are not too far apart, 
 this method is accurate enough for much practical work. The fact 
 that this method is only approximate should be explained, and 
 when the pupils are older and have had sufficient practice with 
 
50 
 
 MECHANICAL DRAWING. 
 
 instruments suitable for accurate work, they should calculate the cir- 
 cumference. This cannot be done before the pupils enter the high 
 school, and not often then, for the instruments generally provided 
 are such that perfect work cannot be obtained by any method. 
 
 57. In Fig. 18 the circle is divided into twelve parts, which for 
 most work in the public schools, wdll be found better than a larger 
 number, since the pupils do not work accurately enough to warrant 
 smaller spaces on the circle. 
 
 If a good pair of spring dividers could be used and the entire circle 
 accurately spaced, twenty-four parts would be preferable to twelve ; 
 but the pupils will do all that can be expected, if they understand 
 the principles and succeed in making models which are neat illustra- 
 tions of the forms. 
 
 The bases of the cylinder are circles. In the development, they 
 may be drawn tangent, at any points, to the sides of the rectangle 
 which represent the development of the edges of the bases. 
 
 58. The Cone. The curved surface of the right cone. Fig. 19, 
 will be developed by rolling the cone and tracing as it moves until 
 all its convex surface has coincided with the plane of the paper. 
 If a straight line, V—^^ drawn on 
 the cone from the vertex to the 
 base, is placed upon the paper, and 
 the cone then rolled until this 
 line returns to the paper, all the 
 curved surface will have rolled over 
 the paper. All the lines that may 
 be drawn from the vertex to the 
 base are the same length. As the 
 cone rolls, it moves about its vertex 
 V, which remains stationary. • One 
 after another the different lines, 
 or elements, of its surface come 
 to the paper, and, as they are of equal length, the development of 
 the circumference of the base is an arc of a circle whose centre is 
 the vertex of the cone. The radius of this arc is the length of the 
 straight line V-4^ from the vertex to the base, and its length is equal 
 to the circumference of the base of the cone. 
 
DE VEL OPMENTS. 
 
 51 
 
 To determine the length of the arc bounding the development of 
 the lateral surface, divide the base of the cone into twelve equal 
 parts, as in the case of the cylinder, and set off, upon the arc, one 
 of these spaces twelve times. 
 
 The development of the base of the cone is a circle equal to the 
 base. This may be drawn tangent to the arc at any desired point. 
 
 59. Pyramids. Fig. 20 represents a square pyramid, whose sur- 
 face may be developed by placing any edge, as i- V, upon the paper 
 
 and revolving the object until 
 its four triangular faces have 
 been traced as they have coin- 
 cided with the paper. The 
 triangles are equal and isosce- 
 les ; their bases form the sides 
 of the base of the pyramid. 
 The lateral edges of the pyra- 
 mid are of equal length, and 
 in the development, extend 
 from the vertex V to equidis- 
 tant points in an arc whose 
 radius is the length of the 
 lateral edges. The distance 
 z-2, 2-j, j-4 and ^f.-i in this 
 
 arc, is equal to the length of a side of the base of the pyramid. 
 
 The development of the square base of the pyramid may be 
 
 placed so that one side of the square coincides with a base of any 
 
 one of the triangles forming the lateral surface of the pyramid. In 
 
 this way, any regular pyramid may be developed. 
 
 The distance from V to i or j will give the real length of the 
 
 lateral edges only when the top view shows that these edges are 
 
 parallel to the front plane. 
 
 60. The Sphere. The solids whose surfaces have been devel- 
 oped illustrate the way in which the development of any solid 
 bounded by faces or by surfaces of one curvature may be obtained. 
 The sphere is bounded by a surface which is curved in every direc- 
 tion ; this surface cannot be laid out upon a plane, for the sphere 
 touches a plane in one point only and as it rolls describes a line 
 
 Fig. 20. 
 
52 
 
 MECHANICAL DRAWING. 
 
 upon the plane. The surface of the sphere is similar to those of 
 many other solids which are curved in more than one direction and 
 thus cannot be developed. 
 
 Though the surface of the sphere cannot be developed, a spheri- 
 cal or other surface of double curvature may be covered with paper, 
 metal; or other thin substance, by using many small pieces, each of 
 which stretches slightly in places and is compressed in other places, 
 so that practically they cover the surface. The globe is thus 
 covered with narrow strips of paper which taper to a point in oppo- 
 site directions. Each of these strips is the development of the 
 curved surface of one of the equal surfaces of a solid bounded by 
 edges which are equal circles intersecting each other at two opposite 
 points, and by curved surfaces which are straight in one direction 
 and connect these edges. These curved surfaces come within the 
 surface of the sphere, but if the surfaces are narrow they are within 
 that of the sphere only a very short distance. In the case of a globe 
 covered in this way the surface of the sphere may be supposed to be 
 intersected at equal intervals by planes passing through a diameter. 
 
 The surface of the sphere may be intersected by parallel planes 
 perpendicular to a diameter of the sphere, and its surface covered by 
 the developments of the surfaces included between the cutting planes. 
 These planes intersect the surface in parallel circles, any two of 
 which may be supposed to be the bases of the frustum of a cone ; if 
 the two circles are near each other the curved surface of the frustum 
 will come but slightly within that of the sphere and in practice 
 a spherical surface may be covered by assuming and developing the 
 surfaces of many frusta. 
 
 To obtain these developments the real dimensions of the assumed 
 surfaces must be determined by measuring the real length of each 
 line by which the real shape is found in the view in which it appears 
 its real length, or, if its real length cannot be seen in any view, by 
 finding it as explained elsewhere. 
 
 Any surface which is curved in two or more directions may be 
 developed by assuming and developing cylindrical or conical sur- 
 faces which approximate the given surface. 
 
 It will often be necessary, instead of proceeding in any of the 
 ways explained, to divide a surface, by means of lines drawn upon it, 
 
DE VEL OPMENTS. 
 
 53 
 
 SO that triangular surfaces, which together form a series of plane sur- 
 faces which approximate the given curved surface, may be supposed 
 to pass through these lines. 
 
 In practical work, sheets of metal cut to form the development of 
 the assumed plane surfaces will bend and stretch to form the curved 
 surface. Such developments require advanced knowledge and are 
 not explained in this book. 
 
 . 6i. The simplest way to present the principles of this subject 
 is to trace around the surface of an object which is placed upon 
 paper as explained in this chapter. 
 
 When developments are made from drawings, it is not important 
 how the surfaces are placed as long as, when folded together, they 
 will form the object. The surfaces may be placed as they would be 
 arranged by tracing around the object, or as they would be arranged 
 by tracing the surfaces, one after the other, upon a transparent 
 plane placed in front of the object. This method is in harmony with 
 the arrangement of views adopted in this book, and therefore the 
 developments in the plates are arranged in this way. 
 
CHAPTER V. 
 SHADOW LINES. 
 
 62. Shadow lines are wider than the regular lines of the draw- 
 ing ; they have the same effect as cast shadows and relieve the pro- 
 jecting parts so as to produce an effect of perspective, even in pro- 
 jection, which is without perspective. They thus make drawings 
 easier to read. If they cannot be applied so as to produce this re- 
 sult, they should not be drawn. 
 
 Some draughtsmen suppose the light to come from behind the 
 left shoulder, in the direction of the diagonal of a cube which is 
 so placed that the front view of the diagonal is a line at 45° ex- 
 tending downward to the right, and the top view of the diagonal is a 
 line extending at 45° upward to the right. The shadow lines in this 
 case will generally be the lower and right hand lines of the front 
 view, and the upper and right hand lines of the top view. 
 
 Suppose a cube resting upon a corner, to be revolved about a 
 vertical axis. As the cube revolves the shape of its cast shadow is 
 continually changing, and, since this cast shadow is composed of the 
 shadows of the different edges which separate the surfaces in light 
 from those in shadow, it is evident that these lines are continually 
 changing. This being the case, no rule or conventional method will 
 enable one to apply shadow lines to any given position of the cube, 
 so that they shall represent the edges which actually separate light 
 from shadow. To find these edges we must obtain the cast shadow 
 of the cube ; for, except in very simple cases, it is impossible to say, 
 without finding the cast shadows, which are the edges that separate 
 the light from the shadow surfaces. This is a very complicated 
 problem, and hence shade-lining the edges that separate light from 
 shadow is not suited to the requirements of practical work. Not 
 only this, but since the edges separating light from shadow surfaces 
 are continually changing as the object moves, the shadow lines when 
 properly placed upon the drawing cannot convey, at a glance, infor- 
 mation of much value. 
 
SHADOW LINES. 
 
 55 
 
 63. Shadow lines will be of little value if they cannot be applied 
 according to some system which does not require much time or 
 knowledge to determine them, and which always represents the same 
 facts of form in the same way. 
 
 Instead of determining the edges which really cast the shadows^ 
 some draughtsmen place shadow lines on the right hand and on the 
 lower lines of the front view, and on the upper and right hand lines 
 of the top view. Conventionally shaded in this way, the result is very 
 different from that given by shading the actual shadow edges. As 
 clearness is the only point sought, it is not wise to follow the assumed 
 direction of the light eveA to this extent if a simpler and clearer 
 method can be used. As already explained, the views may be sup- 
 posed to be made upon one plane by turning the object as explained 
 in Arts. 32 and 43. In this case the light will have one direction in 
 all the views and the shadow lines will come in the same position in 
 all views, and thus be much easier to determine than when they are 
 the upper lines of the top view and the lower lines of the front view. 
 
 This being the case, it is the custom of most draughtsmen to- 
 treat all the views in the same way, as if the light came from behind 
 and at the left, its direction being 45° downward to the right. 
 
 Assuming this direction in all the views, the lower and right hand 
 outlines of all projecting parts will cast shadows, and should be 
 shade-lined as illustrated in the following cases : 
 
 \ 
 
 ^i 
 
 
 A 
 
 A square pipe in different positions is represented by A, B, C, 
 and D. The direction of the light is represented by the arrow. 
 
 64. In A^ the shade lines are the lower and right hand lines of 
 the outside, and the opposite lines of the inside of the pipe. These 
 lines at the inside are the upper and left hand lines of the drawing, 
 but lower and right hand Hnes of the top and left sides of the pipe ; 
 and every part of any object which is thus situated must be shade- 
 lined as if it were a separate object. 
 
56 
 
 MECHANICAL DRAWING, 
 
 65. In B, two sides of the pipe are parallel to the light, and the 
 lines of neither are shaded. The lower line of the upper left and the 
 lower line of the lower right side of the pipe, are right hand lines 
 and are therefore shaded. 
 
 66. In C, 1-2 makes an angle less than 45° with a horizontal 
 line, and is therefore called a lower line and is shaded, as is also the 
 parallel upper line on the inside. Line 2-4 and the parallel upper 
 line on the inside are right hand lines and are shaded. 
 
 67. In Z>, 3-4 makes an angle with a horizontal line greater than 
 45°; it is called a right hand line and is shaded, as is also the lower 
 parallel line on the inside of the pipe. Line 2-4 and the upper 
 parallel line on the inside are lower lines and are shaded. 
 
 68. E represents a cylindrical pipe. Here the out- 
 side circle from i to 2 (points in a diameter at right 
 angles to the light) is shaded below, and the inside circle 
 is shaded above points j and 4^ which are in the same 
 diameter. 
 
 69. 7^ represents a cylinder from whose end projects a 
 smaller cylinder. Here both circles are shaded below the 
 diameter at right angles to the light. 
 
 70. G and ZTare parts of the same object ; ZTis cylindrical, G 
 
 is square, and has its sides at 45° to the 
 plane of the drawing. From ZTa square 
 part projects, and from G a round part. 
 
 ^ ^ It is generally customary not to shade 
 
 the element of a curved surface, and, therefore, if this is understood, 
 a glance at this one view would indicate that the part H is not square, 
 and has no sharp edge represented by its lower outline ; the same 
 is true regarding the projection from G^ and the first impression 
 produced is that these parts are round. If they are not round, this 
 fact will be shown by the other views. The shade lines on the lower 
 part of (9, on the projection from H,, and on the right verticals of the 
 right hand parts, emphasize the fact that there are none on the round 
 parts. The right hand line of Zfis shaded because it represents an 
 edge or a plane surface. 
 
 Study of these illustrations will show how much drawings are 
 improved in effect, and how much easier they are to read, when the 
 
SHADOW LIASES. 
 
 57 
 
 shade lines are given by this simple, conventional method, which 
 may be stated as follows : 
 
 The lower lines and the right ha?id lifies of all objects are shaded 
 when these lines represent edges or surfaces^ but are not shaded when 
 they represeiit the eleme7tts of curved surfaces^ unless there are plane 
 surfaces extendi?tg back from these elements. 
 
 71. In addition it should be said that the draughtsman must use 
 discretion, and give or omit the shade lines as will produce the best 
 effect. The student, however, should shade all drawings as explained. 
 
 The lines of sections should be shaded according to the above, 
 just as if the object shown in section, or in elevation behind the 
 section, were entire and complete in itself. 
 
 Shadow lines should be about twice the width of the regular line, 
 and should come inside the line, so as not to change the dimensions 
 of the parts. 
 
 After the drawing has been completed in light working lines, the 
 pupil should go over all the lines, making them of a width and 
 strength suitable for result lines ; he should then give the required 
 width to the shadow lines by going over them again. The draughts- 
 man may ink a drawing by going over all lines not shade lines, with 
 the regular result line, and not inking the shadow lines until they are 
 put in at once of their proper width. 
 
CHAPTER VI. 
 
 INKING. 
 
 72. Pupils of the public schools, in any except advanced high 
 school classes, should not attempt to finish draw- 
 ings in ink, as they will obtain the best training 
 and best results from the use of the pencil. 
 
 To ink accurately, it is imperative that an 
 exact pencil drawing should first be made, its 
 lines being fine and clear, and the centres of 
 all arcs being carefully placed and marked by a 
 small free-hand circle about them. 
 
 A special pen, called a drawing pen^ and also 
 special ink, are required to ink a drawing. The 
 best pen for inking is one without a hinged joint, 
 having its outer blade more curved than the 
 inner one. 
 
 The illustration represents, full size, the lower 
 portion of a pen suitable for students' use. 
 
 73. India Ink. The ink to be used may be 
 liquid India ink which is sold in bottles, or ink 
 prepared as it is needed, by grinding the solid 
 stick India ink. 
 
 The liquid ink is prepared with chemicals which cause it to enter 
 the paper so that its lines are erased with much more difiiculty than 
 those made from stick ink, which is ground as it is needed. Highly 
 finished ink-drawings will be made more easily if the ink used is pre- 
 pared by grinding, but the trouble of preparing it may render the 
 use of the liquid ink advisable. 
 
 The stick ink may be prepared for use by grinding it in pure water, 
 in an ink slab which may have either of the sections illustrated. The 
 stick must be kept in motion all the time, slight pres- 
 sure being applied, until the ink is thick enough to 
 give, when dry, a perfectly black and solid line. After 
 the ink is prepared, the stick should be carefully wiped 
 to prevent its crumbling. 
 
 m 
 
INKING. 
 
 59 
 
 The slab must be kept covered to prevent evaporation and to 
 keep the ink free from dust. If the ink hardens in the slab, it must 
 be washed out. Fresh ink must be prepared every few days, as it 
 spoils quickly. 
 
 74. Inking a Drawing. The pen may be filled with a quill 
 toothpick. It is not necessary to move the blades to fill the pen ; 
 they should be set for the proper line and remain unchanged until all 
 lines of that width are inked. When filled, the outside surfaces of the 
 blades should be wiped with a cloth or chamois-skin to remove any 
 ink upon them. If any remains upon that of the inside blade it will 
 soon find its way to the straight edge, and then to the paper. The 
 pen must be thoroughly cleaned after using, to prevent the blades 
 from rusting. 
 
 The pen should be held with the inside blade against the straight 
 edge or French curve, and parallel to its edge. If it is turned so 
 that the blades are at an angle, the ink will flow to the straight edge 
 or curve and blot the paper. To keep the pen in the proper posi- 
 tion, the forefinger may be placed upon the set screw, or the thumb 
 may be placed on the edges at one side and the fingers on the 
 edges at the opposite side of the blades. 
 
 The pen should be held with slight and even pressure against the 
 straight edge or curve. If the pressure varies, the blades will spring 
 and the width of the line will change. Both blades of the pen should 
 bear equally upon the paper. A ragged line results when only one 
 blade touches the paper. To correct this defect the pen must be in- 
 clined in the direction of the broken side of the line, until both 
 blades bear equally upon the paper. 
 
 The blades should be of such length that both will bear equally 
 upon the paper when the pen is incUned slightly so as to bring the 
 inner blade near the straight edge. If the pen does not make a good 
 line when held as directed, it must be sharpened ; but this cannot be 
 done by young pupils. 
 
 The angle of the pen with the edge of the straight edge must not 
 be changed while drawing any line, as this will vary the distance of 
 the point from the straight edge and produce a crooked line. 
 
 In going over a line the second time, the pen should be inclined 
 and moved in the same direction as when the line was first drawn. 
 
6o MECHANICAL DRAWING. 
 
 If a pen does not draw a smooth line without pressure, or if it 
 cuts or scratches the paper, it should be sharpened. It will require 
 sharpening often if it is used frequently, for the blades are quickly- 
 dulled by the paper. 
 
 When inking circles or arcs, the pen point must be so inclined 
 that both blades bear equally on the paper. 
 
 To ink very small circles a good bow pen is necessary, which, 
 together with hair-spring dividers, will be required whenever really 
 fine work is desired. 
 
 All circles and arcs should be inked first; next the horizontal lines, 
 beginning with those at the top of the sheet and working downward ; 
 then the vertical lines, beginning with those at the left of the sheet 
 and working toward the right. All parallel lines, when not hori- 
 zontal, vertical, or at any of the angles given by the triangles, should 
 be inked at one time, by means of triangles used as explained in Art. 6. 
 
 If the lines are not inked in order, from the top downward and 
 from left to right, they will be blotted by placing the straight edge 
 upon them before they are dry. 
 
 - To obtain perfectly tangential arcs, the tangent point must be 
 
 found by drawing a pencil line connecting the centres from which 
 
 the arcs are drawn ; to obtain perfectly tangen- 
 
 \ "^^ tial straight lines and arcs, the tangent points 
 
 ° 1 ^^ must be found by drawing a straight line from 
 
 \ the centre of each arc, perpendicular to each 
 
 straight line to which it is tangent. The 
 draughtsman may work without these aids, but they are required 
 by students. 
 
 Hatching lines should be a little finer than outlines, and should 
 not be placed nearer together than is necessary to avoid the effect of 
 a series of bars or wires, which will be given when they are too far 
 apart. They should present the appearance of a tint : from ten to 
 twenty lines per inch will give satisfactory results. 
 
 When many working lines radiate from a point, all should not be 
 inked to the point, as this would form a blot ; they should stop at 
 unequal distances from the point. 
 
 In inking a symmetrical curve, such as the ellipse, part of the 
 curve each side of the axis should be struck with the compasses from 
 
INKING, 
 
 6l 
 
 a centre in the axis of the curve. This should be done, even if no 
 more than i or ^ of an inch can be drawn in this way. 
 
 When the surface of the paper has been roughened by erasing, it 
 may be made smooth by rubbing it with the clean, polished, rounded, 
 ivory handle of a knife or other article. 
 
 75. Different materials are shown in inked drawings by using 
 different colors for the line sectioning, or by tinting the sections with 
 washes of different colors. Centre lines are generally inked full red ; 
 cast iron is sectioned black; wrought iron or steel, blue; brass, yel- 
 low ; other materials are represented by other conventional methods. 
 
 76. Blue Prints. Shop drawings are generally blue prints. 
 These are really photographs printed from a drawing made on 
 tracing cloth. The lines are white on a blue ground. 
 
 When a drawing is finished wholly in black lines, or made for re- 
 production by the blue print process, different materials are shown 
 by different hatchings. The materials commonly used are often 
 represented as follows : 
 
 Cast iron. 
 
 Wrought iron. 
 
 Steel. 
 
 Composition. 
 
 Babbitt metal. 
 
 ^jV- ' \'/.':;v; ! ;Sv^ ' - . ,.-r 
 
 
 Vulcanite. 
 
 Wood. 
 
 Leather. 
 
 Brick. 
 
 Stone. 
 
 77. Erasing and Cleaning. The student should make erasures 
 in an inked drawing wholly by the use of pencil or ink erasers, for 
 when the knife is used for this purpose the surface of the paper is 
 injured, so that the lines cannot be inked again neatly. The pencil 
 
62 MECHANICAL DRAWING. 
 
 eraser will, if used for several minutes, remove ink lines without 
 injuring the surface of the best drawing papers. The ink eraser 
 removes the lines more quickly, and generally gives satisfactory 
 results. 
 
 When the inking is finished the whole drawing may be cleaned 
 by rubbing it wdth 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 draw- 
 ing, 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. 
 
 78. To Sharpen the Pen. — The blades of the pen should be 
 curved at the points, and elliptical in shape. To sharpen the pen, 
 screw the blades together and then move the pen back and forth 
 upon a fine oil-stone, holding it in the position it should have when 
 in use, but moving it so that the points are ground to the same 
 length, and to an elliptical form. When this form has been secured, 
 draw a folded piece of the finest emery paper two or three times be- 
 tween the blades, which are pressed together by the screw. This 
 will remove any roughness from the inner surfaces of the blades; 
 these surfaces should not be ground upon the oil-stone. 
 
 When the blades are ground to the proper shape, they must be 
 placed flat upon the stone and ground as thin as possible without 
 giving them a cutting edge. To do this, the pen should be moved 
 back and forth and slightly revolved at the same time. Both blades 
 must be made of equal thickness. If either blade is ground too 
 thin, it will cut the paper as would a knife, and the process must be 
 repeated from the beginning. In order to see the condition of the 
 blades, they should be slightly separated while being brought to the 
 proper thickness. 
 
 79. Stretching Paper. — When paper is to be stretched it should 
 be dampened and then immediately secured to the drawing board 
 by mucilage. It is not necessary to strain the paper so that it is flat 
 while it is wet ; it is better not to do this, for, if this is done, the 
 tension created in drying may cause the paper to tear if the board 
 
INKING. 
 
 63 
 
 is dropped, and when the paper is cut from the board, it may shrink 
 enough to change the dimensions of the drawing a sixteenth or, if 
 the drawing is large, even an eighth of an inch. 
 
 80. The following are the steps involved in stretching paper : 
 
 1. Turn over and press down for about y^ the edges of the 
 
 paper, all the way around the sheet. 
 
 2. With a clean sponge and water, dampen all the upper surface 
 
 of the paper, including that of the folded edges. Allow^ 
 the sheet to stand, and moisten again if necessary, until 
 the paper has become dampened and swollen throughout. 
 Time may be saved by moistening both sides of the paper. 
 When this is done the side of the folded edge which is to 
 receive the mucilage should be dried with a blotter or 
 cloth. The paper should not be rubbed with the sponge, 
 as this will roughen and destroy the surface. 
 
 3. Apply to the folded edge a thick mucilage made by dissolving 
 
 cheap gum arable in cold water. 
 
 4. Turn the edges over and press them upon the board, begin- 
 
 ning in the centre of each side and working toward the 
 corners. 
 
 5. Press the edges upon the board and rub their surfaces until 
 
 they hold firmly. 
 
 6. See that the mucilage holds firmly before leaving the paper to 
 
 dry, as the strain will pull the paper from the board if the 
 mucilage does not set while the paper is damp. 
 
 7. The board should be left in a horizontal position with no 
 
 water upon the surface of the paper; it should never be 
 placed near a radiator or in the sun. 
 
 8. The edges may be quickly set by rubbing them with a piece 
 
 of polished hot metal. 
 Mucilage is better than glue for stretching paper, as it does 
 not set too quickly, and the edges of the paper may be 
 easily removed from the board when the sheet is cut off. 
 To remove these edges, cover them with water ; after a few 
 minutes the paper will absorb the w^ater and they can 
 readily be removed. 
 
CHAPTER VIL 
 
 MACHINE SKETCHING AND DRAWING. 
 
 8i. Projection forms the basis of practical working drawings ; 
 in order to make technical drawings one who understands projection 
 has simply to become familiar with the different conventional repre- 
 sentations and methods which cause practical drawings to differ 
 from complete projections of the objects. 
 
 The draughtsman finds it necessary to make drawings of machines 
 already made, as well as of those which he designs. To do this he 
 makes free-hand sketches of the machine, measures all its parts, and 
 places the dimensions upon the sketches, never making more views 
 than are necessary to show the construction of the object repre- 
 sented. He then returns to the office and makes, from his sketches, 
 finished drawings to scale. 
 
 This book is intended to explain projection principles and also 
 the making and arrangement of practical working drawings, and 
 illustrates drawings of both kinds. The objects used by pupils for 
 study must be simple and easy to obtain. In order that pupils 
 may understand how to represent more difiicult subjects, it is neces- 
 sary to give more views than would be required to make the objects. 
 This should be remembered when the plates are studied. 
 
 Projecting, or guiding, lines as they are sometimes called, are 
 not given in practical drawings, and are given in this book in the 
 drawings of machine details only when they are necessary to show 
 how projection principles are applied to special points. 
 
 In order to make complete working drawings of the drawings 
 previously made and explained, it will only be necessary to place 
 dimensions upon them and to indicate the material to be used and 
 the amount of finish to be given the different parts. 
 
 In the plates dimensions or dimension lines have been placed 
 upon a sufficient number of drawings to illustrate the way in which 
 dimensions should be placed. 
 
MACHINE SKETCHING AND DRAWING, 65 
 
 82. Sections are of great value to the machine draughtsman, as 
 they enable him to dispense with many comphcated views. They 
 are usually taken horizontally or vertically, but may be taken in any 
 direction ; they are often taken at right angles to parts whose real 
 shapes are not shown by views of the outside of the object. The 
 part of the object behind the cutting plane is generally shown in 
 addition to the surface cut by the plane ; but the surface cut by the 
 plane is often all that is necessary to give the desired information, 
 and is all that is given in many drawings representing sections. 
 The section of the arm on line AB in Fig. 117 illustrates this 
 point. Such a section may be placed as illustrated. It is often 
 found about AB and on the part sectioned ; in this case the part 
 is often represented as broken, thus leaving a space for the section. 
 
 83. When an object is too long to be shown of its entire length, 
 and is of one shape throughout or for any great distance, the central 
 part may be considered removed, each end being represented as 
 broken,, and the entire length shown by the arrow-heads and figures. 
 
 The positions at which sections are taken are usually indicated by 
 dot-and-dash lines. 
 
 84. The surface of the material cut by the cutting plane is cross- 
 hatched with lines which are generally drawn at 45°, and in the same 
 direction on the same piece, wherever it may extend or however 
 much it may be cut up or intersected by other pieces. The distance 
 between the lines must be the same throughout all the surface of any 
 one piece. 
 
 When different pieces are cut by the plane, they should be 
 hatched in opposite directions, especially if there is no space 
 between them. When three or more pieces are cut and come 
 together, they can be distinguished by a difference in the spaces 
 between the hatching lines. 
 
 85. When objects such as bolts, rods, or other solid parts lie in 
 the plane in which a section is taken, they should be shown in eleva- 
 tion ; for time is required to hatch the parts, and nothing is gained 
 by doing this, since they are solid. Fig. 125 illustrates this point. 
 
 86. It is not necessary that the section be taken on one straight 
 line; it may be taken so as to produce the clearest drawing. Thus 
 in Fig. 119 a vertical section through the upper part of the caster 
 
66 MECHANICAL DRAWING. 
 
 would cut the frame so as to show part in elevation and part in 
 section. It is much clearer to suppose the cutting plane to pass 
 obliquely through the centre of the frame, and to suppose the wheel 
 to be cut by a vertical plane, than to adhere to the facts of projection 
 given by the use of any one plane. 
 
 87. If a wheel having arms is situated with one or two arms in 
 the plane of the section, neither arm should be sectioned, for the 
 sectional view would then not differ from that of a solid wheel. 
 Thus in Fig. 116 a horizontal section of the wheel would cut two 
 arms, but should be represented the same as the vertical section. 
 
 88. The aim of the draughtsman is to convey a clear idea of the 
 object to be constructed, and, as certain parts and details which are 
 common to various classes of work are generally perfectly known and 
 always recognized from even one view, he often shows a part (for 
 instance a set screw) in position in one view, and omits it in other 
 views where simply its position is shown. Instead of perfect projec- 
 tions he is content to make conventional drawings ; he is generally 
 considered free to use his judgment, both as to the views w^hich shall 
 be made and as to the essentials of these different views ; he never 
 makes more views of an object than are needed to show its construc- 
 tion. 
 
 89. It is necessary to make drawings which show all parts of an 
 object in their proper positions. These views are called assembly 
 draiuings ; upon them are placed important over-all dimensions, 
 distances between centres, etc. For the workmen, detail drawings 
 are prepared which give as many views as are necessary to show 
 fully all the different parts which make up the complete machine. 
 
 The assembly drawing shows these details in the positions they 
 occupy when in use, and many of them, owing to the positions they 
 occupy, may not be clearly shown. The detail drawings arrange 
 these parts, without regard to the position which any part occupies 
 in the machine, so that their forms are clearest shown by the fewest 
 views. 
 
 Other points are explained on the pages opposite the plates, 
 Chap. XII. 
 
CHAPTER VIII. 
 
 ORTHOGRAPHIC PROJECTION. 
 
 90. Orthographic Projection is the art of representing an 
 object by means of projections or views made upon different planes, 
 at right angles to each other, by the use of projecting lines perpen- 
 dicular to the different planes and passing through all the points of 
 the object. 
 
 The work of the draughting office often requires no thought of 
 the planes of projection ; hence some teachers claim that reference 
 to these planes is unnecessary ; they assert that pupils should not be 
 asked to consider problems of projection which are difficult for most 
 of them to understand. 
 
 It is easier to remember rules than to understand and apply prin- 
 ciples. Working drawings may be made by rule with far^ less effort 
 on the part of the teacher and pupil than when they are made as the 
 result of knowledge of projection principles ; but this does not prove 
 the principles useless or unnecessary. 
 
 To draw from the object is very simple, and is all that can be ex- 
 pected from grammar school pupils ; sometimes little more can be 
 done in high, evening drawing, and in elementary technical schools; 
 but in the high and elementary technical schools, there may often be 
 found students able to reason and to understand the principles of 
 projection well enough to draw the views of a simple object from a 
 written description of the object and its position. Such students 
 should have the benefit of the training given by a course in projec- 
 tion. Many pupils unable to understand projection sufficiently to 
 draw even simple objects from written statements, will nevertheless 
 be interested and benefited by a talk upon its principles ; and 
 teachers should give instruction in the principles of projection when- 
 ever pupils are able to profit by such instruction. 
 
 Any draughtsman ought to be able to draw without the object 
 before him all the time, or without being given in whole, or in part, 
 
68 ' MECHANICAL DRAWING. 
 
 one or two views of the object. Yet the instruction given by many 
 teachers, while it enables pupils to draw from the object, to complete 
 unfinished views, and to add other views when some are given, does 
 not give the capacity for representing objects conceived in the mind 
 in definite positions. 
 
 91. A common method of instruction is for teachers to place upon 
 the blackboard two views of an object and ask the pupils to draw the 
 third ; or to give the axis or a line or two of drawings which are to 
 be completed by the pupils. This method causes the students to re- 
 member the problem, or a similar one, and the steps involved in its 
 working. There are few who with this assistance are unable to make 
 the required drawings. 
 
 If the representation of constructed objects was the only work of 
 the draughtsman these methods woiild always be as satisfactory as 
 they are in the simple work of object sketching and draughting, 
 whether from the object constructed or imagined ; but the draughts- 
 man often has problems which require objects to be placed in certain 
 definite relations to each other. In drawing, he must refer to some- 
 thing, and whatever this is called, it is the plane of the drawing or a 
 parallel plane. If he is unable to draw a line which makes definite 
 angles with the ground and with the vertical plane of the drawing, 
 he cannot represent an object as simple as a cube or cylinder, which 
 is to have some definite position. 
 
 Whatever may be done with the most elementary work, the 
 advanced student should be able to draw simple geometric solids 
 from a description of the solid and its position. To do this the 
 planes must be thought of ; and teachers who are proclaiming 
 against the use of projection methods will do great harm if they con- 
 fine students old enough to understand and profit by these methods, 
 to work from the object, and they must be thus confined when pro- 
 jection methods are not used. 
 
 Models and objects are necessary in the first study of work- 
 ing drawings or of projection, but they should be dispensed with as 
 soon as possible. 
 
 Pupils should learn to see the object mentally when its dimen- 
 sions and relations to the planes are given, and to see also the pro- 
 jections of the object upon these planes. They will then be able to 
 
ORTHOGRAPHIC PROJECTION, 69 
 
 draw without reference to models. Only in this w^ay is it possible to 
 understand the subject of projection; and it should be realized that 
 those who are protesting against such study cannot produce a sub- 
 stitute method which will accomplish the same results. 
 
 92. When pupils are taught to make working drawings solely by 
 observation of the object, it is impossible for them to work in any 
 other way ; they cannot draw even two views of the simplest form, 
 such as a cone, whose axis makes an angle of 30'' with the ground 
 and 45° with the front vertical plane. 
 
 In giving an examination in the subject it is impossible, unless 
 reference is made to the planes of projection, to call for two views of 
 any line or object inclined to both planes, without giving lines or 
 views which tell the pupils what they are to do, so that the work 
 becomes memory instead of reasoning. 
 
 The objection to reference to the planes of projection is absurd 
 when coming from those teachers who give to their classes difficult 
 problerns of intersections and oblique projections, which cannot be 
 solved except by the use of cutting and other auxiliary planes, whose 
 relations are much more difficult to understand than those of the 
 planes of projection. Though not named, die planes of projection 
 must still be practically used; hence, they should be explained and 
 the study carried on by projection methods when pupils .attempt 
 these problems. 
 
 93. Teachers of drawing in any school above the grammar, 
 ought to understand projection and descriptive geometry, even if 
 these subjects are not explained to their classes ; they will then be 
 able to answer questions relating to the representation of form, 
 abstract or concrete, even if they have not studied the various 
 methods in which the principles are applied to the practical work of 
 the mechanic. 
 
 It is true that knowledge of descriptive geometry will not enable 
 one to represent details of construction in the conventional ways 
 peculiar to the architect or the machine draughtsman ; but this 
 knowledge is the foundation for all drawings, and conventionalities 
 are quickly understood and applied by one who is able to represent 
 correctly any form in any desired position. 
 
 94. The principles of projection which have been explained in 
 
70 
 
 MECHANICAL DRAWING. 
 
 connection with the study of working drawings cover the simple 
 work of the draughting office, and are all that the teacher in the 
 grammar grades requires ; but teachers of drawing, advanced stu- 
 dents in the high, and students in technical schools, should carry the 
 study farther. For their use the following drawings and explana- 
 tions are given, and will be all that are required for knowledge of 
 the simple principles of orthographic projection. A thorough under- 
 standing of the subject requires a course in descriptive geometry. 
 
 95. This book, is intended for teachers of public and elementary 
 schools, in which the subject of working drawings is more important 
 than projection. The best arrangement of views for working draw- 
 ings is explained in Chapter III. This arrangement is different 
 from that given by the planes generally used in the study of pro- 
 jection. 
 
 The drawings of the plates of this book are arranged as working 
 drawings should be arranged, and, in order that there may be no 
 confusion, the following notes on projection make use of planes of 
 projection which produce the arrangement of views chosen for use 
 in the study of working drawings. 
 
 Some of the objects represented on the plates, with their rela- 
 tions to assumed planes of projection, are described at the end 
 of this chapter. These planes are not represented in the drawings, 
 and those who study the following notes may test their knowledge of 
 the subject by using the statements as test questions, and making 
 the projections required to represent the objects described. 
 
 96. Projection Principles. — The front, top, side, back, and 
 bottom planes of projection are represented in Fig. 21. 
 
 The drawings or tracings made upon these planes are the ^i^^r- 
 ^Xi\. projections^ or views^ of the object. The perpendiculars to these 
 planes, by means of which the views are obtained, are the projecting 
 lilies. The projections will be called views throughout this chapter. 
 
 In all the drawings a point will be designated by a letter or fig- 
 ure ; its projections by the same letter or figure with that letter as 
 exponent which represents the plane upon which the view is found. 
 Thus 7" means the top, 7^' the front, R the right side, L the left side, 
 B the back, and G the bottom view. To avoid confusion the expo- 
 nents are often placed upon only part of the letters; but in any view 
 
ORTHOGRAPHIC PROJECTION. 
 
 71 
 
 where an exponent is found upon any letter, the same exponent 
 should be understood when reading the other letters. 
 
 97. Axes of Projection. Fig. 22 represents the top, bottom, 
 and side planes when they have been revolved to coincide with the 
 plane of the front vertical plane. 
 
 The lines in which the planes of projection intersect are called 
 axes of projection. The different axes are named in Figs. 21 and 22. 
 
 When the planes are in the positions shown in Fig. 21, and the 
 surface of any one plane is seen, the other planes are seen edgewise 
 and are represented by the axes of projection. Thus, when the front 
 plane is seen, the front horizontal axis at the top represents the top 
 horizontal plane ; the front horizontal axis at the bottom represents 
 the bottom plane, and the right and left side planes are represented 
 by the right and left vertical axes. 
 
 In the first study of projection the front vertical plane, the top 
 horizontal plane, and the two side vertical planes are the only planes 
 of projection which are generally required. Their axes of projection, 
 will be designated by the following abbreviations : F, H. Axis ; R,. 
 H. Axis; L. H, Axis ; R, V. Axis and Z. V. Axis. The lower hori- 
 zontal plane not being used, the terms are understood to refer to the 
 upper axes. 
 
 98. Views of a Point. Fig. 21 represents a point, A^ situated 
 between the planes of projection. A projecting line from A to the 
 front plane gives A^^ which is the front 
 
 view of point A ; 2. projecting line from A 
 to the left side plane gives A^^ which is 
 the left side view of A; a projecting line 
 from A to the top plane gives A^^ the top 
 view oiA; projecting lines to the right 
 side plane, to the back plane, and to the 
 lower plane will give the views of A upon 
 these planes. These views may be desig- 
 nated by A^, A^^ and A^ respectively. 
 
 99. The projecting Unes from A to the vertical planes are 
 horizontal and in a horizontal plane passing through A. This plane 
 intersects the different vertical planes in four horizontal lines, which 
 are at equal distances from the top plane, and form a rectangle. 
 
 Fig. 
 
72 
 
 MECHANICAL DRAWING. 
 
 When the planes of projection are revolved into the plane of the 
 front plane, these horizontal lines form one straight line which 
 passes through A^ and contains the different vertical views of A. 
 The projecting lines from A to the front, back, top, and bottom 
 planes are in a vertical plane which is perpendicular to the front 
 plane and parallel to the side planes. This vertical plane intersects 
 the front, back, top, and bottom planes in a rectangle whose sides 
 develop into a vertical line when the planes are revolved into the 
 plane of the front plane. Thus the views ^^, A^^ and A^ are in a 
 vertical line passing through A^. 
 
 >^L>^^ 
 
 iCS 
 
 ^ Lef-t l/or.9titi: 
 
 "A^L 
 
 lp//J/i'/-V/»r//fa/ 
 
 ^.H-h bdr. ax/i .8. 
 
 Too "Horijepnfi^l Pldnp 
 
 /A" 
 
 Front- ^or 9Aij T. 'VI 
 
 
 yj* 
 
 Solfom Hi/mionM/ Plane 
 
 \ 
 
 Ri^hf- f?or ^^is.To/>. 
 I 
 
 \ PUnr 
 
 Jii^hh bar- » Alii. B. 
 
 Fig. 22. 
 
 100. Position of a Point (Figs. 21 and 22). Point ^ is a cer- 
 tain distance from each of the planes of projection. In the front 
 view A^-i is the distance of A above the bottom plane ; A^-2 is the 
 distance of A below the top horizontal plane ; A^-j and A^-4 are the 
 distances of A from the side vertical planes. In the top view the re- 
 lations of A to the side vertical planes are seen, also the distance 
 A'^-2, of A behind the front vertical plane. Projecting lines from 
 ^^ to the right and left horizontal axes give the top projections A\ 
 A" of A Upon the side vertical planes. When the left and right 
 horizontal axes are revolved about points 5 and 6 respectively, points 
 
ORTHOGRAPHIC PRO/ECTIOA\ 73 
 
 ^'" and ^"" are obtained. The side views of A are in verticals from 
 these points and in a horizontal line through A^. The distance A^-j 
 and A^-4 is thus the same as A'^-2^ and shows the distance of A from 
 the front vertical plane. In the same way the view of A upon the 
 bottom plane agrees with the other views. 
 
 101. Having the front and top views of any object or point 
 given, the side views can always be obtained by projecting from the 
 top view to the left and right horizontal axes, revolving these axes 
 until they coincide with the front horizontal axis, then drawing verticals 
 from the points in the horizontal axes and intersecting them by hori- 
 zontals from the points of the front view. Having the front and side 
 views given, the reverse of the above process will give the top or 
 bottom view. When the planes are seen from above, and the points 
 of the top view have been projected to the side planes (that is to the 
 right and left horizontal axes) these projected points describe, when 
 the planes (axes) revolve, arcs of circles whose centres are in the ends 
 of the axes (5 or (5, Fig. 22.) In the same way, the points of either 
 side view, when they have been projected to the horizontal axis, 
 describe, when the axis revolves, arcs of circles whose centre is an 
 end of the axis. 
 
 102. In order to understand the subject, students must accus- 
 tom themselves to looking at the different views separately. When 
 looking at A^ only the front view should be seen. The lower edge 
 of the front plane should represent to their minds the bottom plane, 
 which seen edgewise appears a line ; the upper edge of the front 
 plane should represent the top plane, of which the edge only is seen, 
 and the vertical edges of the front plane should represent the vertical 
 side planes, which, seen edgewise, appear vertical lines. 
 
 In the same way, when the top plane or a side plane is seen, the 
 edges of these planes should represent the planes at right angles to 
 them. It is possible to memorize rules and methods and in this way 
 to make drawings which are correct ; but this does not give the 
 power to do original work, or the best work, or to attain real under- 
 standing of the subject. The lines must represent planes and solids 
 and space, and only by means of this mental construction of the 
 actual conditions, can drawings mean more than the lines of a com- 
 plicated plane geometrical construction. 
 
74 
 
 MECHANICAL DRAWING. 
 
 103. Views of a Straight Line. Fig. 23 represents the planes 
 of projection and a triangular prism formed by bisecting a cube as 
 explained in Art. 27. The edges of this object represent the lines to 
 be studied. 
 
 The views are obtained by means of projecting lines, which are 
 perpendicular to the planes. It follows that a line which is perpen- 
 
 
 
 TOP 
 
 / 
 2 
 
 
 3 
 
 
 
 
 
 
 
 
 LEFT 
 
 
 
 FRONT 
 
 Si 
 
 
 ■3 
 
 J{iGnT 
 
 A 
 
 
 
 
 
 \ 
 
 Fig. 23. 
 
 dicular to any plane will be projected upon that plane as a point, for 
 the projecting lines from its two ends coincide with each other and 
 with the line. Thus points 2 and j are the upper ends of vertical 
 edges which are perpendicular to the top plane and are represented 
 in the top view by points 2 and j ; points i and 4 are the back ends 
 of edges which are perpendicular to the front plane and are projected 
 
ORTHOGRAPHIC PROJECTION. 
 
 75 
 
 upon it as points i and 4. Edges i-j and 1-4 are perpendicular to 
 the side planes and are represented in the side views by points 2, j 
 and 7, 4, 
 
 The projecting lines to any one plane are parallel to each other; 
 the distance between them must be measured perpendicularly, that 
 is by a line which is parallel to the plane to which the projecting 
 lines extend. It follows that the view of a line on a plane to which 
 it is parallel must be parallel and equal to the line. Edges 2-j and 
 1-4 are parallel to both top and front planes and their views upon 
 these planes are thus parallel and equal to each other and to the 
 lines. 
 
 Edge 1-2 is oblique to the top and front planes, therefore the pro- 
 jecting lines from its ends to these planes make its view^s shorter 
 than its actual length. The edge 1-2 is parallel to the side planes, so 
 its views upon these planes give its real length. 
 
 A line connecting the corners i and j is a diagonal of the 
 sloping face of the prism ; it is oblique to all the planes and there- 
 fore in all the views the distance 1-3 is less than the actual distance 
 between the points i and 3. 
 
 Fig. 24. 
 
 '* lop Plil 
 
 ¥3 
 
 heffSik Pkiif. 
 
 Fi^oiit Fkne. 
 
 "IT 
 
 F iinr.»\-is. 
 
 Fig. 25. 
 
 104. Views of a Plane Surface. Figs. 24 and 25 represent 
 three planes of projection, and a rectangular card, i, 2, 3, 4, parallel 
 to the front vertical plane and perpendicular to the top and side 
 planes. Those who understand the previous figures will understand 
 these drawings at a glance. The front view gives the real dinicn- 
 
76 
 
 MECHANICAL DRAWING. 
 
 sions of the card and the positions of its edges with reference to the 
 top and side planes ; the top view gives the width of the card, its 
 relation to the front and side planes, and its distances from these 
 planes ; the side view gives the length of the card, its relation to the 
 front and top planes, and its distances from these planes. 
 
 105. Views of a Solid. — Figs. 26 and 27 represent a rectan- 
 gular pyramid w^hose axis is vertical and at given distances from the 
 planes of projection. Two edges of the base of the pyramid are 
 parallel and two are perpendicular to the front plane. 
 
 Fig. 26 is a perspective view which represents the pyramid and 
 the planes of projection, with the views of the pyramid upon them. 
 
 
 
 "f 3 
 
 
 
 I 1 
 
 
 '^^>^^^^ 
 
 
 
 
 .^^-. 
 
 . 
 
 J 
 Top Pkine. 
 
 1 z 
 
 A 
 
 
 y^ 
 
 
 z^ 
 
 \ 1 / L.d.PiAiw 
 
 Trout Fl^ne. \ i / 
 
 
 Fig. 26. 
 
 Fig. 27. 
 
 Fig. 27 represents the planes and their respective views when the 
 top and side planes have been revolved into the plane of the front 
 plane. 
 
 The base of the pyramid is at right angles to the front plane, 
 upon which it is projected as a horizontal line 1^-2^ \ it is parallel to 
 the top plane, upon which its real shape is given. 
 
 The axis, being perpendicular to the top plane, is projected upon 
 it as a point 5'^, at the centre of the base, and to this point the top 
 views of the lateral edges extend and form the diagonals of the 
 figure. The right and left triangular faces of the pyramid are per- 
 pendicular to the front plane, and are projected upon it as lines 
 2^-5^ and i^S^' The two wide triangular faces of the pyramid 
 are oblique to the front plane, and the two narrow triangular faces 
 
ORTHOGRAPHIC PROJECTION. 
 
 17 
 
 are oblique to the side plane ; thus the real shapes of these faces are 
 not given in either view. 
 
 1 06. Simple geometric forms are used in the first study of projec- 
 tion. Generally the views of these forms upon the front, top, and one 
 side plane are all that are required to describe the object. It is not 
 customary to limit these planes of projection except by the three 
 lines, or axes, in which they intersect. These axes are the front 
 horizontal axis, a left or right horizontal 
 axis, and a vertical axis at the left or right, 
 and are represented in the illustration. 
 
 The F. hor. axis is the only one re- 
 quired to locate objects with reference to 
 the front and top planes. Figs. 28, 29, 
 and 30 show how the views of objects 
 may be obtained, using only this axis. 
 
 107. Views of a Rectangular Card. — Fig. 28 gives the front 
 and top views of a rectangular card, 2^X4'', which is parallel to, and 
 
 Ttont hof.^j'ii- 
 
 JUi^Ilt- 7;af'.?//S- 
 
 Scale 
 
 Fig. 28. 
 
 Fig. 29. 
 
 Fig. 30. 
 
 2'' behind the front plane ; its short edges are parallel to the top 
 plane, the upper one being i" below the plane. 
 
 The front view is a rectangle 2^X4", its long sides vertical, 
 and its upper short side i" below the axis of projection. The top 
 view is a horizontal line 2" above the axis of projection. 
 
7 8 MECHANICAL DRAWING. 
 
 108. Fig. 29 represents the same card as Fig. 28, when it is 
 parallel to the front plane, and the same distance behind it as in 
 Fig. 28, but with its long edges at 60° to the top plane. The front 
 view gives the real shape of the card, and the real angles which 
 its edges make with the top plane. The top view is a horizontal 
 line. 
 
 109. Fig. 30 represents the same card when perpendicular to 
 the top plane, and the same distance from it as in Fig. 29 ; its 
 edges are at 60° and 30° to the top plane as in Fig. 29, but its 
 surface is at 30° to the front plane, instead of parallel to the plane 
 as in Fig. 29. 
 
 When the card is in the position shown by Fig. 29, its top view 
 is a horizontal line whose length is obtained by projecting from 2 
 and J of the front view. If the card is revolved about a vertical axis 
 passing through point z, the angles of its edges and its surface with 
 the top plane not changing, the top view will show the angle of the 
 card with the front plane, but its length will not change ; therefore, 
 for all positions of the card during a complete revolution, the only 
 change in the top view is in the angle which it makes with the front 
 horizontal axis. 
 
 Suppose the card to revolve about point i as described, points 2, 
 J, and 4 describe horizontal circles whose centres are in a vertical 
 line passing through i. These circles appear circles in the top 
 , view, and as horizontal lines in the front view. It will be seen that 
 points 2, J, and 4 of the front view, which represents the card when 
 its surface is at an angle to the front plane, must be in horizontal 
 lines drawn through the corresponding points of Fig. 29. Hence, to 
 obtain the front view of the card when at any given angle, it is 
 necessary simply to place the top view of Fig. 29 at the required 
 angle and draw vertical projecting lines from all its points, and 
 intersect them by horizontals from the corners of the front view of 
 Fig. 29. 
 
 The views of Fig. 30 cannot be obtained without making use of 
 those of Fig. 29, or of substitute drawings. The views of a solid 
 oblique to one or both the principal planes of projection may be 
 determined in the same way as those of the card. 
 
 no. Views of a Pyramid. — Fig. 31 represents a pyramid 
 
ORTHOGRAPHIC PROJECTION. 
 
 79 
 
 whose base is a rectangle 2"X3"; its axis is 5" long, is vertical, 
 and 2" behind the front plane ; the vertex of the pyramid is i" from 
 the top plane, and the long edges of the base are parallel to the front 
 plane. " 
 
 III- Fig. 32 represents the same pyramid after it has been 
 revolved to the right through an angle of 45°, about the right edge 
 of the base ; all the edges of the object are situated with reference 
 to the front plane as they are in Fig. 31. 
 
 Suppose the object to be moved to the right of its position in 
 Fig. 31, its distance from the front plane not changing, and then 
 
 SC/^LE. 
 
 Fig. 
 
 v''- / 
 
 \ i \ i 
 
 ._^ 
 
 
 Fig. 32. 
 
 Fig. 33. 
 
 I'lG. 34. 
 
 revolved to the right about the edge 2-4. As the object revolves, 
 all points in it move in arcs of circles whose centres are in 2-4 and 
 which are parallel to the front plane and appear horizontal lines in 
 the top view. The points in the top view of Fig. 32 must then be 
 in horizontal lines drawn through the corresponding points of the top 
 view of Fig. 31. 
 
 The object may revolve through a complete circle, and cause no 
 change in the size or shape of the front view; thus the front view of 
 Fig. 32 will be the front view of Fig. 31, with the axis at an angle of 
 4S° instead of vertical. The top view will, however, present an 
 infinite number of different appearances, as the object revolves. 
 
8o MECHANICAL DRAWING. 
 
 To obtain the views of the object when its axis is parallel to the 
 front plane as in Fig. 31, and at any angle to the top plane, it is 
 simply necessary to place the front view of Fig. 31 wdth its axis at 
 the required angle ; to draw vertical projecting lines from all its 
 corners and intersect these verticals by horizontals from the corre- 
 sponding points of the top view of Fig. 31. The front view in Fig. 
 32 might be drawn without reference to Fig. 31, but it will often be 
 necessary to draw the views of an object situated as in Fig. 31, in 
 order that the shape of the front view for Fig. 32 may be known. 
 
 112. Fig. -^Ty represents the object when its axis is at an angle 
 with both planes. 
 
 Suppose the pyramid placed as in Fig. 32, with its axis at 45° to 
 the top plane, to be moved horizontally to the right and then revolved 
 about a vertical axis passing through its vertex, through a complete 
 circle, the angle of the axis with the top plane not changing, and 
 the distances of all points of the object from the top plane remaining 
 as in Fig. 32. The top view of the pyramid 'in any and all of its 
 positions must be the same in form and size as the top view in Fig. 
 32'. Hence, to obtain the projections of the object when its axis is 
 at 45"^ to the top plane, as in Fig. 32, and the vertical plane con- 
 taining the axis is at any given angle, say 45*^, to the front plane, we 
 have simply to place the top view of Fig. 32 so that the axis of the 
 pyramid makes an angle of 45° to the front axis of projection, and 
 draw verticals from the points of this view, and intersect them by 
 horizontals from the corresponding points of the front view of Fig. 
 32. Thus the front view of any point, as i, will be in a vertical 
 from I in the top view of Fig. 2)2>i ^^^d in a horizontal from i of the 
 front view of Fig. 32. 
 
 In this way, by considering one point at a time, the views of the 
 most complicated objects can be obtained. 
 
 Similarly, the front view of Fig. ^iZ "^^Y be revolved through any 
 desired angle and a top view to correspond be found by projecting 
 from the top view of Fig. ^^ ; this process of revolving first one view 
 and then the other may be repeated as many times as is desired. 
 
 Generally no more than the revolution of one top view and 
 one front view will be necessary to give views of any object at angles 
 to both planes of projection. 
 
ORTHOGRAPHIC PRO/ECTIOA'. 8 1 
 
 Fig. 34 shows the top view of Fig. 31 revolved through an angle 
 of 45°. The front view in Fig. 34 may be revolved as desired and 
 thus any appearance of the object be obtained. 
 
 113. All the vertical and horizontal projecting lines may be 
 drawn before any of the points of the required view are determined ; 
 but it is much simpler for elementary pupils to draw the projecting 
 lines for the different points, one at a time, and mark the points of 
 the view as they are determined. 
 
 The points of solids studied should be numbered, and the num- 
 bers carefully placed in all the views so that they indicate cor- 
 responding points. When this has been done, to obtain the points 
 for any required view, as the top view of Fig. 32, it is simply neces- 
 sary to draw a vertical from any point, as 5, of the front view and in- 
 tersect it by a horizontal from 5 of the top view of Fig. 31. 
 
 114. True Length and Position of a Straight Line. The true 
 length of a line is given by any view only when the line is parallel to 
 the plane upon which the view is made ; when thus situated the view 
 gives not only the true length of the line, but its true positions with 
 reference to the planes at right angles to the one upon which the 
 view is made. Frequently a line is oblique to all the planes, and its 
 real length is not shown by any view ; hence it is important that a 
 simple method of determining the real length of a line be found. 
 
 This problem is solved in the figures of this book as follows : 
 Drawing A represents a line a b which is situated so that its 
 front view makes an angle of 30° with the top plane (F. H. axis), 
 and its top view makes an angle of 45° with the front plane (F. H. 
 axis). To find the real length of ^ ^ and the angle it makes with 
 the top plane, the line must be revolved until it is parallel to the 
 front plane, when it will be represented in the top view by ^^//. 
 A.S ab revolves, b moves in an arc which, in the top view, is repre- 
 sented by the arc b'^ b\ and in the front view by the horizontal line 
 b^ b^\ Line a^ b^^ is the real length of a b, and gives the real angle 
 of ^ ^ with the top plane. 
 
 If the angle of ab with the front plane is desired, a b must be 
 revolved until it is parallel to the top plane, as illustrated by drawing 
 B. As ab revolves, b moves in an arc which is represented in the 
 front view by the arc b^^ b' ; and in the top view by the horizontal 
 
82 
 
 MECHANICAL DRAWING. 
 
 ABC 
 line b^ h^\ Line a^ U^ is the real length of ab^ and gives its real 
 angle with the front plane. 
 
 In di^awiiig A, the real length of 2ih Is equal to the hypoteiiuse of a 
 right-angled triangle.^ of ivhich the base is equal to the length of the top 
 view of 2ih, and the altitude is equal to the differe?ice in the distances of 
 points a a7id b from the top plane. 
 
 ■ In drawing B, the real leiigth of 2ih is equal to the hypote?iuse of a 
 right-angled tria?igle whose base is equal to the le?igth of the fro?it view 
 of 3.h, and whose altitude is equal to the differe7ice iii the distances of 
 points a and h from the f'ont plane. 
 
 The real lengths of a number of lines of different lengths will be 
 obtained most quickly by constructing, at any convenient place out- 
 side the drawing, right-angled triangles according to the above para- 
 graphs ; these triangles may have a common right angle. 
 
 115. When, instead of both views being given and the true 
 length and angles being required, the angle of one view, and the real 
 length of the line, and its real angle with the other plane are given, 
 the views of the line will be found as follows : 
 
 Suppose that ab\'$> the length a^" b^^ (drawing ^) and makes the 
 angle m with the top plane, while its top view makes the angle 45° 
 with the front plane. Draw first <2^//' of the proper length, and at 
 the angle ;;/ with the top plane ; then project and obtain a^b\ which 
 is the length of the top view. From a^ draw the top view at the 
 given angle, 45°, and make its length equal to a^ V by drawing the 
 arc V li^. Project from b'^ to the level of V^ and obtain Z;^, join b^ 
 and a^'^ and the front view is driven. 
 
ORTHOGRAPHIC PROJECTION. 83 
 
 The views will be obtained in the same way if <^ ^ is the length 
 a^ V^ (drawing ^), and at the angle n with the front plane, and hav- 
 ing its front view at the angle 30° with the top plane. 
 
 116. Drawing C illustrates the way in which the views of a line 
 are found, when, instead of the angles of the views, the real angles 
 with both planes and the real length of the line are given. 
 
 Line ^^ is the distance c^ d^ in length ; it makes an angle of 
 30° with the top plane, and one of 45° with the front plane. To 
 obtain the views, whose angles are not known, draw first c^ d^ equal 
 in length to cd and at the angle 30° with the top plane; then draw^ 
 c^ d^^ equal in length to c d^ and at the angle 45° with the front 
 plane. From c^ d^ project to obtain c'^ d^^\ which is the real length 
 of the top view of <;^when it is at the angle 30° with the top plane. 
 From c^^ d^^ project to obtain c^ d^^^\ which is the length of the front 
 view of <r^when it is at the angle 45° with the front plane. Point 
 ^ must be as far from the top plane as is d\ and as far from the 
 front plane as is ^"; hence, ^^ must be in an arc whose radius is 
 c^ d^^^\ and in a horizontal hne through d^ ; and point d^ must be in 
 an arc whose radius is c^ d^^\ and in a horizontal line through ^". 
 
 From the preceding pages, the following principles may be 
 noted. They are also illustrated by the drawings of the previous 
 chapters. 
 
 Views of a Point. 
 
 117. Any view of a point is a point. 
 
 118. The top, front, and bottom views of a point are in the same 
 vertical line. 
 
 119. The front, side, and back views of a point are in the same 
 horizontal line. 
 
 Views of a Straight Line. 
 
 120. Any view of a straight line is a line or a point. 
 
 121. Any view of a line on a plane to which it is parallel is par- 
 allel and equal to the line. 
 
 122. Any view of a line on a plane to which it is perpendicular 
 is a point. 
 
 123. Any view of a line on a plane to which it is obHque is 
 shorter than the line itself. 
 
84 MECHANICAL DRAWING. 
 
 124. Parallel and equal lines are represented on any plane by 
 parallel and equal lines. 
 
 125. A straight line parallel to the fro?it pla?ie and perpendicular to 
 the top pla?ie. 
 
 The front view is a vertical line whose length equals that of the 
 line ; the top view is a point ; either side view is a vertical line whose 
 length equals that of the line. 
 
 126. A straight line parallel to the fro7it plane and perpendicular to 
 the side pla?ies. 
 
 The front view is a horizontal line whose length equals that of the 
 line ; the top view is a horizontal line whose length equals that of the 
 line ; either side view is a point. 
 
 127. A straight liiie parallel to the front pla?ie and oblique to the top 
 and side planes. 
 
 The front view is an oblique line which gives the real length and 
 the angles of the line with the top and side planes ; the top view is 
 a horizontal line shorter than the line itself ; either side view is a ver- 
 tical line shorter than the line itself. 
 
 .128. A straight li?ie perpendicular to the front plane. 
 
 The front view is a point ; the top view is a vertical line whose 
 length equals that of the line itself ; either side view is a horizontal 
 line whose length equals that of the line itself. 
 
 129. A straight line oblique to the front plane a7id parallel to the top 
 pla?ie. 
 
 The front view is a horizontal line shorter than the line itself; 
 the top view is an oblique line which gives the real length and the 
 angles of the line with the front and side planes ; either side view is 
 a horizontal line shorter than the line itself. 
 
 130. A straight line oblique to the fv?it plane and parallel to the 
 side planes. 
 
 The front view is a vertical line shorter than the line itself ; the 
 top view is a vertical line shorter than the line ; either side view is 
 an oblique line which gives the real length of the line and the angles 
 of the line with the front and top planes. 
 
 131. A straight line oblique to the frorit, top, and side planes. 
 
 The front, top, or side view is an oblique line shorter than the 
 line itself. 
 
ORTHOGRAPHIC PROJECTION, 85 
 
 Views of Plane Surfaces or Figures. 
 
 132. Any view of a plane surface or figure is a figure or a line. 
 
 133. A plane surface or figure parallel to the front plane. 
 
 The front view gives the real size and shape of the figure, and 
 the real angles that its edges make with the top and side planes ; 
 the top view is a horizontal line ; and either side view is a vertical line. 
 
 134. A plane surface or figure perpendicular to the front plane and 
 parallel to the top plaite. 
 
 The front view is a horizontal line ; the top view gives the real 
 size and shape of the figure and the real angles that its edges make 
 with the front and side planes ; and either side view is a horizontal 
 line. 
 
 135. A plane surface or figure perpendicular to the fro7it plane and 
 parallel to the side planes. 
 
 The front view is a vertical line ; the top view is a vertical line ; 
 either s'ide view gives the real size and shape of the figure and the 
 real angles that its edges make with the front and top planes. 
 
 136. A plane surface or figure perpendicular to the front plane and 
 oblique to the top and side planes. 
 
 The front view is an oblique line which shows the real angles of 
 the plane with the top and side planes ; the top view is a figure 
 which is foreshortened in one direction ; either side view is a figure 
 which is foreshortened in one direction. 
 
 137. A pla7te surface or figure oblique to the f'ont and side planes 
 and perpendicular to the top plane. 
 
 The front view is a figure which is foreshortened in one direc- 
 tion ; the top view is an oblique line which shows the real angles of 
 the plane with the front and side planes ; either side view is a figure 
 which is foreshortened in one direction. 
 
 138. A plane surface or figure oblique to the front and top planes 
 and perpendicular to the side planes. 
 
 The front view is a figure which is foreshortened in one direc- 
 tion ; the top view is a figure which is foreshortened in one direction ; 
 either side view is an oblique line which shows the real angles of the 
 plane with the front and top planes. 
 
86 MECHANICAL DRAWING. 
 
 139. A plane surface or figure all of whose edges are- oblique to the 
 fro?it^ top, and side planes. 
 
 The front, top, or either side view is a figure all of whose hnes 
 are shorter than the Hnes of the figure they represent. 
 
 140. Parallel and equal plane surfaces or figures, whose corre- 
 sponding edges are parallel, are represented on any plane by similar 
 figures whose corresponding lines are equal and parallel. 
 
 Locations of the Views. 
 
 141. The top and bottom views are always respectively above 
 and below the front view. 
 
 142. The side views are always on the same level as the front 
 view. 
 
 143. The view of the right side is always at the right of the 
 front view, and the view of the left side is always at the left of the 
 front view. 
 
 144. The line of any view which is nearest the front view repre- 
 sents the front face or front line of the object. 
 
 PROJECTION PROBLEMS. 
 
 These problems are arranged for those who wish to work by pro- 
 jection methods. They are not intended for public school pupils. 
 
 All polygons and solids referred to are regular, unless otherwise 
 stated. When two views are asked for, the top and front views are 
 desired. 
 
 1. Two views of a line 2" long. It is parallel to the top plane 
 and i" below it, and at 45° to the front plane, with its nearest end 
 5^" behind the front plane. 
 
 2. Two views of a line 3" long. It is parallel to the front plane, 
 and ^" behind it, and at 60° to the top plane, from which its upper 
 end is ^" distant. 
 
 3. Two views of a square card whose edges are 2" long. It is 
 parallel to the front plane and has two edges vertical. It is ^" 
 behind the front plane, and its upper edge is 54^" below the top 
 plane. 
 
PROJECTION PROBLEMS. 87 
 
 4. Two views of a circular card 2" in diameter. It is parallel to 
 the front plane, with its centre i" behind the front plane and i^" 
 below the top plane. 
 
 5. Two views of the same card, when it is parallel to the top 
 plane, its centre being i^^" behind the front plane and J^" below the 
 top plane. 
 
 6. Two views of an equilateral triangular card whose edges are 
 2" long. It is parallel to the front plane ; its lowest edge is horizon- 
 tal and 2}^" below the top plane and ^" behind the front plane. 
 
 7. (a) Two views of an hexagonal card whose edges are i" long. 
 It is parallel to the front plane and j^" behind it, with two edges 
 horizontal and the upper one ^" below the top plane. 
 
 (p) Two projections of the same card in the same position with 
 reference to the top plane, but at 45° to the front plane, the nearest 
 point being ^" behind the front plane. 
 
 8. Two views of a prism 4" long, whose bases are squares of 2". 
 The lo«g edges are vertical. The upper base is 5^ " below the top 
 plane, and the nearest lateral face is parallel to and i" behind the 
 front plane. 
 
 9. Two views of a cone whose axis is 4" long and whose base is 
 2" in diameter. Its axis is vertical and i^'' behind the front plane, 
 the vertex is ^" below the top plane. 
 
 10. Two views of a pyramid whose axis is 4" long, and whose 
 base is a square of 2". The axis is vertical and 3" behind the front 
 plane; its upper end is i" below the top plane. The edges of the 
 base are at 45° to the front plane. 
 
 11. Two views of an hexagonal prism 4'' long. The prism is 
 vertical, its nearest lateral face being parallel to and ^ " behind the 
 front plane. The edges of its bases are i" long, and its upper base 
 is 1^4^" below the top plane. 
 
 12. Two views of an hexagonal pyramid of the same dimensions 
 as the prism of Problem 11, the centre of the base being 5^^-" below 
 the top plane and 2^" behind the front plane. A long diagonal 
 of the base is parallel to the front plane. 
 
 13. Two views of a triangular prism 4" long, whose bases are 
 equilateral triangles of 2" sides, when the lateral edges are horizontal 
 and parallel to the front plane. One lateral face is horizontal and 
 
88 MECHANICAL DRAWING. 
 
 3" below the top plane. The prism extends above this face, whose 
 nearest edge is ^" behind the front plane. 
 
 14. {a) Two views of a prism 4" long, whose bases are squares 
 of 2" side, when its lateral edges are parallel to the front plane and 
 at 60° to the top plane. Two lateral faces are parallel to the front 
 plane ; the nearer one is 3" behind the front plane, and its upper 
 corner is % " below the top plane. 
 
 (/?) The same prism, the same distance below and at the same 
 angle with the top plane ; but its vertical faces at 30° to the front 
 plane. The nearest corner of the object is J^" behind the front 
 plane. 
 
 15. (a) Two views of a pyramid 4" long, whose base is a square 
 of 2". The centre of the base is 2}'(" behind the front plane and 
 6}4" below the top plane, and the axis of the pyramid extends to the 
 right parallel to the front plane and upward at 45° to the top plane. 
 A vertical plane through the axis contains two of the lateral edges of 
 the object. 
 
 (/^) Project the pyramid upon the right side plane, which is i" to 
 the right of the vertex of the pyramid. 
 
 16. Two views of a cylinder 4" long and 2" in diameter, whose 
 uxis is parallel to the front plane, at 45° to the top plane and i}^" 
 behind the front plane. The upper end of the axis is 2j^" below the 
 top plane. 
 
 17. (a) Two views of a cone w^hose axis is 4" long, and whose 
 base is 2" in diameter. The lowest element of the cone is horizontal 
 and parallel to the front plane ; it is 2}^" behind the front plane and 
 3>^" below the top plane. 
 
 (I?) Two views of the same cone at the same level, when the end 
 of the horizontal element at the base of the cone is i" behind the 
 front plane, the element extending from this point to the right from 
 the front plane at 45° with it. 
 
 18. The front view of a line is 2" long and is at 60° to the front 
 axis of projection. The top view is at 45^ to this axis. Give the real 
 length of the line and measure the actual angles made by the line 
 with the two planes. 
 
CHAPTER IX. 
 
 SECTIONS. 
 
 145. Suppose a plane to pass through an object in any direc- 
 tion, the part of the object in front of the plane to be removed, and 
 the projection of the part of the object behind the cutting plane to 
 be made on a parallel plane. Such a projection is generally called 
 a section, though section sometimes means a drawing which gives 
 simply the actual shape of the section given by the cutting plane. 
 
 Sections are taken to show the interior of hollow objects, or the 
 shape of solid parts which are not clearly represented by views of 
 the outside of the object. Sections are most necessary and most 
 used in practical working drawings ; but the principles involved can 
 best be studied by the use of simple geometric solids. 
 
 146. Sections may be drawn in place of the principal views, or 
 they may be taken in any direction whatever. The position of a 
 section should be indicated by a dot-and-dash line in the view in 
 which the plane of the section is seen edgewise ; but if the view 
 represents the object when the part in front of the cutting plane is 
 removed, the line of the section should be represented by a full line 
 upon the object, and by a dot-and-dash line outside. 
 
 In practical working drawings, the objects are often shown entire 
 in one view and in section in another. In the study of projection, 
 or working drawings, by the use of the geometric solids, all views 
 should agree in representing the object with the part in front of the 
 cutting plane removed. In this case the part removed may be 
 shown, if desired, by dotted or by dot-and-dash lines. 
 
 147. If it is desired to represent only the line of intersection 
 given by passing any plane through any solid, all views should agree 
 in representing the entire solid with the line of intersection upon its 
 surface. 
 
 148. When the part in front of the cutting plane is removed, it 
 is customary to show the surface cut by the plane, by hatching it 
 with parallel equidistant lines as explained in Art. 84. 
 
90 
 
 MECHANICAL DRAWING. 
 
 149. Sections of the Sphere. — The simplest 
 sections are those of the sphere, for every section 
 made by a plane is a circle, whose diameter 
 ranges from that of the great circle, given by a 
 plane passing through the centre of the sphere, 
 to that of a circle as small as can be imagined^ 
 given by a plane barely cutting the sphere. 
 
 The centre j of the circle C, given by a plane 
 which cuts the sphere and does not pass through 
 its centre, is in a line passing through the centre 
 of the sphere perpendicular to the cutting plane. 
 
 150. Sections of the Cube. — Sections of a 
 cube are polygons, for the surfaces of the cube are plane, and when 
 intersected by a plane must produce straight lines. 
 
 If a plane cuts three faces of the cube, the section is a triangle ; 
 if it cuts four faces the section has four sides. The section may 
 be a polygon of three, four, five, or six sides, according to the num- 
 ber of faces which are cut by the plane. 
 
 151. To obtain the views of the section given by any cutting 
 plane, the plane must first be represented in the view in which it 
 appears a line, for only in this view are the points in which it inter- 
 sects the edges, or elements, of the object seen. The points of 
 intersection, having 
 been determined in 
 this view, are readily 
 projected to the other 
 views. 
 
 152. Suppose a 
 cube, placed so that 
 two vertical faces are 
 at 45° to the front 
 plane, to be inter- 
 sected by a cutting 
 plane at 30° to the 
 top plane and perpen- 
 dicular to the front 
 plane. 
 
SECTIONS. 
 
 91 
 
 The top view of the cube is a square whose sides are at 45^ ; 
 the front and side views are equal rectangles. Line EF in the front 
 view represents the cutting plane. We will suppose the part of the 
 cube above this plane to be removed, and will represent it by dotted 
 lines. In the top view, the line 2-j on the top face of the cube is 
 seen of its true length and is the only edge of the section which is 
 not represented by the square. The side view of 2-j will be 
 obtained by setting off one half the distance 2-j of the top view each 
 side of the centre of the top line of the side view, and the other 
 points in the section will be obtained by projecting from the points, 
 in the front view, where the plane cuts the vertical edges. 
 
 The cutting plane is oblique to the top plane and also to the side 
 plane. The section appears of different shapes in these two views, 
 and neither gives the real dimensions of the figure. Line i-o of the 
 top view is the centre line of the section and is parallel to the front 
 plane ; therefore the front view gives the real length of the section 
 and the distance of line 2-j from a line connecting 4 and 5. The 
 top view gives the real lengths of lines 2-j and ^-5, and by com- 
 bining the lengths of the front view with the widths of the top view, 
 the real shape of the section will be obtained. 
 
 A simple way to obtain the true shape is to draw LM parallel to 
 EF, and from points 7", 2, and 4 in EF, draw perpendiculars to EF; 
 these perpendiculars in- 
 tersecting LM give the 
 lengths ; the widths are 
 obtained by setting off each 
 side of LM the distances 
 of the points in the top 
 view from i-o. Join 1-4, 
 4-2,2-j,j-s, and 5-7, and 
 the real shape of the sec- 
 tion is obtained. 
 
 153. The true shape 
 of the section can be 
 obtained as just explained 
 only when line i-o of the 
 top view is parallel to the 
 
92 
 
 MECHAXICAL BRA IVI.VG. 
 
 front plane. When the sides of the cube are not at 45° to the front 
 plane the section will not be symmetrical, the front view will not give 
 the real length of the section, and line i-o will pass through only one 
 of the corners of the top view. The real shape of the. section in this 
 case can be obtained by measuring, in the top view, the distances of 
 points 4, 2, J, and 5 from i-o, by means of perpendiculars to 7-0, 
 and setting these distances off on the proper lines, measuring from 
 ZJ/, which as before is drawn parallel to 1-2 of the front view and 
 represents line i-o of the top view. 
 
 154. The opposite sides of the section are parallel, and when- 
 ever a plane intersects an object whose opposite surfaces are parallel, 
 the opposite sides of the section must be parallel ; they will be of 
 equal length when the intersected surfaces are equal in width and 
 the intersecting plane cuts the entire width of the surfaces. 
 
 155. Sections of the Cylinder. A section of a cylinder is a 
 circle when the cutting plane is perpendicular to the axis of the 
 cylinder, a rectangle when the plane is parallel to the axis of the 
 cylinder, and an ellipse when it is oblique to the axis of the cylinder. 
 
 Sections A and B require no further explanations. 
 
 A 
 
 156. Section C is an ellipse which appears a straight line in the 
 front, and a circle in the top view ; its real length is seen in the front 
 
SECTIONS, 
 
 93 
 
 view. The short axis of the ellipse bisects its long axis, and intersects 
 the axis of the cylinder ; it appears a point {i^ 2^'^) in the front view, 
 and the line 1-2 in the top view. 
 
 To obtain the real shape of the section, draw a I) parallel to 
 a^ b^'^ and set off upon it the real length of the ellipse by means of 
 perpendiculars to a^ b^ from the points a^^ b^ ; then set off each 
 side of 0^ on the perpendicular to ab, one half of 1-2 of the top view. 
 This gives the short axis of the ellipse. 
 
 To obtain other points, j and 4, assume a point ( j^^ 4^^) in a^ b^. 
 This represents point j at the front and point 4 at the back of the 
 section. Project from j^ 4^ to the top view and also to the real 
 shape of the section, where the distance 3-4 is to be made equal to 
 j-4 of the top view. In this way any number of points in the real 
 shape may be obtained. 
 
 157. If the cylinder is placed so that it does not appear a circle 
 in any view, its section by a plane may be obtained by assuming 
 elements upon the cylinder, and finding the points in which they are 
 intersected by the cutting plane. These points in the elements are 
 seen in the view in which the cutting plane appears a line, and can 
 be projected from this view to the other views. 
 
 158. Sections of a Pyramid. A section of a pyramid is a 
 figure similar to the base, when the cutting plane is parallel to the 
 base ; a triangle, when the plane passes through the vertex and the 
 base of the pyramid, or when it intersects the base and two of the 
 triangular faces of the pyramid. When the pyramid is intersected in 
 any other way the section will be a polygon having a side upon each 
 face of the pyramid cut by the plane. 
 
 If the preceding articles are understood, these statements and 
 drawings A and B will be clear to all. 
 
 159. In C, plane AB cuts the lateral edges of the pyramid in 
 four points, z, 2, j, 4, which are determined in the front view. 
 Points I and 2 may be projected from this view to the top view. 
 Points J and^ are in lateral edges which are represented by vertical 
 lines in both views, and therefore these points cannot be obtained in 
 the top view by projecting from the front view. 
 
 To obtain j and 4 in the top view of C, it is necessary to take an 
 auxiliary cutting plane through j and 4. A horizontal plane through 
 
94 
 
 MECHANICAL DRAWING. 
 
 these points gives a square section whose opposite corners a^ and 
 b^ can be projected to the top view, thus giving a'^ and b'^. The 
 corners j and 4 of this square are the points required to enable the 
 top view of the section to be completed. 
 
 160. Sections of the Cone. Any section of a right circular cone 
 made by a plane parallel to its base is a circle. This section is 
 illustrated by Fig. '^Z. 
 
 161. If a cutting plane passes through the vertex of the cone, it 
 intersects the base of the cone in a chord of the circle, and. the 
 curved surface of the cone in two elements ; the section is thus a tri- 
 angle. See Fig. 96. 
 
 162. If the plane intersects all the elements of the cone, the sec- 
 tion is an ellipse. Fig. 98. 
 
 163. If a cone is intersected by a plane which is parallel to one 
 of the elements of the cone, the section is a parabola. Fig. 99. 
 
 164. If a cone is intersected by a plane which makes a greater 
 angle with the base than do the elements, the section is an hyper- 
 bola. Fig. 97. 
 
 165. When a solid, bounded by plane surfaces, is cut by any 
 plane, the edges of the solid are seen piercing the plane, in the view 
 in which the cutting plane appears a line ; these points of intersec- 
 
SECTIONS. 
 
 95 
 
 tion are readily projected to the lines of the other views, and thus the 
 angles of the section are obtained. 
 
 l66. When curved bodies or those with curved surfaces are in- 
 tersected by a cutting plane, only the points of intersection in the 
 contour elements are seen in the view in which the cutting plane 
 appears a line. To obtain other points in the section, it is necessary 
 to assume elements or other lines upon the curved surface ; their 
 intersections will be found in the same way as the intersections of 
 the edges of solids bounded by plane surfaces. When possible the 
 lines assumed should be elements of the surface. Fig. 98. 
 
 Instead of assuming elements in order to obtain the points in a 
 section, it is often easier and more accurate to pass through the 
 object a number of auxiliary cutting planes whose sections are 
 simple. These planes intersect the plane of the section in lines, 
 whose ends must be points in the required section. See 4 and 5, 
 Fig. 97. 
 
 If pupils can obtain the sections of the regular geometric solids 
 they will readily find the sections of any solid which may be con- 
 structed. 
 
CHAPTER X. 
 
 INTERSECTIONS. 
 
 167. In practical work it is necessary to represent all kinds of 
 regular and irregular bodies which intersect or penetrate each other. 
 The knowledge required to do this is best obtained by study of the 
 geometric solids. 
 
 Simple intersections are produced when a body small enough 
 to pass through another enters one plane surface and leaves by 
 another. The large object being bounded by plane surfaces, or 
 at least by the two mientioned, the intersections are simply sections 
 of the smaller body made by the plane surfaces of the larger, and 
 have been explained. 
 
 Simple intersections are also given by a cone or cylinder which 
 penetrates a sphere in such a way that its axis passes through the 
 centre of the sphere. In this case the plane of the intersection 
 must be at right angles to the axis of the penetrating body and the 
 lines in which the cylinder or cone enters and leaves the sphere must 
 be circles. Fig. 100. 
 
 If the axis of the cone or cylinder does not pass through the 
 centre of the sphere, the intersections will not be circles, and must be 
 obtained as explained later. 
 
 168. The principles of sections enable us to find all intersec- 
 tions ; for if the lines of intersection are not, in part or in whole, 
 some of the regular sections explained, they can be determined by 
 means of points situated in sections given by auxiliary cutting 
 planes. 
 
 169. When bodies bounded by plane surfaces intersect, the lines 
 of intersection will be straight, and must connect the points in which 
 the edges of each solid penetrate the other solid ; the problem is thus 
 to find these intersections. 
 
 170. When curved bodies intersect or are intersected, there are 
 elements, instead of edges, which penetrate the surfaces and must be 
 treated as if they were edges. Thus problems in intersections, as 
 
INTERSECTIONS. 
 
 97 
 
 they are generally solved, may be reduced to the simple problem of 
 finding the intersection of a line and a plane, or of a line and a 
 curved surface. 
 
 Intersections of a Line and a Plane Surface. 
 
 171. Drawing A represents a cube pierced by an inclined line 
 which enters the left side at a and leaves the top of 
 the cube at b. The front view determines both a 
 and b^ for it represents both the left and the top sur- 
 faces by straight lines. The top view determines 
 only one point (a), for in it the top of the cube 
 appears a surface, and when a plane appears a surface 
 its intersectio?i by a line cannot be deterviined by means 
 of this view alone. Point b must be obtained in the 
 top view by projecting from the front view. 
 
 Note. — The positions of the intersecting lines in all the problems are 
 
 assumed. , 
 
 173. 
 
 172. Drawing B represents a cube and a 
 line I which is in the same plane as the right 
 and left vertical edges of the cube, and there- 
 fore intersects these edges. Line 2 is in front 
 of the right and left vertical edges ; its intersec- 
 tions with the faces of the cube are seen in the 
 top view, and may be projected from this view 
 to the front view. 
 
 Drawing C represents a cube intersected by a line which 
 enters the left visible vertical surface, and leaves 
 at the top of the cube. The front view represents 
 the top of the cube by a horizontal line, and in this 
 view the intersection b^^ of the line with the top is 
 seen. 
 
 The intersection a of the line with the left ver- 
 tical surface of the cube is seen in the top view. 
 The points determined by each view can be pro- 
 jected to the other view, and in this way each view 
 completes the other 
 
98 
 
 MECHANICAL DRAWING. 
 
 
 
 
 
 u 
 
 
 
 
 q} 
 
 
 
 
 D 
 
 
 
 ^ 
 
 A 
 
 a'- 
 
 
 ^^^^^ 
 
 
 /'■ 
 
 R--^ 
 
 ^^' 
 
 ' 
 
 
 174. Drawing D shows a line which pierces the left end and the 
 
 front inclined surface of a triangular 
 prism. The intersection of the line 
 and the inclined surface cannot be 
 determined in either the front or the 
 top view, for in neither view does the 
 surface appear a line. In the side 
 view the inclined surface appears a 
 line, and the intersection a^ is seen and 
 may be projected to the other views. 
 
 175. Drawing E represents a square pyramid and an inclined 
 line L which pierces the pyramid in front of the axis. 
 The intersections of the line and the lateral faces of 
 the pyramid cannot be seen in either view, for the 
 faces appear surfaces in both ; and for the same 
 reason a side view is of no assistance. To find the 
 points, an auxiliary cutting plane must be taken 
 through the line. If a vertical plane is chosen it 
 will appear the line Z^, in the top view, and in this 
 view its intersections, i and 2, with the edges of the 
 base are seen. It intersects the central lateral edge 
 in J, which is found as explained in Art. 159. In 
 
 the front view the triangle z, 2, j is the section made upon the pyra- 
 mid by the vertical cutting plane passing through Z. The points a 
 and b^ w^here Z intersects the sides of the triangle, are the points in 
 which Z enters and leaves the surface of the pyramid. 
 
 176. Drawing F represents the same pyramid and line as draw- 
 ing Z", but instead of choosing a vertical plane through 
 Z as the auxiliary plane, a plane is passed through 
 Z, perpendicular to the front plane. The front view 
 gives the points 7^, 2^, j^, and 4^ where the lateral 
 edges intersect this cutting plane, whose figure of 
 intersection is readily found as explained in Art. 159. 
 In the top view this section appears a surface, and 
 the points in it, a^ and /;^, in which Z enters and 
 leaves the pyramid, are seen and may be projected to 
 the front view. 
 
IN TERSE C TIONS. 
 
 99 
 
 177. Drawing G represents the same conditions as D. 
 
 The intersection may be found without drawing 
 the end view, by means of a cutting plane used as 
 explained in E and F, 
 
 The cutting plane used in the illustration is 
 perpendicular to the front plane ; and in the front 
 view points i, 2, j of the section, which are in the 
 edges of the prism, are seen. When these points 
 are projected to the top view and connected, the 
 triangular section is represented by a triangle, and 
 the point ^^, where Z^ intersects the triangle, is 
 the top view of the required point a. 
 
 Intersections of a Line and a Curved Surface. 
 
 178. Drawing ZT represents a sphere intersected by a horizontal 
 line Z, in front of the centre of the sphere. 
 
 If the line were in a vertical plane passing 
 through the centre of the sphere and parallel to 
 the front plane, the front view would give the in- 
 tersections of the line and the sphere. If the line 
 were in a horizontal plane passing through the 
 centre of the sphere, the top view would give the 
 intersections of the line and the sphere. As the 
 line is not situated in either of these positions, the 
 intersections cannot be in the circle of either view, 
 and must be found by means of an auxiliary cut- 
 ting plane. 
 To obtain the points in which Z enters and leaves the sphere, a 
 horizontal cutting plane may be passed through the line. This plane 
 gives a circular section whose diameter 1-2 is seen in the front view. 
 The circle appears a circle in the top view; here its intersections 
 a'^ and b'^ with Z^ are seen, and may be projected to the front 
 view. 
 
 179. Drawing Z solves the same problem, except that the line is 
 parallel to the front plane only. An auxiliary cutting plane parallel 
 to the front plane is used. This plane gives a circle on the sphere 
 
lOO 
 
 MECHANICAL DRAIVIXG. 
 
 whose diameter, 7-2, is seen in the top view ; its real shape is seen 
 in the front view, where a and b are determined, 
 and from which they may be projected to the top 
 view. 
 
 When L is parallel to only one plane, the sec- 
 tion must be taken through L parallel to this plane. 
 If L is not parallel to either plane the cutting plane 
 may be perpendicular to either plane and will give 
 a circle wdiich in one view will appear an ellipse. 
 
 180. Drawing J represents a sphere and two 
 inclined lines w^hich are represented in the top 
 view by the line z^-2^, and in the front view 
 by two lines, the upper of which intersects the 
 \ sphere in points a and b and has its ends un- 
 __^ marked ; the lower line intersects the sphere 
 I in points c and d^ and its ends are represented 
 by points i^ and 2^. The upper line 1-2 
 intersects the sphere at points a and b^ which 
 are upon the upper surface of the sphere, 
 point a being upon the front surface. Neither 
 a nor b appears, in either view, in the contour 
 of the sphere, and to determine these points, 
 a vertical cutting plane is taken through the 
 line. This plane intersects the sphere in a circle, which in the front 
 view appears an ellipse. Only the upper half of this ellipse is 
 drawn, as it contains both a and b. (When follow^ing this solution, 
 cover the lower half of the front view with paper, to hide the second 
 solution of the problem.) 
 
 181. If the lower half of the ellipse, which represents the section 
 given by the vertical plane through the lines, is drawn, it will con- 
 tain points c and d, in which the lower line intersects the surface. 
 These points may also be found by revolving the vertical plane taken 
 through 7-2, until it is parallel to the front plane. The section given 
 by it will then appear a circle in the front view. As the plane 
 revolves about a vertical axis passing through the centre of the 
 circular section, the points of the section and points i and 2 
 describe arcs of circles, which appear arcs in the top view, and hori- 
 
 ~^'3r 
 
INTERS E C TIONS. 
 
 lOI 
 
 zontal lines in the front view. (The upper half of the front view is 
 to be covered when following this description.) 
 
 As the vertical cutting plane passing through line 1-2 revolves 
 into a position parallel to the front plane, points i and 2 move, in 
 the front view, in horizontal lines. In these lines points z" and 2" 
 must be given by projecting lines from z' and 2' which represent the 
 ends of the line 1-2 when it is parallel to the front plane. When the 
 vertical plane passed through 1-2 has been revolved parallel to the 
 front plane, the circle which it gives upon the sphere will appear a 
 circle, whose lower half is shown in the drawing ; the line z"-^" in- 
 tersects this circle in points c^ and d\ These points move, when the 
 plane of the section is revolved back into its original position, in arcs 
 which appear horizontal lines in the front view, and which intersect 
 iF_2F in c^ and d^ ; these are the front views of points c and d^ in 
 which the lower line 1-2 intersects the surface of the sphere. These 
 points are not shown in the top view, but would be given by verticals 
 from c^ and d^^ intersecting i^^-2^. 
 
 182. Drawing A" represents a cylinder intersected 
 by a line Z. 
 
 The intersection of a line and the curved surface 
 of a cylinder is determined in the view in which the 
 curved surface appears a circle. If the curved sur- 
 face is not seen edgewise in any view, the cylinder 
 must be cut by a plane passing through the line, in 
 order that the points of intersection may be deter- 
 mined ; this applies to any and all positions of the 
 cylinder and the line. 
 Drawing L represents a cone, and a line which intersects it 
 in front of its axis. 
 
 If the line were in the plane of the contour elements, 
 the points of intersection would be seen in the front 
 view, but, as this is not the case, to obtain the points, 
 an auxiliary plane must be used. A horizontal cutting 
 plane, A^ intersects the cone in a horizontal circle, 
 which appears a circle in the top view. In this view 
 the intersections, a and /^ of the line A and the circle 
 are determined, and may be projected to the front 
 view. 
 
I02 
 
 MECHANICAL D^RAWING. 
 
 184. Drawing M represents a cone intersected by an inclined 
 line. 
 
 To obtain the points of intersection, a cutting plane 
 perpendicular to the front plane may be used ; this 
 will cut the cone in an ellipse, which will appear an 
 ellipse in the top view. A cutting plane perpendicular 
 to the top plane is, however, used here. This cuts the 
 cone in an hyperbola, whose real shape is seen in the 
 front view. Art. 164. The intersections, a and b^ of 
 L and the cone are seen in the front view, and from 
 this view may be projected to the top view. 
 
 185. When auxiliary cutting planes are used, it is not necessary 
 to find the complete section given by the plane, as the part near the 
 point of intersection is all that is required. 
 
 Intersections of Solids. 
 
 186. All the points necessarily involved in intersections are 
 explained above. If understood they can be readily applied when, 
 instead of a single line penetrating a solid, one solid penetrates 
 another. In this case, when the points of intersection of the various 
 lines of the penetrating solid cannot be seen in one view or the 
 other, it is necessary to use auxiliary cutting planes. These planes 
 should be so chosen as to give simple sections upon both solids. 
 The circle and the rectangle are the simple sections of the cylinder ; 
 the triangle and the circle are the simple sections of the cone ; and 
 in many problems it will be possible to obtain these sections instead 
 
 of ellipses, hyperbolas, and parabolas. 
 
 187. When a cutting plane intersects two 
 intersecting bodies, it gives upon each a cer- 
 tain figure. These figures intersect in points, 
 which must be points in the lines in which 
 the solids intersect. 
 
 • This is illustrated by N and O. In iV, a 
 vertical cutting plane intersects the horizontal 
 prism in a rectangle, and the sphere in a 
 circle. These figures intersect in four points, 
 which are points in the lines of intersection 
 of the prism and the sphere. 
 
 
 ^^ 
 
 ~^\ 
 
 < 
 \ 
 
 c 
 
 V 
 
 .K 
 
 
 ^4--, 
 
 
IN TERSE C TIONS. 
 
 103 
 
 O 
 
 \) 
 
 =^ 
 
 'h=4 
 
 In O, a horizontal cutting plane gives a rectangle upon the cylin- 
 der and a circle upon the cone. These fig- 
 ures intersect in four points, which are points 
 in the lines of intersection of the cylinder and 
 the cone. 
 
 The same problem in intersections may be 
 solved in many different ways, but always by 
 means of auxiliary cutting planes, when the 
 required points are not seen at once in one of 
 ,' the given views. These planes may be taken 
 J in so many different positions that to explain 
 all would be impossible in the limits of this 
 book. If the principles of the subject are 
 understood, pupils will have no trouble in 
 deciding what planes will give the simplest sections, or in determin- 
 ing the points of these sections. 
 
 Instead of using parallel cutting planes, as explained in the pre- 
 ceding articles, one plane maybe supposed to be hinged, or to swings 
 upon a given line as an axis. In the case of two intersecting pyra- 
 mids or cones, this axis should pass through the vertices of both, 
 solids, thus the sections of both will be triangles. If one solid is a, 
 cylinder or prism the plane should swing upon a line parallel to the 
 axis of this solid. This method is not illustrated, but is often of 
 great value. 
 
CHAPTER XI. 
 
 ARRANGEMENT AND NAMES OF VIEWS. 
 
 i88. In descriptive geometry the planes of projection are 
 supposed to be indefinite in extent, one horizontal and the other 
 vertical. See Fig. 35. Four dihedral angles are formed by 
 these planes. The eye is supposed to be always 
 in front of the vertical plane and above the horizon- 
 tal plane. Objects may be placed in any of these 
 angles and projected upon the two planes ; in pro- 
 jection and descriptive geometry all four angles are 
 used. 
 
 The projection of an object upon the front vertical plane is called 
 its vertical projection ; that upon the horizontal plane is called its 
 horizontal projection. These drawings are also called elevation and 
 plan, and front view and top view. All these terms are applied in 
 practical work, but the preference seems to be to call shop drawings 
 views. In the study of projection and descriptive geometry, the 
 drawings are generally called projections. 
 
 189. If an object is in the first angle, it is between the planes 
 and the eye, and covers its projections upon the planes. 
 
 Fig. 35. 
 
 Fig. 36. 
 
 Fig. 37. 
 
ARRANGEMENT AND NAMES OF VIEWS. 105 
 
 Fig. 36 represents the vertical and horizontal planes forming the 
 first angle, with a side vertical plane placed at the right. The drawing 
 also shows a pyramid placed within this angle, and its projections 
 upon the three planes. 
 
 When the planes are revolved, as in Fig. 37, the top view of the 
 pyramid is found below the front view, and the view of the left side 
 comes at the right of the front view. 
 
 This arrangement is very different from that due to the use of the 
 third angle, which is explained in Chap. Ill, and in which the top 
 view is above the front view. 
 
 190. Draughtsmen have no uniform way of working. Some 
 arrange the views according to the use of the first angle, and some 
 according to that of the third angle. To those who do not read 
 drawings easily this lack of uniformity results in confusion. To 
 those who are accustomed to reading them, the positions of the 
 views are of less importance, and the convenience of the draughts- 
 man in making the drawings should determine largely the method of 
 arrangement. 
 
 It may often happen that it is not easy or convenient to adhere 
 to any system, and it is certainly reasonable that a rule which ham- 
 pers should not be followed. Generally, however, some uniform ar- 
 rangement of views may easily be given, and most draughtsmen use 
 either that given by the first or that given by the third angle. The 
 chief points to decide the arrangement to be adopted should be the 
 ease and accuracy of making and of reading the drawing. 
 
 191. If an object similar to that illustrated by Fig. 38 be placed 
 in the first angle, the draughtsman will be obliged to project from 
 the right end of the front view across its entire length to the space at 
 the left of the front view, to make the view of the right end of the 
 object ; and in the same way across the entire length of the front view 
 to give the view of the left end. He will project from the top of the 
 front view to the space below this view to draw the top view ; and 
 from the bottom of the front view to the space above this view to 
 draw the bottom view. To make drawings thus arranged must take 
 much longer than to make them arranged as in Fig. ^i"^, with the view 
 at the left of the front view showing the left end of the object ; that 
 at the right, the right end; that below, the bottom of the object, and 
 
io6 
 
 MECHANICAL DRAWING. 
 
 SO on. Not only will it take longer to make the drawings, but the 
 inaccuracies of drawing boards and T-squares will cause the drawings 
 to be less exact than those whose arrangement is due to the use of 
 the planes placed in front of the object. 
 
 The views due to the use of the first angle are not so easy to read 
 as those due to the use of the third, for the third angle places the 
 different views of the same part as near as possible to each other, 
 while the first angle places them as far apart as possible. These 
 considerations make the use of the third angle desirable for all prac- 
 
 Bottom View. 
 Fig. 38. 
 
 A 
 
 HJUi 
 
 I 
 Right Side View. 
 
 tical working drawings ; and therefore, this arrangement is adopted 
 throughout this book. 
 
 192. In the study of projection with advanced classes the choice 
 of angle is of little importance. The objects may be placed in 
 either of the angles, and there will be little difference in the ease (or 
 perhaps, difficulty) with which the students understand the subject. 
 A point in favor of the first angle for use in the study of projection 
 is that books on the subject of projection generally use this angle ; 
 another point is that when the third angle is used, and glass planes 
 and expensive models to place behind them are not provided, the 
 
ARRANGEMENT AND NAMES OF VIEWS. 
 
 107 
 
 problems cannot be easily illustrated. When the first angle is used 
 two blackboards may be hinged together, as illustrated in Fig. 36 ; 
 they should be arranged so that they may revolve while fixed at right 
 angles to each other, or revolve independently so as to form one 
 vertical surface. Thus the students may see the horizontal board 
 edgewise and the vertical one as a surface ; or the horizontal one as 
 a surface and the vertical one edgewise ; or with the plane of the 
 horizontal one coinciding with that of the vertical, as in Fig. 37. A 
 vertical blackboard may be attached at either side of the vertical 
 board, as illustrated in Fig. 36, to represent a side vertical plane. 
 By means of these boards and of models which can be procured 
 with little expense and held as desired in front of the planes, the 
 problems may be much more easily illustrated than when the third 
 angle is used. These boards are readily managed and can be seen by 
 all the occupants of a large class-room, so that for advanced classes 
 who are studying the theory of projection, the first angle seems 
 preferable, especially when the course is preparatory to one in 
 descriptive geometry. But although books on this subject use 
 principally the first angle, all are necessary, and a thorough course 
 will deal with points, lines, planes, and objects in all the angles. 
 
 The third angle as well as the first might be used principally in 
 the study of descriptive geometry. The fact that the arrangement of 
 views given by the first angle is different from that which we have 
 decided to be the best for shop drawings should not, however, be of 
 the slightest consequence to the advanced student, for he should 
 study the relations of points in all the angles ; and he should be as 
 familiar with the arrangement due to the third angle as he is with 
 that due to the first. When his course is completed and he makes 
 practical drawings, he will then make them with equal ease, whatever 
 arrangement of views he may choose. 
 
 Projection and descriptive geometry are means by which the 
 mind is trained to conceive any form in any position ; their aim is 
 not the simpler and less important one of enabling young students to 
 make views or working drawings of concrete objects. 
 
 The question as to whether to use solids, planes, lines, or points 
 for the first study, has caused some discussion. From what lias 
 been said, it will be seen that for the first work in the public scliools 
 
I08 MECHANICAL DRAWING. 
 
 and for elementary work generally, it will be better to begin with 
 solids ; for to deal with lines, etc., is more in accordance with pro- 
 jection than with the making of views from simple objects. 
 
 In the theoretical study of projection it makes no difference what 
 is taken first ; for if the solid is taken the lesson still must deal with 
 the relations of points and lines to each other and to the planes of 
 projection, so that in reality these are studied first. 
 
CHAPTER XIL 
 
 PLATES AND EXPLANATIONS. 
 
 The plates of this chapter illustrate work from that suited for the 
 youngest pupils of the subject, to that which advanced pupils of high 
 and elementary technical schools may require. Teachers are to 
 select work suitable for their pupils and make it as practical as 
 possible by following the directions given in the preceding chapters. 
 
 The principles of working drawings may be taught by means of 
 free-hand sketches. Free-hand sketches may also be made and 
 dimensioned in order that finished drawings may be made from 
 them. All other drawings should be made by the use of instruments, 
 and to some fixed scale. In the lower grades of the public schools 
 an architect's scale cannot be provided, and, when the objects 
 studied cannot be represented full or half size, a special scale must 
 be drawn as explained in Art. lo. 
 
 The drawings of the plates are small, and, in order that they may 
 not be obscured, working or projecting lines are given only when 
 necessary to illustrate important principles or ways of working ; and 
 dimensions are given only in a few cases, but sufficient to show clearly 
 how dimensions should be placed. In some of the drawings, dimen- 
 sion lines and arrow-heads are placed where dimensions should be 
 given ; this is done so that the drawings may not be used for copies, 
 and that objects similar to those illustrated may be studied, measured, 
 and drawn to scale. 
 
 The models represented are often different in proportion from 
 the regular drawing models ; the proportions are chosen to present 
 the principles by drawings as large as the plates will allow. 
 
 Those who understand tl\e preceding chapters will require no 
 further explanation of the points there considered ; therefore explana- 
 tions of such points, if given at all, will be stated very briefly, and 
 often simply by reference to preceding articles. 
 
no MECHANICAL DRAWING. 
 
 The plates are reproduced from drawings twice their size. The 
 lines of the original drawings are suitable in width for practical 
 drawings. The lines of the plates are finer than is required for 
 any except the most highly finished drawings made by advanced 
 pupils. 
 
 The lines of the drawings upon pages 6i and io6 are suitable 
 in width for practical drawings and for pupils' work. 
 
112 MECHANICAL DRA WING, 
 
 PLATE I. 
 
 Fig. 39. Front and top views of a sphere. 
 
 Any view of a sphere must be a circle, whose diameter is equal to that 
 of the sphere. The centres of these views are in a vertical line. 
 
 Fig. 40. Front aitd top views of an he^nisphere whose plane surface 
 is horizontal and uppermost. 
 
 The horizontal circle is represented by a circle in the top view, and by 
 a horizontal line in the front view (Art. 134) ; the curved surface of the 
 hemisphere is represented in the front view by a semi-circle extending from 
 the horizontal line downward. 
 
 Fig. 41. Front and top views of a cube, two faces being horizontaly 
 and one face appearing a square in the front view. 
 
 The top face appears a square in the top view. 
 
 For the development and that of any simple solid in the following fig- 
 ures, see Chap. IV. 
 
 Fig. 42. Front and top views and developrnent of a right square 
 pris7n. 
 
 Fig. 43. Fro7tt, top, and right side views of a vertical square tablet 
 so placed that the front view gives its real shape. 
 
 See Art. 133. 
 
 Fig. 44. Fro7tt a^td top views of a vertical oblong tablet, which ap- 
 pears its real shape in the front view. 
 
 See Art. 133. 
 
 Fig. 45. Front and left side views, and develop7nent of a horizontal 
 cylinder, whose axis is parallel to the front plane. 
 
 The front view is a rectangle and represents the circles by vertical lines ; 
 the distance between these verticals is equal to the length of the cylinder. 
 The side view is a circle, and should be drawn first. 
 
 The form of the laps by which the parts are fastened together is imma- 
 terial ; that shown in the figure may, in the case of the cylinder or cone, 
 give the most satisfactory results. 
 
PLATE 
 
 
 
 
 
 y' 
 
 
 
 
 
 
 ■x 
 
 
 V AO 
 
 
 <. 
 ^ 
 
 
 Fig 43 
 
 Fig. 45 
 
114 
 
 MECHANICAL DRAWING, 
 
 PLATE II. 
 
 Fig. 46. F?'07it and left side views^ and development of an equilateral 
 t7'ia?igular pris7n,, placed hori207itally and so that the side view is a tri- 
 a7igle. 
 
 The side view gives the real shape of the triangle and should be drawn 
 first (Art. 135). 
 
 The front view is an oblong, whose length is equal to that of the prism, 
 and whose height, given by projecting from the side view, is equal to 1-2 of 
 that view (Art. 138). 
 
 Fig. 47. Fro7it a7id top views of a ci7'cular tablet which appears its 
 real shape i7i the front view. 
 
 See Art. 133. 
 
 Fig. 48. Front a7id top views of an equilateral triangular tablet 
 which appears its real shape in the fro7it view. 
 
 See Art. 133. 
 
 Fig. 49. Fro7it atid top views and develop7nent of a7i 7ipright square 
 pyra77iid^ placed so that the edges of its base are at ^5° to the front pla7ie. 
 
 The top view is a square and should be drawn first (Art. 134). The 
 lateral edges are represented in this view by the diagonals of the square. 
 In the front view the length of the axis is seen, also the real length of the 
 right and left lateral edges ; the nearest lateral edge appears a vertical line 
 as long as the axis, and thus shorter than its actual length (Art. 130). 
 
 Fig. 50. Front and top views of a vertical tu77ibler. 
 
 At the top and bottom of the object are horizontal circles, which appear 
 concentric in the top view. Draw the top view of the outside upper edge of 
 the tumbler ; then the front view of the tumbler ; and from the dotted line, 
 showing the thickness of the glass, obtain the dimensions of the inside 
 circles of the top view. 
 
 Fig. 51. Front a7id top views of a ti7i cookie-ciitter^ placed so that in 
 the front view the handle is a se77ii-circle. 
 
 The top view of the cutter is a circle and should be drawn first ; then 
 draw the front view of the cutter and add the handle ; draw last the handle 
 in the top view. 
 
 Fig. 52. Top and front views of a cyli7idrical box. 
 
 The thickness of the top and bottom is shown by the dotted lines. 
 
 Draw the top view first (Art. 134). 
 
 Fig. 53. Front a7id top views of a ti7i dipper. 
 
 Draw the top view of the dipper first ; then the front view ; then the 
 front view of the handle ; and lastly the top view of the handle. 
 
plAte II. 
 
 / \. 
 
 Fig. 46. 
 
 Fig. 50. 
 
 Fig. 51 
 
 Fig. 49. 
 
 Fig. 47. 
 
 Fig. 48. 
 
 Fig. 52. 
 
 F,g. 53. 
 
Il6 MECHANICAL DRAWING 
 
 PLATE III. 
 
 Fig. 54. Top and front views of an tipt'ight hexagonal prism ^ a7id 
 the development of the same. 
 
 Fig. SS' ^^P atid front views and developme?it of an up7'ight hexag- 
 onal pyramid. 
 
 Fig. 56. F?'ont and top views and develop7nent of a?i upright cone. 
 
 Fig. 57. Fronts top, afid right side views of a7i hexagonal tablet. 
 
 Fig. 1%. Front and top views of a tin tiDinel^ placed so that its circu- 
 lar edges appear circles in the top view. 
 
 Fig. 59. Top and front views of a ti7i grater. 
 
 In all the figures of this plate, except Fig. ^^, the top view should be 
 drawn first. In Fig. 57 the front view should be drawn first. 
 
PLATE llf. 
 
 / \/ \x 
 
 FigJ5 
 
 \/ \ 
 
 1 1 
 
 I 
 I I 
 
 I I 
 
 I I 
 
 I I 
 I I 
 V—-\ 1 
 
 4 \ 
 
 Fig. 59. 
 
 Fig. 57. 
 
 Fig. 56. 
 
 Fig. 58. ^^ 
 
I 1 8 MECHANICAL DRA WING. 
 
 PLATE IV. 
 
 Fig. 6o. Fronts top^ and right side views of a knife-box. 
 
 Fig. 6i. Top and front views of an oilcan. 
 
 Draw the top view first. 
 
 Fig. 62. Top and front views of a caU-bell. 
 
 Fig. 63. Front and left side views of a horizontal hollow cylinder; a 
 horizontal section on AB, and a cross section of the cylinder on CD, 
 
 Draw the side view or the cross section first. 
 
 Fig. 64. Top and front views of a dish with a handle. 
 
 Draw the top view of the dish first ; then the front view ; then draw the 
 handle. 
 
 Fig. 65. Front and top views of a mallet. 
 ■ Draw the front view of the head first, and then carry the drawing of the 
 two views along together, as explained in Art. 36. 
 
 All the above drawings may be made most advantageously by carrying 
 all the views of each object along at the same time, as explained in Art. 36. 
 It is, however, not necessary or important that young pupils should work in 
 this way. With them the question is not speed, but accuracy and under- 
 standing of the principles : hence they may begin with the view which is 
 easiest to draw, and may carry it as far as possible, and even finish it,., 
 before beginning any other view. 
 
PLATE IV. 
 
 ■? 
 
 1^-^ 
 
 
 Fig. 60. 
 
 Fig. 64. 
 
 Fig. 65. 
 
I20 MECHANICAL DRAWING. 
 
 PLATE V. 
 
 Fig. 66. Top and front views of a ti7i coffee-pot. 
 
 Draw the top view of the pot first ; then the front view, adding the 
 handle and nose, first in this view, and then in the top view. 
 
 Fig. 67. Front., and left side views., and a longitudinal section of a 
 tool handle. 
 
 Draw the side (end) view first, and the other two views as explained 
 in Art. 36. The handle, being of wood, may be represented, if desired, as 
 shown on page 61. 
 
 Fig. 68. Fronts top^ and left side views of a flatiron. 
 
 Fig. 69. A front view., a section on CD., and a horizontal section at 
 AB 'through a horizontal spool. 
 
 Draw the section on CD first. 
 
 Fig. 70. Front and top views of a vertical circular tablet^ attached to 
 the back edge of a horizontal square tablet. 
 
 Fig. 7 1 . Front and top views of a vertical hexagonal tablet, attached . 
 to the back edge of a horizontal square tablet. 
 
PLATE V. 
 
 Fig. 70. 
 
 TT 
 
 Fig. 71 
 
 .na 
 
 SECTION AT A B 
 
 Fig. 69. 
 
 riii:/ 
 
 SECTION AT C D 
 
122 MECHANICAL DRAWING. 
 
 PLATE VI. 
 
 Fig. 72. Top^ fronts and right side views of a horizontal oblo7ig 
 tablet^ with a vertical triangular one attached to its right edge. 
 
 Draw the top or side view first. 
 
 Fig. 73. The same views of the same combination as in Fig. 72, but 
 with a vertical pentagonal tablet attached at the back edge of the horizontal 
 tablet. 
 
 Represent the pentagonal tablet in the front view first. 
 
 Fig. 74. Fronts top^ and left side views of a chair., represe7ited by co7n- 
 bini7tg a horizo7ital square tablet for the seat with vertical square tablets 
 for the front a7id back of the lower part, a7id a vertical square tablet for 
 the back of the chair. 
 
 Draw the top view first. 
 
 Fig. "j^. Fro7it, top, and right side views of a square pris7n, support- 
 ing an oblo7tg tablet at a7i a7igle of 4^^. 
 
 Draw the side view of the prism first ; then the front and top views. 
 Draw the front view of the tablet first ; then the side and top views. 
 
 Fig. jG. Fro7it, top, and right side views of a circular plinth, or of 
 tablets arra7iged in the for 771 of a plinth, supporti7ig a square pli7ith, or 
 tablets arra7iged i7i the for77i of a plinth. 
 
 Draw the top view of the circular plinth first ; then the front and side 
 views. Draw the front view of the square plinth first, and then the top and 
 side views. 
 
 Fig. "J']. Top a7td fro7it views of a square plinth with a circular 
 plinth resti7tg upo7i it. 
 
 Draw the top view first. 
 
PLATE VI, 
 
 Fig. 74. 
 
 / 
 
124 MECHANICAL DRAWING, 
 
 PLATE VII. 
 
 Fig. ']^. Top, fronts and right side views of an hexagonal plinth with 
 a circular plinth up07i it. 
 
 Draw the top view first. Make the width of the side view equal to 1-2 
 of the top view. 
 
 Fig. 79. Top, front, afid right side views of a circular plinth with a 
 square plinth upon it. 
 
 Draw the top view first. 
 
 Fig. 80. Front, top, and right side views of a triangular pris?n sup- 
 porting an oblong tablet at an angle of 4J^. 
 
 Draw the side view first. 
 
 Fig. 81. {A) F^-ont and top views of a7i hexago7ial tablet, when it is 
 parallel to the front plane. 
 
 (JB^ Front a7id top views of the sa7ne hexagonal tablet when it is at an 
 angle of 60° to the front plane a7td perpendicular to the top plane, as i7i A, 
 
 Draw the front view of A first ; then the top view. Place the top view 
 at 60° and project from it and from the front view of ^, to complete B, as 
 explained in Art. 109. 
 
 Fig. 82. {A^ Fro7it and top views of a square pris7n, two of its ob- 
 long faces being at ^ j° to the top plane and the other two being parallel to 
 the fro7it pla7ie. 
 
 {B^ Fro7it a7id top views of the sa77ie square pris7n when two of its 
 oblo7ig faces are at ^j° with the top pla7ie, and the other two are at jo^ 
 to the front pla7ie. 
 
 Draw the front view of A first ; then the top view. For the top view of 
 B place the top view of A at 30° and then obtain the front view of B, as 
 explained in Arts. 112 and 113. 
 
 At C are given front and top view^s of the prism when in an upright 
 position. 
 
 The exponents F, T, and S used in this and following figures indicate 
 the different views, — front, top, and side. To avoid confusion they are not 
 placed upon all the figures and some of the points are not numbered or 
 lettered. 
 
PLATE VII. 
 
 Fig. 78. 
 
 Fig. 79. 
 
 Fig. 80. 
 
 Fig. 81. 
 
 Fig. 82. 
 
126 MECHANICAL DRAWING, 
 
 PLATE VIII. 
 
 Fig. 83. Fronts top^ and right side views of the lower portion of an 
 upright square prisni^ cut by a plane at an angle of 4^° to its base ; also 
 the real shape of the section and the develop77ient of the surface of the object. 
 
 Draw the top view first, next the front and side views, then, in the front 
 view, the line of the cutting plane at 45°. Project the section to the side 
 view and obtain its real shape as explained in Art. 152. 
 
 Develop the lateral surface of the prism as explained in Art 54. To 
 obtain the line of intersection upon it, measure (in the front or side view) 
 the distances of points 7, 2, j, and 4 from the base of the object and set these 
 distances off on the development, obtaining 7^, i'^, J^, and ^^. Joining 
 these points in the lines defining each lateral surface gives the line of inter- 
 section. Place the bases upon any desired lines of the development of the 
 bases. 
 
 Fig. 84. Top., fronts and right side views of an upright square prism 
 whose lateral faces are at ^5° to the front plane., and which is intersected 
 by a plane at an angle with its base; also the real shape of the section and 
 the develop7Jient of the lateral surface of the object. 
 
 See Arts. 152 and 54. 
 
 Fig. %^. Fronts top., and right side views of a horizontal triangular 
 prism J a section of the same by a plane at 60° to its horizontal face j also 
 the develop7nent of its lateral faces. 
 
 Draw the side view first, then the front and top views, then the line of 
 the section at 60° in the front view, then the top view of the section, and 
 last the real shape of the section. The lengths A-2D^ ^-J^, etc., for the 
 development are seen in the front and top views. The distance between A 
 and B^ etc., is seen in the side view. 
 
PLATE VIII. 
 
 Fig. 83 
 
 1 
 
 
 
 \ 
 
 — X 
 
 1° 
 
 2" 
 
 j^ 
 
 / 
 
 r 
 
 A 
 
 8 C 
 
 D 
 
 A 
 
 F\g. 84. 
 
 \ 
 
 \i-^ 
 
 Fig. 85. 
 
 
 
 
 
 i 
 
 ^/<D 
 
 
 /D 
 
 ^^^^^^^ 1 
 
 ^ 
 
 
 
 
 2^ 
 
 i 
 
 A 
 
 £ 
 
 3 
 
 c 
 
 A 
 
128 MECHANICAL DRAWING, 
 
 PLATE IX. 
 
 Fig. Z(y. Top, front, and right side views of a vertical hexagonal 
 prison cut by a plane at ^5° to its base; also the real shape of the section 
 and the developiJient of the surface. 
 
 Fig. ^"j. Top and front views of the frustum of a square pyrainid^ 
 a7id the development of its surface. 
 
 Draw the top view of the complete pyramid first ; next its front view ; 
 and then the section in the front view; project from the front view the 
 points for the square which is the top view of the section and shows its real 
 shape. 
 
 The length of the lateral edges of the entire object is seen in the front 
 view from /^ to ^^, and the length of the part below the cutting plane is seen 
 from 2^ to j^. The sides of the base appear their real length in the top view. 
 
 See Art. 59. 
 Fig. 88. Top and front views and develop7nent of the frustum of a 
 cone. 
 
 To develop, divide one quarter of the base into three equal parts. 
 Draw an arc with radius i^-gF, Upon this arc set off 21^-4^ twelve times. 
 The development of the line of the section may be obtained by considering 
 the section as the base of a cone extending from the vertex to the cutting 
 plane. The elements of this cone are equal in length to the distance i^-j^. 
 
 Fig. 89. Top, front, and right side views and development of an up- 
 right square pyramid, intersected by a plane at 43^ to its base. 
 
 The lateral edges of the pyramid are not parallel to the front or side 
 planes, and do not appear their real lengths in the front or side view. To 
 develop the surface, the length of the lateral edges must be found by revolv- 
 ing one of them, as 1-2, until in the top view it is horizontal and at i'^-2'. 
 As the pyramid revolves upon its axis until the lateral edge comes to this 
 position, point 2 moves in an arc which appears an arc in the top view and a 
 horizontal Hne in the front view, and 2" will be under 2', and on the level of 
 2P^, and the distance 2''-i^ must be the real length of the edge 1-2. Point j, 
 in 1-2, also describes an arc in the top view, but it is not necessary to draw 
 the arc, as f must be in 1F-2'' and on the level of J^. The distance 1^-3'' 
 is the real distance from the vertex to the points of the section which are 
 in the two left lateral edges. The real length of the distance, 1-4, from the 
 vertex to the points in the two right lateral edges is found in a similar way. 
 
PLATE IX. 
 
I30 
 
 MECHANICAL DRAWING. 
 
 PLATE X. 
 
 Fic;. 90. (y/) Front and top views of a vertical circidar card, par- 
 allel to the front plane, 
 
 (B) The same views of the same card when perpendicular to the top 
 plane and at an angle of 43^ with the front plane. 
 
 The views of the first position are necessary to obtain those of the second, 
 and the top view of A, when it is placed at 45°, will be the top view of B. 
 
 Draw the horizontal and vertical diameters in the front view of A and 
 mark their ends /, j and 2, 4. Place these points in both the top views 
 and obtain them in the front view of B by verticals from the top view of B 
 intersected by horizontals from the front view of A. These points are the 
 ends of the diameters of the ellipse, which is the front view of B, To find 
 other points, place any point as C^ in the front view of A. Then place C^ in 
 both top views, and find C in the front view of B as explained. If more 
 than four points are desired it is best to take eight or twelve equidistant points. 
 
 FUi. 91. (y^) Front a7id top views of a cone resting upon an eleine^it 
 which is horizontal and parallel to the front plane, 
 
 {B) Top and front views of the sa?ne cone, when the element upon 
 which if rests is horizontal and at an angle ofjo^ to the front plane. 
 
 The front view of A will be an isosceles triangle of which the lower of the 
 two equal sides is horizontal ; the base of the triangle represents the base of 
 the cone. If the dimensions of the base and axis of the cone are given, the 
 length of the elements must be found by drawing an isosceles triangle 
 whose base is equal to the diameter of the base of the cone and whose alti- 
 tude is equal to the axis of the cone. 
 
 To obtain the front view of A, draw the element i-g horizontal and its 
 real length. With q^ as a centre and g^-i^ as radius, describe an arc 
 through point i^. With i^ as a centre and the diameter of the base of the 
 cone as a radius, describe an arc to intersect the arc from g^ in point j^. 
 Join 7^-5^ and j^-p^y this gives the front view of the cone. Bisect the 
 line i^-J^ and obtain the centre of the base of the cone. 
 
 The top view of the base is an ellipse which may be found as explained 
 in Fig. 90. To obtain the points by which the ellipse may be determined, 
 the circle of the base must be revolved until it is parallel to the front plane : 
 it may then be divided as desired and the points revolved back to the line 
 i^-j^. As the circle revolves the points in it move in lines which in the 
 front view appear parallel to the axis of the cone. 
 
 In the top view the circle must be drawn w^hen revolved until parallel to 
 the top plane ; the points marked in the circle of the front view can then be 
 placed in that of the top view. As the circle revolves these points move, in 
 
132 
 
 MECHANICAL DRAWING. 
 
 this view, in lines parallel to the axis of the cone, and these lines intersected 
 by verticals from the points of the front view give the points in the ellipse. 
 
 The top view of B is the top view of A with its axis at an angle of 30°. 
 Having drawn this top view the points of the front view may be found as 
 explained in Art. 112. 
 
 Fig. 92. Front a7id top views of a cylinder whose axis is at 4^^ to 
 the top plane and parallel to the front plafie. 
 
 Draw the front view first. Obtain the top view of the circles by means 
 of points in the circles, as in Fig. 91. 
 
 To place more than four points, draw a circle upon 1-2 of the front 
 view and upon j-4 of the top view ; divide these circles as desired, and so 
 number the divisions that any point in the base is represented in both circles 
 by the same figure or letter. 
 
 Project the points in the circle drawn in the front view to line 1^-2^, 
 and from this line project the points to the top view, where they will be 
 situated in horizontal lines through the points of the circle there drawn. 
 This amounts to revolving the circular base so that it is parallel first to the 
 front plane and then to the top plane, as explained in Fig. 91. 
 
 Fig. 93. Front., top., and right side views of a square pyra?nid^ whose 
 axis is parallel to the front plane and at ^5° to the top plane. The edges 
 of the base are at 43^ to the front pla^ie. 
 I The edges of the base, being at angles to both the front and the top 
 
 planes, do not appear their real length in either view. One diagonal of the 
 base is parallel to the front plane, the other diagonal is parallel to the top 
 plane ; each diagonal appears its real length upon the plane to which it is 
 parallel, and thus the base of the pyramid, which is perpendicular to the 
 front plane, is represented in the front view by a line whose length is equal 
 to the diagonal of the base. If the length of the edge of the base is given, 
 that of the diagonal must be found by drawing the square, or one half of it, 
 . as at ^. The front view should be drawn first, then the top or side view. 
 I Fig. 94. (y^) Fro7it^ top^ and right side views of a square plintJi sup- 
 porting a triangular prism. 
 
 (^B) Top a7id front vic%vs of the same objects when the vertical faces of 
 the plinth are at 30P and 60^ with the frofit plane. 
 
 (A) First draw the views of the plinth; then draw the prism in the 
 front view. The distance /-2 of this view is equal to the altitude of the 
 triangular end^^of the prism. To obtain 1-2, supposing that the length of 
 the side of the triangle is given, draw the real shape of the triangle as at C. 
 From the front view of the prism obtain the other views. 
 
 (Z?) Place the top view of A at the required angle, and obtain the front 
 view, as explained in Arts. 1 12 and 1 13. 
 
A -f' 
 
 Fig. 92. 
 
 PLATE X. 
 
 Fig. 91. 
 
 Fig. 94. 
 
 
134 
 
 MECHANICAL DRAWING. 
 
 PLATE XI. 
 
 Fig. 95. Top, front, and right side views of an hexagonal pyramid 
 intersected by a plane oblique to the base; the true shape of the sectio7i 
 77tade by the plane, and the developtJient of the lateral surface of the py ra- 
 ni id. 
 
 All the points of intersection of the lateral edges with the cutting plane 
 are seen in the front view, and may be projected to the other views. The 
 width of the side view is the distance j^-j^. The lateral edges i-y and 
 4-7 appear their real length in the front view ; therefore the distance from 
 the vertex, 7, to the points c and d of the section, may be measured from 
 7^ to c^ and from yF to d^. 
 
 To obtain the distances from the vertex 7 to the points of the section in 
 the other lateml edges, draw, in the front view, horizontal lines through 
 points a and b to the left or the right lateral edge, and measure the distances 
 from point 7 upon these lines, as explained under Fig. 89. 
 
 Fig. 96. Top and front views of a cone, showing the section 77tade by 
 a pla7te passing through its vertex. 
 
 The line i^-J^ is the real length of the line represented by ii-aT, Line 
 iF^jF is equal to the altitude, and line 2I-JI equal to the base of the 
 isosceles triangle which is the section made by the plane. Art. 161. 
 
 Fig. 97. Top and fro7it views of a vertical cone i7itersected by a plane 
 parallel to the front plane and ifi front of the axis of the conej also the 
 develop77ie7it of the lateral surface. 
 
 The circle of the base is cut in points i and 2, which are determined in 
 the top view. To obtain j, the highest point in the section, measure 
 V^-jT^ the distance of the plane from the axis of the cone, and draw a 
 vertical line 3' -3'' the distance V^-3^ from the axis. This represents a 
 side view of the plane, and intersecting the element of the cone at 3'', gives 
 the level of the highest point. 
 
 To obtain two other points, intersect the cone by a horizontal plane B. 
 This gives a circle which appears a circle in the top view, where it inter- 
 sects the cutting plane in 4^ and 5^, points in the required Hne of intersec- 
 tion ; 4F and 5^ must be in B^ and in projecting lines from these points 
 of the top view. 
 
 Points 4 and j are in circle B, which is the base of a cone, just as i 
 and 2 are in the base of the original cone. Any number of planes parallel 
 to B may be taken, and each will give two points in the section, which is an 
 hyperbola. 
 
136 MECHANICAL DRAWING. 
 
 To develop the lateral surface of the cone proceed as explained in Art. 
 58 and under Fig. ZZ. To show the line of intersection upon it, measure, 
 upon the arc, the distance of /^ and 2T^ from b'l, and set this distance off 
 from bi> in the development. The circle given by plane B becomes, in the 
 development, an arc parallel with the one which bounds the lateral surface 
 of the cone. In the circle of plane B are points 4 andjy their distances 
 from each other and from C are seen in the top view. These points will be 
 placed in the development by measuring the distance C-47 and C-j^ on the 
 arc of the top view, and setting it off from C in the development. In this 
 way all the points by which the section is obtained may be placed in the 
 development. 
 
PLATE. Xi; 
 
 ?' 
 
 
 3^ 
 
 
 eV 
 
 
 / ^ 
 
 
 mA 
 
 \ ^"'"^^ 
 
 
 ^ / 
 
 XjT 
 
 1 
 
 jj<^. 
 
 6-^ . 1 
 
 1 
 
 -^5i 
 
I^S MECHANICAL DRAWING, 
 
 PLATE XII. 
 
 Fig. 98. Top^ fronts aiid left side views of a vertical cone intersected 
 by a plane at an angle with its base and cutting all the ele7nentsj the real 
 shape of the section j also the develop7nent of the lateral surface of the 
 C07ie. 
 
 The points in which the contour elements, V-i and F-7, of the front view 
 intersect the cutting plane are seen in the front view, and may be projected 
 to the top and side views. The points f and g^ in which the contour ele- 
 ments, V-4 and V-io^ of the side view intersect the cutting plane are seen in 
 the front view, and from this view may be projected to the side view. To 
 obtain f and g in the top view the distance between these points may be 
 measured in the side view and then set off in the top view ; or a horizontal 
 cutting plane may be taken through f and g; this gives a circle which in 
 the top view gives /"^^ and^^. To obtain other points in the section, other 
 horizontal cutting planes may be used, as explained under Fig. 97, or equi- 
 distant elements may be placed on the cone, and their intersections seen in 
 the front view. As the surface is to be developed, the latter method is 
 preferable. The lengths of these elements are found as if they were the 
 edges of the pyramid of Fig. 95. The lateral surface should be divided into 
 at least twelve equal parts. 
 
 To obtain in the development the direction of the curve at its termina- 
 tion on lines VD-ii>^ add one twelfth at either end of the arc which is the 
 development of the base of the cone, and obtain lines F^-^ and F^-zy. 
 From VD^ on these lines set off the distance Vi>bl> and Vl>ci>. Trace 
 the curve of intersection through these points. 
 
 Fig. 99. Top and front views of a cone cut by a plane parallel to an 
 eleifient; also the real shape of the parabola which is the section. 
 
 This section may be obtained as explained under Figs. 97 or 98. The 
 drawing makes use of the cutting plane explained under Fig. 97. 
 
 Fig. 100. Top, front, and right side views of a sphere, intersected by 
 a vertical cone, and a horizontal cylinder. 
 
 In this figure the axes of the cylinder and cone pass through the centre 
 of the sphere ; therefore the lines of intersection are circles. The planes 
 of these circles are perpendicular to the front plane and are therefore 
 represented by straight lines in the front view. The circles in which the 
 cone and sphere intersect are parallel to the top plane and are represented 
 by circles in the top view. 
 
PLATE XII, 
 
 Fig. 98. 
 
 ~\'/W\' 
 
 II !Z I' Z 
 
 W ^N ^-'^ i \ V 
 
 \\ N m<^'\ L \ 
 
 %. , 4 i J. M A\ 
 
 Fig. 100. 
 
 j 
 
140 MECHANICAL DRAWING, 
 
 PLATE XIII. 
 
 Fig. 1 01. Top aiid front views of an elbow exte7iding froin a conical 
 support. 
 
 If a cylinder or cone is cut by a plane oblique to its axis, one part may be 
 revolved upon the other until the ellipses of the parts coincide in a second 
 position. This will happen when an arc of 180° has been formed by the 
 revolution of the parts. After such revolution the angle between the axes 
 of the two parts is twice that of the angle of the cutting plane and the axis 
 of the original solid. 
 
 The elbow illustrated may be formed by cutting a cylinder at 45° to its 
 axis, and revolving the parts as explained. 
 
 The plane of the section is perpendicular to the front plane and there- 
 fore the ellipse in which the vertical and horizontal cylinders intersect ap- 
 pears a straight line in the front view ; it appears a circle in the top view, 
 for the surface of the vertical cylinder is perpendicular to the top plane. 
 
 To develop the cylinder, divide its surface into any number of equal 
 parts by elements. To do this in the case of the horizontal cylinder the 
 circles must be revolved to appear their real shapes and divided as shown 
 by the dotted lines. The elements having been drawn, their real lengths 
 are seen in both views and may be set off upon the respective lines of the 
 development. 
 
 When a cylinder is to be developed neither of whose bases is at right 
 angles to the axis of the cylinder, it is necessary to assume a circle at right 
 angles to the axis ; this will develop into a straight line. Thus, suppose the 
 horizontal part to be represented in the front view by {a b de d)^. To 
 develop this cylinder, it will be necessary to develop a circle which is at 
 right angles to the axis. This circle may be assumed anywhere, as at a c, 
 and develops into a straight line a^cD. The development of the part at 
 the right of the circle should be placed at the right of a^^c^^ just as the 
 part at the left is placed at the left of a^c^. 
 
 The frustum of the cone is developed as explained under Fig. Z%, 
 
 Fig. 102. Top and front views of an upright square prism inter- 
 sected by a horizontal square prism at the rights and by a triangular 
 prism at the left; also the develop?ne7tts of the lateral stirfaces of the 
 vertical and triangular pris7ns^ and half that of the horizontal square 
 pris7n. 
 
 The points in which all the edges of both horizontal solids intersect the 
 surface of the vertical one are seen in the top view and are readily pro- 
 jected to the front view. 
 
142 MECHANICAL DRAWING. 
 
 Develop the lateral surface of the upright prism as explained in Arts. 
 51 and 54. The horizontal faces of the square block intersect the upright 
 prism in horizontal lines which are parallel to its bases, and which in the 
 development must be parallel to MN and OP. The vertical faces inter- 
 sect the prism in vertical lines parallel to the edges B and Dj and the line 
 of intersection of the horizontal and vertical square prisms on the develop- 
 ment of the vertical prism is a rectangle whose length is the distance 
 jT-ci-jij its width is the distance 1^-2^. 
 
 The horizontal face of the triangular prism intersects the vertical square 
 prism in lines which develop into a straight line parallel to OPj its 
 length is the distance 7-5-6 of the top view ; the positions of this line and 
 also that of point 5 are seen in the front view. 
 
 The other developments require no special explanation. In all the prob- 
 lems it is simply necessary to remember that the development gives the real 
 length of every line of every surface and that the length of any line must be 
 taken from the view in which it appears its rear length, or, if not seen of its 
 real length in any view, this must be determined as explained in Art. 114. 
 
 The development of any surface is most easily obtained by placing it 
 so that one set of dimensions may be projected from the front view. 
 
PLATE XIII. 
 
 Fig. 102 
 
 
 
 
 6' 
 
 \b 
 
 / 
 
 ^2' 
 
 
 
 
 
 S' 
 
 
 
 3 
 
 l*^ 
 
 
 
 1 
 
 / 
 
 
 \ 
 
 
 / 
 
 \ 
 
 
 
 
 
 
 / 
 
 \ 
 
 
 
 
 
 
 / 
 
 \ 
 
 
 
 
 
 
 / 
 
 N 
 
 
 
 
 
 
 / 
 
 
 s 
 
 
 
 
 
 / 
 
 
 \ 
 
 
 
 
 
 / 
 / 
 / 
 
 
 .\ 
 
 ., 
 
 /•f 
 
 Z' 
 
 
 
 
 A' 
 
 
 B' 
 
 
 C 
 
 
 V 
 
 
 
 
 
 
 
 
 
 N 
 
 "5 
 
 --l 
 
 1 
 
 \r 
 
 
 1 " 
 Z" 
 
 
 
 
 -/ 
 
 ) 
 
 
 
 16 
 
 1 
 
 
 
 
 
 / f 
 
 
 
 A^ 
 
 ! 
 1 
 1 
 
 
 
 C" 
 
 
 
 D" \ 
 
 Pi 
 
144 MECHANICAL DRAWING. 
 
 PLATE XIV. 
 
 Fig. 103. An upright square prism intersected at the left by a hori- 
 zontal square prisjH^ whose lateral faces are at 43^ to the top and front 
 planes^ and at the right by an hexagonal prism with two of its lateral 
 faces vertical; also the develop77tent of the lateral surface of the hexagonal 
 prison, and half that of the horizontal square prism. 
 
 The intersections are found as previously explained. 
 
 To develop the lateral surfaces of the penetrating prisms, measure the 
 widths of the faces in the end views, and the lengths of the lateral edges 
 in the front or top view, and combine these dimensions as explained. 
 
 Fig. 104. An upright square pris7n intersected at the right by a hori- 
 zontal hexagonal pris7n^ two lateral faces bei7ig horizontal^ and at the left 
 by a horizontal cylinder; also the develop77ie7tts of the lateral surface of the 
 square prism ^ and half those of the hexagonal pris7n a7td of the cyli7ider. 
 
 In the front view, the intersections of the upper and lower elements 
 I and 7 of the cylinder with the left edge of the prism are seen. In the top 
 view, the intersections of the front and back elements 4 and 10 are seen. 
 To obtain other points assume elements, to place which a side view is nec- 
 essary ; when determined in this view they may be transferred to the top 
 view by revolving the circle in the top view until it appears a circle, 
 dividing it, and numbering the points, to correspond with the end view ; or 
 projection methods may be used and the distances of the points 2, 12, etc., 
 from a vertical line through the centre of the cylinder may be taken in the 
 compasses from the side view and set off in the top view, one half each side 
 of the axis. 
 
 The intersections of the elements and the prism are seen in the top view 
 and may be projected to the front view. 
 
 The intersections of the cylinder and the faces of the prism are semi- 
 ellipses which appear a semi-circle in the front view. 
 
 The intersection of the hexagonal and square prisms is found as explained. 
 
 To develop the lateral surface of the square prism and show the lines 
 of intersection upon it, first develop the entire lateral surface ; then draw, in 
 the front view, verticals upon the surface of the square prism through the 
 points of the intersections. Place these lines in the development by setting 
 off from AD and CD the distances, seen in the top view, of the lines from 
 Ai and C^; thus to obtain aD and eD draw a parallel to AD at the distance 
 A'lai from it, and then set off in this line the distances of a^^ and e^^ from 
 AP'C^. In this way all the points necessary to determine the lines of inter- 
 section in the development may be obtained and the lines drawn through 
 the points. 
 
PLATE'Xll/. 
 
146 MECHANICAL DRAWING. 
 
 PLATE XV. 
 
 Fig. 105. Top a?id front views of an upright cylinder intersected at 
 the left by a horizo7ital square prisjn^ and at the right by a horizontal 
 cylinder J also the development of the lateral surface of the cylinder and 
 half that of each intersecting body. 
 
 Both intersections and developments are the same in principle as those 
 already explained. The line in which the square prism intersects the cylin- 
 der develops into a rectangle in the development of the cylinder, because 
 the horizontal faces intersect the cylinder in circles which develop into 
 straight lines, and the vertical faces intersect elements of the cylinder. 
 
 Fig. 1 06. Top and front views of a vertical cylinder intersected at the 
 left by a square prisjn^ and at the right by a triangular prism at 4^° with 
 the cylinder; also developments. 
 
 In the front view the distance E^F^ is equal to the distance E'k of 
 the view, F'E'G\ of the end of the prism, which must be drawn before the 
 front view can be completed. It is not necessary to make this separate 
 drawing, as the dotted lines in the front view which show half the end give 
 all that is necessary. 
 
 The points in which the lateral edges of the prisms intersect the cylin- 
 der are seen in the top view. The inclined sides of the prisms intersect the 
 cylinder in ellipses, points in which may be found by assuming lines on the 
 surfaces of the penetrating solids. A Hne from i^ intersects the cylinder 
 at ai., from which the positions of a^ and c^ may be obtained. A line from 
 d^ gives the position oi g^. In this way any number of points may be ob- 
 tained. 
 
 The development of all the lateral surface of the triangular prism is 
 given ; point ^ is in a parallel to the lateral edges, and distant from E the 
 distance E^d'j point // is in a parallel to the lateral edges, whose distance 
 from F is the distance F^^eT, 
 
 In the development of the square prism, points a^ and cD are in par- 
 allels to the lateral edges, and the distances B^-i^ and B^-2^ from Bl^. 
 
 In the development of the hne of intersection upon the cylinder, point 
 gD is in a parallel to fl>ml>, whose distance from it is seen in the top view, 
 from /"^ to ^^, measured on the arc. Point n^ is in this same line, and 
 hD is in a parallel iof^m^^ whose distance from it is the distance /^^/z^, 
 measured on the arc. 
 
PLATE XV. 
 
 Fig. 105 
 
 Fig. 106. 
 
 D' 
 
 / 
 
 rt^ 
 
 / 
 
 
 Q 
 
 r 
 
 ■ V"' 
 
 B^ 
 
 ■- 1\ 
 
148 MECHANICAL DRAWING. 
 
 PLATE XVI. 
 
 Fig. 107. Front., top., aiid right side views of a horizontal square 
 prism i7ttersected by a vertical square pyrainidj also develop7nents. 
 
 The intersections of lines A V and CV with the edges of the prism are 
 seen in the front view, and the intersections of lines B V and DV with the 
 prism are seen in the side view. Project these points of intersection from 
 the side to the front view and from the front to the top view. The inter- 
 secting surfaces are plane and therefore intersect in straight lines, which 
 join the points of intersection. 
 
 To obtain the lines of intersection in the development of the pyramid, 
 measure the distances from the vertex V to the points of intersection in VA 
 and VC in the front view, and the distances from the vertex to the points 
 of intersection in VB and VD in the side view. Set these distances off on 
 the proper lines of the development and join the points. 
 
 To obtain the points for the lines of intersection upon the lateral sur- 
 face of the prism place points k and d in edge 2^ and e and f m edge 4; 
 their positions, in the edges, are seen in the front view. Points ^, /?, g^ and b 
 are in a plane perpendicular to the axis of the prism and must, in the devel- 
 opment, be in a straight line perpendicular to the lateral edges. The dis- 
 tances of a and h from edge 2., and of b and g from edge 4 are seen in the 
 side view. 
 
 Fig. 108. Top^ front., and left side views of a horizontal cylinder in- 
 tersected by an upright square pyramid; also develop77ients. 
 
 This problem is solved in the same way as Fig. 107, but the intersec- 
 tions are not straight lines. Each lateral face of the pyramid cuts the cyl- 
 inder in lines which are parts of the complete ellipse which will be given by 
 the plane of each face cutting through the cylinder. 
 
 The intersections of the lateral edges are seen in the front and side 
 views. To obtain other points in the lines of intersection assume any line, 
 as V-i., on the surface of the pyramid. This line intersects the surface of 
 the cylinder in points b and g; these points are readily obtained in the front 
 and top views from b^ and^^. In the side view V-i also represents the line 
 V-2 on the right face of the pyramid in which points h and o are situated. 
 In this way any number of points in the curves may be found. 
 
 To develop the pyramid and show the line of intersection, the distance 
 from V to the points of the intersection in VD and VB must be measured 
 in the front view ; the distances from V to the points of the intersection in 
 VA and VC must be measured in the side view. 
 
I50 
 
 MECHANICAL DRAWING. 
 
 To place the points 3,^ and o^ h in lines V-i and V-2y draw through these 
 points lines parallel to the base of the pyramid ; the line through o inter- 
 sects V^BF in point ^^. The distance of 4 from V is seen in the front 
 view and 4D is readily placed. A line through 4P parallel to ADB^ and 
 intersecting VD-2D must give oDj in this way all the points may be found. 
 
 To obtain the lines of intersection in the development of the cylinder, 
 measure, in the end view and around the arc, the distances apart of the ele- 
 ments containing the points ; determine the positions of the points in the 
 elements in the front view. 
 
PLATE XVL 
 
 Fig. 107. 
 
 
 / .yx. 
 
 N^T 
 
 ^^^^ 
 
 
 ' A 
 
 'W^ 
 
 
 Fig. 108. 
 
 D ''A* D' / A^ -i B 
 
152 MECHANICAL DRAWIXG. 
 
 PLATE XVII. 
 
 Fig. 109. Top and fro7it views of a ve?'tical cone intersected at the 
 left by a ho?'izontal triangular pris7n^ a?id at the right by a sqjiare prism. 
 
 The points in which edge D of the triangular prism and edges 2 and ^ 
 of the square prism intersect the cone are seen in the front view, for these 
 edges are in the plane of the contour elements. To find other points in the 
 curves in which the cone is intersected by the lateral faces of the prisms, 
 assume any horizontal cutting plane, as AB ; this gives, upon the triangular 
 prism, a rectangle whose width is j-8^ and upon the square prism, one whose 
 width is 5-dy the plane intersects the cone in a circle C. The circle and these 
 rectangles intersect in points e^ d and ^, b, which must be points in the lines 
 of intersection of the cone with the triangular and square prisms. As many 
 points as are desired may be obtained by means of other cutting planes. 
 
 Fig. II o. Top a7id fro7it views of a vertical co7ie i7itersected by a 
 horizo7ital sqjcare pyramid. 
 
 Edges E and F intersect the contour elements of the cone. To find the 
 intersections of edges G and H, and also to obtain other points in the inter- 
 sections, intersect the solids by cutting planes. CD is a horizontal cutting 
 plane and gives four points, e,f g, h, in the lines of intersection. 
 
 Fig. III. A vertical cyli7ider i7itersected by a ho7'izo7ital cone; also 
 develop77ie7its. 
 
 The principles involved have been previously explained. 
 
 To develop the cone, find the distances of points b^ c^ d^ e^ and/" from 
 the vertex F, by measuring upon the contour elements, as explained under 
 Fig. 98. 
 
 Obtain the line of intersection in the development of the cylinder by 
 measuring, in the top view and around the arc, the distances between the 
 elements containing the points, and by measuring, in the front view, the 
 positions of the points in these elements. 
 
PLATE XVII. 
 
154 MECHANICAL DRAWING. 
 
 PLATE XVIII. 
 
 Fig. 112. Front., top., and left side views of a loose joint hinge; also 
 front and top views of the two parts forming the hinge., when they are 
 separated from each other. 
 
 Fig. 113. Fronts top., and left side views of a sash lift; also a section 
 on AB. 
 
 Fig. 114. Fro?it, bottom^ ajid left side views of a drawer pull j also a 
 section on CD. 
 
 Most of the objects shown on Plates XVIII to XXII inclusive do not 
 require all the views given to make them complete working drawings. In 
 Fig. 113 or 114, for instance, simply the front and top views and the sec- 
 tion furnish all the information necessary to make the object. The side 
 views are given to familiarize the student with the arrangement of all the 
 views. 
 
PLATE* XV|[ 
 
 Fig. 112, 
 
 LOOSE JOINT HINGE. 
 
 Lip ^ 
 
 1 
 ' ' ^ 
 
 >- 
 
 -^ 
 
 '^ 
 
 .- n. I 
 
 SASH LIFT 
 
 Fig, 113. 
 
 0- 
 
 -^ 
 
 SECTION AT A B' 
 
 DRAWER PULL 
 
 Fig. 114. 
 
 SECTION AT C D 
 
Is6 MECHANICAL DRAWING. 
 
 PLATE XIX. 
 
 Fig. 115. Fronts top^ and right side views of an iron-cased bolt^ with 
 details of the saine. 
 
 Dotted lines are very confusing when numerous ; therefore when many- 
 would be required it is customary to represent the different parts of an ob- 
 ject separately. 
 
 In this figure all the views are needed to make the construction clear. 
 
 The ornamental tracery upon the object is not represented. Such de- 
 tails should be omitted when pupils make working drawings of common 
 objects. 
 
PLATE XIX. 
 
 Fig. 115. 
 
 DETAILS OF AN IRON CASED BOLT. 
 
 THREE FOURTHS SIZE. 
 DRAWING 19 BOSTON DEC.'94. 
 
 ^ 
 
 D 
 
 -#4 
 
 
 y//V>y//OV//>y/y//y/>vv<>vv>V/vy//yyyyi__. 
 
 •:SSZZSZZ2ZZZZ22m 
 
 SECTION OF CASE ON A B ^ 
 
 SECTION OF CASE ON C D 
 
 SECTION AT E F 
 
 i 
 
 '{WfiWiWMlU^ 
 
 BRASS SPRING 
 
 SECTIONS OF STRIKER 
 
158 MECHANICAL DRAV/ING. 
 
 PLATE XX. 
 
 Fig. 116. Details of a7i iron side pulley. 
 
 A is a front view of the piilley. 
 
 B is a side view of the pulley, 
 
 C is a section of the f rain e on GH. 
 
 D is a back view of the frame. 
 
 E is a front view and a section on EF of the wheel. 
 
 Fig. 117. Fronts top, and side views of an iron bracket. 
 
 Fig. 118. Fronts top., and bottom views of an iron clamp. 
 
PLATE XX. 
 
l6o MECHANICAL DRAWING, 
 
 PLATE XXI. 
 
 Fig. 119. Free-hand sketches of an iron caster^ from which finished 
 drawings are to be made. 
 
 In making such sketches the proportions of all the parts should be rep- 
 resented as truly as possible ; but the parts should be drawn free-hand, 
 and the drawings completed before the object is measured. It is not 
 necessary that the proportions be exact, but it is better to have them so, as 
 errors in measuring or in figuring the sketches may be made. If the 
 sketches are correct in their proportions, these errors will be shown when 
 the drawings are made to scale, and compared with the free-hand sketches. 
 
PLATE XXI. 
 
 PLATE. 
 
 FRAME. 
 
I 62 MECHANICAL DRAV/ING, 
 
 PLATE XXII. 
 
 Fig. 120. Views of a wooden faucet. 
 
 A is the front view. 
 
 B is the top view. 
 
 C is the left side view. 
 
 D is a section at KL. 
 
 E is a sectiofi at J/yV. 
 
 The plug is surrounded by cork, which gives a tight joint. The cork is 
 shown in the sections by the dotted parts. 
 
 The cut surface of the wood may be shown as illustrated on page 6i. 
 
 The lines in which the conical parts of the faucet intersect the faces of 
 the octagonal part are the only points requiring explanation ; these lines 
 are really hyperbolas (Fig. 97). It is customary to represent such curves 
 by arcs of circles of which the highest points and the points in the edges 
 are found exactly ; the lines of intersection upon the narrow faces are 
 so slightly curved that straight lines may be, and generally are, substi- 
 tuted for the arcs. 
 
 To determine the points for the arcs, notice that, in the front view, the 
 contour elements of the conical parts must intersect the upper and lower 
 lines at the central or highest points of the curves in two wide faces. As the 
 wide faces are of equal width, the highest points in the curves of these faces, 
 in both front and top views, will be in vertical lines drawn through the 
 points of intersection. To find the points in the edges, an edge must be 
 represented so that its real distance from the centre of the faucet shall be 
 seen ; to do this revolve, about the centre of the end view, point a^ into the 
 position a'; project from a' a horizontal line which represents an edge, and 
 intersecting element 2-1 of the cone at a'\ gives the position of the point 
 where the conical surface cuts the edges of the faucet. These edges are 
 equally distant from the centre of the faucet ; therefore in both front and 
 top views all the arcs must end at the edges in points which are in a vertical 
 line drawn through a'\ 
 
u 
 C) 
 
 
 UJ 
 Q 
 
 
 3 
 
 
 2: 
 
 O 
 
 
 <. 
 
 
 1- 
 
 o 
 
 h 
 
 LlI 
 
 co 
 
 CM 
 
 
 —1 
 
 
 
 
 <c — 
 
 - m 
 
 
 ^ 
 
 o 
 
 
 hn 
 
 UJ 
 
 00 
 
 Csi- 
 
 Li_ 
 
 Q 
 
 
 CM 
 
 
 O 
 
 
 O 
 
 
 O 
 
 
 > 
 
 
 o 
 
164 MECHANICAL DRAWING. 
 
 PLATE XXIII. 
 
 Fig. 121. A front view and section of a washer. 
 
 Fig. 122. Front and top views of the end of a rod adapted to hold a 
 lever., which is pivoted on screws inserted one in each part of the fork. 
 
 The forked part is cast and is screwed to the end of a round rod, of 
 which a part is shown. 
 
 The information which an end view would give is necessary to make 
 the drawings give all the facts of form, but the placing of the letter D after 
 the dimension which shows the diameter of the round part of the fork, 
 makes the end view unnecessary. 
 
 Threaded holes are best shown as illustrated ; in one view two circles 
 are drawn, the outer being dotted and the inner full. Some draughtsmen 
 represent part of the outer circle by a full line which becomes tangent to 
 the inner circle. This is more as the thread appears, but the simpler rep- 
 resentation is the better. In the other view the dotted lines which show 
 the section of the thread are at an angle of 60° with each other ; they 
 should not join each other, as this will destroy the effect of a dotted line ; 
 they should be drawn free-hand with a writing-pen. Only the outline of the 
 thread should be represented, for dotted lines in place of all the full lines 
 shown on Plate XXVI will give an unsatisfactory drawing. 
 
 Fig. 123. A top view arid a front view., 07ie half of which shows a 
 sectioji through the ceritre^ of a machine detail called a gland^ and used for 
 compressing packing about a piston •to prevent the escape of steam or 
 water. 
 
 It is customary to show in section one half (or any other part) of the 
 view of a symmetrical object ; this saves the time required to make com- 
 plete views of the outside and of the inside of the object. 
 
 In any working drawing no more lines sfeould be given than are neces- 
 sary to show the construction ; dotted Hues are especially to be avoided. 
 
I 66 MECHANICAL DRAWING, 
 
 PLATE XXIV. 
 
 Fig. 124. Top a7id side views ^ and a front view^ one half of ivhich 
 shows a section, of a bearing for the end of a shaft which moves back and 
 forth in a plane, and so requires a bearing that rotates. The shaft rests 
 lipon a box of composition B, which is replaced when worn. 
 
 Fig. 125. A front view, of which one half is in section, and a verti- 
 cal cross sectio7i, of a bearing used upori a locomotive.. 
 
PLATE XXIV. 
 
 SHAFT BEARINGS. 
 
 SCALE, 3in. = 1 ft. 
 DRAWING 24. BOSTON, DEC 94. 
 
1 68 MECHANICAL DRAWING. 
 
 PLATE XXV. 
 
 THE HELIX. 
 
 If a point moves in two directions about a given line as axis, — that is 
 around and along the line at the same time, — a helical curve results. 
 
 The simplest form of this curve is found in a common spring. The 
 lines which bound the threads of screws are helices ; these curves are found 
 in many constructions. 
 
 The motion of the point in its different directions may be uniform or 
 variable ; in the common forms of the curve both motions are uniform, and 
 are produced by a point which moves uniformly around and along a cylin- 
 der at the same time. 
 
 The distance which the point travels along the cylinder, in going once 
 around it, is called the pitch. 
 
 To draw the common helix, divide the circle which is a view of the line 
 in which the generating point moves about the axis, into any number of 
 equal parts, and divide the pitch into the same number of equal parts. 
 When the point has moved upon the circle of the end view over one of 
 the equal spaces into which the circle is divided, it has moved in the other 
 view, along the axis from its starting point, a distance equal to one of the 
 equal spaces into which the pitch is divided ; when it has moved one 
 quarter around the circle it has moved one quarter of the pitch ; when it 
 has moved one half around the circle it has moved one half of the pitch, 
 and so on for the whole revolution of the generating point. Hence to ob- 
 tain all the points for the view of the helix, draw parallels to the axis from 
 the points of division in the circle, and intersect these lines by perpendicu- 
 lars to the axis from the equal divisions of the pitch. If the points of the 
 circle and the pitch are numbered in order from the first point, the inter- 
 sections of lines of the same number will be points in the required 
 curve. 
 
 Fig. 1 26. A helical curve Mpo7i a cyli7ider. 
 
 The curve is found as explained above. The division of the pitch into 
 twelve equal parts is advisable for pupils. To avoid error, the points of the 
 bottom view and also those of the pitch may be numbered from i to 12. 
 The curve is symmetrical; its vertices 12 and/' must be curved. 
 
 If the cylinder is developed, the helix will become the hypothenuse of a 
 right-angled triangle A, of which one side of the riglit angle is equal to the 
 circumference of the cylinder and the other is equal to the pitch. 
 
 When many curves of the same pitch are desired, a templet should be 
 
lyO MECHANICAL DRAWING. 
 
 made of thin wood to fit the first one drawn ; by this the others can be 
 quickly and accurately drawn. 
 
 Fig. 127. A cylinder with a helical blade or surface., formed by the 
 revolittio7i of a line perpe7idicular to the cylinder around and along the 
 cylinder at the sa7ne time. 
 
 The two ends of the generating line describe helices of the same pitch, 
 but on cylinders of different diameters. Each helix is found as already ex- 
 plained. 
 
 Instead of revolving from left to right the line revolves from right to 
 left, and the helices advance along the cylinder in the opposite direction to 
 that of the helix in Fig. 126. 
 
 Threads are called right-handed ^h^nXh^y advance as in Fig. 126, and 
 left-handed when they advance as in Fig. 127. The most common ex- 
 amples of both kinds of threads are found on the axles of all wagons. The 
 threads on the right hand ends are right-handed ; the threads on the left 
 hand ends are left-handed, that is, the nuts are screwed upon them by the 
 motion which unscrews those on the right hand ends. 
 
 A helical curve may be generated by a point which moves along the 
 surface of a cylinder at a varying rate of speed, while its motion around the 
 cylinder is uniform ; or it may be generated by a point whose motion in 
 both directions is variable. A helical curve may be traced upon a conical 
 or spherical surface. A helical surface may be generated by a line which 
 moves as in Fig. 127, or by a line which is inclined to the axis of the cylinder. 
 These problems are of too advanced a nature to be given in this book. 
 
 Fig. 128. A V-threaded bolt and nut., the upper part of the bolt arid 
 the nut showing a section through the centre. 
 
 The section of this thread is an equilateral triangle. The size of the 
 thread depends upon the diameter of the bolt ; there are regukr standard 
 threads for bolts of all diameters, and other standard threads for pipes. 
 Iron bolts of the following diameters have threads as specified : 
 
 Diatneter of Bolt. Threads per inch. 
 
 % inch .20 
 
 ^ " ...... 16 
 
 yi '' 12 
 
 ^ " 10 
 
 I " 8 
 
 i>^ - ...... 6 
 
 We will suppose the screw shown to be of wood ; the thread is there- 
 fore large in proportion to the bolt. First draw the section of the lower 
 thread and divide the pitch into twelve equal parts. As only half of the 
 
172 
 
 MECHANICAL DRAWING. 
 
 bottom view is shown, this is divided into six equal parts. The outer and 
 inner helices are obtained as explained under Fig. 126, and the drawing of 
 the bolt is completed by drawing the section of the thread which connects 
 the outer and inner helical curves. 
 
 In exact drawings of bolts whose threads are large, the F-shaped outline 
 of the section of the thread will come inside the correct projection of the 
 helical surfaces forming the thread ; but to obtain the exact projection is 
 so complicated a problem that generally the thread is represented as illus- 
 trated. A straight line tangent to the two helices approximates the actual 
 projection. As the helices are not angular at their vertices it is sometimes 
 necessary to represent the thread in this way instead of by the line of the . 
 section which is given in the figure. 
 
 When the first outer and inner helices are complete, the others may 
 be obtained by setting off the pitch, with the dividers, from the points of 
 the first lines as many times as other curves are desired. 
 
 The hnes of the thread in the interior of the nut have the opposite 
 direction to those upon the bolt ; they correspond to the dotted lines of the 
 bolt and are obtained in the same way. 
 
PLATE XX\^ 
 
 -i- 1 -f — 
 
 ----- r— -h- 
 
 — I — 
 1 
 
 
 
 T 
 
 
 ^-^r" 
 
 ^^^ 
 
 1 
 
 -~^^Y\ 
 
 i i i 
 
 ■i 
 
 1 
 
 1 
 1 
 
 f 
 
 
 A 
 
 i [-- 
 
 
 
 ~? 
 
 ' \. 
 
 1 
 1 
 
 1 
 1 J 
 
 77 
 
 
 ^ 
 
 
 9 
 Fig. 126. 
 
 FigJ27. 
 
I 74 MECHANICAL DRA WING. 
 
 PLATE XXVI. 
 
 Fig. 129. A square-threaded bolt and nut, the upper part of the bolt 
 and the nut being in sectio7i. 
 
 In the square thread both the thread and the space are square, each 
 occupying one half the pitch. There are two heHcal curves at the inside of 
 the thread, and two at the outside ; the points for these curves are found as 
 already explained. 
 
 Figs. 1 28 and 1 29 give the actual projections of the common forms of 
 threads. These threads are cut in a lathe, and drawings for them are un- 
 necessary, as all that the workman requires is to know the diameter of the 
 bolt, the pitch of the thread, and its shape. These facts might be written 
 upon the drawing, but it is the custom to represent threads conventionally 
 by means of straight lines, which require less time to draw than do the 
 curves. 
 
 Fig. 130. A bolt with V and square threads^ and a nut in section for 
 each. 
 
 At the right a F-threaded bolt and nut are shown. The drawing is 
 given by substituting straight lines for the curves of Fig. 128. 
 
 Two methods of representing the thread of a square-threaded bolt and 
 nut are shown at the left. The threads at the left end of the bolt and in 
 the nut are obtained by substituting straight lines for the curves of Fig. 1 29. 
 The threads near the centre of the bolt are shown more simply by straight 
 lines representing only the outer edges of the thread. 
 
 Fig. 131. Three different ways of representing a V thread. 
 
 At A is shown a method suitable for use when the screw is very small ; 
 the lines of the bolt in such a case may be omitted, especially when the end 
 of the bolt shows for a little distance outside of its nut. 
 
 At B is shown the representation used by many draughtsmen. Some- 
 times the long lines are made heavy instead of the short lines. Heavy 
 short lines give the best effect of the thread. 
 
 At C is shown a satisfactory representation, which requires less time to 
 produce than the other methods. 
 
 It is not necessary that there be as many long lines per inch as there 
 are threads, though it is well for pupils to represent each thread by a long 
 line, because they are not able to space by eye so as to produce a satisfac- 
 tory result. 
 
 The angle at w^hich the lines of these conventional representations are 
 drawn, should be determined by the spacing of the lines. In all common 
 
176 MECHAXICAL DRAWIXG. 
 
 threads point j of Fig. 131 should be half-way between points / and 2, 
 This drawing represents correctly a single-threaded right-handed bolt. The 
 opposite direction of the long line, that is from j to 2^ will represent a left- 
 handed thread. 
 
 Fig. 132. A triple-threaded bolt. 
 
 A bolt nKiy have any number of parallel threads. In this case there 
 are tw^o threads, B and C, between the turns of the thread A, One turn of 
 the bolt moves it the distance of the pitch AAj the threads A^ B, and C 
 give three times the strength that would be given by thread A, To draw 
 this bolt, divide the pitch into as many equal parts as there are threads, and 
 place one thread in each space, as in a single-threaded bolt. 
 
 Fig. 133 shows the conventional shading sometimes added to a draw- 
 ing representing cylindrical objects. Part of Fig. 132 shows how a vertical 
 cylinder is shaded. 
 
PLATE XXVI. 
 
178 MECHANICAL DRAWING. 
 
 PLATE XXVII. 
 
 Fig. 134. Three views of a square bolt-head. 
 
 Fig. ^135. Three views of a square bolt-head^ whose corners have been 
 cha^nfered {bevelled). 
 
 The surface of the chamfer is really that of a cone, which is shown by 
 the dotted lines. This cone will be intersected by each of the four faces of 
 the head in an hyperbola. In practice this is never drawn, but is repre- 
 sented by an arc of a circle, whose points are found as follows : The con- 
 tour elements of the cone intersect the vertical lines which represent the 
 right and left sides of the head, and give the highest points of the curves 
 of intersection. To obtain the lowest points, which must be in the edges 
 since they are farther from the centre than any other lines of the faces, 
 revolve ^^ to ^ and draw from 2' a vertical line to intersect the element of 
 the cone at 2"'. This vertical represents an edge of the head, when its real 
 distance from the centre of the bolt is seen, and 2'' must be the point in 
 which the edge is cut by the chamfer. The highest points in the curves of 
 intersection must be at the same level upon all the faces ; the lowest points 
 must also be at one level ; thus the arcs which represent the chamfer must 
 be tangent to a horizontal line drawn through i and must end in the edges 
 in points in a horizontal line drawn through 2". 
 
 The angle of the chamfer ranges usually from 30° to 45°, and much or 
 little may be cut from the head ; thus the circle left upon the top may be 
 tangent to the sides, or smaller than in Fig. 135. 
 
 Fig. 136. Three views of a square bolt-head with a spherical top. 
 
 The top is a portion of a sphere, and its section by any one of the faces 
 of the bolt-head is an arc of a circle. The highest point is seen at 7^, and 
 the lowest point is obtained as in Fig. 135. 
 
 The radius of the spherical top is usually about twice the diameter of 
 the bolt. 
 
 Fig. 137. Three views of a square bolt-head when its vertical faces 
 are at an ajigle of 4^^ with the front plane. 
 
 The arcs of circles at the top of each face appear ellipses but are repre- 
 sented by arcs of circles. The lowest points in the arcs are in the edges 
 and are seen in both views. To obtain the level of the highest point re- 
 volve the centre of any face, as j>^, to 3' and project to the front view as 
 explained in Fig. 135. The highest points are in the centre of each face ; 
 their level is thus given at 3"' The arcs of both views must be contained 
 between horizontal lines through i^ and j^, as in Fig. 135. 
 
l8o MECHANICAL DRAWING. 
 
 Fig. 138. Three views of a sqtcare nut so placed as to show two faces 
 in the front view. 
 
 The nut is shown upon a bolt, which should always be represented as 
 projecting slightly beyond the nut. The points for the curves representing 
 the chamfer are found as has been explained. 
 
 A square nut or bolt-head should be so placed as to show one face only, 
 as in Fig. 135. 
 
 The thickness of the square nut and bolt-head is equal to the diameter 
 of the bolt. The distance between the parallel sides varies ; it is often i ^ 
 times the diameter of the bolt plus y^ of an inch. 
 
 Fig. 139. Three views of an hexagonal bolt-head. 
 
PLATE XXVII. 
 
 Fig. 134. 
 
 Fig. 135. 
 
 zT 
 
 ■"t' 
 
 z'' 
 
 ry- 
 
 Fig. 136. 
 
 Fig, 138. 
 
 Fig. 139. 
 
I 82 MECHANICAL DRAWING. 
 
 PLATE XXVIII. 
 
 Fig. 140. Three views of a bolt having an hexagonal head with a 
 spherical top. 
 
 The lowest points in the curves of the chamfer, which are really arcs in 
 this figure, are in the edges and are determined in the front view ; the highest 
 points are in the centres of the faces and are determined in the side view. 
 The arcs which are drawn to represent the chamfer must be tangent to 
 a line drawn through p, and have their lowest points in a line drawn 
 through 7. 
 
 Fig. 141. Three views of a bolt., with an hexagonal head and nut^ 
 whose corners are cha7nfered. 
 
 The circle on the top of the head given by the chamfer appears, in the 
 front and in the side view, a straight line whose length is equal to the diame- 
 ter of the circle. Drawing from the extremities of these lines the elements 
 of the cone (generally at 30°) determines, in the front view, the lowest points 
 in the curve by the intersections of these lines with the edges ; and the 
 highest points are determined in the side view, where the lines are seen inter- 
 secting the centres of the faces. In this figure, and also Fig. 140, the arcs 
 must end in the edges at points given by a line drawn through point /, and 
 must be tangent at the centre of each face to a line drawn through point 2, 
 The curves of the chamfer in this figure are hyperbolas, as they are in Fig. 
 
 135- 
 
 The thickness of the hexagonal nut and bolt-head is equal to the diame- 
 ter of the bolt ; the distance between the parallel sides varies. It is not 
 necessary to represent the exact proportions ; therefore many draughtsmen 
 make the conventional drawings shown in Figs. 139, 140, and 141, in which 
 the long diagonal of the head, or nut, is made twice the diameter of the bolt. 
 This is larger than the standard sizes, but is the best representation for 
 practical drawings. 
 
 When only one view of an hexagonal bolt-head or nut is shown, it 
 should represent three faces of the object. 
 
 The draughtsman does not need to find the exact points for the curves 
 of the chamfer, as he knows the appearance and can give it by eye ; but it 
 is well that students should know how to obtain the correct points. 
 
 Fig. 142. A spring or a pipe of helical form. 
 
 Make the drawing of the helical curve, which is the centre line of the 
 spring, as explained in Fig. 1 26. With different points in the curve as cen- 
 tres, draw circles whose diameters are equal to that of the spring. The 
 outline of the spring must be drawn tangent to these circles. 
 
PLATE XKyilL 
 
DEFINITIONS. 
 
 Altitude. The perpendicular distance between the bases, or between 
 the vertex and the base, of a soUd or plane figure. 
 
 Angle. The difference in 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 angle. 
 
 An angle is measured by means of an arc of a circle described from its 
 vertex as a centre and included between its sides. The centre of the arc is 
 the vertex of the angle. 
 
 If the radius of the circle moves through -^\-^ of the circumference, 
 it produces an angle which is taken as the unit for measuring angles, 
 and is called a deg7'ee. 
 
 The degree is divided into sixty equal parts called mmiites^ and the 
 minutes into sixty equal parts called seconds. 
 
 Degrees, minutes, and seconds are denoted by symbols. Thus 5 de- 
 grees, 13 minutes, 12 seconds, is written 5° 13' \2'\ 
 
 A Right Angle is one which is formed by the radius 
 moving through i of the circumference. It is an angle of 
 90°. A straight angle is formed when the radius has 
 moved over \ of the circumference. It is an angle of 180°. 
 
 Acute Angle. An angle less than a right angle. 
 
 Obtuse Angle. An angle greater than a right angle. 
 
 Oblique Angle. One which is not a right or a straight angle. 
 
 Reflex Angle. One which is greater than 180°. 
 
 Adjacent Angle. Two angles are adjacent when 
 they have the same vertex and a common side. 
 
MECHANICAL DRAWING. I 85 
 
 Dihedral Angle. The opening between two intersecting 
 planes. 
 
 Solid Angle. One formed by planes which meet at a point. 
 Apex. The summit or highest point of an object 
 Arc. See Circle. 
 
 Axis of a Solid. An imaginary straight line passing through its centre 
 and about which the different parts are symmetrically arranged. 
 
 Axis of a Figure. A straight line passing through the centre of a 
 figure, and dividing it into two equal parts. 
 
 Axis of Symmetry. A straight line so placed in a solid or a plane 
 figure that every straight line meeting it at right angles and extending in 
 each direction to the boundary of the solid or figure is bisected at the 
 point of meeting. In many solids and plane figures an axis of symmetry 
 cannot be drawn. 
 
 Base. 'The opposite parallel polygons of prisms. The polygon oppo- 
 site the vertex of a pyramid. The plane surfaces of cyhnders and cones. 
 The opposite parallel sides of a parallelogram or trapezoid. The shortest 
 or longest side of an isosceles triangle, and any side in any other triangle, 
 but usually the lowest. 
 
 Bisect. To divide into two equal parts. 
 
 Bisector. A line which bisects. 
 
 Cinquefoil. A figure composed of five leaf-like parts. 
 
 Circle. A plane figure bounded by a curved line, called a 
 circumference, all points of which are equally distant from a a 
 point within called the centre. ^ 
 
 The boundary line is called the Circumference. 
 Diameter. A straight line drawn through the centre, and con- 
 necting opposite points in the circumference, as a b. 
 
 Radius. The distance from its centre to the circumference, 
 as c e. 
 
 Semi-circle. Half a circle, formed by bisecting it with a diam- 
 eter, as a db a. 
 
 Arc. Any part of the circumference, as c b. 
 
1 86 . DEFINITIONS. 
 
 Chord. A straight line whose ends are in the circumference, 
 as/^. 
 
 Segment. The part of a circle bounded by an arc and a chord, 
 
 as/^^/. 
 
 Sector. The part of a circle bounded by two radii and an arc, 
 as b e c b. 
 
 Quadrant. A sector bounded by two radii and one fourth of 
 the circumference, as a c d a. 
 
 Tangent. A straight hne which meets a circumference, but 
 being produced does not cut it, as k d. The point of meeting is 
 called ih.t point of contact ox point of tangency. 
 
 Circumscribe. A polygon is said to be circumscribed about a circle 
 when each side of the polygon is a tangent to the circle ; and a circle is said 
 to be circumscribed about a polygon when the circumference of the circle 
 passes through all the vertices of the polygon. 
 
 Concave. Curving inwardly. 
 
 Cone. A solid bounded by a plane surface called the base^ which is a 
 circle, ellipse, or other curved figure, and by a lateral surface which is 
 everywhere curved, and tapers to a point called the vertex. Its base names 
 the cone. Thus a circular cone is one whose base is a circle. 
 
 A Right Circular Cone is generated by an isosceles triangle 
 which revolves about its altitude as an axis. The equal 
 sides of the triangle in any position are called ele7nents of A 
 
 the surface. The length of an element is called the slant / j \ 
 height of the cone. Unless otherwise stated " cone " ^ri^ 
 means a right circular cone. 
 
 A Frustum of a Cone is the part included between the base 
 and a plane parallel to the base and cutting all the elements of the 
 cone. 
 
 A Truncated Cone is the part included between the base and 
 a plane oblique to the base and cutting all the elements of the 
 cone. 
 
 Concentric. Having a common centre. 
 
 Conic Section. A section obtained by cutting a cone by a plane. 
 
 Construction. The making of any object. 
 
 Construction Lines. The lines by which the desired result is obtained. 
 
MECHANICAL DRAWING. 
 
 187 
 
 Constructive Drawing. A drawing intended for the workman who is 
 to make the object. 
 
 Contour. The outline of the general appearance of an object. 
 Contour Element. An element which is in the contour of an object. 
 
 Convergence. Lines extending toward a common point, or planes 
 extending toward a common line. 
 
 Convex. Rising or swelling into a spherical or rounded form. 
 
 Corner. The point of meeting of the edges of a solid, or of two sides 
 of a plane figure. 
 
 Cross-hatched. In mechanical drawing, a half tinting placed upon parts 
 cut by a cutting plane. In free-hand drawing, the use of lines crossing each 
 other and producing light and shade effects. 
 
 Curvature. Variation from straightness. 
 
 Curve. A Hne of which no part is straight. 
 
 Reversed. One whose curvature is first in one direction 
 • and then in the opposite direction. 
 
 Spiral. A plane curve which winds about and 
 recedes, according to some law, from its point of begin- 
 ning, which is called its centre. 
 
 Cylinder. A solid bounded by a curved surface and by tw^ 
 opposite faces called bases ; the bases may be ellipses, circles, or 
 other curved figures, and name the cylinder. Thus a circular 
 cylinder (the ordinary form) is one whose bases are circles. 
 
 A Right Circular Cylinder is generated by the revolution 
 of a rectangle about one side as an axis. The side about which 
 the rectangle revolves is called the height of the cylinder, also its 
 axis. The side opposite the axis describes the curved surface of the 
 cylinder, and in any of its positions is called an element of the surface. 
 
 Cylindrical. Having the general form of a cylinder. 
 
 Degree. The 360th part of a circumference of a circle. 
 
 Describe. To make or draw a curved line. 
 
 Design. Any arrangement or combination to produce desired results in 
 industry or art. 
 
 Develop. To unroll or lay out upon one plane the surface of an object. 
 
I 8 8 DEFINITIONS. 
 
 Diagonal. A straight line in any polygon which connects 
 vertices not adjacent. 
 
 In regular polygons, diagonals are called long when they 
 pass through the centre, as c d^ and sho7't when they extend 
 between parallel sides, as a b. 
 
 Diameter. See Circle. In a regular polygon with an even 
 number of sides a line joining the centres of two opposite sides e 
 is often called a diameter. 
 
 Edge. The intersection of any two surfaces. The boundary line. 
 Edges are straight or curved, and are represented by lines. 
 
 Elevation. A drawing made on a vertical plane by means of projecting 
 lines perpendicular to the plane from the points of the object. The terms 
 elevation, vertical projection, and front view all have the same meaning. 
 
 Ellipse. A plane figure bounded by a line such that the sum 
 of the distances of any point in it, as c^ from two given points /^^"^\\ 
 e and f, called foci^ is equal to a given line, as a b. The point VjHx 
 midway between the foci is called the centre. 
 
 The Transverse Axis of an eUipse is the longest diameter that 
 can be drawn in it, 2iS a b. It is also called the major or long axis. 
 
 The Conjugate Axis is the shortest diameter which can be 
 
 drawn, as c d. It is also called the minor or short axis. The 
 
 foci, e and /', are two points in the long diameter whose distance 
 
 from c ox d\^ equal to one-half a b. 
 
 Face. One of the plane surfaces of a soHd. 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 accenting where this is 
 required. 
 
 Foreshortening. Apparent decrease in length, due to a position oblique 
 to the visual rays. 
 
 Free-hand. Executed by the hand, without the aid of instruments. 
 Frustum. See Cone and Pyramid. 
 Generated. Produced by. 
 Geometric. According to geometry. 
 
 Half-tint. The shading produced by means of parallel equidistant 
 lines. 
 
MECHANICAL DRAWING. 1 89 
 
 Hemisphere. Half a sphere, obtained by bisecting a sphere by 
 a plane. 
 
 Horizontal. Parallel to the surface of smooth water. 
 
 In drawings, a line parallel to the top and bottom of the sheet is called 
 horizontal. 
 
 Inscribe. A polygon is said to be inscribed in a circle when all its 
 
 vertices are in the circumference of the circle ; and a circle is said to be 
 
 inscribed in a polygon when the circumference of the circle is touched by 
 each side of the polygon. 
 
 Instrumental. By the use of instruments. 
 
 Lateral Surface. The surface of a solid excluding the base or bases. 
 
 Line. A line has length only. In a drawing its representation has 
 width but is called a line. 
 
 Straight. One which has the same direction through- 
 out its entire length. 
 
 Curved. One no part of which is straight. 
 
 Broken. One composed of different successive 
 straight lines. 
 
 Mixed. One composed of straight and curved lines. 
 Centre. A line used to indicate the centre of an object. 
 Construction. A working hne 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 hne upon which a dimension is placed. 
 
 Full. An unbroken line, usually represent- 
 ing 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. 
 
IQO DEFINITIONS, 
 
 Oblique. Neither horizontal nor vertical. 
 Oblong. A rectangle with unequal sides. 
 
 Oval. A plane figure resembling the longitudinal section of an 
 ^gg ; or elliptical in shape. 
 
 Overall. The entire length. 
 
 Ovoid. An egg-shaped soUd. \ jjj 
 
 Parallel. Having the same direction and everywhere equally """ 
 
 distant. I I AX 
 
 Parallelogram. See Quadrilateral. 
 
 Pattern. That which is used as a guide or copy in making anything. 
 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. The art of making upon a plane, called \\\^ picture plane ^ 
 such a representation of objects that the lines of the drawing appear to 
 coincide with those of the object, when the eye is at one fixed point called 
 the station poi7it. 
 
 Diagram. An exact perspective drawing obtained scientifically 
 by perspective methods. It is often very false pictorially when not 
 seen from the station point. 
 
 Parallel. Diagram perspective which represents a cubical 
 form by the use of one vanishing point, and represents by its real 
 shape any face parallel to the picture plane. 
 
 Angular. Diagram perspective in which two sets of horizontal 
 edges of a cubical form are at angles to the picture plane, and the 
 object is thus represented by the use of two vanishing points. 
 
 Oblique. Diagram perspective in which, none of the edges of 
 a cubical form being parallel to the picture plane, it is represented 
 by the use of three vanishing points. 
 
MECHANICAL DRAWING. 
 
 191 
 
 Free-hand or Model Drawing. A drawing which, without 
 confining the eye to the station point, represents as far as possible 
 the actual appearance of objects. It is made free-hand, and is for 
 most purposes more satisfactory than an exact diagram perspective. 
 Plan. Plan, horizontal projection, and top view have the same meaning, 
 and designate the representation of an object made on a horizontal plane 
 by means of vertical projecting lines. In architecture it means a horizon- 
 tal section. 
 
 Plane Figure. A part of a plane surface bounded by lines. 
 A plane figure is called rectilinear if bounded by straight lines, citrvi- 
 li7tear if bounded by curved lines, and mixtilinear if bounded by both 
 straight and curved lines. 
 
 Similar figures are those that have the same shape. 
 
 Plinth. A cylinder or prism, whose axis is its least dimen- / — -~^ 
 sion. It is circtilar^ triangidar^ square^ etc., according as it has L^]^J/^ 
 circles, triangles, squares, etc., for bases. 
 
 Polygon. A plane figure bounded by straight lines. 
 
 An Equilateral Polygon is one whose sides are all equal. 
 
 An Equiangular Polygon is one whose angles are all equal. 
 
 A Regular Polygon is one which is equilateral and equiangular. 
 
 Parallel Polygons are those whose sides are respectively 
 parallel. 
 
 Triangle. A polygon having three sides (i). 
 Quadrilateral. A polygon having four sides (2). 
 Pentagon. A polygon having five sides (3). 
 Hexagon. A polygon having six sides (4). 
 Heptagon. A polygon having seven sides (5). 
 
192 
 
 DEFINITIONS. 
 
 Octagon. A polygon having eight sides (6). 
 
 NoxAGOX. A polygon having nine sides (7). 
 
 Decagox. a polygon having ten sides (8). 
 
 UxDECAGOX. A polygon having eleven sides (9). 
 
 DoDECAGOX. A polygon having twelve sides (10). 
 
 The centre of a regular polygon is the common intersection of 
 perpendiculars erected at the middle points of its sides. 
 
 The polygons represented in the figures are regular polygons. 
 
 A Polyhedron is a soHd bounded by planes. It is regular when its 
 faces are regular equal polygons. 
 
 There can be but five regular polyhedrons : 
 
 1. The Tetrahedrox, or Pyramid, which has four triangular 
 faces. 
 
 2. The Hexahedrox, or Cube, which has six square 
 faces. 
 
 3. The OcTAHEDROX, which has eight triangular faces. 
 
 4. The DoDECAHEDROX, which has twelve pentagonal faces. 
 
 5. The IcosAHEDROX, which has twenty triangular faces. 
 
 The term hexahedron is applied only to a regular polyhedron : 
 the other terms may be applied to irregular polyhedrons. 
 
 An infinite number of irregular polyhedrons, also an infinite 
 number of other solids bounded by plane or curved surfaces, may be 
 conceived. 
 
 Prism. A solid bounded by two equal parallel polygons, having their 
 equal sides parallel, and by three or more parallelograms. 
 
 The polygons are called the bases of the prism, the parallelograms the 
 lateral faces^ the intersections of the lateral faces, the lateral edges. 
 
 Prisms are called triangular^ square^ pentagonal^ etc., according 
 as the bases are triangles, squares, pentagons, etc. 
 
 A Right Prism is one in which the edges connecting the 
 bases are perpendicular to the bases. 
 
193 
 
 MECHANICAL DRAWING. 
 
 An Oblique Prism is one in which the edges connecting 
 the bases are not perpendicular to the bases. 
 
 A Regular Prism is a right prism whose bases are regular 
 polygons. 
 
 A Truncated Prism is the part of a prism included [T — V 
 between the base and a section m^de by a plane inclined to 
 the base, and cutting all the lateral edges. 
 
 The Altitude of a prism is the perpendicular distance between 
 the bases. 
 
 The Axis of a regular prism is a straight line connecting the 
 centres of its bases. 
 
 A Right Section of a prism is a section made by a plane 
 perpendicular to its lateral edges. 
 
 A Parallelopiped is a prism whose bases are parallelograms 
 
 Produce. To continue or extend. 
 
 Profile. The contour outline of an object. 
 
 Projection. Orthographic. The view or representation of an object 
 obtained upon a plane by projecting lines perpendicular to the plane. 
 
 Pyramid. A solid of which one face, called the base^ is a polygon, and 
 the other faces, called lateral faces^ are triangles having a common vertex 
 called the vertex of the pyramid. The intersections of the lateral faces are 
 called the lateral edges. 
 
 A pyramid is called triangular., square., etc., according as its 
 base is a triangle, square, etc. 
 
 A Regular Pyramid is one whose base is a regular polygon 
 and whose vertex is in a perpendicular erected at the centre of the 
 base. Its other faces are equal isosceles triangles. The altitude of 
 any of these triangles is called the slant height of the pyramid. 
 
 A Frustum of a pyramid is the part included between ^^:--^ 
 the base and a plane parallel to the base and cutting all the / / \ 
 lateral edges. 
 
 A Truncated Pyramid is the part included between 
 the base and a plane oblique to the base and cutting all the 
 lateral ed<2:es. 
 
1 94 DEFINITIONS, 
 
 The Axis of a pyramid is a straight line connecting the vertex 
 and the centre of the base. 
 
 The Altitude of a pyramid is the perpendicular distance from 
 the vertex to the base. 
 Quadrant. See Circle. 
 
 Quadrilateral. A plane figure bounded by four straight lines. These 
 lines are the sides. The angles formed by the lines are the angles., and 
 the vertices of these angles are the vertices of the quadrilateral. 
 
 A Parallelogram is a quadrilateral which has its opposite 
 sides parallel. 
 
 A Trapezium is a quadrilateral which has no two sides r [ 
 
 parallel. L J 
 
 A Trapezoid is a quadrilateral which has two sides, 
 and only two sides, parallel. 
 
 A Rectanglji: is a quadrilateral whose angles are 
 right angles. 
 
 A Square is a rectangle whose sides are equal. 
 
 A Rhomboid is a parallelogram whose angles are > y 
 
 oblique angles. L / 
 
 A Rhombus is a rhomboid whose sides are equal. / / 
 
 The side upon which a parallelogram stands and the opposite side are 
 called respectively its lower and upper bases. 
 
 Quadrisect. To divide into four equal parts. 
 
 Quatrefoil. A figure composed of four leaf-like parts. C^ j 
 
 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 cross-hatching. 
 
 Sectional. Showing the section made by a plane. 
 
 Sector and Segment. See Circle. 
 
 Shadow. Shade and shadow have about the same meaning, as gen- 
 erally used ; but it will be well to designate by shadow those parts of an 
 
MECHANICAL DRAWING. 
 
 195 
 
 object which are turned away from the direct rays of light, while those sur- 
 faces which receive less direct rays and are intermediate in value between 
 the light and the shadow are called shade surfaces. 
 
 Cast. The shadow projected on any body or surface by some 
 other body. 
 
 Similar Figures are those which have the same shape. 
 
 Solid. A sohd has three dimensions, length, breadth, and thickness. 
 It may be bounded by plane surfaces, by curved surfaces, or by both plane 
 and curved surfaces. As commonly understood, a solid is a limited portion 
 of space filled with matter, but geometry does not consider the matter and 
 deals simply with the shapes and sizes of soHds. 
 
 Sphere. A solid bounded by a curved surface every point f \ 
 of which is equally distant from a point within called the centre. V _JS 
 
 A sphere may be generated by the revolution of a circle about a diameter 
 as an axis. 
 
 Spheroid (Ellipsoid). A solid generated by the revolution of 
 an elHpse about either diameter. When revolved about the 
 long diameter, the spheroid is called prolate^ or the long 
 spheroid ; when about the short diameter, it is called oblate^ 
 or the flat spheroid. The earth is an oblate spheroid. 
 
 Spiral. See Curve. 
 
 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. 
 
 Symmetry. Design. A proper adjustment or adaptation of parts to 
 one another and to the whole. 
 
 Bilateral. Having two parts in exact reverse of each other. 
 
ig6 ' DEFINITIONS. 
 
 Symmetry. Geometry. If a solid can be divided by a plane into two 
 parts such that every straight line, perpendicular to the plane and extending 
 from the plane in each direction to the surface of the solid, is bisected by 
 the plane, the solid is called a sy7)i7netrical solid, and the plane is called a 
 pla7ie of sy7n7netry. If two planes of symmetry can be drawn in a solid, 
 their intersection is called an axis of symmetry. See Axis of Symmetry. 
 
 The line about which a plane figure revolves when it generates a solid 
 of revolution is an axis of symmetry for the solid ; it is also called the axis 
 of revohition. 
 
 Tangent. A straight line and a curved line, or two curved lines, are 
 tangent when they have one point common and cannot intersect ; lines or 
 surfaces are tangent to curved surfaces when they have one point or one 
 line common and cannot intersect. 
 
 Trefoil. A figure composed of three leaf-like parts. r \ 
 
 Triangle. A plane figure bounded by three straight lines. These lines 
 are called the sides. The angles that they form are called the angles of 
 the triangle, and the vertices of these angles, the vertices of the triangle. 
 
 Triangles are named by their sides and angles. 
 
 A Scalene Triangle is one in which no two sides are equal. 
 
 An Isosceles Triangle is one in which two sides are 
 equal. 
 
 An Equilateral Triangle is one in which the three 
 sides are equal. 
 
 A Right Triangle is one in which one of the angles 
 is a right angle. 
 
 An Obtuse Triangle is one in which one of the 
 angles is obtuse. 
 
 An Acute Triangle is one in which all the angles 
 are acute. 
 
 The Hypotenuse is the side of a right triangle opposite the 
 right angle. The other sides are called the legs. 
 
MECHANICAL DRAWING. 
 
 197 
 
 An Equiangular Triangle is one in which the three angles 
 are equal. The value of each angle is 60°. 
 
 The Base is the side on which the triangle is supposed to stand. 
 In an isosceles triangle, the equal sides are called the legs^ the other 
 side the base; in other triangles any one of the sides may be called 
 the base. 
 
 The Altitude is the perpendicular distance from the vertex to 
 the base. Except in the isosceles triangle, there are three altitudes. 
 
 The vertex of the angle opposite the base is often called the 
 vertex of the triangle. 
 
 Trisect. To divide into three equal parts. 
 
 Truncated. A truncated solid is the part of a solid included between 
 the base and a plane cutting the solid obhque to the base. 
 
 Type Form. A perfect geometrical plane figure or solid. 
 
 Vertical. 
 
 Upright or perpendicular to a horizontal plane or line. 
 
 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. 
 
 VieTAT. See Elevation. Views are called front, top, right or left side, 
 back, or bottom, according as they are made on the different planes of pro- 
 jection. 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. 
 
 "Working Lines. See Lines.