A manual of elementary geometrical drawing, involving three dimensions. Designed for use in high schools, academies, engineering schools, etc.: and for the self-instruction of inventors, artisans, etc. ... In five divisions ..

Front matter

A MANUAL

ELEMENTARY

GEOMETKIOAL DRAWING,

Chapter I. — First Principles. § I. — The purely geometrical or rational theory of projections 1

§ U. — Of the relations of lines to their projections 8

§ III. — Physical theory of projections 5

§ lY. — Conventional mode of representing the two planes of projection, and the two projections of any object upon one plane ; viz.

the plane of the paper 5

§ V. — Of the conventional direction of the light; and of the position

and use of heavy lines 6

§ YL — Notation 7

§ YII. — Of the use of the method of projections 8

Chapter II. — Projection of lines. Problems in right projection ; and in oblique projection, shoioing two sides of a solid right angle. (Thirtytwo probleins.)

§ I. — Projections of straight lines , 9

§ II. — Right projections of solids 12

§ III. — Oblique projections showing two sides of a solid right angle.. . 14 § lY. — Special Elementary Intersections and Developments 23

DIYISION SECOND.

DETAILS OP CONSTRUCTIONS Ei MASONRY, WOOD, AND METAL.

Chapter I. — Constructions in Masonry.

§ I. — General definitions and principles applicable both to brick and

stone work 32

XV CONTENTS.

PAGK

§ II.— Brick work 32

§ III.— Stone work 35

Chapter II. — Constructions in Wood. §L — General remarks. (Explanation of Scales.) ST

§ II. — Pairs of timbers whose axes make angles of 0° with each other. 40 § III. — Combinations of tunbers whose axes make angles of 90° with

each other 43

§ rV. — Miscellaneous combinations. (Dowelling, &c.) 40

§ Y. — Pairs of timbers which are framed together obliquely to each

other 4t

§ VI. — Combinations of timbers whose axes make angles of 180° with

each other 48

Chapter III. — Constructions in Metal. 53

Cage valve of a locomotive pump. Metallic packing for stufiTing boxes, &c. 54

DIVISION THIRD.

RUDIMENTS OF SHADES AKD SHADOWS.

Chapter I. — Shadows. § I. — Observed facts and practical rules 59

§ II.— Problems. (Twelve.) 63

Chapter II. — Shading 10 (Hexagonal prism ; cylinder; cone, and sphere.)

DIVISION FOURTH.

ISOMETEICAIi DRAWING AlsTD CABINET PROJECTIONS.

Chapter I. — First Principles of Isometrical Drawing 77

" II. — Prohlems involving only isometric lines 81

" III. — Prohlems involving non-isometrical lines 84

" IV. — Problems involving the construction and equal division of circles in

Isometrical Drawing 88

(Approximate construction of the isometrical drawing of a circle.) (Nineteen problems.) " V. — Cabinet Projections , 95

CONTENTS. V

PAOI

DIVISION FIFTH.

ELEMENTARY STRUCTURAL DRAWIKO.

Chapter I. — Stone structures. 101

A brick segmental arch , 101

A semi-cylindrical culvert, with vertical cylindrical wingwalls 103

Chapter II. — Wooden structures. 107

A king post truss 107

A queen post truss-bridge 108

An architectural model, with shadows. 110

Chapter III. — Iron constructions. , 114 A railway track — frog, chair, Ac , 114

A hydraulic ram ,. , 116

NOTE TO THE THIRD EDITION.

At the suggestion of several friends, 1 have at last been induced to add a fourth section to Division I., Chapter II., embracing some useful special intersections, &c., and one additional plate. I have waited, partly, till a vision of the seemingly best selection of figures as to novelty, variety, and utility, for a single plate only, should present itself; and partly, doubting whether all such problems should not be referred only to the introduction to a contemplated work on Machine Drawing. But a few, of the larger number necessary there, may be found useful here, especially to sheet metal workers, and j^oung machinists, and are accordingly inserted. I am not aware that the pleasing and useful constructions — Figs. 8 and 9, PI. III. A, — can anywhere else be found in a work so readily obtainable as the present.

An important addition has also been made to Division lY., in the chapter on " Cabinet Projections ;" which are often preferable to Isometrical Drawings. And special pains have been taken to improve the treatment of the subject of Shadows.

Besides these principal changes, numerous minor corrections and improved paragraphs, have been scattered through the present edition : the result of a careful revision, designed to make my work, as a whole, more acceptable to all having occasion to use it.

Preface

Experience in teaching shows, that correct conceptions of the forms of objects having three dimensions, are obtained with considerable difficulty by the beginner, from drawings having but two dimensions, especially when those drawings are neither "natural" — that is "pictorial" — nor shaded, so as to suggest their form ; but are artificial, or "conventional," and are merely " skeleton," or unshaded, line drawings. Hence moderate experience suggests, and continued experience confirms, the propriety of interposing, between the easily understood drawings of problems involving two dimensions, and the general course of problems of three dimensions, a rudimentary course upon the methods of representing objects having three dimensions. This much may be said in defence of Division I. of the present volume.

Experience again proves, in respect to the drawing of any engineering structures that are worth drawing, that it is a great advantage to the draftsman to have — 1st, some knowledge of (he thing to he djxiwn, aside from his knowledge of the methods of drawing it ; and 2d, practice in the leisurely study of the graphical construction of single members or elements of a piece of framing, or other structure.

The truth of the second of the preceding remarks, is further apparent, from the fact that in entering at once upon the drawing of whole structures, three evils ensue, viz. — 1st, Confusion of ideas, arising from the mass of new objects (the many different parts of a structure) thrown upon the mind at once ; 2d, Loss of time, owing to repetition of the same detail many times in

Viii PREFACE.

the same structure ; and 3d, Waste of drawings, as well as of time, through poor execution, which is due to insufficient previous practice.

Hence Division II. is filled up with a tolerably liberal and comprehensive collection of details of structures, each one of which affords a useful problem, illustrative of some one feature of constructive art and of graphical procedwrc.

Here it may be said, that the department of instruction, part of which is embodied in this work, has involved in it a subdepartment, viz., that of Descriptive Constructions, i.e. a popular descriptive study of the ways in which many common things are made and put together, or in which they operate.

The classification of Chap. II,, Div. II., is believed to be a happy one, as readily suggesting various forms of framings.

Since my object is not so much to acquaint the student solely with the works of the authors or artizans of other nations, as to give them just those specimens of American methods of construction which are not usually found in books, I must refer them to foreign works for examples of foreign modes of construction— not that such examples should be disparaged, but because it is intended that this work shall add something to our stock of book knowledge of the subjects of which it treats, rather than merely repeat what has been before published.

Division III. contains a short special study of Shadows, taken up in the form of the study of a few individual problems of a practical character, assuming that the light comes only in the simple conventional direction, commonly agreed upon by draftsmen, and saying nothing of the possibility of allowing the light to be taken in any other direction. The problems of this Division are intended to be sufficient, when full}^ understood, to enable the student to construct accurately, or by a sufficiently near approximation, all the shadows he will meet with in Geometrical Drawing, till he shall pursue a general course on shades and shadows.

Division lY. contains a sufficient exposition of the few and

PKEFACE. IX

simple principles of Isometrical projection and dratuing, to meet all the ordinary wants of the student or draftsman.

Division Y. includes examples of a few simple structures, to fulfil the threefold purpose of affording occasion for learning the names of parts of structures; for practice in the combination of details into whole structures; and for profitable review j^ractice in execution.

Finally, since experience has dictated so much of this preface, it shall dictate also these closing remarks. First : Classes will generally be found to take a lively interest in the subjects of this volume — because of their freshness to most learners, as new subjects of interesting study — because of the variety and brevity of the topics, which causes frequent change of the subject of study (such change being thought quite desirable by most learners) — and because of the compactness, neatness, and beaut} of the volume which is formed by binding together all the plates of the course, when they are well executed. Second : As to the use of this volume, it is intended that there should be formal recitations of the problems in the 1st, 3d, and 4th divisions, with graphical constructions of a selection of the same or similar ones ; and occasional interrogations mingled with the graphical constructions of the practical problems of the remaining divisions. Remembering that excellence in mere execution, though highly desirable and to be encouraged, is not, at this stage of the student's progress, the sole end to be attained, a small nunv ber of graphic exercises will be sufficient. The student may, in place of a tedious course of finished drawings, be called on frequently to desGi'ibe, by the aid of pencil or blackboard sketches, how he would construct drawings of certain objects — either those given in the several Divisions of this volume, or other similar ones proposed by his teacher.

In a course of school education, the full delivery to the student of the greatest number of distinct ideas, with the least amount of ijieTely manual labor, should be the teacher's general aim.

PEELlMIMPiY NOTES.

As beginners not seldom find peculiar difficulties at the outset of the study of projections, the removal of which, however, makes subsequent progress easy, the following special explanations are here prefixed.

I. Figs. 1, 2, 3, 5, and 15, of PI. I. are pictorial diagrams, used for illustration in place of actual models. Thus, in Fig. 3, for example, MH^ represents a horizontal square cornered plane surface, as a floor. MGV represents a vertical square cornered plane surface, as a wall, which is therefore perpendicular to MH^. P represents any point in the angular si)ace included by these two planes. Vp represents a line from P, perpendicular to the plane MH^, and meeting it at jo. Vj^' represents a line from P, perpendicular to the plane MGV, and meeting it at p'. Then Pp and Vp' are called the projecting lines of P. The point p is called the horizontal projection of P, and p' the vertical projection of P. The projecting lines of any point or of any body, as in Fig. 1, are perpendicular to the planes, as MHz! and MGY, which are called planes of projection.

II. In preparing a lesson from this work, the object of the student is, by no means, to Commit to memory the figures, but to learn, from the first principles^ and subsequent explanations, to see in these figures the realities in space which they represent ; so as to be able, on hearing the enunciation of any of the problems, to solve it from a clear understanding of the subject, and not "by rote" from mere memory of the diagrams. The student will be greatly aided in so preparing his lessons, by working out the problems, in space, on actual planes at right angles to each other, as on the leaves of a folding slate, when one slate is placed horizontally and thp other vertically.

A MANUAL

ELEMENTAEY GEOMETPJCAL DRAWING.

DIVISION FIRST,

PROJECTIONS.

Chapter I. First Principles

§ I. The purely Geometrical or Rational Theory of Projections,

1. It is always easy to make a graphic solution of any problem mvolving but two dimensions, or a graphical construction of any object having but two dimensions, upon a plane surface, as a sheet of paper ; since, upon a surface having tioo dimensions, any figure having but two dimensions can have both of those dimensions represented in their true size and relative position.

2. The preceding article leads to the conclusion, that upon a plane surface, which has but ttoo dimensions, no more than two dimensions can be represented in their true size and relative position. But when we consider the vast number of geometrical magnitudes, as cylinders, cones, spheres, &c., which have three dimensions, it appears probable, and actually is true, that the study of those magnitudes gives rise to a great many problems involving three dimensions. It appears, then, that there are problems of three as well as of two dimensions, and also that more than one plane surface is necessary to enable us to represent, both completely and accurately, bodies having three dimensions,

3. What then is the number and the relative position of the planes which will enable us to represent all the dimensions of any geometrical solid^ in their real size, on those planes ? To assist in answering this question, reference may be made to PI. I. Fig. 1. Let ABCFED be a regular square-cornered block, whose length is AB ; breadth, AD ; and thickness, AC ; and let MN be any horizontal

2 FIRST PRINCIPLES.

plane below it and parallel to its top surface ABED. If now tbu four vertical edges of the block, at A, B, E, and D, be produced to meet this plane, they will meet it in points — as a, h, e, and d. By joining these points, it is evident that a figure — abed — will be formed, which will be equal to the top surface of the block, and Avill be a coi'rect representation of the length and breadth of that toj surface — i. e. of the length and breadth of the block. Similarly, let MP be a vertical plane, behind the block and parallel to its front face, ABCF. Produce the four horizontal edges, through A, B, C, and F, till they meet the vertical plane, MP, and connect the points, a', b\ c', and f\ thus found. The figure, a'c'b'f, so found, will evidently be equal to the front face ABCF of the block, and will therefore correctly represent the length and thickness of the block.

4. From the foregoing article, the following fundamental principles are deduced. First: Two planes, at right angles to each other, are necessary to enable us to represent, fully, the three dimensions of a solid. Second: In order that those dimensions shall be seen in their true size and relative position, they must be parallel to that plane on which they are shown. Third: Each plane shows tw^o of the dimensions of the solid; viz., the two which are parallel to it, and that dimension which is thus shown twice, is the one which is parallel to both of the planes. Thus AB, the length, and AD, the breadth, are shown on the plane MN ; and AB, the length again, and AC, the thickness, are shown on the plane MP.

5. From Art. (3), the following definitions naturally arise. If a body be thrown directly forward, as across a table, all its points will describe parallel straight lines, while the body will be said to be projected. Hence, as the lines of each group which are drawn through the edges of the block, PI. I., Fig. 1, to produce the figures abed and a'b'c'f\ are parallel, the block is naturally said to be projected upon the planes MN" and MP; and the figures ahed and a'b'c'f are called the projections of the block. The planes, containing these figures, are hence naturally called js^aJics of p)r ejection — MIST, being horizontal, is called the horizontal plane of projection^ ^\\(\. MP, being vertical, is called the vertical plane of projection.

The projections take their particular names from the planes in which they are found. Thus, abed is the horizontcd projection of the block, and a'b'cf is its vertical projection. The same idea is expressed otherwise, by saying, that the block is horizontally projected in the figure abed., and that it is vertically projected in the figure a'b'cf.

The system of lines, of Avhich Aa is one, may be called horizon-

FIRST PEIXCIPLES. 3

tally j?rojecting lines, and the group of wliicli Aa' Is one, vany he called vert icalli/ 2)}'oJectiriff lines. These projecting lines are always perpendicular to the planes of projection. Finally, ilR, the intersection of these planes, is called the ground line.

6. Tlie preceding definitions enable us to express a fourth fundamental principle, more briefly than could otherwise be done ; viz. The height of the vertical projection of a point above the ground line, is equal to the height of the j^oint itself, in space, above the horizontal pdane y and the perpendicular distance of the horizontal p>rojection of a point from the ground line, is equal to \\\Qi perpendicular distance of the point itself, in front of the vertical plane. Thus; PL I., Fig, 1, aa"—Aa' and a'a"^=Aa.

7. These four fundamental principles, with the accompanying definitions, are the foundation of the subject of projections, but, by attending carefully to PI. I., Fig. 2, some additional special principles may be discovered, which are frequently applied in practice. PI. I., Fig. 2, is a pictorial model of a pyramid, Ycdeg, and of its two projections. The face, Ycd, of the pyramid, is parallel to the vertical plane, and the triangle, XaJ, is equal and parallel to Ycd, and a little in front and at one side of it. By first conceiving, now, of the actual models, which are, perhaps, represented as clearly as they can be by mere diagrams, in PI. I., Figs. 1 and 2 ; and then by attentive study of those figures, the flicts stated in the next two articles will be fully comprehended.

§ II. — Of the Relations of Lines to their Projections.

8. Relations of single lines to their projections.

a. A vertical line, as AC, PI. I., Fig. 1, has, for its horizontal projection, a pomt, a, and for its vertical projection, a line a'c\ perpendicular to the ground line, and equal and parallel to the line AC, in space.

b. A horizontal line, as AD, which is perpendicidar to the vertical plane, has, for its horizontal projection, a line, ad, perpendicular to the ground line, and equal and parallel to the line, AD, in space ; and for its vertical projection a point, a' .

c. A horizontal line, as AB, which is parallel to both planes of projection, has, for both of its projections, lines ab and a'b', which are parallel to the ground line, and equal and parallel to the line, AB, in space.

d. A horizontal line, as BD, which makes an acute angle icith the vertical plane, has, for its horizontal projection, a line, bd, which makes the same angle with the gi"ouud line that the Ihic, BD,

4 FIEST PRINCIPLES.

makes with the vertical plane, and is equal and parallel to the line itself (BD) ; and has for its vertical projection a line b'a\ which is parallel to the ground line, but shorter than BD, the line in space. e. An oblique line, as BC, PI. I., Fig. 1, or Yd, PL I., Fig. 2, which h parallel to the vertical plane, has, for its vertical projection, a line J'c', or v'd\ which is equal and parallel to itself, and for its horizontal projection, a line ha or vd, parallel to the ground Ihie, hut shorter than the line in space.

f. An oblique line, as V^, PI. I., Fig. 2, which is oblique to both planes of project ion^h^L?, both of its projections, v'd' and vg, oblique to the ground line, and shorter than the line itself.

g. Aa oblique line, as AH, PI. I., Fig. 1, which is oblique to both planes of projection, but is in a plane ACDH, perpendicular to both of those planes, has both of its projections, a'c' and ad, perpendicular to the ground line, and shorter than the line itself.

h. A line, lying in either plane of projection, coincides with its projection on that plane, and has its other projection in the ground Ihie. See cd — c'd', the projections ol cd, PI. I., Fig. 2.

9. JRemarh. A general principle, which it is important to be perfectly familiar with, is embodied in several of the preceding examples; viz. When any line is pjaraUel to either pAane of p>rojeC' tion, its projection on that plane is equal and parallel to itself, and its projection on the other plane is parallel to the ground line.

10. The preceding remark serves to show how to find the true length of a line, when its projections are given. When the line, as V^, PI. I., Fig. 2, is oblique to both planes of projection, its length, Yg, is evidently equal to the hypothenuse of a right-angled triangle, of which the base is vg, the horizontal projection of the line, and the altitude is \v, the height of the upper extremity, V, above the horizontal plane. When the line, as AH, PI. I., Fig. 1, does not touch either plane of projection, it is evidently equal to the hyjiothenuse of a right-angled triangle, of which the base, CH, equals the horizontal projection, ad, and the altitude, AC, equals the difference of the perpendiculars, A« and H<7, to the horizontal plane.

In the same way, it is also true that the line, as Ng, PI. I., Fig. 2, is the hjqiothenuse of another right-angled triangle, whose base equals the vertical projection, v'd' , and whose altitude equals the difference of the perpendiculars, Vw' and dlg^ from the extremities of the line to the vertical plane of projection.

11. Relations o'i p)airs of lines to their 2jrojections. These relations, after the full notice now given of the vai-ious positions of •single lines, may be briefly expressed as follows.

FIKST PRINCIPLES. 5

a. A pair of lines Avliicli are equal and parallel in space, aJidalso parallel to a plane oi projection, as AB and CF, PL I., Fig. 1, or Vc and Xa, PI. I., Fig. 2, have their projections on that plane — a'h' and c'f\ PI. I., Fig. 1, or v'c' and x'a', PI. I., Fig. 2 — equal and parallel — to each other, and to the lines in space.

b. A pair of lines which are equal and parallel in space, hut not parallel to a plane of projection, will have their projections on that plane equal and parallel to each other, but not to the lines in space.

c. Parallel lines make equal a^igles with either plane of projection ; hence it is easy to see that lines not parallel to each other — as \d and Vc, or 'Yg and Ve, PI. I., Fig. 2 — but which make equal angles with the planes of projection, will have equal projections on both planes — i.e. v'd' =:v'c' and vg—ve, also vd—vc.

§ III. — Physical Theory of Projections.

12. The preceding articles comi^rise the substance of tlie purely geometrical or rational theory of projections, which, strictly, is sufficient ; but it is natural to take account of the physical fact that the magnitudes in space and their representations, both address themselves to the eye, and to inquire from rohat distance and in what direction the magnitudes in PI. I., Figs. 1 and 2, must be viewed, in order that they shall appear just as their projections represent them. Since the vertically-projecting perpendiculars, Q, regarded as rays of light, reflected from the block, Fig. 1, could only meet in the eye at an infinite distance in front of the vertical plane, we conclude that the vertical projection of an object rejjresents that object, in respect to form, as it would appear to the eye situated at an infinite distance from it, and looking in a direction perpendicular to the vertical plane of projection. Reasoning similarly in reference to the group, S, we conclude that the horizontal projection of an object presents the same appearance, as respects form, that would be afforded by the object itself, when viewed by the eye at an infinite distance above it, looking in a direction perpendicular to the horizontal plane.

§ IV. — Conventional Mode of representing the two Planes of Projection, and the two Projections of any Object upon one plane — viz. the Platie of the Paper.

13. In practice, a single flat sheet of paper represents the two planes of projection, and in the following manner. The vertical plane, MV, PI. I., Fig. 3, is supposed to revolve backwards, as

6 FIRST PEINCIPLES.

shown by the ai'cs r>i and V^, till it coincides Avith the horizontal plane i^roduced at M w ^ G. Hence, drawing a line from right to left across the paper, to represent the ground line, MG, all that part of the paper above or beyond such a line will represent the vertical plane of jirojection, and the part below it the horizontal plane of projection.

14. Elementary Rational Geometry teaches that two lines, as PP', Yp^ PI. I., Fig. 3, which intersect, fix the position of the plane containing them. It also teaches that if from a point in space, as P, two lines P — P', P — ■/), be drawn — one and the other perpendicular to one and the other of two planes, MV and MH, which are at right angles to each other — then the plane, PP' ^j>"j!?, of these lines, will be perpendicular to the two given planes, and will intersect those planes, in lines P' — p" and j-/ — -p^ which Avill be perpendicular to the intersection, MG, of the planes, at the same point, p" .

This conclusion being reached, it can now be expressed more briefly by saying, that \}i\% plane of the projecting lines of any point, intersects the planes of projection^ in lines ichich are perpendicular to the ground li?ie at the same point.

15. Kow, when the vertical plane revolves about the line MG, as an axis, the point p'\ being in the axis, must remain fixed, and any other point, as P', must revolve in a plane perpendicular to MG. Hence the line F'-^f will revolve — with the plane MV-^to the position p' — 2^"t still remaining perpendicular to the ground line. Hence we have this fundamental principle of graphic constructions, viz. 27ie two lyi'ojections p>, p\ PI. I., Figs. 3 or 4, of a point in space^ are always in a line, pp\ perpendicidar to the ground line.

§ V. — Of the Conventional Direction of the Light / and of the Position and Use of Heavy Lines.

16. A single topic from Divisiox HI. must here be anticipated, viz., that wliich refers to the assumption of a certain direction for the light. Without going into the subject fully, it will be sufficient to say, that as one stands facing the vertical plane of projection, the light is assumed to come from behind, and over the left shoulder, in such a direction that each of its projections makes an angle of 45° with the ground line, as shown in PI. I., Fig. 6.

17. The practical effect of the preceding assumption in reference to the light, is, that upon a body of the form and position shown in PI. I., Fig. 5, for example, the top, front, and left hand surfaces — i.e. the three seen in the Fig. — are illuminated, while the other three faces of the body are in the shade.

FIRST PKIXCirLES. 7

18. The practical rule by -svljich the direction of the light and its eiFect are indicated in the projections, is, that all those visible edges of the body in space, which divide tlie light from the dark surlaces, are made heavy in projection.

19. To illustrate : The edges BC and CD of the body in space, Pi. I., Fig. 5, divide liglit from dark surfaces, and are seen in looking towards the vertical plane, and hence are made heavy in vertical projection, as seen at h'c' and c'd'. BK and KF divide illuminated from dark surfaces, and are seen in looking towards the horizontal plane, and are therefore made heavy in the horizontal projection, as shown at hJc and Jcf.

20. By inspection, it will be seen that the following simple rule in reference to the position of the heavy lines on the drawings, may be deduced, as an aid to the memory. In all ordinary four-sided prismatic bodies, placed with their edges respectively parallel and perpendicular to the planes of projection, or nearly so, the right hand lines, and those nearest the ground line, of both projections, are made heavy.

21. Heavy lines are of considerable use, in the case of line drawings particularly, in indicating the forms of bodies, as will be seen in future examples.

§ VI. — Notation.

22. Under the head of dotation, two points are to be considered, the manner of indicating the various lines of the diagram, and the lettering. As will be seen by examining PI. I., Figs. 1, 2 — see Ve, eg, &c. — and 5, the visible lines of the object represented are indicated by full lines ; lines of construction and invisible lines of the object, so far as they are shown, are made in dotted lines. Tlie intersections of auxiliary 23lanes with the planes of projection, called traces, are represented by broken and dotted lines, as at P'QP", PI. I., Fig. 16.

23. Unaccented letters are used to indicate points of the horizontal projection; and the same Avith a single accent, denote the vertical projections of the same points. A strict adherence to the simple rule of lettering the same point with the same letter, wherever it is shown, will afford a complete key to very complicated diagrams, as will be shown as the course proceeds.