■■":'^"'*-«! 1 J / v>w/. z, ^>o|J;,^_ WORKS ON ^ DESCRIPTIVE GEOMETRY, AND ITS APPLICATIONS TO ENGINEERING, MECHANICAL AND OTHER INDUSTRIAL DRAWING. By S. EDWARD WARREN, C.E. I. ELEMENTARY WORKS. Primary Geometry. An introduction to geometry ao usually presented ; and designed, first, to facilitate an earlier beginning of the subject, and, second, to lead to its graphical applications in manual and other elementary schools. With numerous practical examples and cuts. Large 12mo, cloth, 75c. Industrial Science Series Comprising: 1. Free-hand Geometrical Drawing, widely and variously useful iu training the eye and hand in accurate sketching of i)lane and solid figures, lettering, etc. 12 folding plates, many cuts. Large 12mo, cloth, $1.00. 2. Drafting Instruments AND Operations. A full descrip- tion of drawing instruments and materials, with applications to useful examples ; tile work, wall and arch faces, ovals, etc. 7 folding plates, many cuts. Large 12mo, cloth, $1.25. 3. Elementary Projection Drawing. Fully explaining, in six divisions, the principles and practice of elementary plan and elevation drawing of simple solids ; constructive details ; shadows ; isometrical drawing ; elements of machines ; simple structures. 24 folding plates, numerous cuts. Large 12mo, cloth, 11.50. This and No. 3 are especially adapted to scientific, preparatory, and manual-training industrial schools and classes, and to all mechanics for self-instruction. 4. Elementary Perspectiye. With numerous practical examples, and every step fully explained. Revised Edition (1891). Numerous cuts. Large 12mo, cloth, $1.00. 5. I'LANE Problems on the Point, Straight Line, and Circle. 225 problems. Many on Tangencies, and other useful or curious ones. 150 woodcuts, and plates. Large 12mo, cloth, $L25. n. HIGHER WORKS. 1. The Elements of Descriptive Geometry, Shadows AND Perspective, with brief treatment of Trehedrals ; Trans- versals; and Spherical, Axonometric, and Oblique Projections; and many examples for practice. 24 folding plates. 8vo, cloth, $3.50. 2. Problems, Theorems, and Examples in Descriptive Geometry.. Entirely distinct from the last, with 115 problems, embracing many useful constructions ; 52 theorems, including examples of the demonstration of geometrical properties by the method of projections ; and many examples for practice. 24 fold- ing plates. 8vo, cloth, 62.50. 3. General Problems in Shades and Shadows, with practical examples, and including every variety of surface. 15 folding plates. 8vo, cloth, |>3.00. 4. General Problems in the Linear Perspective of Form, Shadow, and Reflection. A complete treatise on the jirinciplcs and jjractice of perspective by various older and recent methods ; in 98 problems, 24 theorems, and with 17 large plates. Detailed contents, and numbered and titled topics in the larger problems, facilitate study and class use. Revised edition, correc- tions, changes and additions. 17 folding plates. Bvo, cloth, $3.50. 5. Elements of Machine Construction and Drawing. 73 jiractical examitles drawn to scale and of great variety ; besides 30 problems and 31 theorems relating to gearing, belting, valve- motions, 6crew-])ropellcrs, etc. 2 vols., 8vo, cloth, one of text, one of 34 folding plates. $7.50. 0, Problems in Stone Cutting. 20 problems, with exam- jjle.s for j)ractice under them, arranged according to dominant surface (plane, devclojjablc, warped or double-curved) in each, and enibraciiig every variety of structure ; gateways, stairs, arches, dometr, winding ])assages, etc. Elegantly printed at the Riverside Press. 10 folding plates. 8vo, cloth, $2.50. INDUSTRIAL SCIENCE DRAWING. ELEMENTART PROJECTION DRAWING. THEOJRY AND PK.ACTIOE. KOR PREPARATORY AND HIGHER SCIENTIFIC SCHOOLS ; INDUSTRIAL, AND NORMAL CLASSES ; A_ND THE SELF- INSTRUCTION OF TEACHERS, INVENTORS, DRAFTSMEN, AND ARTISANS. Sn Six Diuisions. DIV. I. ELEMENTARY PROJECTIONS. DIV. II. DETAILS OF BIASONRY, WOOD, AND METAL CONSTKUCTIONS. DIV. ni. ELEMENTARY SHADOWS AND SHADING. DIV IV. ISOMETRICAL AND OBLIQUE PROJECTIONS. DIV. V. ELEMENTS OF BIACHINES. DIV. VI. SIMPLE STRUCTURES AND MACHINES. By S. EDWAED WAEEEI^, C.E., FORMER PROFESSOR IN THE RENSSELAER POLYTECHNIC INSTITUTE, MASS. INST. OPTECHNOIOGE AND BOSTON NORMAL ART SCHOOL; AND AUTHOR OF A SERIES OP TEXT-BOOKS IN DESCRIPTIVK GEOMETRY AND INSTRUMENTAL DRAWLNQ. THIRTEENTH EDITION. FIFTH THOUSAND, NEW YORK : JOHN WILEY AND SONS, 53 East Tenth Street, 1893. Copyright, 1880, By JOHN WILEY & SONS. raiM or I. I. iiTTii ii CO., KOI. tfi 10 t6 A^TOR PLACC, NIW TMI, CONTENTS. Note to the Fourth Edition, ...... vi Note to the Fifth Edition, ....... vii From the Original Preface, ix Preliminary Notes — Drawing Instruments and Materials, . xi DIVISION I.— PROJECTIONS. Chapter I. — First Principles. § I. — The purely geometrical or rational theory of pro- jections, ........ 1 § II. — Of the relations of lines to their projections, . 3 § III. — Physical theory of projections, .... 5 § IV. — 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 § VI.— Notation, 7 § VII. — Of the use of the method of projections, . . 8 Chapter II. — Projection of Li7ies. Problems in Eight Projection ; and including Projections showing two Sides of a Solid Right Angle. (Thirty-two Problems.) § I. — Projections of straight lines, .... 9 § II. — Right projections of solids, 13 § III. — Projections showing two sides of a solid right angle 14 § IV. — Special Elementary Intersections and Developments 23 General ExamjAes, 31 DIVISION II.— DETAILS OF MASONRY, WOOD, AND METAL CONSTRUCTIONS. Chapter I. — Constricctions in Masonry. § I. — General definitions and principles applicable both to brick and stone work, 33 § II. — Brick work, 33 § III.— Stone work, 36 A stone box-culvert, ...... 37 2016052 1" CONTEXTS. PAG 15 Chapter II. — Constructions in Wood. 1 1. — General remarks. (Explanation of Scales.) . 3i) § II. — Pairs of timbers whose axes make angles of 0° with each other, 42 § III. — Combinations of timbers whose axes make angles of 90^ with each other, 45 § IV. — Miscellaneous comljinations. (Dowelling, &c.) . 48 § V. — Pairs of timbers wliich are framed together obliquely to each other, 49 § VI. — Combinations of timbers whose axes make angles of 180° with each other, 50 Chapter III. — Constructions in 3fetal, ...... 55 Cage valve of a locomotive pump. Metallic pack- ing for stuffing-boxes, &c., . . . . 56 Boiled-iron beams and columns, .... 60 DIVISION in.— ELEMENTARY SHADOWS AND SHADING. CuAPTKU I. — ShmJmcs. § I. — Facts, Principles, and Preliminary Problems, . 66 § II. — Practical Problems. (Twelve, with examples.) . 70 Chapter II. — Shading, 77 Hexagonal prism; cylinder; cone, sphere, and model. DIVISION IV.— ISOMETRIC AL AND OBLIQUE PROJEC- TIONS. Chapter I. — First Principles of Jsometrical Drawing, . . 87 Chapter II. — Problems involving only Isometric Lines, . . 90 ('n.KPTKH III. —Problems invoicing ]^on-iso7netri^al Lines, . . 95 Chaptku IV. — Problems intohing the Construction and Equal Division of Circles in Isometrical Draicing, ... 99 Chapter V. — Oblique Projections, 106 DIVISION v.— ELEMENTS OF MACHINES. Chapter I.—Principhs. Supporters and Crank Motions, . . 114 Pillow-block; standard; bed and guide -rest; crank; ril)bed eccentric; grooved eccentric. Chapter U. — Oearing, 126 Spur-wheel; bevel wheels; screws and serpentines; worm- wheel. CONTENTS. DIVISION VI. -SIMPLE STRUCTURES AND MACHINES. Chapter I. — Stone Structures, . A brick segmental arch, . A semi-cyliadrical culvert, with vertical cylindri- cal wing-walls, Chapter II, — Wooden Structures, A king-post truss, A queen-post truss- bridge, Chapter III. —Iron Constructions, A railway track— frog, chair, &c., A hydraulic ram, . Exercises.— A stop-valve; a Whipple truss-bridge; a. vertical boiler; a Knowles steam-pump. 140 140 112 111) 14G 147 151 151 153 NOTE TO THE FOURTH EDITION. This edition is improved cliicfly by the extension of the chapter on obUque, or pictorial projections, also called mechanical per- ppective, with the addition of a new plate. Numerous niuior corrections, and new or improved paragraphs have beeti scattered through the work, as suggested by further ex- perience. Yet it is not expected that tliis edition, however im- proved, can supersede either careful and thinking study on the pait of the willing student, or ample, repeated^ and varied instruc- tion on tlie p:irt of tlie willing teacher. Many minds require variously changed and amplified statements of the same thing before tht-y are ready to exclaim heartily, "I see it now;" and the teacher must be ready to meet, with many forms of instruction, the conditions presented to him by different minds. A very moderate collection of such objects as are illustrated in Division II., and such as can be made by a carpenter or turner, or m machine, gas-fitting, or pattern shops, will very usefully supple- ment the text, and add to the pleasure and benefit derived by the Btudcnt; and will be better than an increased size of the book, which is meant to be rather suggestive than exhaustive, or to be too closely followed, in respect to the examples chosen for practice. Finally, the author's chief desire in relation to this, as well as the rest of his '• elementaiy works," is, to see them so generally used in hi'jhtr preparatory instruction as to give due place to higher •ludies in the same department in the strictly Technical Schools. H. 1'. I., TuoY, July, 1871. NOTE TO THE FIFTH EDITION. The call for a new edition of this manual has led the Author, after the lapse of ten years since the last revision, to make such further improvements in a new edition, as additional and varied experience and reflection have suggested. A few paragraphs m Division I. have been re-written. A few but valuable additions have been made to the text and plates of Div. II. Div. III. contains one more plate, taken from Div. VI., and examples of finished shading, not before provided. Div. IV. has been improved by the addition of a few figures and by partial re-writing. The most important change is the addition of Div. V., em- bracing the more important and universal elements of machines. The volume is thus made more complete, both in itself, and as an introduction to higher works both theoretical and practical.* Div. VI. (V. in previous editions) has been slightly enlarged by a few new and valuable practical examples. In general: while the subjects of all the Author's " Elementary Works " have been largely taught in the earlier classes of Poly- technic Schools, of whatever name, it is to be hoped that by the increasing development of scientific instruction, they will all ultimately be included in Preparatory Scientific Instruction, and in Special Normal Classes; and in behalf of the many pupils whose education ends in preparatory schools, but to whom an exact knowledge of elementary instrumental or mathematical drawing would be highly useful. The explanations of first principles have purposely been made very complete, in behalf of all classes of self-instructors, and because what is not thus printed must be said, and often repeated, * The beautiful plates XVII., XVIII., XIX., modified however to suit a full explanatory text, are from the Cours de dessln lineaire, par Delaistke, a work which every draftsman would do well to possess. viii XOTE TO THE FIFTH EDITION". It) ensure that full understanding of the subject, the test of which is the ready performance of new examples. At this point the testimony of an evidently experienced and faithful English author and teacher may well be noted. Speak- ing of the copying system, he says, "If, however, at the end of one or two years' practice, the copyist [though able to make a highly finished copy] is asked to make a side and end elevation and longitudinal section of his lead-pencil, or a transverse section of his instrument-box, the chances are that he can do neither the one nor the other. Strange as it may appear, this is a state of things which I have had frequent opportunities for witness- ing. . . . The remedy has been to commence a course of study from the very beginning. ... He has made from the copy a highly finished drawing, with all the shadows admirably projected, being at the same time, however, perfectly ignorant of the rules for projecting such shadows. This is the true picture of a student who had a course of two years' study where mechanical drawing was taught " [by merely copying successively more and more elaborate drawings].* With these remarks the present edition, in its final form as now intended, is committed to the favor of Schools and Self- Instructors. Newton, Mass., October, 1880. * Preface to Bln'n's Orthographic Projections. London, 1867. FEOM THE OPtlGINAL PEEP ACE. 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. Ilence moderate experi enc-e suggests, and continued experience confirms, the propriety of interposing, between the easily understood drawings of pro blems involving two dimensions, and the general course of pro- blems of three dimensions, a rudimentary course upon the methoda of representing objects having three dimensions. 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 the thing to he drawn^ 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 draw- ing 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, Losi of time, owing to repetitirn of the same detail many times \v PREFACH. the same structure ; and 3d, Waste of drawings, as well as of time, through poor execution, which is due to insufficient pre vious practice. Hence Divisions II. and Y. contain a liberal collection of elements of structures and machines, each one of which affords a useful problem, while Division YI. includes examples of a few simple structures, to fulfil the threefold pur- pose 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 practice in execution. 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 — and because of the compact- ness and beauty of the volume which is formed by binding to- gether all the plates of the course, when they are well executed. As to the use of this volume, it is intended that there should be formal interrogations upon 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 remainini^ divisions. Ilememberino: 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, the student may, in place of a tedious course of finished drawings, be called on frequently to describe, by the aid of pencil or blackboard sketches, hoto he would construct drawings of certain objects — cither those given in the several Divisions of this volume, or other similar ones proposed by his teacher. PRELIMINARY 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 explana- tions are here prefixed. I. Figs. 1, 2, 3, 5, and 15, of PI. I. are pictorial diagrams, used for illustration in place ofactital models. Thus, in Fig. 3, for example, MHi represents a horizontal square cornered plane sur- face, as a floor. MGY represents a vertical square cornered plane surface, as a wall, which is therefore perpendicular to MH^. P represents any point in the angular space included by these two planes. Pj:; represents a line from P, perpendicular to the plane MH<, and meeting it at^. Vp' represents a line from P, perpendicular to the plane AIGV, and meeting it at^'. Then Vp 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 hody, as in Fig. 1, are perpendicular to the planes, as MH^ and MGV, 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 tliey 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 the other vertically. In the construction oi\c\& plates, he should also test his knowledge oi \\iq. principles, \)'^ varying the form of the examples, though without essentially changing their character. Xll PRELIMINARY NOTES. Drawing Instruinents and Materials., This volume is intended to be the immediate successor of my " Drafting Instruments and Operations," which is there- fore supposed to have been read first,' by students of tliis one. But as some self-instructors and other students may desire to acquire a knowledge of projections as quickly as possible, for j^ractical use, and without regard, at first, to finished execution of their drawings, the following condensed information is here inserted for their convenience. To abridge the descriptions to the utmost, it may first be stated that dealers in Drawing Instruments and Materials are found in all large cities, who will send descriptive catalogues on a])plication. Such are Frost & Adams, Boston ; W. & L. E. Gurley, Troy, N. Y.; Keuffel & Esser, New York City; James "W. Queen, Philadelphia; and others, doubtless, whose advertisements can be found in educational and popular me- chanical periodicals. The necessary articles are : — 1. A good^a27' of cotnpasscs, with tliQiv furniture; that is, a pen, pencil, and needle point to replace the movable steel points, when drawing circles in pencil or ink. 2. A good drawing pen. o. A (b-awing board 20 x 30 inches. 4. A T square ; that is, a hard-wood rnler, having a stout M'oodcn cross-piece about 2|-x 9 inches, and half an inch thick, at one end, njion the fiat side of which the blade is firmly fastened, truly at right angles. The blade may be about 30 inches long. n PKELIMINART NOTES. XIU 5. A pair of hard-wood riglit-angled triangles, the longest side about 10 inches long ; one with the two acute angles of 45° each, the other with acute angles of 30° and 60°, 6. A triangular scale, graduated into tenths or twelfths of the unit as may be preferred ; or, a flat ivorj " protractor scale." 7. Buff manilla office, or "detail" paper, or, if preferred and it can be afforded, Whatman's rough drawing paper, of con- venient size, from "medium," 17" x 22", to "imperial," 21" X 30". 8. Hard lead-pencils. 9. A cake of Indian ink — Chinese the best for shadina:, the Japanese for Imes. Where the utmost economy is sought, a very cheap, fair quality of brass instruments can be had in boxes, or the draw- ings can even be made with pencil only ; any neat worker in hard wood can make the drawing board, T square, and triangles, and a foot-rule may be made to serve as a scale. When drawings are not to be colored, the paper can be lightly gummed or tacked to the drawing board at the corners. Other- wise, the sheet should be well wet by sponging with clean water and, while wet, fastened to the board by means of thick mucilage applied along the edge of the paper. Indian ink is prepared for use by rubbing it on a bit of china, with a few drops of water. It is then applied between the blades of the drawing pen by a small feather or slip of paper. Pen and ink should be wiped dry when done with. ELEMENTARY PROJECTION DRAWING. DIVISION FIRST. PROJECTIONS. CHAPTER I. FIRST PRINCIPLES. § 1 . Tlic purely Geometrical or Rational Theory of Projections. 1. Elementary Projection Drawing is an introduction to Descriptive Geometry, and shows how to represent simple solids, singly and in combination, upon plane surfaces, yet so as to show their real dimensions. 2. If ten feet of 5-inch stove-pij)e were wanted, a circle five inches in diameter, drawn on paper, would be all the pattern the workman would need. But if the desired length were forgotten, or if the pipe were to be conical, the circular drawing would no longer be sufficient. That is, as a plane surface has but two di- mensions, no more than two dimensions of any object can be ex- actly shown in one figure on that plane. But practical toork, and geometrical problems for study, are both continually arising, which require, for conveniejit execution in one case, and proper solution in the other, that we should be able, in some way, to truly show all the dimensions of solid bodies upon plane surfaces. 3. What, then, is the number and the relative position of the planes which will enable us to rejjresent all the dimensions of any geometrical solid, in their real size, on those planes? To assist in an- swering this question, reference maybe made to PL 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 plane below it and parallel to its top surface ABED. If now from the four points A, B, D and E, perpendiculars be let fall r.pon the horizontal plane MN, they will meet it in the points a, b, d and e. 2 FIRST PRINCIPLES. Bv joining these points, it is evident that a figure — abed — will be formed, avIucIi a\i11 be equal to the top surface of the block, and will be a correct representation of the lerigth and breadth of that top surface — i. e. of the length and breadth of the block. Simi- larly, if MP be a vertical plane, parallel to the front, ABCF, of the block; and if perpendiculars, Aa', etc., be let fall from A, B, C, F, upon MP; the figure, a'b'c'f, will be equal to ABCF, and hence will show the length and thickness of the block. 4. From the last article the following definitions arise. The point a, PI. I., Fig. 1, is where A would arrive if thrown, that is, projected, vertically doAvnwards along the line Aa. Likewise, a' is where A would be, if thrown or projected along Aa', from A to a', perpendicularly to the plane MP. Hence a is called a horizontal projection of A ; and a' is called its vertical projection. Also, conversely, A is said to be horizontally i^rojected at a, and vertically projected at a'. The plane MN is thence called the horizontal plane of projec- tion ; and the plane MP, the vertical plane of pi'ojection. Aa and Aa' are called the two projecting lines of the point A relative to the planes of projection, MN and MP. Hence we have this definition. If from any given point a per- pendicular be let fall upon a plane, the point where that perpen- dicular meets the plane will be the projectio7i of the point upon the plane, and the perpendicular will be the projecting line of the point. 5. Similar remarks apply to any number of points or to objects limited by such points; and to their projections upon any other planes of projection. Thus abed is the horizontal projection of the block ABCD, and a'b'c'f is its vertical projection, and, generally, the projecting lines of objects are perpendicular to the planes of projection employed. Finally, the intersection, as MR, of a horizontal, and a vertical plane of projection, is the ground line for that vertical plane. 6. From the foregoing articles the following principles arise. 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 jjo.sition, they must be parallel to that plane on which they arc shown. Third: Each plane shows two of the di- FIRST PRINCIPLES. 3 mcnsions of the solid, viz., the two which are panillel 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. Fourth : The height of the vertical projection of a point above the ground line, is equal to the height of the point itself, in space, above the horizontal plane; and the perpendicular distance of the horizontal projection of a point from the ground line, is equal to the perpendicular distance of the point itself, in front of the vertical plane. Thus : PI. I., Fig. 1, aa" = Aa' and a'a" = Aa. 7. The preceding principles and definitions are the foundation ot the subject of projections, but, by attending carefully to Pi. I., Fig. 2, some useful elementary applications of them may be discovered, which are frequently applied in practice. PI. I., Fig. 2, is a pic- torial 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, X«5, 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, perhnps, represented as clearly as they can be by mere diagrams, in PL I., Figs. 1 and 2 ; and then by attentive study of those figures, the next two articles may be easily imderstood. § II. — Of the Melations 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 point, a, and for its vertical projection, a line a'c', perpendicvxlar to the ground line, and equal and parallel to the line AC, in space. b. A horizontal lijie, as AD, which is perpendicidar to the verti- cal plane, has, for its horizontal projection, a line, ad, perpeU' dicular 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 parcdld 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, bdy v^hich makes the same angle with the ground line that the line, BD, 4 FIRST PRINCIPLES. makes willi the vertical plane, and is equal and parallel to the line itself (BD) ; and has for its vertical projection a line h'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 \d, I'l, I., Fig. 2< which is parallel to the vertical plane, has, for its vertical projection, a line he, or v'd\ which is equal and parallel to itself, and for its horizontal projection, a line ha or vd, parallel to the ground line, but shorter than the line in s])ace. f. Au oblique line, as V^, PI. I., Fig. 2, ichich is oblique to both planes of projection, has both of its projections, v'd' and vg, oblique to the ground line, and shorter than the line itself. lane 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 angles with either plane of projec- tion ; hence it is easy to see that lines not parallel to each other — as Yd and Vc, or Yg and Ve, PI. I., Fig. 2 — but which make equal angles with the planes of projection, will have equal projec- tions on both planes — i.e. v'd' =.v'c' and vg—ve, also vd=vc. § III. — Physical Theory of Projections. 12. The preceding articles comprise the substance of the purely geometrical or rational theory of projections, which, strictly, ia 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 lohat distance and in what direction the magnitudes in PL I., Figs. 1 and 2, must be viewed, in order that they shall appear just as their projections represent thera. Since the projecting lines, Q, regarded as rays, reflected from the block. Fig. 1, to the eye, are parallel, they could only meet the eye at an infinite distance in front of the vertical plane. Plence the vertical projection of an object represents it 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 pro- jection. Likewise, the* projecting lines, S, show that tlie horizontal projection of an object represents it as seen from an infinite distance above it, and looking perpendicularly down upon the horizontal plane. Thus, the projecting lines represent the direction of vision^ which is perpendicular to the plane of projection considered. § IV. — Conve7itional Mode of representing the two Planes of Pro- jection, and the two Projections of any Object upon one plant — viz. the Plane 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, IMY, PL I., Fig. 3, is supposed to revolve backwaids, as U FIRST PRINCIPLES. shown by the arcs rii and V^, till it coincides with the horizontal plane produced at M u t G. Ilence, drawing a line from right to loft across the paper, to represent the ground line, MG, all that joart of the pajDcr above or beyond such a line will represent the vertical plane of projection, and the part below it the horizontal plane of projection. 14. Elementary geometry shows that the plane, as PP' p"2'>, PI. I., Fig. 3, of the projecting lines, Vp and PP', (3, 4) is per- pendicular to both of the planes of projection, and to the ground line MG. Hence it intersects these j^lanes in lines, as j^lJ" and P' p", both of which arc i)crpendicular to the ground line at the same j^ointjy". 15. If, now, as exiDlained in (13) the vertical plane MY, PI. I., Fig. 3, be revolved about MG, to coincide with the horizontal plane, the point p" will remain in the axis MG, and the lines 23'j>" and V'2}" will unite to form one line 2^p'> perpendicular to MG. That is: Whenever two points are the projections of one point tJisjMce, the line joining them iviU he perpendicular to the ground line. § Y. — Of the Conventional Direction of the Light ; and of the Position and Use of Heavy Lines. 10. Without going into this subject fully, as in Div. III., it is suflicicnt to say here that, as one faces the vertical plane of ])ro- jection, tlic light is assumed to come from behind, and over the left shoulder, in such a direction that eacli 2)rojectio7i of a rag (but not the ray itself) malces an ayigle ofAo° with theground line, as shown in PI. I., Fig. G. And note that the light is supposed to turn vuth the observer, as he turns to face any other vertical plane. 17. T\\Q practical effect ot the preceding assumption in refer- ence to the light, is, that upon a body of the form and position fihown in PI. I., Fig. 5, for example, the top, front, and left liand surfaces — i. e. the three seen m the Fig. — are illuminated, wliile the other three faces of the body are in the shade. FIRST PRINCIPLES. 7 18. The practical rule by Avliicli tlie diroction of {\h\ liglit,and is effect, are indicated in the projections, is, tliat :ill those visible edparent forms and relative sizes of objects. 2G. Drawings of the former kind are often called, on account of the uses to which they are applied, '•'■mechanical''^ or '■'• xoork'mg '''' drawings. Those of the latter kind are commonly called pictures ; and here it is to be noticed that if "working" drawings are to Bhow the tnie^ and not the upjjxtreiU^ proportions of all parts of an object, they must, all and always, conform to this one rule, viz. All those lines wldch are equal and similarly situated on the oljoi't, must be equal and similarly situated on the drawing. But, as is now abundantly evident, drawings made according to the method of projections, do conform to this rule; hence their use, a? aboN e described. CHAPTER n. PROJECTIONS OF LINES: PROBLEMS IN" RIGHT PROJECTION; AND INCLUDING PROJECTIONS SHOWING TWO SIDES OF A SOLID RIGHT ANGLE. 27. The style o? execution of the following problems is so simple, and so nearly alike for all of them, that it need not be described for each problem sepai'ately, but will be noticed from time to time. In the solution of problems, Imes are considered as wdimiied, and may be produced mdefij^itely in either direction. § I. — Projections of Straight Lines. 28. Prob. 1. To construct the jyojections of a vertical straight line, 1^ inches long, xohose lowest point is \ an inch from the horizontal plane, and all of whose points are ^ of an inch from the vertical plane. Jiemarks. a. The remaining figures of PI. I. are drawn just half the size indicated by the dimensions given in the text. It may be well for the student to make them of full size. b. Let MG be understood to be the ground line for all of the above problems, without further mention of it. 1st. Draw, very lightly, an indefinite line perpendicular to the ground line, PI. I., Fig. 7. 2d. Upon it mark a point, a', two inches above the ground line, and another pohit, 6', half an inch above the ground line. 2d. Upon the same line, mark the point a,b, three-fourths of an lich below the ground line. Then a' i' will be the vertical, and ab the horizontal projection of the required line. (8 a) 29. Prob. 2. To construct the projections of a horizontal line, H inches long, Ij inches above the horizontal plane, perpendicular to the vertical plane, andxoith its furtherm,ost point — from the eye — \ of an inch from that j)lane. PL I., Fig. 8, in connection with the full description of the preceding problem, will afford a sufficient explanation of this one. Remark. It often happens that a diagram is made more intel li) TKOJECTIOXS OF LINES ligiUe by lettering it as at aZ», PI. I., Fig. *", and at c'd\ PI. I. Fig 8, for thus the notation shows unmistakably, that ah or c'd' are not the projections of points but of lines. 30. PnoBLEMS 3 to 8, inclusive, need now only to be enunciated witli references to their constructions, in PI. I. Fig. 9 shows the projections of a line, 2\ inches long, parallel tc the ground line, H inches from the horizontal plane, and 1 inch from the Acrtical plane. Fig. 10 is the representation of a line, 2 inches long ; parallel to the horizontal piano, and 1 inch above it ; and making an angle of 30° with the vertical plane. Fig. 11 represents a line, 2^ inches long, i)arallel to the vertical plane, and 1| inches from it, and making an angle of 60° with the horizontal plane. Fig. 12 gives the projections of a line, 1^ inches long, lying in the horizontal plane, parallel to the ground line, and 1^ inches from it. The jirojcction a'h' shows the lino to be in II (G, ^tli). Fig. 13 shows the projections of a line, 1^ inches long, lying in the vertical plane, parallel to the ground line, and 1 inch from it. Fig. 14 indicates a line, 2^ inches long, lying in the vertical plane, and making an angle of 60° with the horizontal plane. 31. Projections of the revolution of a 2^oint about an axi- When a point revolves about an axis, it describes a circle, or arc, whose plane is perpendicular to the axis. Tiius a point, revolving al)out an axis which is pe7'pendicular to the vertical plane, describes an arc, parallel to that plane. The vertical projection of such an arc is an equal arc. Its horizontal projection (6) is a straight line parallel to the ground line. Tims, Pi. I., Fig. loa, Ca represents a perpendicular to the ver tical plane, V. Tlie point, A, by revolving a certain distance about this axis, describes the arcAB; whose vertical projection is the iq>ial arc, a'b'; and whose horizontal projection is a5, a straight line parallel to the ground line. Likewise, briefly, in Figs. \5b and 15c, XYjs a vertical axis. The point A, revolving abcmt it, describes a horizontal arc, AB; wliose horizontal projection, ai, is an equal arc ; and whose verti- cal projection, a'i' is a straight line jjarallel to the ground line. 32. PiiOH. 9. 7b construct tliC jyrojections of a line lohich is i?i a plane perpendicular to both pAanes of projection^ the line being oblique to both planes of j/rojection. Plate I., Fig. 15, represents a PROJECTION OF LINES. 11 model of this problem, AB represents the line in space; ab iln horizontal projection; a'b' its projection on the vertical plane MP'; and A'B' its projection on an auxiliary vortical plane PQP'; which is parallel to AB, and perpendicular to the ground line. Ilenoe A'B'=AB. Now in making these three planes of projection coincide with the paper, taken as the horizontal plane of projection, the plane PQP' is revolved about P'Q as an axis, till it coincides with the primitive vertical plane, MP', produced, as at P'QV", and then the united vertical planes, MPV", are revolved backward about MH' as an axis into the horizontal plane. In the first revolution. A' describes, according to the last article, the horizontal arc, A'a", (31) about m as a centre, and whose projections are «""«'", liaving its centre at Q, and wia". Also B' describes the horizontal arc, B'J", about 71 as a centre, and whose projections are h""h"\ whose centre is Q, and nh" . Thus wc see that two or more differe^it vev' tical 2^roJections, as a' and a", of the same j^oint, are in the same jmrallel, a'a", to the ground line; that is, the^/ are at the same lieight above that line. Hence a" is at the intersection of a'a", parallel to the ground line, MIP, with «'''«'■', perpendicular to MH'. 33. a. Notice further that a""h"" is the horizontal projection of A'B', and that it coincides with the projection of ab upon PQP' Likewise, that mn is the vertical projection of A'B', and that it co incides with the projection oi a'b' upon PQP'. b. Note that B^', for example (6), is equal to bt, and that bt=b"n the distance of the auxiliary vertical projection, b", of B, from the trace, or axis, P'Q, of the auxiliary plane. c. Note that a"b" shows the true length and direction o? A\^ ; that is, the angles made by a"b" with ll'Q and P'Q, respectively, are equal to those made by AB with the planes of projection. 34. To construct PL I., Fig. 15, in 2^yojection. See Pi. L, Fig. IG, where, to make the comparison easier, like points have the same letters as in Fig. 15. Supposing the length and direction of the line given, we begin with a"b", which suppose to be 2" long, and to make an angle of 60'' with the horizontal plane. Suppose the line in space to be \\ inches to the left of the auxiliary vertical plane P'QP then a' b', its vertical projection, will be perpendicular to the ground line, between the i>arallels a"a' and b"b' (32), and U inches from P'Q. The horizontal projection, ah, will be in a'b' produced; b"n — b"'b"" are the two projections of the arc in which ihe point b" revolves back to its position, n — b"", in the plane P'QP, and b""b — nb' is the line in which nb"" is projected back 12 PBOJECnOK OF LTSTES. t to it5 primitive position h'b. Therefore, h is at the intersection of i""6 with afb' proiiuced. a is similarly found, giving ab as the liorizontal projection of the given line. An. 32 shows suflSciently how to/V«t? the length a^'b" \£ ab — a^b' were given. Example. Construct the figure when a',b is the highest point. 35. Exf.cution. The foregoing problems are to be inked with very black ink; the projections of given lines, and the ground line, m. heavy full lines; and the lines of construction vafine dotted lines as shown in the figures. Lettering is not necessary, except for purposes of reference, as in a text book, though it affords occasion for practice in making small letters. On the other hand, lettering, if poorly executed, disfigures a diagram so much that it should be made only after some pre vie ua pract.oe, and then carefiilly ; making the letters small, fine, and regular. § IL — Right Projections of Solid*, Remark. The tenn " right projection " becomes significant only when it refers to bodies which are, to a considerable extent, bounded by straiglit lines at right angles to each other. Such bodies are said to be drawn in right projection when their most important lines, and faces, are parallel or perpendicular to one or the other of the planes of projection. 36. Peob. 10. — To construct the projections of a vertical right prism, having a square base; standing upon the horizontal plane, and icith one of its faces paralltl to the vertical plane. PI. U., Fig. 17- Let the prism be 1 inch square, H inches high, and ^ of an inch from the vertical plane. \st. Tiie square ABEF, \ of an inch from the ground line, is the plan of the prism, and stiijtly represents its upper base. 2d. A'B'C'D', 1^ inches high, is the elevation of the prism, and strictly represents its front face. Ill this, and in all similar problems, it is useful to distinguish the positions of the p'jints, lines, and faces, in words ; as ripper aud lotcer / front and back / right and left / just as is done in speaking of the bodies which (23) the projections represent. Thus, Ist. AA' is the fn)nt, upper, left hand comer of the prism. 2d. EF — A'C i.s the back top edge ; BF — D' is the lower riglit hand edge ; each corner of the plan is the horizontal projection of a vertical edge ; etc. PIi.1. c PROJECTIONS OP SOLIDS. 13 Sd. AE — A'C is the left hand/ace; etc. 37. Prob. 11. — To construct the plan and two elevations of a vrism having the proportions of a brick, and placed with its length (>a7'allel to the ground line. Plate II., Fig. 18. Ist. abed is the plan, f of an inch broad, twice that distance in length, and f of an inch from the ground line, showing that the prism in space is at tlie same distance from the vertical plane of projection. 2nd. a'b'ef is the elevation, f of an inch thick, and as long as the plan ; and ^ of an inch above the ground line, showing that the prism in space is at this height above the horizontal plane. ^rd. If a plane, P'QP, be placed jDcrpendicular to both of the principal planes of projection, and touching the right hand end of the prism, it is evident that the projection of the prism upon such a plane will be a rectangle, equal, in length, to the width, bd^ of the plan, and, in height, to the height, bf of the side elevation. This new projection will also, evidently, be at a distance from the primitive vertical plane, i.e. from P'Q, equal to JQ, and at a dis- tance from the horizontal plane equal to Q/". When, therefore, the auxiliary plane, P'QP, is revolved about P'Q into the primitive vertical i^lane of j^rojection, the new projection will appear at a"e"c"g". ith. dc'" is the horizontal, and b'o" the vertical projection of the arc in which the point db' revolves into the primitive vertical plane. Ja'", b'a% are the two projections of the horizontal arc in which the corner bb' of the prism revolves. Example. — Let the auxiliary plane PQP' be revolved about PQ into the horizontal plane. a"c" will then appear to the right of PQ and at a distance from it equal to Qb'. 38. Prob. 12. — To const7'uct the two 2y'>'ojections of a cylinder which stands upon the horizontal jolane. PI. IL, Fig. 19. The circle AaB5 is evidently the plan of such a cylinder, and the rectangle A'B'C'D'its elevation. Observe, here, that while the elevation, alone, is the same as that of a prism of the same height. Fig. 17, tlie plan shows the body represented, to be a cylinder. Any point as a in the plan, is the horizontal projection of a ver tical line lying on the convex surface, and called an element. A — A'C, and B — B'D', which limit tlie visible part of the convex sur- face, are called the extreme elements. 39. As regards execution, the right hand line B'D' of a cylindef 14 PROJECTIONS OK SOLIDS. or cone may l)e made less Iieavy than tlie line B'D', Fig, IV ; and in the plan, the semicircle, aVtb, convex towards the ground line, and limited by a diameter ab, which makes an angle of 45° with the ground line, is made heavy, but gradually tapered, into a fine line in the vicinity of the points a and b, 40. Prob. 13. — To construct the projections of a cylinder whose axis is 2)lciced parallel to the f/round line. PI. 11., Fig. 20. Let the cylinder be 1^ inches long, f of an inch in diameter, its axis f of an inch from the horizontal plane, and ^ an inch from the vertical plane. The principal projections will, of course, be two equal rectan<:^les, geh/ and a'h'c'd\ since all the diameters of the cylinder are equal. The centre lines, cfh' and a5, are made at the same distances from the ground line, that the axis of the cylinder is from the planes of projection (G). The end elevation, knowing its radius, which is equal to half of the diameter ge^ or aV, of the cylinder, may be made by revolving the jirojection of its centre a/j\ only, upon PQP', around P'Q as an axis. 41. Standing with a horizontal cylinder before one, with its axis lying from right to left, and parallel to the ground line, one of its elements is its highest one, that is the highest above the ground, or the horizontal plane ; another is the loicest ,' another, i\\Q foremost^ that is the one nearest to one, and another the hindmost, or the one furthest from one. Ti-ansfering the same terms to the jyrojections of the same elements, by (23) we have ab — a' h' — h" [the three pro- jections of j the higliest element; ab — c'd' — c?'', the loxoest element; (jh — g'h' — /i", the foremost element; and ef — g'Ji' — -/*", the hiiid- most element. In inking, the end elevation, b"f"d", is made heavy at nf"d"p, and tapered into a fine line in the vicinity of n and ^9/ because, by (16) when the observer turns to face the plane PQP', looking at it ill the direction hg (12), the light turns with him. 42. We have now three ways of distingui-;iiing the projections of a horizontal cylinder from those of a square prisin of equal dimensions. First, Ijy medium instead of fully heavy lines on ff and c^d'. Seco)id,hy the lettering of the principal elements, a? just explained. Third, and most clearly, by the circular end ele vation. § in — Projections showing two sides of a Solid Right Angle. 43 A solid right angle is an angle such as that at any corner of 7& 2D. pr.if. ■b" A' 22. B" A' 2S. B" B"'0"- 26! D" E^' F" P'" G d 25. !»• P' f" : 27. PROJECTIONS OF SOLIDS. If, A cube, or a square prism, and is therefore bounded by three plane riglit angles. When two faces of such a body are seen at once they will be seen obliquely, and neither will appear in its true size. Hence only o?ie of the projections of the object will show iioo o( its dimensions in their real size. Hence, we must always malie first, that projection^ whichever it be, which shotcs two dimensious in their real size. 44. Peob. 14. — To construct the plan and ttco elevations of a Vtyrtical jyrism, xoith a, square base ; resting on the horizontal 2^1 ane, and having its vertical faces inclined to the vertical plane of pro- jection. PL n., Figs. 21 — 22. \st. ABCG is the plan, wdiich must be made first (43) and with its sides placed at any convenient angle with the ground line. Id. A'B'D'E' is the vertical projection of that vertical face whose horizontal projection is AB. 3f?. B'C'E'F' is the vertical projection of that face whoso hori- zontal projection is BC. This completes the vertical projection of the visible parts of the prism, when we look at the prism in the direction of the lines CF', &c. 4?/t. Let gh be the horizontal trace of an auxiliary vertical plane of projection, which is perpendicular to both of the principal j^lancs of ])rojcction. In looking perpendicularly towards this jilane, i.e. /n the directions G^, &c., AG and AB are evidently the hoiizontal projections of those vertical faces that would then be visible; and the projecting lines, Gy, Aa, and B^ determine the widths ga and ah of those faces as seen in the new elevation. Now the auxiliary plane gh is not necessarily revolved about its vertical trace (not shown), but may just as well be taken up and transferred to any position where it will coincide with the primitive vertical plane : only its ground line gh must be made to coincide with the principal ground line, as at WYJ' . Hence, making H"D" and D"£" respec- tively equal to ga and ah, and by drawing H"G", &c., the new elevation will be completed. 45. The two elevations — PI. H., Figs. 21, 22 — appear exactly alike, but the faces seen in Fig. 22 are not the same as the equa. ones of Fig, 21. The different projections of the same face maybe distinguished by mnrks. Thus the surfaces marked 1^ are the two elevations of Ihe same face of the prism; the one maiked ^ is visible only on the first elevation, and the one marked x is visible only on the second felevation — Fi^. 22. 16 TBOJECTIONS OF SOLIDS. 46. PI. II., Fig. 23, represents a small quadrangular prism in twc elevations, the axis being horizontal in space, so that the left hand elevation shows the base of the ])rism. In the practical applica- tions of this construction, the centre, s, of the square projection is generally on a given line, not parallel to the sides of the square. Hence tliis construction afibrds occasion for an application of the problem : To draw a square of given size, with its centre on a given line, and its sides 7iot jxirallel to that line. The following solution should be carefully remembered, it being of frequent application. Through the given centre, s, draw a line, L, in any direction, and another, L', also through s, at right angles to L. On each of these lines, lay off each way from 5, half the length of a side of the square. Through the points thus formed, draw lines parallel to the lines L and L' and they will Ibrm the required square whose centre is s. 47. Proi5. 15. — Ih construct the plan and several elevations of a vertical hexagonal prism, lohich rests upon tlie horizontal plane oj projection. PI. II., Figs. 24, 25, 26. Tlie distinction between bodies as seen perpendicularly, or ob- liquely, becomes obscure as we pass from the consideration of bodies whose surfaces are at right angles to each other. Figs. 24 and 25 show a hexagonal prism as much in right prrvjection as such a body c:in be thus shown, but, as in both cases a majority of ita surfaces aie, considered separately, seen obliquely, its construction is given here. In Fig. 24 the hexagonal prism is, as shown by the ]ilan, placed so that two of its vertical faces are parallel to the vertical plane of ])rojection. Observe that where the hexagon is thus placed, three of its faces will be visible, one of them in its real size, viz., r>C, B'C'F'G', and that the extreme width, E'tP, of the eleva- tion, equals the diameter, AD, of the circumscribing circle of the plan. This is therefore tlie loidest possible elevation of this prism. Xolicc, also, that as BC equals half of AD, while AB and CD are equal, and equally inclined to the vertical plane, the elevations, A'F' and G'D', of these latter faces, %oill he equal, and each half as wide as the middle face. This Ihct enables us to construct the elevation of a hexagonal prism situated as here described, without construct- ing the plan, provided we know the width and height of one face of the prism. This last construction should be remembered, i1 being of fi-cquent and convenient application in the drawing of Quts, bolt-heads, tfec, in machine drawing. PBOJECTIONS OF SOLIDS. 11 48. PI. n., Fig. 25 shows the elevation of the same prism on a plane which originally was placed at ib^ and perpendicular to the horizontal plane ; whence it nppears, that if a certain elevation of a hexagonal prism shows three of its faces, and one of them in its full size, anotlier elevation, at right angles to this one, will show but two faces, neither of them in its full size ; the extreme width, I"B", of the second elevation being equal to the diameter of tho uiscribed circle of the plan. This is therefore the narrowest pos- sible elevation of this prism. 49. PI. II., Fig. 26 shoios the elevation of the same prism as it a}ypears when projected upon a vertical plane standing on jb'% and then transferred to the principal vertical plane, at Fig. 26. In this elevation, none of the faces of the prism are seen in their true size. The auxiliary vertical plane, owjb", could have been revolved about that trace, directly back into the horizontal plane, causing the corresponding elevation to Appear in the lines Df?, &c., produced to the left of jV as a ground line. Elevations on auxiliary vertical planes can always be made thus, but it seems more natural to see them side by side above the principal ground line, by transferring the auxiliary planes as heretofore described. 50. Fig. 27 represents two elevations of a hexagonal prism, placed so as to show the base in one elevation, and three of its faces, unequally, in the other. The centre of the elevation which shows the base, may be made in a given line perpendicular to o'fj\ by placing the centre of the circumscribing circle used in con- structing the hexagon, upon such a Ihie. Having constructed this elevation, project its points, a,6, &c., across to the other vertical plane, P', which is in space perpendicular to the plane, P, at the line, o'g'. By representing the elevation on P' as touching o'g\ we indicate that the prism touches the plane, P, just as the elevation in Fig. 24, indicates that the prism there shown rests upon the horizontal plane. 51. Peob. 16. — To construct the plan and two elevations of a pile of blocks of equal widths, but of different lengths, so placed as to form a symmetrical body of uniform width. PI. HI., Figs. 28, 29. Here for example afg is the plan of the lowest step; kbe is that of the middle step, and cdh that of the upper step (43). The auxiliary vertical plane of projection, perpendicular to the horizontal plane at hf"f"\ is made to coincide with the principal vertical plane by direct revolution. The point a'"a"'\ the projec 18 PROJECTIONS OF SOLIDS. tion of an' on the auxiliary vertiral pl;ine, revolrcs in a liorlzonta) arc, of wliich a"'a" is ihe horizontal, aiul a""a*' the vertical pro- jection (31), giving a", a point of the second elevation. Other points of this elevation are found in tlie same way. This figure differs from Figs. 18 and 20, of Plate II., only in presenting more points to be constructed. If the student finds any difficulty witli this example, let him refer to those just mentioned, and to first principles. Example. — Construct an elevation on a plane parallel to af. 52. Peob. 17. — To constnict the vertical projection of a verticai circle, seen ohliquehj. Pi. III., Fig. 30. Let BF be the given projection of the circle. It is required to find its vertical projection, A'B'D'F'. For this purpose, the circle must be first brought into a position parallel to a plane of projec- tion, since we can then make both of its projections, and hence can then take both projections of any point upon it. Let the circle be made parallel to the vertical plane. To do this, it only need be revolved about any vertical axis. In the figure, the axis is the vertical tangent, Y—f'V. After this I'evolution, the projections of the circle are bV — i'c'FV/. Now taking any j)oint on this circle, as aa\ it returns about the axis F — -f'Y' in the horizontal arc ff A — a' A! (31), giving A' by projecting A upon a' h! . Likewise bb' returns in the arc bVt — J'B to BB'; and cc', which is vertically under aa\ returns in the arc cC — c'C to CC. Thus all the points A',B', C, etc., being found and joined, we have A'B'C'II', the required oblique elevation of the vertical circle FB. Examples. — \st. Let the circle be revolved about its vertical diameter IID, or any vertical axis between F and B. Id. About any vertical axis ; in the plane BF produced ; or only parallel to it. Zd. Let the ciicle be made perpendicular to the vertical plane, and oblique to the horizontal plane. 63. Prob. 18. — To construct the projectiom of a cylinder whose convex surface rests on the horizontal plane, and whose axis is in' dined to the vertical plane. PI. III., Fig. 31. As may be learned ivvm Fig. 19, PI. II., the projection of a right cylinder upon any plane to which its axis is parallel, will be a rectangle. Therefore let CSTV, PI. III., Fig. 31, be the plan of the cylinder. Since it nsts uj)on the lioiizontal ])lane, q'u\ in the ground line, is the vertical j)rojection of its line of contact with that PROJECTIONS OP SOI.ros. Ij plane, nnd^'A'is the vertical projection of joA, the highest element of the cylinder, as it is at a height above the ground line, equal to the diameter, TV, of the cylinder. The vertical projection of either base may be found by the last problem. In the figure, the left hafid base is thus found, and the construction, being fully given, needs no further explanation. 54. The vertical projection of the right hand base TV is found somewhat differently. It is revolved about its horizontal diameter, TV — T'V, till parallel to the horizontal plane. It will then appear as a circle, and a line, as n"n^ will show the true height oin above the diameter TV. So, also, o"o will show the true distance of o below TV. Therefore the vertical piojections of the points n and o, will be in the line n — n\ perpendicular to the ground line, and at distances above and below T'V, the vertical projection of TV, equal, respectively, to nn" and oo". Having, in the same manner, found t' and ;;', the vertical projections of two points whose com- mon horizontal projection t — r is assumed, as was n — o, the vertical projection of the base TV can be drawn by the help of the irregu- lar curved ruler. 55. In the execution of this figure, SV is made slightly heavy, and TV fully heavy, and the portion, n'T't\ of the elevation of the right hand base, and the small portion, DV, of the left hand base, are made heavy. Suffice it to say : First. That a part of the convex surface is in the light, while the right hand base is in the dark. * Second. niT't' divides the illuminated half of the convex surface, from the base at the right, which is in the dark ; and D'u' divides the illu- minated left hand base from the visible portion of the darkened htilf of the convex surfoce (18-20). Example. Let the axis of the cylinder be parallel to the vertical plane, only. 66. Prob. 19. To construct the two projections of a right cone with a circular base in the horizontal plane ; and to construe either p>rojection of a line, drawn from the vertex to the circum- ference of the base, having the other projection of the same lint given. JPl. III., Fig. 32. Remark. When the axis of a cone is vertical, perpendicular to the vertical plane, or parallel to the ground line, the cone is sho^vD in right projection as much as such a body can be, but as all the straight lines upon its surface are then inclined to one or both planes of projection, the above problem is inserted here among roblems of oblique projections. 80 PROJECTIONS OP SOLIDS. liCt VB be the radius of the circle, which, with the pciint V, is the horizontal projection of tlie cone. Since the base of the cone rests in tlie horizontal plane of projection, C'B' is its vertical projection. Since the axis of the cone is vertical, V, the vertica' projection of the vertex, must be in a perpendicular to the ground line, through V, and may be assumed, unless the height of tho cone is given. V'C and VB', the extreme elements, as seen in elevation, are parallel to the vertical plane of projection, hence their horizontal projections are CV and BV, parallel to the ground line (8 e). Let it be required to find the horizontal projection of any element, whose vertical projection, V'D', is given. V is the horizontal projection of V, and D', being in the circumference ol the base, is horizontally projected at D, therefore VD is the hori- zontal projection of that element on the front of the cone, whose vertical projection is V'D'. V'D' is also the vertical projection of an element behind VD, on the back of the cone. Having given, VA, the horizontal projection of an element of the cone, let it be required to iind its vertical projection. V is the vertical projec- tion of V, and A, being in the circumference of the base, is verti- cally projected at A'. Therefore V'A' is the required vertical pro- jection of the ju'oposcd line. In inking the fignre, no part of the plan i^ heavy lined, and in the elevation, only the element VB' is slightly heavy. Examples. — 1st. Construct three projections of a cone placed as the cylinder is in Prob. 13. « 2d. As the cylinder is in Prob. 18. 57. Pkob. 20. To construct the projections of a right hexagonnl j^rism / whose axis is oblique to the horizontal plane, and parallel to the vertical plane. PI. III., Figs. 33, 34, \st. Commence by constructing the projections of the same prism as seen when standing vertically, as in Fig. 33. The plan only ia strictly needed, but the elevation may as well be added here, for completeness' sake, and because some use can be made of it. 2nd. Draw J'G", making any convenient angle with the ground line, and set off upon it spaces equal to G'J', J'H', and J'l', from Fig. 33. Zrd. Since the i)rism is a right one, at J", &c., draw perpen- diculars to J"G', make each of them equal to J'C, Fig. 33, and draw F'C, which will be parallel to J'G", and will complete the second elevation. Ath. Let us suppose that the prism was moved from its first PUOJECTIONS, OP SOLIDS. 21 position, Fig. 33, parallel to the vertical plane, and towards tha right, and then inclined, as described, with the corner, C.T', of the base, remaining in the liorizontal plane. It is clear that all points of the new plan, as B"', would be in parallels, as BB"', to the ground line, through the primitive plans, as B, of the same points. It is equally true that the points of the new plan will be in perpen- diculai's to the ground line through the new elevations B", '>'ojections of a regular hexof go7ial 2iy'>'aniid^ whose axis is inclined to the horizontal plane only. PI. III., Figs. 35, 36. \st. Commence, as with the prism in the last problem, by repre- senting the pyramid as having its axis vertical. Ind. Draw a"d'\ equal to a'd\ and divided in the same way. At w", the middle point of a'V?", draw ?i"V" perpendicular to «"«', and make it equal to w'V, v^'hich gives V" the new elevation of the vertex. Join V" with a", i", c", and d'\ and the new elevation will be completed. 3rc?. Supposing the same translation and rotation to occur to the primitive position of the pyramid, that was made in the case of the prism (57, 4?/t), the points of the ^le\v plan, Fig. 36, will be found in a manner similar to that shown in Fig. 34. V" is at the inter- section of VV" with V'V"; c'" is at the intersection cc" with c"c"'; d'" is at the intersection ofdd'" with d"d"\ h'k' is a plane p>erpendicidar to the ground line, MG, and, therefore, to both planes of projection, and wo sed that its two traces, hk' and h'h', are perpendicular to the ground line at ^'. Likewise in PI. I., Fig. 15, kaa't is a plane perpendicu- lar to the ground line MQ, and its traces at and at are perpendi- cular to MQ. That is: if a plane is perpendicular to the ground line, its traces icill also he p)crpendicidar to that line. This is seen in regular projection, in PI. I,, Fig. 10, where PQ ia the horizontal trace, and P'Q, the vertical tiace, of such a plane. In PI. I., Fig. 5, ¥\\fk is ^ plane, parallel to the vertical plane, and it has only a horizontal trace, fk, which is parallel to thi ground line. The same is true for all such planes. Likewise, 'lA ELEME>TARY IXTERSECTIONS. ABa'b' is a horizontal plane. All such planes have only a I'criicai trace, as a'b\ parallel to the growid line. In PI. I., Fig. 2, the plane Yv'd'd is perpendicular only to the vertical plane, and, as the figure shows, tlie horizontal trace only^ as dd\ of such a plane, is perpendicular to the ground line. Also the angle v'd'b\ between the vertical trace, v d\ and the gromid line, is the angle made by the plane with the horizontal plane. In like manner, it can easily be seen that, if a plane be perpen dicular only to the horizontal plane, as in case of a partly open door, its vertical trace only (the edge of the door at the hinges) will 1)6 perpendicidar to the ground line, and the angle between its hori- zontal trace and the ground line, will be tlie angle nude by the plane with the vertical plane of projection. Finally, if a plane is ohllque to both planes of projection, both of its traces will be oblique to the ground line, and at the same point. Thus, PI. I., Fig. 6, may represent such a plane, having LF for its horizontal, and L'F for its vertical trace. All the principles just stated can be simply illustrated by taking a book, half open, for the planes of projection, and either of the triangles for the given movable ]ilane ; and when clearly under- stood, the following problems can also be easily comprehended. Pkob. 24. — To find the curve of intersection of a cylinder with a plane. PI. R'., Fig. 1. Let the cylinder, ADBG — A'B", be vertical, and the cutting plane, PQP', be per})endicular only to the vertical plane. All points in such a plane must have their vertical projections (that is, must be vertically projected) in the vertical trace, QP', of the plane, but the required curve must also be embraced by the visible limits, A'A" and B'B", of the cylinder. Hence, a'b' is the verti- cal projection of this curve. ' Again, as the cylinder is vertical, all points on its convex surface must be horizontally projected in ADBG. Hence, this circle is the horizontal projection of the required cuive. Prob. 25. — To revolve the curve found in the last p>roblem, so as to show its true size. "When a plane revolves about any line in it as an axis, every point of it, not in the axis, moves in a circular arc, whose radii are all jjcrpendicular to the axis. Tlie representation of the revolutioc is mucli simplified by taking the axis in, parallel to, or j^erpendicu lar to, a plane of projection (31). pi-.in. K^ I c ELEMENTARY INTERSECTIONS. 25 Let AB — a'J>\ tlie longer axis of the curve, and which is parallel to the vertical plane of projection, be taken as the axis of revolu- tion. The curve may then be revolved till parallel to that plane, when its real size and form will appear. Then, nt c', <:?', &c., the vertical projections of C and H,_D and G, &c., draw perpendicu- Jars, as c"A", to a'h\ and make c'c" ~c'h" —nC Proceed likewis at d\ &c., since the lines, as nC, being parallel to the horizonta, plane, are seen in their true size in horizontal projection ; and join the points a'h"(j'\ tfcc., which will give the required true form and size of the curve of intersection before found. Example.— This curve is an oval, called an ellipse. Its true size could have been shown by revolving its original position about DG as an axis, till parallel to the horizontal plane. The student may add this construction to the plate. Prob. 2G. — To dcvclope the portion of the cylinder, PI. IV., Fig. 1, below the cutting plane^ PQP'. The convex surface of a cylinder is wholly composed of straight lines, called elements, parallel to its axis. The convex surface of a cone is composed of similar elements, all of which meet at its vor- tex. Hence, each of these surfaces can evidently be rolled upon a plane, till the element first placed in contact with the plane, returns into it again. The figure, thus rolled over on the plane, is called the development of the given convex surface, and its area equals the area of that surface. Suppose the cylinder to be hollow as if made of tin, and to be cut open along the element B'5'. Then suppose the element A'a' to be placed on the paper, as at A'a', Fig. 2, and let each half be rolled out upon the paper. The part ADB will appear to the left of A'a', and the part AGB, to the right. The base being a circle, perpendicular to the elements, will develope into a straight line B B", Fig. 2, found by making A'c = AC, Fig. 1, cd—CT>, Fig. 1, &c., and A7i=AH, Fig. 1, &c. B'B" may also, for convenience, be A'B', Fig. 1, produced. Then the parallels to A'«', through c rf, &c., will be developments of elements standing on C, D, &c^ Fig. 1, and by projecting over upon them, a' at a', c' at c' and h' ; (V at d' and g\ B' at b' and b'\ and joining the points, the figure B'B"6"a'i', will be the required development of the cylinder. ^ Remark. — If, now, a flat sheet of metal be cut to the pattern just found, it will roll up into a cylinder, cut olF obliquely as by the plane PQP'. By making the angle P'QA' of any desired size, the corresponding flat pattern can be made as now explained. 26 ELEMENT AKT INTEKSECT10N8. Prob. 27. — To find the intersection of a vertical cone, icith a plane, perpendicular to the vertical 2)lane of projection. PL IV., Fig. 3. Let V— ADBC he the plan, and A'B'V the elevation of tbe cone, and PQ and P'Q the traces of the given cutting plane; whose horizontal trace, PQ, shows it (G2) to be perpendicular to the ver- tical plane. For the reasons given in Problem 25, ci'b' will be the vertical projection of the required curve. The convex surface of the cone not being vertical, the horizontal projection of the inter- section will be a curve, whicli must be found by constructing it* points as follows. First. The method by chmients. Any line, as VE', is the verti- cal projection of two elements whose horizontal projections are VE andVF (Prob. 19). Therefore e', where it crosses the vertical projection, a'b\ of the intersection, is the vertical projection of t\vo points of the required curve. Their horizontal prp-ections, e ami/', are found by projecting e' down upon VE and VF. Other points can be found in the same manner, except d and //, since the ]»ro- jectiug line d'd coincides with the elements VD and VG. The horizontal projections of a' and b' are a and b. Second. The method by circular sections. Let JNFN' be the ver- tical trace of a horizontal auxiliary plane through d' . This plane will cut from the cone the circle in'n' — dnxg, on which d' can be projected at d and o cones, a plane must simply contain both vertices. Examples. — \st. Thus, in Fig. 10, all planes cutting elements, both from cone W, and cone AA', will contain the line VAB, hence their traces on the horizontal plane will merely pass through B. Thus the plane BD cuts from the cone, V, the elements Y'a' — Va, and W — Yc ; and from the cone, A, the elements A'D' — Ad, and A'd' — AD. The student can complete the solution, the remainder of which is very similar to the two preceding. 2d. To find the intersection of +1 sphere and cone, PI. IV., Fig. 1 1 , auxiliary planes may most conveniently be placed in two ways. First, horizontally. Then each will cut a circle fiom the sj)here, Efnd one from the cone ; whose horizontal projections will be circles, and whose intersections will be points of the intersection of the cone and sphere. Second, vertically. Then each plane must con- tain the axis of the cone, from which it will cut two elements. It will also cut the sphere in a circle, and by revolving this plane about the axis of the cone till parallel to the vertical plane, as in Prob. 17, the intersection of the circle with the revolved elements, see Prob. 27, may be noted, and then revolved back to their true position. The student can readily make the constructicm, after due familiarity with preceding problems has made the apprehension of the present article easy. Pkob. 31. — To find the intersection of a vertical hexagonal prism with a sphere, whose centre is in the axis of the prism, PI. IV., Fig. 8. Let O — ABC be part of the sphere, and DGIIK the prism, showing one lace in its real size, and therefore requiring no ])lan (47). Draw dg parallel to AC, and the arc eA/'with O as a centre, and through e and/. Tiiis arc is the real size of the intersection of tlic middle face of the prism with the surface of the sphere. AU the faces, being equal, have circular tops, equal to ehf ; but, being ELEMEXTARY INTERSECTION'S. 31 seen obliquely, they would be really elliplical in ijrojectioii. It h oi'dinarily sufficient, however, to represent them by circular arcs, tangent to hn^ the horizontal tangent at 4, and containing tiic points d and e, and ^Z" and g, as shown. Hemark. — The heavy lines here, show the ynX of the prism within the sphere, as a spherical topped bolt h«.-ij. To make Df?=EF, draw Od at 45° with AC, to locate dg. To make the sphei'ical top flatter, for the same base DG, take a iyrgm' sphere, and a plane above its centre for the base of the prism. Prob. 32. — To construct the intersection of a vertical cone tcith a vert iced hexagonal prism j both having the same axis. PI. IV., Fig. 9. Let YAB be the cone, and CFGII, the prism, whose elevation can be made without a plan (48), since one face is seen in its real size. The semicircle on cf is evidently equal to that of the cir- cumscribing circle of the base of the prism, and ct is the chord of two thirds of it. Then half of ct, laid ofi' ou either side of O, the middle of CF, as at 0«, will give ?ijo.the pj-ojection of the middle face EDc? after turning the prism 90° about its axis. This done, np Avill be the height, above the base, of the highe.'it point at which this and all the faces will cut the cone. A vertical plane, not through the vertex of a cone, cuts it in the curve, or " conic sec- tion," called a hyperbola. The vertical edges of the j^rism cut the cone at the height F/, hence, drawing the curves, as dse., sharply curved as at s, and nearly straight near d and e, we shah have a sufficiently exact construction of the required intersection. Remark. — ^The heavy lines represent the part of the prism within the cone, finished as a hexagonal head to an iron " bolt," such as is often seen in machinery. The horizontal top, hg., of the head, may be drawn by bisecting pr at g. To make Cc=ED, as is usual in practice, simply draw Oc at an angle of 45° a\ itli AB, to locate cf By making VAB = 30° perhaps the best proportions will be found. (34. In the subsequent ajiplications of projections hi practical problems, the ground line is very generally omitted; since a know- ledge of the object represented makes it evident, on invspection, which are the plans, and which the elevations. General Examples. The careful study of the detailed explanations of the preced- ing problems, will enable the student to solve any of the follow- ing additional examples. 32 ELEMENTARY INTERSECTIONS. Ex. 1. — In Prob. 24, substitute for the cylinder any pr dn, find the intersection with the plane PQP', and, by Prob. 25, -rind the true form and size of this intersection. Ex. 2. — In Prob. 27, substitute for the cone any pyramid. Vary this and Ex. 1 by dillcrcnt positions of PQP', cutting both hoses in tlx. 1. Ex. 3.— In Ex. 2, find, by Prob. 25 or by Prob. 27, Ex. 2d, the true form and size of the intersection and, by Prob. 28, the development of tlie convex surface of the pyramid. Ex. i. — In Probs. 22, 23, substitute for the pyramid a cone whose convex surface, rolling on H (23), shall be shown, first, with its axis parallel to V; and, second, with its axis oblique to V. Ex. 5. — In Ex. 4, find the intersection of the cone with any plane parallel to II; and show the curve on both positions of the cone. Ex. C. — In Ex. 5, let the cutting plane be vertical but ob- lique to V, and not containing the cone's vertex. Ex. 7. — In Prob. 20, let the horizontal cylinder be the larger one, and, after finding its intersection with the vertical one, de- velope it. Ex. 8. In Probs. 22, 23, substitute for the pyramid a cylinder. Ex. 9. — In Probs. 22, 23, substitute for the pyramid a hollow hemisphere. Ex. 10. — In Prob. 29, let the axis of the horizontal cylinder be inclined first to II only, and then to both H and V. Ex. 11. — In Probs. 22, 23, let the pyramid, when in the posi- tion shown in Fig. 3G (but more inclined), rest its edge V"a'" against an upper edge of a cube standing on 11. Ex. 12. — Find the four following sections of a sphere: one by a horizontal plane, one by a plane parallel to V, one by a vertical plane ol)liquc to V, and one by a plane perpendicular to V and oblique to n. Ex. 13. — Cut a regular hexagon from a cube. Ex. 14. — Cut a rhombus and an isosceles triangle from the square prism. PI. II., Fig. 17. Ex. 15. — Construct the ])rojections of the. cylinder, PI. IV., Fig. 1, after rotating it and PQP', together, 45° on its axis. Ex. IC— Sul)stitute for the blocks, PI. III., Figs. 28, 29, a pile of thin cylinders of unequal diameters, but with a common axis placed obliquely to V. PL.iy DIVISION SECOND. DETAILS OF MASONRY, WOOD, AND METAL CONSTRUCTIONS. CHAPTER I. CONSTRUCnOXS IN MASONRY. § 1. — General Definitions and Principles aj^plicable both to JBrick and &tone-work. 65. A horizontal layer of brick, or stone, is called a course. The seam between two courses is called a coursing-joint. The seam between two stones or bricks of the same course, is a vertical or heading-joint. The vertical joints in any course sliould abut against the solid stone or brick of the next courses above and below. This arrangement is called breaking joints. The particular arrangement of the pieces in a wall is called its bond. As far as possible, stones and bricks should be laid with their broadest surfaces horizontal. Bricks or stones, whose length is in the direction of the length of a wall, are called stretchers. Those whose length is in the direction of the thickness of a wall, are called headers. § 11.— Brick Work. 06. If it is remembered that bricks used in building have, usually, n invariable size, 8" x 4" X 2" (the accents indicate inches), and bat in all ordinary cases they are used Avhole, it will be seen that brick walls can only be of certain thicknesses, while, in the use of Btone, the wall can be made of any thickness. Thus, to begin with the thinnest house wall which ever 0C(;ur8, viz. one whose thickness equals the lengtii of a brick, or 8 inches ; the next size, disregarding for the present the thickness of mortar^ would be the length of a brick added to the width of one, or e4ua] to the width of three bricks, making 12 inches, a thickness empK/yed in the ]>artition avails and upper stories of first class houses, oi tbo COXSTRL'CTIOXS IX MASOXRY, outside walls of small houses. Then, a wall whose thickness ia equal to the length of two bricks or the wiilth of four, making 16 inches, a thickness proper for the outside walls of the lower stories of first class liouses ; and lastly, a wall whose thickness equals the length of two bricks added to the width of one; or, equals tlic width of five l)ricks, or 20 inches, a thickness proper for the base- ment walls of first class houses, for the lower stories of few-storied, heavy manufactory buildings, &c. G7. In the common bond, generally used in this country, it may be observed — a. Tliat in heavy buildings a common rule appears to be, to have one row of headers in every six or eight rows of bricks or courses, i.e. five or seven rows of stretchers between each two successive rows of headers ; and, b. That in the 12 and 20 inch walls there may conveniently be a row of headers in the back of the wall, intermediate between the rows of headers in the face of the wall, while in the 8 inch and 16 inch walls, the single row of headers in the former case, and the double row of headers in the latter, would take up the whole thick- ness of the wall, and there might be no intermediate rows of headers. c. The separate rows, making up the thickness of the wall in anj one layer of stretchers, are made to break joints in a liorizontal direction, by inserting in every second row a half brick at the end of the wall. 68. Calling the preceding arrangements connnon bonds, let us next consider the bonds used in the strongest engineering works which aie executed in brick. These are the £Jnglish bond and. the Fleynish bond. The J^nfjUsh Bond. — In this form of bond, every second course, as seen in the face of the wall, is composed wholly of headers, the intermediate courses being composed entirely of stretchers. Hence, in any practical case, we have given the thickness of the wall and the arrangement of the bricks in tlie front row of each course, and are required to fill out the thickness of the wall to tlie best advantage, llic Flemish Bond. — In this bond, each single course consistfc of alternate headers and stretchers. The centre of a header, in any course, is over the centre of a stretcher in the course next al)0vc or below. The face of the wall being thus designed, it remains, as before, to fill out its thickness suitably. 69. Example 1. To represent an Eight Inch Wall in Eng-- lish Bond. Let each course of stretchers consist of two rows, sidfl 3 CONSTRUCTIONS IN MASONRY. 35 bv side, the bricks in wliicli, break joints with each other 1, ori- zoiitally. Tlien the joints in the courses of headers, will be distant half the width of a brick from the vertical joints in the adjacent courses of stretchers, as may be at once seen on constructing a diagram. VO. Ex. 2. To represent a Twelve Inch Wall in English Bond. Sec PL V., Fig. 37. In the elevation, four courses are §hoAvn. The upper plan represents the topmost course, and in the lower plan, the second, course from the top is shown. Tlie courses having stretchers in the foce of the wall, could not be filled out by two additional rows of stretchers, as such an arrangement would cause an unbroken joint along the line, «J, throughout the Avhole height of the wall — since the courses having headers in the face, must be filled out with a single row of stretchers, in order to make a twelve inch wall, as shown in the lower plan. In order to allow the headers of any couise to break joints with the stretchers of the same coui'se, the row of headers may be filled out by a brick, and a half brick — split lengthwise — as in the upper plan ; or by two three-quarters of bricks, as seen in the lower plan. 71. Ex. 3. To represent a Sixteen Inch Wall in English Bond. The simplest plan, in which the joints would overlap pro- perly, seems to be, to have every second course composed entirely of lieaders, breaking joints horizontally, and to have the intermediate courses composed of a single row of stretchers in the front and back, with a row of headers in the middle, which would break joints with the headers of the first named courses. If the stretcher courses were composed of nothing but stretchers, there would evidently be an unbroken joint in the middle of the wall extending through its who-le height. 72. Ex. 4. To represent an Eight Inch Wall in Flemish Bond. PI. v., Fig. 38, shows an elevation of four courses, and the plans of two consecutive courses. The general arrangement of both courses is the same, only a brick, as AA', in one of them, is set sis inches to one side of the corresponding brick, B, of the next course — measuring from centre to centre. 73. Ex. 5. To represent a Twelve Inch Wall in Flemish Bond. PI. v., Fig. 39, is arranged in general like the preceding figures, with an elevation, and two plans. One course being arranged as indicated by the lower plan, the next course may be made up in two ways, as shown in the upper plan, where the grouping shown at the right, obviates the use of half bricks in every second course. 36 coxsTRrcrio.vs in masoxiiy. riiere seems to be no other simple way of combining ilie Inicks ir this wall so as to avoid the use of half bricks, without lea\iny ujicn spaces in some parts of the courses. 74. Kx. 0. To represent a Sixteen Inch Wall in Flemish Bond. PI. v., Fig. 40. The ligure explains itself sutHciently. IJrieks may not only be split crosswise and lengthwise, but even thicknesswise, or so as to give a piece 8x4x1 niches in size. Alihough, as has been remarked, whole bricks of the usual dhuen- sions can only foi-m walls of certain sizes, yet, by inserting frag- ments, of jiroper sizes, any length of wall, as between windows and doors, or width of jiilasters or panels, may be, and often is, con structed. By a similar aititice, and also by a skilful disposition of the mortar in the vertical joints, tai)ering structures, as tall cliim- neys, are formed. § UL— Stone Work. 75. The following examples will exhibit the leading varieties of arrangement of stones in walls. Example 1. Regular Bond in Dressed Stone. PI. YL, Fig. 41. Here the stones are laid in regular courses, and so that the middle of a stone in one course, abuts against a vertical joint in the course above and the course below. In the present example, those stones whose ends appear in the front face of the wall, seen in elevation, take up the whole thickness of the wall as seen in plan. The right hand end of the wall is represented as broken down in all the figures of this plate. Broken stone is represented by a smooth broken line, and the under edge of the outhanging part of any stone, as at w, is made heavy. 7G. Ex. 2. Irregular Rectangular Bond. PI. YL, Fig. 4:.\ In this example, each stone has a rectangular face in the front of the wall. These faces are, however, rectangles of various sizes and proportions, but arranged with their longest edges horizontal, and also so as to break joints. 77. That horizontal line of the ))lan which is nearest to the lower border of the plate, is evidently the plan of the top line of the elO' vation, hence all the extremities, as a', b\ «fec., of vertical joints, found on that line, must be horizontally projected as at a and ^, in the horizontal jji-djection of" (he same line. 78. Ex. 3. Rubble Walls. The remaining figures of PI. YL, represent various forms of " rubble " wall. Fig. 43 represents a wall of broken boulders, or loose stones of all sizes, such as are found abundantly in New England. Since, of course, such stones PLV f J" 1 — 1 ' 1 1 1 1 , ', 1 \ ' ,) 1 1 1 1 II 1 1 1 1 1 1 1 1 1 1 11 I I I T—1 ih' 'i i' ' i i'=^=^^ I I A- I II L — 1 1 1 1 1 1 1 1 1 ' 1 1 1 1) — '' i' 'i 1 ! !i 1 ' !i/ -^ — ,W-n^ 1 — — — 1 i' 'i i' 'i i' 'i i' 'i' i ' ' i i ' ' 1 1 ' ' . 1 ' ' i' 1 — — — 1 — — 1 — — CONSTRUCTIONS IX ilASOXRT, 37 would not fit together exactly, the "chinks" between them are filled with small fragments, as shown in the figure. Still smaller irregularities in the joints, which are not thus filled, are repre- sented after tinting by heavy strokes in inking. Fig. 44 repre- sents the plan and elevation of a ruljble wall made of slate ; hence, in the plan, the stones appear broad, and in the elevation, long and thin, with chink stones of similar shape. Fig. 45 represents a rubble wall, built in regular courses, which gives a i^leasing effect, particularly if the Avail have cut stone corners, of eqiud thickness with the rubble courses. Ex. 4. A Stone Box-culvert. PI. YL, Figs. C, D, E. Scale tV of an inch to 1 ft. Fig. C is a longitudinal section; D, jjart of an end elevation; and E, jjart of a transverse section. Waste water flowing over the dam del', into the well a between the wing- walls a and b' and the head h, escapes by the culvert cc — c", which is strengthened by an intermediate cross- wall '»i'', occurring in the course of its length. The masonry rests on a flooring of 2-inch planks lying trans- versely on longitiidinal sills, which, in turn, rest on transverse sills. Thus a firm continuous bearing is formed which prevents un- equal settling of the masonry, while washing out underneath is provided against by sheet piling partly shown at /;, j/, p" , and extending six feet into the ground. The student should construct this example on a larger scale, from 4 to G sixteenths of an inch to a foot; and should add a jilan, or a horizontal section, both of which may easily be con- structed from the data afforded by the given figures. 79. Execution. — Plate A^I. may be, \st, pencilled; 2f/, inked in fine lines; Zd, tinted. The rubble walls, having coarser lines for the joints, may better be tinted, before lining the joints in ink. Also, in case of the rubble walls, sudden heavy strokes may be made occasionally iii the joints, to indicate slight irregularities in their thickness, as has already been mentioned. The right hand and lower side of any stone, not joining an- other stone on those sides, is inked heavy, in elevation, and on the plans as usual. The left-hand lines of Figs. 43 and 44 are 38 COXSTRUCTIONS IX MASOXRY. tangent at various points to a vertical straight line, walls, such as are represented in those figures, being made vertical, at the fin- ished end, by a plumb line, against "uiiich the stones rest. The shaded elevations on PI. YI. may serve as guides to the depth of color to be used in tinting stone work. The tint actu- ally to be used, should be very light, and should consist of gi'ay, or a mixture of Ijlack and white, tinged with Prussian blue, to give a blue gray, and carmine also if a purplish gi-ay is desired. Remarks. — a. A scale may be used, or not, in making this plate. The number of stones shown m the width of the plans, shows that the walls are quite thick. I. liubble walls, not of slate, are, strictly, of two kinds: first, those formed of small boulders, used whole, or nearly so ; and second, those built of broken rock. Each should show the broad- est surfaces in ^j)/a«. c. After tinting, add pen-strokes, called hatchings, to repre- sent the character of the surface; as in Fig. A., for rough, or un- dressed stone; in waving rows from left to right of short, fine, equal, vertical strokes, for smooth stone; and in a mixture of numerous fine dots and small angular marks, for a finely picked- up surface. (See actual stone work, good drafting cppies, and my "Drafting Instruments and Operations.") PJD.VT- c^ "7 (D) 7~n" .JM I , - r :J I , I fc) IE) ■^ •_r CHAPTER II. OONSTRTTCriONS IN WOOD § I. — General Memarks. 80. Two or more beams may be framed together, so as to make any angle with each other, from 0° to 180° ; and so tliat the plane of two united pieces may be vertical, horizontal, or oblique. 81. To make the present graphical study of framings more ful'iy rational, it may here he added, that pieces may be framed with reference to resisting forces which would act to separate them in the direction of any one of the three dimensions of each. Follow- ing out the classification in the preceding article, let us presently l)roceed to notice several examples, some mainly by general de- scription of their material construction and actiofi, and some by a complete description of their graphical construction and execution^ also, 82. Two other points, however, may here be mentioned. First: A pair of pieces may be immediately framed into each other, or tliey may be intermediately framed by "bolts," "keys," &c., or buth modes may be, and often are, combined. Second: Two com- binations of timbers which are alike in general appearance, may be adapted, the one to resist extension, and the other, compression, and may have slight corresponding differences of construction, 83. JSfote. — For the benefit of those who may not have had access to the subject, the following brief explanation of scales, &c,, is here inserted, (See my "Drafting Instruments and Operations.") Drawings, showing the pieces as taken apart so as to show the mode of union of the pieces represented, are called '"'• Details.''^ Sections^ are the surfaces exposed by cutting a body by planes, and, strictly, are in the planes of section. Sectional elevations^ or plans, show the parts both ««, and be- yond, the planes of section. Drawings are made in plan, side and end elevations, sections and details, or in as few of these as will show clearly all parts of the obj'ict represented. 84. In respect to the instrumental operations, these drawings are 40 COXSTRUCTIOXS IN -WOOD. STijiposed to 1)0 " lunde to sealc^," from mcasurcment.s of models, or fivMii assii.ned incnsui'eiiuMits. It will, therefore, Lc necessary, I)cfore beginning the drawings, to explain the maimer of sketching the oliject, and of taking and recording its measurements. 85. In sketching the object, make the sketches in the same way in which th.ey are to be drawn, i.e. in 2^la>^ <^''>'d elecation, and not in pers]iective, and make enough of them to contain all the mea surements, i.e. to show all parts of the object. In measuring, take measurements of all the parts which are to be shown ; and not merely of individual parts alone, but such con- necting measurements as will locate one pait with reference lo another. 86. The usual mode of recording the measurements, is, to indi- cate, by arrow heads, the extremities of the line of which the figures between the arrow heads show the length. 87. For brevity, an accent (') denotes feet, and two accents (") denote inches. The dimensions of small rectangular pieces are indicated as in PI. VII., Fig. 50, and those of small circular pieces, as in Fig. 51. 88. In the case of a model of an ordinary house framing, such as it is useful to have in the drawing room, and in wdiich the sill is represented by a piece whose section is about 2^ inches by 3 inches, a scale of one inch to six inches is convenient. Let us then describe this scale, which may also be called a scale of two inches to the foot. The same scale may also be expressed as a scale of one foot to two inches, meaning that one foot on the object is represented hy two inches on the drawing ; also, as a scale of ^, thus, a foot being equal to twelve inches, 12 inches on the object is re])resented by two inclies on the drawing ; therefore, one inch on the drawing represents six inches on the object, or, each line of the drawing is \ of the same line, as seen upon the object ; each line^ for we know from Geometry that surfaces arc to each other as the squares of their homologous dimensions, so that if the length of the lines of tlie drawing is one-sixth of the length of the same lines on the object, the area of the drawing would be one thirty-sixth of the area of the object, but the scale always refei's to the relative lengths of tlie lines only. 89. In constructmg the scale above mentioned, upon the stretched drawing paper, .see PI. VI., Fig. B '[St. Set off ujioa a tine straight pencil line, two inches, say thro* times, mal ing four points of division. CONSTRUCTIONS IN WOOO. 41 2d. Xumber the left hand one of tliese points, 12, the next, 0, , the next, 1, the next, 2, &c., for additional points. Sc7: Since each of tliese spaces represents a foot, if any one of I hem, as tlie left hand one, he divided into twelve equal parts, those ])arts will be representative inclies. Let the left hand space, from (12) to (0) be thus divided, by fine vertical dashes, into twelve equal parts, making the three, six, and nine inch marks longei", so as to catch the eye, when using the scale. 4th. As some of the dimensions of the object to be drawn are measured to quarter inches, divide the first and sixth of the inches, already found, into quarters ; dividing two of them, so that each may be a check upon the other, and so that there need be no con tinual use of one of them, so as to wear out the scale. 5th. When complete, the scale may be inked ; the length of it in fine parallel lines about ^V of an inch apart. 90. It is now to be remarked that these spaces are always to be called by the names of the dimensions they represent, and not according to their actual sizes, i. e. the space from 1 to 2 repre- sents a foot upon the object, and is called a foot; so each twelfth of the foot from 12 to is called an inch, since it represents an inch on the object; and so of the quarter inches. 91. Next, is to be noticed the directions in which the feet and inches are to be estimated. The feet are estimated from the zero point towards the right, and the inches from the same point towards the left. Thus, to take off 2' — 5" from the scale, place one leg of the dividers at 2, and extend the other to the fifth inch mark beyond 0, to the left ; or, if the scale Avere constructed on the edge of a piece of card-board, the scale being laid upon the paper, and with its graduated edge against the indefinite straight line on which the given measurement is to be laid off, place the 2' or the 5" mark, at that point on the line, from which the measurement is to be laid oft', according as the given distance is to be to the left or right of the given point, and then with a needle point mark the 5" point or the 2' point, respectively, which will, with the given point, include the required distance. 92. Other scales, constructed and divided as above described, only smaller, are found on the ivory scale, marked 30, &c., mean- ing 30 lect to the inch when the tenths at the left are taken as feet ; and meaning til ree feet to the inch when tlie larger s|)aces — three of which make an inch — are called feet, and the twelfths of the left hand space, inches. Intermediate scales are marked 13 CONSTRUCTIONS IN -VTOOIi. 85, etc. Thus, on tlie scale marked 45, four and a half of the hirtrer spaces make one inch, and the scale is therefore one of four and a lialf feet to one inch, wlien these spaces rei)resent feet ; and of forty-five feet to one inch, when the tenths represent feet. In like manner the other scales may be exjjlained. So, on the other side of the ivory, are found scales marked |, Ac, meaning scales of -| inch to one foot, or ten feet, accoi'ding as tlu! whole left hand space, or its tenth, is assumed as lepreseniing one foot. Kote that | of an inch to a foot is | of a foot to the inch, I of an inch to te7i feet, is 16 feet to an inch, &c. 94. Of the immense superiority of drawing by these scales, over drawing without them, it is needless to say much : without them, we should have to go through a mental calculation to find the length of every line of the drawing. Thus, for the piece which is two and a half inches high, and drawn to a scale of two inches to a foot, we should say — 2\ inches = j| of a foot^j^ ^^ ^ ^ooX.. One foot on the object =two inches on the drawing, then ^'y of a foot on the object=2y of 2 inches=32_ = J^ of an inch, and gV of a foot (=2|- inches) =/y of 2 inches^/j of an inch. A similar tedious calculation would have to be gone through with for every dimension of the object, while, by the use of scales, like that already described, we take off the same number of the feet and inches of the scale, that there are of real feet and inches in any given line of the object. § II. — Pairs of Timbers wJiose axes make angles qfO° loith each other. The student should be required to vary all of the remaining con- structions in this Division, in one or more of the following ways. First, by a change of scale ; Second, by choosing other examplet from models or otherwise, but of similar character; or, Third, by a cliange in the number and arrangement of the iwojectioris em- ployed in representing the following examples. 95. Example 1. A Compound Beam bolted. PL VII., Fig. 46. Medtanical Construction. The figure represents one beam as laid on top of another. Thus Bituate^ arc represented, and from which the rest can be under StOC'd,] 40 CONSTRUCTIONS IX WOOD. Wlien the mortise is surrounded on tliree or 0:1 two side«, par- ticularly in the latter case, the framed pieces are said to b«3 "/i«^«e(?" together, more especially in case they are of equal thick- ness, and have half tlie thickness of each cut away, asat PIA^II, , Fig. 52. 105. Example 1. Two examples of a Floor Joist and Sill. (From a Model.) PL YIl., Fig. 53. Mechanical Co7istruction. A — A' is one sill, B — B' another. CC is a floor tnnber framed into both of them. At tlie left hand end, it is merely " dropped in," with a tenon ; at the right hand end, it is framed in, with a tenon and " tusk," e. At the right end, therefore, it cannot be lifted out, but must be drawn out of the mortise. The tusk, e, gives as great a thickness to be broken off, at the insertion into the sill, and as much horizontal bearing surface, as if it extended to the full depth of the tenon, ^, above it, while less of the sill is cut away. Thus, labor and the strength of the sill, are saved. 106. Ghxqyldcal Construction. — \st. Draw ah. 2d. On ab con- struct the elevation of the sills, each 2^ inches by 3 inches. 3d. ]\Iake the two fragments of floor timber with their upper surfaces flush with the tops of the sills, and 2 inches deep. 4th. The mor- tise in A', is f of an inch in length, by 1 inch in vertical depth. 5th. Divide cd into four equal parts, of which the tenon and tusk occupy the second and third. The tenon, ^, is ^ of an inch long, and the tusk, e, 5 of an inch long. Let the scale of } be used. 107. Execution. — The sills, appearing as sections in elevation, are shaded. In all figures like this, dotted lines of construction should be freely used to assist in " reading the drawing," i.e. in com- prehending, from the drawing, the construction of the thing repre- sented. 108. Ex. 2. Example of a "Mortise and Tenon," and of "Halving." (From a Model.) I'l. VJl., Fig. 54. 3fecha)iicitl Construction. In this case, the tenon, AA', extends entirely through the piece, CC, into which it is fi-araed. B and C aro halved together, by a mortise in each, A\hose depth equals half the thickness of B, as shown at B" and C", and by the dotted line, ab. Graphical Construct io7i. — Make, l5^, the elevation, A' ; 2d, the plan ; 3c?, the details. B" is an elevation of B as seen when looking in the direction, BA. C" is an elevation of the left hand portion of CC, showing the mortise into which B is halved. The dimensions may be assumed, or found by a scale, as noticed below. 109. Execution. — The invisible parts of the framing, as the halv COXSTllUCTIOXS IN WOOD. 47 ing, as seen at ah in elevation, are shown in doited lines. The brace and the dotted lines of construction serve to show what separate figures are comprehended under tlie general number (54) of the diagram. Tlie scale is }. From this the dimensions of the j)ieces can be found on a scale. 110. Ex. 3. A Mortise and Tenon as seen in tv7o sills and a post. Use of broken planes of section. (From a Model.) PI. VII., Fig. 55. Jleckanical Construction. — The sills, being liable to be drawn apart, are pinned at a. The post, B13', is kept in its mortise, bb", by its own weight ; m is the mortise in which a vertical wall joist rests. It is sliown again in section near ni'. 111. Graphical Construction. — The plan, two elevations, and a broken section, show all parts fully. The assemblage is supposed to bo cut, as sliown in the plan by the broken line AA'A"A"', and is shown, thus cut, in the shaded figure, A'A'A"'m'. The scale, which is the same as in Fig. 53, indicates the measurements. At B", is the side elevation of the model as seen in looking in the direction A' A. In Fig. 55 a, A's obviously equals iVs, as seen in the plan. 112. Execution. — In the shaded elevation, Fig. 55a, the cross-sec- tion, A'A'", is lined as usual. The longitudinal sections are shaded by longitudinal shade lines. The plan of the broken upper end of the post, B, is filled with arrow heads, as a specimen of a way sometimes convenient, of showing an end view of a broken end. Sometimes, though it renders the execution more tedious, narrow blank spaces are left on shaded ends, opposite to the heavy lines, 8o as to indicate more plainly the situation of the illuminated ed^-ea (100). The shading to the left of A', Fig. 55a, should be placed so as to distinguish its surface from that to the right of A'. 113. Ex. 4. A Mortise and Tenon, as seen in timbers so framed that the axis of one shall, -when produced, be a tliagonal diameter of the other. PL VII., Fig. 56. Mechani- cal Construction. — In this case the end of the inserted timber is not square, and in the receiving timber there is, besides the moitisc, a tetraedron cut out of the body of that timber. 114. Graphical Construction. — D is the plan, D' the side eleva- tion, and D" the end elevation of the piece bearing the tenon. F' and F are an elevation and plan of the piece containing the mor- tise. Observe that the middle line of D, and of D', is an axis of symmetry, and that the oblique right hand edges of D and D' an parallel to the correspunding sides of the incision in F'. 48 CONSTRUCTIONS IN WOOD. § IV. — Miscellaneous Combinations. 115. Example 1. Dowelling-. (From ;i Model.) Pl.VIT., Fig.57, Mechanical Construction. — DoxoelUng is a mode of fastening by pins, projecting usually from an edge of one piece into correspond- ing cavities in another piece, as seen in the fastening of the pails of the head of a water tight cask. The mode of fiistening, how- ever, rather than the relative position of the pieces, gives the name to this mode of union. The example shown in PI. VTT. . Fig. 57, represents the braces of a roof flaming as dowelled together Avith oak pins. 1 1 6. Graphical Construction. — This figure is, as its dimensions indicate, drawn from a model. The scale is one-third of an inch to a." inch. t. Draw acJ, witli its edges making any angle with the imagi- nary ground line — not drawn. 2d. At the middle of this piece, draw the pin or dowel., pp., ^ of an inch in diameter, and projecting f of an inch on each side of the piece, ach. This pin hides another, supposed to be behind it. ^d. The pieces, d and d"., are each 2^ inches by 1 inch, and are shown as if just drawn oif from the dowels, but in their true direc- tion, i.e. at right angles to acb. Ath. The inner end of f?is shown at f?', showing the two holes, 1^ inches apart, into which the doicels fit. Execution. — The end view is lined as usual, leaving the dowel holes blank. 117. Ex. 2. A dovetailed Mortise and Tenon. PI. VII., Fig. 58. 3fechanical Construction. — This figure sliows a species of joining called dovetailing. Here the mortise increases in width as it becomes deeper, so that pieces wliich are dovetailed togetlier, either at right angles or endwise, cannot be pulled directly apart. The corners of drawers, for instance, are usually dovetailed ; and Bometiines even stone structures, as lighthouses, which are exposed to furious storms, have their parts dovetailed together. 118. Graphical Construction. — The skct(;hes of this framing are arranged as two elevations. A bears the dovetail, B shows the length and breadth of the mortise, and "B" its depth. A and B belong to the same elevation. E:cecution. — In this case a method is given, of representing a hidden cut surface, viz. by dotted shade lines, as seen hi the hidden faces of the mortise in B". 119. Leaving now the examples of pieces framed together at righ^ angles let us consider : — Pi.VII > rW^^ So SJ. \-^ •v^X/^w B* 3' ; 1 1 c r OONSTRUCriOXS IN WO >D. 49 g V, —Pairs of Timbers which are framed together obliquely to each other. Example 1. A Chord and Principal, (From a Model.) PI. V^III., Fig. 59. Mechanical Construction. — The oblique piece (" principal ") is, as the two elevations together show, of equa width with the horizontal piece ("chord," or "tie beam"), and i fi am(>d into it so as to prevent sliding sidewise or lengthwise. Neither can it be lifted out, on account of the bolt which is made to pass perpendicularly to the joint, ac, and is "chipped up" atj!?p, so as to give a ilat bearing, parallel to ac, for the nut and bolt-head. 120. Graphical Construction. — \st. Draw^jfZe/ 2d. Lay oflf cfe ^13 inches; Sd. Make e'ea — ^0'^\ 4th. At any point, e', draw a perj)endicular to ee\ and lay off upon it 9 inches — the perpendicular width of e'ea/ 5th. Makeec=:4 inches and perpendicular to e'e/ bisect it and complete the outlines of the tenons, and the shoulder anc'; Qth. To draw the nut accurately, joroceed as in PI. VII., Fig. 4<;-47, placing the centre of the auxiliary projection of the nut in the axis of the bolt produced, &c. (40) (96). b represents the bolt hole, the bolt being shown only on one elevation. 121. Ex. 2. A Brace, as seen in the angle bet-ween a "post" and "girth." (From a Model.) PL VIIL, Fig. 60. Jfechanical Construction. — PP' is the post, GG' is tKe girth, and Pi'B" is the brace, having a truncated tenon at each end, Avhich rests in a mortise. When the brace is quite small, it has a shoulder on one side only of the tenon, as if B'B" were sawed lengthwise on a line, oo'. 122. Graphical Construction. — To show a tenon of the brace clearly, the girth and brace together are represented as being drawn out of the post. 1st. Draw the post. 2d. Half an inch below the top of the post, draw the girth 2^ inches deep. 3c?. Frora a, lay off ab and a each 4 inches, and draw the brace 1 inch wide„ UJi. Make cd equal to the adjacent mortise; viz. 1^ inches; make r, produced. 9^7i. "With s' as a centre, draw the arc rt'. \Qth. At h and h' erect perpendiculars, each one fourth of an inch high. Wth. Draw quadrants, as q't.^ tangent to these perpendiculars and of one fourth of an inch radius. \2th. Draw the horizontal line ^y. \Mh. Make ni'=nu and describe the arc t'v. lith. Repeat these operations on the other side of the centre line, AA'. Mcecutiofi. — Let the construction be fully shown on one side of the centre line. 140. Ex.2. An end elevation ofa Compound Rail. PI. VIIL, Fig. 72. '3fechanical Construction. — The compound rail, is a rail formed in two parts, which are placed side by side so as to break joints, and then riveted together. As one half of the rail is Avhole, at the points where a joint occurs on the other half, the noise and jar, observable in riding on tracks built in the ordinaiy manner, are both obviated ; also " chairs," the metal supports which receive the ends of the ordinary rails, may be dispensed with, in case of the use of the compound rail. In laying a compound rail on a curve, the holes, through which the bolts pass, may be drawn past one another by the bending of the rails. To allow for this, these holes are " slotted,'' as it is termed, i. e. made longer in the direction of the length of the rail. 141. Graphical Construction. — \st. Make , is made at pleasure, being roughly hammered down while the rivet is hot, during the process of track-laying. A thin washer is shown under this head. 142. Ex. 3. A " Cage Valve," from a Locomotive Pump. ri. A'lll., Fig. 73-74. — Mechanical Construction. — Tliis valve is made in three pieces, viz. the valve proper, Fig. 74 ; the cage con- taining it. Abb'; and the flange bb'c ; whose cylindrical aperture — shown in dotted lines — being smaller than the valve, confines it. The valve is a cup, solid at the bottom, and makes a water tight joint with the upper surface of the flange, inside of the cage. The whole is inclosed in a chamber communicating with the pump barrel, and with the tender, or the boiler, according as we suppose it to be the inlet or outlet valve of the pump. This chamber nukes a water tight joint with the circumference of the flange bb.' Suppose the valve to be the latter of the two just named. The " plunger" of the pump being forced in, the water shuts the inlet valve, and raises the outlet valve, and escapes between the bars of the cage into the chamber, and from that, by a pipe, into fhe boiler, 143. Graphical Construction. — Scale full size. Make the plan Brst, where the six bars are equal and equidistant, with radial sides. Project them into the elevation; as in Prob. 14, Div. 1.; taking care to note whether any of the bars, as E, on the back part of the cage can be seen above the valve, CD, and between the front bars, as F and G. The diameters of the circles seen in the cage are, in order, from the centre, 1^, 2J, 2||-, and 4 inches. The thickness of the valve. Fig. 74, is -j\ of an inch, its outside height If inches, and the outside diameter 2^ inches. The diameter of the aperture in the flange is 1| inches, its length |- of an inch, and the height of the whole cage is 3-^^ inches. 144. Execution. — Observe carefully the position of the heavy lines. The section, Fig. 74, being of metal, is finely shaded. J CONSTRUCTIONS IN METAL. 57 745. Ex. 4. An oblique elevation of a Bolt Head. PL VIII., Fig. 75. Let PQ be the intersection of two vertical planes, at right angles with each other; and let RS be the intersection, with the vertical plane of the paper to the left of PQ, of a ])lane wliich, in space, is parallel to the square top of the bolt head. On such a plane, a plan view of the bolt head may be made, showing two of its dimensions in their real size; and on the plane above RS, the thickness of the bolt head, and diameter of the bolt, are shown in their real size. Below RS, construct the plan of the bolt head, with its sides making any angle with the ground line RS. Project its corners in perpendiculars to RS, giving the left hand elevation, whose thick- ness is assumed. 146. The fact that the projecting lines of a point, form, in the drawings, a perpendicular to the ground line, is but a special case of a more general truth, which may be thus stated. — When an object in space is projected upon any two planes which are at right angles to each other, the projecting lines of any point of that object form a line, in the drawing, perpendicular to the intersection of the two planes. 147. To apply the foregoing principle to the present problem; it appears that each point, as a", of the right hand elevation, will be in a line, a'a!\ perpendicular to PQ, the intersection of the two vertical planes of projection. Remembering that PQ is the intersection of a vertical plane — perpendicular to the plane of the paper — with the vertical plane of the paper, and observing that the figure represents this plane aa being revolved around PQ towards the left, and into the plane of the paper, and observing the arrow, which indicates the direction in which the bolt head is viewed, it appears that the revolved ver- tical plane, has been transferred from a position at the left of the plan acne^ to the position, PQ, and that the centre line ^?f", must appear as far from PQ as it is in front of the plane of the paper — i. e. e'V=ew, showing also, that ase — e' is in the plane of the paper, its projection at e" must be in PQ, the intersection of the two ver- tioal planes. Similarly, the other corners of the nut, as c", w", &c., are laid off either from the centre line txi'\ or from PQ. Thus v"c''=vc, oi v"'c" = sc. The diameter of the bolt is equal in both elevations. 148. Other supposed positions of the auxiliary plane PQ may be assumed by the student, and the corresponding construction worked out. Thus, the primitive position of PQ maybe at the right of the 58 CONSTRUCTIOXS IX METAU Dolt head, and that may be viewed in the opposite direction frona ihal indicated by tlie arrow. 149. Ex. 5. A " Step" for the support of an oblique tim bex. Pi. V'lll., Fig. 7G. MecJianical Construction. — It will be trec^iiently obsei'ved, in the framings of bridges, that there are certam timbers whose edges have an oblique direction in a vertical plane, while at their ends tliey abut against horizontal timbers, not directl)^, for that would cause them to be cut off obliquely, but through the medium of a prismatic block of wood or iron, so shaped that one of its faces, as ab — n'b\ Fig. 76, rests on the hori- zontal timber, while another, as ac — e"d''c", is perpendicular to the oblique timber. To secure lightness with strength, the step is hollow underneath, and strengthened by ribs, r)\ The holes, h'h", allow the passage of iron rods, used in binding together the partsof the bridge. These holes are here prolonged, as at h, forming tubes, which extend partly or wholly through the horizontal timber on which the step rests, in order to hold the step steadily in its place. 150. Hemarh. When the oblique timber, as T, Fig. 76 (a), seta into^ rather than upon, its iron support S, so that the dotted lines, ab and ci, represent the ends of the timber, the support, S, is called a shoe. 151. Graphical Construction. — In the plate, ahc is the elevation, and according to the usual arrangement would be placed above the plan, e"n"c\ of the top of the step, a'b'e is the plan of the under Bide of the step, showing the ribs, Sec. A line through nm is a centre line for this plan and for the elevation. A line througli the middle point, r, of m?i, is another centre line for the plan of the bottom of the step. Having chosen a scale, the position of the centre lines, and the arrangement of the figures, the details of the construction may be left to the student. 152. Ex, 6. A metallic steam tight "Packing," for the "stuffing boxes" of piston rods. Tl. Mil., Jig. 77. Meclianical Construction. — Attached to that end, J, of Fig. 77 (a), of a steam cylinder, for instance, at which the piston rod, p, entera It, is a cylindrical projection or " neck," n, having at its outer end a flange, f/\ through which two oi- more bolts pass. At its inner end, at^:), this neck fits the piston rod quite close for a short space. Tlie internal diameter of the remaining portion of the neck is suffi- cient to receive a ring, rr, which fits the piston rod, and has on its outer edge a flange, <, by which it is fastened to the flange, Jf, on the neck of the cylinder by screw bolts. The remaining hcillow CONSTRUCTIONS IN METAL. 59 space, s, bet\veen the ring or " gland," ^r, and the inner end of tlie neck, is usually filled with some elastic substance, as picked hemp, which, as held in place by the gland, tr, makes a steam tight joint; which, altogether, is called the "stuffing box." 153. The objection to this kind of packing is, that it requires so frequent renewals, that much time is consumed, for instance in raii road repair shops, in the preparation and adjustment of the packing. To obviate this loss of time, and perhaps because it seems mon neat and trim to have all parts of an engine metallic, this metallic packing. Fig. V7, was mvented. ABC — A'B' is a cast iron ring, cylindrical on th(? outside, and having inlaid, in its circumference, bands, tt\ of soft metal, so that it may be squeezed perfectly tight into the neck of the cylinder. The inner surface of this ring is coni- cal, and contains the packing of block tin. This packing, as a whole, is also a ring, whose exterior is conical, and fits the inner side of the iron ring, and whose interior, yA;c, is cylindrical, fitting the piston rod closely. 154. For adjustment, this tin packing is cut horizontally into three rings, and each partial ring is then cut vertically, as shown in the figure, into two equal segments, ahcd — a'b'c'd', is one segment of tl e upjDer ring; efgh — e'f'f"g'h\ is one segment of the middle ring, ind rjM — r'fk'n'l, is one segment of the lower ring. The segments of each ring, it therefore appears, break joints with each of the other rings. Three of the segments, one in each partial ring, are loose, while the other three are dowelled by small iron pins, parallel to the axis of the whole packing. 155. Operation. — Suppose the interior, ^/tc, of the packing to bo of less diameter than the piston rod, which it is to surround. By drawing it partly out of its conical iron case, the segments forming each ring can be slightly separated, making spaces at a5, &c., which will increase the internal diameter, so as to receive the piston rod. When in this position, let the gland, ti\ be brought to bear on the packing, and it will be firmly held in place ; then, as the packing gradually wears away, the gland, by being pressed further into the neck, will press the packing further into its conical seat, which Avill close up the segments round the piston rod. 156. Graphical Construction. — Let the scale be from one half to the whole original size of the packing for a locomotive valve chest. C is the centre for the various circles of the plan, and DD'. projected up from C, is a centre line for portions of the elevation C/=-j2g- of an inch; Ce=::l-^ inches; CA=1^ inches. A'n=2 inches, and K's'—-^ of an inch. The iron case being constructec' 60 COXSTKUCTIONS IN ^^ETA1. from tliese inoasurements, the rings must be located so tliaty*/)', foj instance, shall be = j\ of an inch ; and then let tlie thickness, /'/"', of cacli segment be \} of an inch. Tliese dimensions and the con- sequent ai-i-angenients of the rings \v ill give spaces between the seg. ments, as at ab, of i of an inch ; though in fact, as this space is variable, there is no necessity for a precise measurement for it. In the plan, there are shown one segment, and a fragment, abef^ of another, in the u]iper ring; one segment and two fragments of tht middle ring, and both segments of the lower ring, with the whoh^ of the iron case. The elevation shows one of the halves of eacb ring, viz., acd — a^c'd', the upper one; ef ffh — e"f^ ifh'^ the middle one; and rjkl — r'fk'l\ the lower one. The circles of the plan are found by projecting the points as/?' upon the diameter AC. Then by comparing kk', IV, and nn\ for example, we see how the vertical projections, as k'l'n\ of the ends of the half rings are found. loY. Execution. — The section lines in the elevation indicate clearly the situation of the three segments, ahcd^ €,fgh^ and ijkl^ there shown. The dark bands on the case at t and t\ indicate the inlaid bands of soft metal already described. The studt'nt can usefully multiply these examples by constructing pla?is, elevations and sections, i'rom measurement, of such objects as steam, water, and gas cocks, valves, or gates, raihoay joints, and chairs, and other like simple metallic details; actual exam])lee of which can be easily obtained as models almost anywhere. Ex. 7. Rolled Iron Beams and Columns. Wood is peri.shalile and growing scarce. Stone is comparatively costly and cumbrous. Cast iron is suspected as treacherous. Fig. 1. Hence, of late years, much attention Las l)ecn paid tol)oams and colurauu of rolled wrought iron. Various figures of such work arc therefore given as an appropriate concluding general example for the ])rcsont chajjter. PL.VIll. COXSTRUCTIOXS IX MKTAL. 61 Rolled iron in its elementary commercial forms, for architec- tural and engineering purposes, is principally known as beam, plate, angle, T (Fig. 1), V (Fig. 2), and chaimel iron, the latter Fvj. 3. ■ ||'^"' ' M Fuj. 5. either in polygonal or circular segments (Fig. 3). All of these are used in building up compound beams, braces, or columns. Figs. 4-10, of those here given,* may all be taken as on a scale 1 ij. u. Fig. 7. of one fourth the full size, and Fig. 9, as a practical example of oblique projection (see Div. IV.), of one eighth the full size. * From the Uniou Iron Works, Buffalo, N. Y. 63 CONSTRUCTIONS IN METAL. Figs. 4-13;, excciot 9, are all transverse sections oi the beams or columns which they represent. .. \_ Fig. 8, Simple beams being made of all sizes from four io fifteen inches in depth, and some of the sizes either light or heavy, Figs. 4-8 represent compound beams of depths greater than fifteen inches. Fig. 4 represents the lower half of a beam formed of riveted plate and angle irons. CONSTRUCTIONS IX METAL. 63 Fkj. 10. Fig. 11. Fig. 12. 64 CONSTRUCTIOXS IN METAL. Fig. 5 represents the left-hand half of a holloTV beam of hori- zontal plate, and Tertical rectangular channel iron. Fig. 6 shows the lower end of a very deep beam, the ends of which are symmetrically placed, and formed of plate and curved channel iron. Fig. 7 is the lower half of a beam composed of a simple beam riveted to horizontal plates. Fig. 8 represents the lower half of a beam, 16" wide at base, composed of plate, beam, and rectangular channel irons. Fig. 9, an oblique projection, given here for convenience in anticipation of Div. IV., shows how beams at right angles to each other may l)e riveted together by angle plates. Leaving ])eams for the highly interesting and practically very important subject of iron columns, Fig. 10 is the partial hori- zontal section of a column curiously formed of two flanged beams, bent at right angles, and riveted along the angle, through an intermediate doubly concave bar, which gives a firmer bearing for the rivets. Fig. 11 is nearly a half section of a column composed of six curved channel irons, disposed with the convexities inwards, and riveted in the angles of the flanges. Fig. 12 shows two examples of the trvie hollow column * as made in segments, consisting of channel irons whose fl.anges are radial and whicli are placed with their convexities outward, and are riveted through the flanges. The inner figure represents a " column of four channel irons, the outer one a larger column of six flanges ; the number varying from four to eight in different cases. Fig. 13 is a section of a column of German design, made of plate and polygonal channel irons. The ideal of Fig.. 12 is a smooth hollow column iu one piece, a given amount of matter having much greater strength, in this form, to resist crushing or bending than when in a solid bar. But this ideal, though easily realized in cast iron, is economical! j impracticable in wrouglit iron; hence the external radial flanges, though giving additional strength by increasing the average dis- tance of the material from the centre, are subordinate to the main idea. In Fig. 11, on tlic contrary, the heavy flanges are thejirimary feature as a means of gaining strength l)y a circumferential dis- * ^ladp at Plia>nixville, Pa. CONSTKUCTIONS IN METAL. 65 position of material, while the curved parts form a meang of connecting them, and of also leaving a hollow interior. A plate alone easily "buckles" in the direction of its thick- ness. Each half of Fig. 10 is therefore a plate, braced to prevent this by bending at right angles, and by means of flanges which also make the circumferential strength prominent. Fig. 13 is also essentially a braced plate column, the primary office of the channgl irons bein^j to unite the plates, leaving their effect on the circumferential strength incidental. If a column could be formed by placing the channel irons of Fig. 11 so as to join them by their flanges, which, in Fig. 13, would then radiate inward from the circumference of the column, this circumference might then be of slightly greater diameter for a given amount of matter. But the riveting would be inaccessible. It may be a question, however, whether collars might not be clamped or shrunk on, numerous enough to make the column essentially solid, and at no more than the cost of riveting. This question appears to have exercised other minds, and Fig. 14 indicates one solution of it,* consisting of cylindrical segments bearing dovetailed in jDlace of rectangular flanges on their edges, and united by clamping bars in place of riv- ets ; as one would hold two boards laid one upon the other, by clasping each hand over the two edges of the boards, while the riveting plan may be repre- sented by thrusting the fingers through holes near the edges of the boards. * Manufactured l)y Carnegie Brothers & Co., Pittsburg, Pa. DIVISION TIIII113. IK ELEMENTARY SHADOWS AND SHADING. CHAPTER I. SHADOWS. § I, — Fads, Principles, and Preliminary Problems. 158. The shadow of a given opaque body, B, PI. IX,, Fig. 78, upon any surJace S, is the portion of S from which hght is excluded by B. A shadow is known when its boundary, splc, called the line of shadow, is known. Hence, to find the shadow of a given body upon a given surface is, practically, to find the boundary of that shadow, 159. The boundar}^ sph, PI. IX,, Fig, 78, of the shadow of a body, B, is the shadow of the line of shade, hrn, which divides the illuminated from the unilluminated part of B, For if a ray, od, pierces S, as at d, ivithm the area of the shadow, it must pierce the body B in its illuminated part as at o; but if another ray pierces S, as at q, without the shadow, it cannot meet the body B. Hence rays, as hs, cp, nh, which meet S in the li7ie of shadow, must be tangent to B at points, as ?i, c, n, of its line of shade. 160. Since the line of shadoio of anybody B on any surface S is the shadow of the line of shade of B, the line of shade on the body casting a required shadow must, in general, be found first in problems of shadows. On plane-sided, that is flat-sided bodies, this line of shade con- sists simply of the edges which divide its faces in the light from those in the dark; that is, it consists of the shade lines (21) of the body. These may in many cases be found by simple inspection, 101. Rays of liyht are here assumed to he parallel straight lines, as they practically are when proceeding from a very distant source, as the sun, to any terrestrial object. \st. It will be thus observed that the shadow of a vertical edge ab, PI. IX., Fig. 78, of the body of the house will be a vertical line, a'h', on the front wall of the wing behind it; that the shadow SHADOWS. G7 of a horizontal line, as he — the arm for a swinging sign — which is parallel to the wing wall, will be a horizontal line, b'e\ parallel to be; that a horizontal line, be, which is perpendicular to the wing wall, will have an oblique shadow, cb', on that wall, commencing at c, where the line pierces the wing wall, and ending at b', where a ray of light through b pierces the wing wall; and finally that the shadow of a point, b, is at b' where the ray bb' , through that point, pierces the surface receiving the shadow. 2c?. Passing now to PL IX., Fig. 79, which represents a chim- ney upon a flat roof, we observe that the shadows of he and ed— lines parallel to the roof — are b'c' and c'cV, lines equal to, and parallel to, the lines be and cd; and that the shadows of ab and ed are ab' and ed' — similar to the shadow eb' m Fig. 78 — i. e. com- mencing at a and e, where the lines casting them meet the roof, and ending at b' and d' , where rays through b and d meet the roof. 1G2. The facts just noted may be stated as elementarij general principles by means of which many simple problems can be solved. 1st. The shadow of a point on any surface, is where a ray of light through that point meets that surface. 2rf. The shadow of a straight line, upon a plane parallel to it, is a parallel straight line. 3f7. In like manner, the shadow of a circle upon a plane parallel to it is an equal circle ; whose eentre, only, therefore need be found. A:th. The shadow of a line upon a plane to which it is perpen- dicular, will coincide with the projection of a ray of light upon that plane. Thus ba, Fig. 78, being perpendicular to H, its shadow upon that plane coincides with the horizontal projection, aa' , of the ray of light bb'. Likewise be, being perpendicular to the vertical plane cb'a', its shadoAV coincides with cb', the projection of bb' on that plane. bth. The shadow of a line upon a surface may be said to begin where that line meets that surface, either or both being produced, if necessary. See a, Fig. 79. %th. When the shadow, as aa'b', of a line, as ah. Fig. 78, falls upon two surfaces which intersect, the partial shadows, as aa' and h'a', meet this intersection at the same point, as a'. Ith. The shadow of a straight line, upon a curved surface, or of a circle, otherwise than as in (3fZ) will generally be a curve, whose points, separately found by {1st), must then be joined. 68 SHADOWS. 163. In applying these principles to the construction of shad- ows, three things must evidently be given, viz. — Isf. The body casting the shadow. 2d. The surface receiving the shadow. 3d. The direction of the light. And these must be given, not in reality, but in projection. i 164, The light is, for convenience and uniformity, usually as- sumed to be in such a direction that its projections make angles of 45° with the ground line (16). For the direction of the light it- self, corresponding to these projections of it, see PI. IX., Fig. 80. Let a cube be placed so that one of its faces, h'hj), shall coin- cide with a vertical plane, and another face, L"aLi, with the horizontal plane. The diagonals, L'L, and L"Li, of these faces, will be the projections of a ray of light, and the diagonal, LLi, of the cube will be the ray itself ; for the point of Avhich L' and U' are the projections must be in each of the projecting perpendicu- lars L'L' and L"L ; hence at L, their intersection. Now the right-angled triangles, LL"Li and LL'Lj, are equal, but not isos- celes ; hence the angles LLjL" and LLiL', which the ray itself, LLi, makes with the planes of projection are equal, but less than 45°. Again: in tlie triangle LLiG, the angle LLjG is that made by the ray LLj with the ground line, and is 77iore than 45°. 165. Having now stated the elementary facts and principles concerning shadows themselves, we proceed to show how to find the'iv jjroject ions, by which they are represented. By (162) we have first to learn how to find where a given line, as a ray of light, pierces a given surface. It will be sufficient here to show how to construct the points in which a line pierces the 2:)lanes of projection. Prelimi7iary Problems. Problem \st. To find where a line drawn through a given point, pierces the horizontal plane. The construction is shown pictorially in Fig. 1, and in actual projection in Fig. 2. Let A, Fig. 1, be the given jioint, wliose projections are repre- sented by a and «'. If a line ])asscs through a point, its projec- tions will pass through the projections of that point; thoi'efore if AB be the line, ab and a'l/ will be its projections. 'Sow, first, the 7'equired {nAnt, C, being in the horizontal plane, SHADOWS. 09 its vertical projection, c', will be on the ground line; second, the same point being a point of the given line, its vertical projection wil! be on the vertical projection, a'h'c', of the line, hence at the intersection, c', of that projection with the ground line; third, the horizontal ])rojection, c, of the point, which is also the point itself, C, will be where the perpendicular to the ground line from c' meets abc, the horizontal projection of the line. This explanation may be condensed into the following rule. l5^ Note where the vertical ])rojection of the given line, pro duced if necessary, meets the ground line. 2d. Project this point into the horizontal projection of the line; which will give the required point. Fig. 2, shows the application of this rule in actual projection. ab — a'b' is the given line, and by making the construction, we find c, wliere it pierces the horizontal plane. Fig. 1. Fig. 2. Example. — If bd (both figures) had been less than b'd, the line would have pierced the horizontal plane behind the ground line, tliat is in the horizontal plane produced bachoards. Let this con etruction be made, both pictoi'ally, and in projection. Pboblem 2d. — To find where a line, drmcn through a given point., pierces the vertical plane. See Figs. 3 and 4. The explanation is entirely similar to the preceding. The requiied point, c', being somewhere in the vertical plane, its horizontal projection, c, will be on the ground line. It being also on the given line, its horizontal projection is on the horizontal projection, nhc. of the line. Hnnce. nt once, che rule 70 SHADOWS. lat. Note where the horizontal projection of the given line mreti the ground line. Pio. 3. Fia. 4. '2d. Project this point into the vertical projection of the line, which will give c'=C, the vertical projection of the point, which is also the point itself. Example. — Here likewise, if b'd be ^55 than bd, the line will pierce the vertical plane below the ground line, that is in the verti- cal plane extended downward. Let the construction be made. 166. The ground line is, really, the horizontal trace of the verti- cal plane; also the vertical trace of the horizontal plane. Hence the traces of any other horizontal or vertical planes may be called the ground lines of those planes ; and therefore the preceding con- etrnctions may be applied to finding the shadows on such planes, as will shortly be seen in the solution ot" problems. 107. In respect to the remaining principles of (162) it only re- mains to note that, as two points determine a straight line, it is sufficient to find the shadow of two jooints of a straight line, when its shadow falls on a plane. But if the direction, of the shadow is known as in (1G2 — 2f?, \th) or if we know where the line meets a plane surface receiving the shadow, (162 — hth) it will be suflScient to construct one point of the shadow. § II. — Practical Problems. 168. FnoB. 1. To find the shadow of a vertical beam., upon a i^tical v:all. PI. IX., Fig. 81. Let AA' be the beam, let the SHADOWS. 71 vertical plane of projection be taken as tlie vertical wall, and let the light be indicated by the lines, as ah — a"b'. The edges wliich cast the visible shadow are cf. — a'a" ; ac — a"/ and ce — a"e'. The sha- dow of a — a'a'' is a vertical line from the point b\ which point is where the ray from a — a" pierces the vertical plane, ab — a"b' pierces the vertical plane in a point whose horizontal pi-qjection ia b. b' must be in a perpendicular to the groimd line fioni b (Art. ] 6), and also in the vertical projection, a"b\ of the ray, hence at i'. Ub is therefore the shadow of a— a^'a'. The shadow of ac — a" is the line b'd\ limited by d\ the shadow of the point c — a" (162). The shadow of ce — a"e,' begins at d\ and is parallel to ce — a"e\ but is partly hidden. 169. Execution. — The bonndary of a shadow being determined, its surfice is, in practice, indicated by shading, either with a tint of indian ink, or by parallel shade lines. The latter method, affording useful pen practice, may be profitably adopted. 170. Prob. 2. To find the shadow of an oblique timber, lohich is parallel to the vertical plane, upon a similar timber resting against the back of it. PI. IX., Fig. 82. Let AA' be the timber which casts a shadow on BB', which slants in an opposite direction. The edgeac — aV of A A', casts a shadow, parallel to itself, on the front face of BB'; hence but one point of this shadow need be con- structed. Two, however, are found, one being a check upon the other. Any point, aa\ taken at pleasure in the edge ac — aV casts a point of shadow on the front plane of BB', whose horizontal projection is J, and whose vertical projection (see Prob. 1) must be in the ray a'b\ and in a perpendicular to the ground line at b ; hence it is at b'. The shadow of ac — a'c' being parallel to that line, b'd' is the line of shadow. d\ the shadow of c — c\ was found in a similar manner to that jnst described. It makes no difference that b' is not on the actual timber, BB'; for the face of that timber is but a limited physical plane, forming a portion of the indefinite immaterial plane, in which bb' is found ; hence the point b' is as good for finding the direction of the indefi- nite line of shadow, b'd\ as is d\ on the timber BB', for finding the real portion only of the line of shadow, viz. the part which lies across BB'. Here, bd is taken as a ground line (166). Observe that the shade line Ab, of the timber AA' should end at bb', where the timbers intersect. Example. — Make the timbers larger, A A' more nearly hori- zontal than vertical, and then, in both figures, find the shadow on the plan. 72 SHADOWS. 171. Prob. 3. To find the shacloxo of a fragment of a horiT/m- tal timber, vpon the horizontal top of an abutment on which tht timber rests. PI. IX., Fig. 83. Let AA' be the timber, and BB' the abutment. The vertical edge, a — «"a', casts a shadow, a6, in the direction of the horizontal projection of a ray (102, 4^/f.), and limited by the shadow of the point a — a'. The shadow of a — a' is at bb\ where ttie ray ab — a'b' pierces the top of the abutment ; b' being evidently the vertical projection of this point, and b being both in a jierpendicular, b'b^ to the ground line and in ab, the hori- zontal projection of the ray. The shadow o^ ad — a' is J5", parallel to ad — a\ and limited by the ray a'b' — db". The shadow of de — aV is b"c^ parallel to de — a'e'^ and limited by the edge of the abutment (162, 2d). Here, a"b' is used as a ground line (166). 172. Pkob. 4. To find the shadow of an oblique timber., upon a horizontal timber into xchich it is framed. PL IX., F'ig. 84. The upper back edge, ca — c'a', and the lower front edge through ee', of the oblique piece, are those which cast shadows. By con- sidering the point, c, in the shadow of be, Fig. 78, it appears that the shadow of ac — aV, Fig. 84, begins at cc\ where that edge pierces the upper sui-faco of the timber, BB', which receives the shadow. Any other point, as aa', casts a shadow, bb% on the plane of the upper suifxce of BB', whose vertical projection is evidently b', the intersection of the vertical ))rojection, a'b\ of the ray at — a'b' and the vertical projection, eV, of the upper surface of BB', and whose horizontal projection, b, must be in a projecting line, b'b, and in the horizontal projection, ab, of the ray. Likewise the line through ee', and parallel to c5, is the shadow of the lower front edge of the oblique timber upon the top of BB'. This shadow is real, only so far as it is actually on the top surface of BB', and is visible and therefore shaded, only where not hidden by the oblique piece. Where thus hidden, its boundary is dotted, as shown at ea. The point, bb', is in th« plane of the top surface of BW, produced. Remark. — Thus it appears that when a line is oblique to a plane containing its shadow, the direction of the shadow is unknown till found. Let this and the Ibllowing iigures be made much larger. Examim.es. — l,s/. Find the shadow when the oblique timber is more nearly vertical tlian horizontal. 2d. Let the oblique timber ascend to the right. 1 73. Pkou. 5. To find the shadoio of the side waU of a flight of tteps upon the faces of the steps. PI. IX., Fig. 85. The stepi can be easily constructed in good pro])ortion, without measure SIIADOAVS. 73 mcnts, by making the height of each step two-thirds of its width, taking four steps, and making the piers rectangular prisms. The edges, aa" — a' and a — ra', of the left hand side wall are those which cast shadows on the steps, Tlie former line casts horizontal shadows, as hh" — h\ parallel to iliself, on the ?o/as of the steps (162, 2c?), and shadows, as he" — h'c\ on the fronts of the steps, in tlie direction of the vertical jirojectioii, a'd\ of a ray of light (162, ^th) — from the upper step down to tlie shadow of the point aa' . Likewise, the edge a — ra' casts vertical shadows, as g — g'h\ on \\\(i fronts of the steps (161), and shadows on their tops, parallel to ac?, the horizontal projection of a ray (IG'2, Mli) from the lower step, iip to (7^^, the shadow o'i aa' ; which is tlierefore, where the shadows of aa" — a', and a — ra', meet, a'd' is the vertical projection of all rays thi'ougii points of aa" — a' j hence project down h', etc., to find the parallel shadows, b"b, etc. Likewise project up g^ etc., to find the shadows, ^7/, etc., ac? being the horizontal projection of all rays through points of a — ra'. ExAMPi>ES. — \st. Vary the jiroportions of the ste2ys and the direc- tion of the light/ and in each case, find the shadows, as above. 2d. Bj pj-eUmi?mr7/ problems 1st and 2d, find directly where the ray through aa' pierces the steps, only remembering that the projections, d and d', must be on the same surface. 3c?. Let the piers be cut off by a plane parallel to that of the front edges of the steps. (Use an end elevation.) 174. Prob. 6. To find the shadoxo of a short cylinder^ or washer^ ripon the vertical face of a board. PL IX.. Fig. 86. Since the circidar face of the washer is i)arallel to the vertical face of the board, BB', its shadow will be an equal circle (161, 8c?), of which .we have only to find the centre, 00'. Tiiis point will be the sha- dow of the point CC of the washer, and is where the ray CO — CO' pierces the board BB'. The elements, rv — r' and tu — t'., which separate the light and shade of the cylindrical surface, have the tangents r'r" and t't" for their shadows. These tangents, with the semicircle t"r'\ make the complete outline of the required shadow. 175. Pbob. 7. To find the shadow of a nut., iipoyi a vertical sinface, the nut having any j^ositioji. PI. IX., P^ig. 86. Lefc a'c'e' — ace be the projections of the mit, and BB' the projections of the surface receiving the shadow. The edges, a'c' — ac and ce — c'e', of the nut cast shadows parallel to themselves, since they are parallel to the surface which receives the shadow, a'n' — an 74 SHADOWS. are the two projections of the ray whicli determines the joint o\ shadow, wn'/ cV — co are the projections of the ray used in lind ing oo', and eV — er is the ray which gives the point of shadow n^. The edges at aa' and ee% whicli are perpendicular to BB', cms; shadows a'n' au<\ e'/, in the direction of tlie projection of a ray of light on BB'. (See cb\ the shadow of cb, PI. IX., Fig. 78.) 176. PuoB. 8. To Jin d the shadow of a vertical cylinder^ on a vertical plane. PI. IX., Fig. 87. The lines of the cylinder, CC, which cast visible shadows, are the element a — a"a\ to which the rays of light are tangent, and a part of the upper base. The shadow of a — a."a\ is gg\ found by the m:>thod given in Prob. I. At g\ the curved shadow of the upper base begins. This is found by means of the shadows of several points, hd\ cc\ dd\ &c. Each of these points of shadow is found as g was, and then they are connected by hand, or l)y the aid of the curved ruler. It is well to construct one invisible point, as ?/, of the shadow, to assist in locating more accurately the visible portion of the curved shadow line. 177. Prob. 9. To find the shadow of a horizontal beam^ upon the slanting face of an oblique abutment. PI. IX., Fig. 88. The simple facts illustrated l)y Pl. IX., Figs. 78-79, have no reference to the case of surfaces of shadow, other than vertical or horizontal. But they illustrate the fact that the point where a shadow, as aa\ PI. IX., Fig. 78, on one sui'face, meets another surface, is a point {a') of the shadow a'b' upon that surface. Tims this problem may be solved in an elementary inanner by proceeding indirectly, i. e., by finding the shadoAvs on the horizontal top of the abutment and on its horizontal base. The points where these shadows meet the front edges of this top and this base, will be points of the shadow on the slanting face, que. By Prob. 3 is Ibuiid gc^ the shadow of the upper back edge, ax — a'x\ of the timber, AA', upon the top of the abutment, c, the point where it meets the front edge, cc, is a point of the shadow ol AA' on the inclined face. By a similar construction with any ray, as b2y — b'p\ IS found qp^ the shadow of c'b' — db upon the base of the abutment; and 2', where it intersects Jig-, is a j)ointof the shadow of db — c'b' on the face, qne. The point, dd\ where the edge, db — c'J', meets f?c, is another point of the shadow of that edge; hence dq — d'q' is the shadow of the fiont lower edge, db — c'b\ on the inclined face of the abutment. The line through cc\ parallel to ^.q — d'q\ is the shadow of the upper back edge, ax — aV, and com- tes the solution. SHADOWS. 75 178. Prob. 10. To find the shadoxo of a pair of horizontal tim- bers^ which are inclined to the vertical plane, ujyon that plane. P!. IX., Fig. 89. Let the given bodies be situated as shown in the dingram. In the elevation we see the thickness of one tiinl)er only, because the two limbers are supposed to be of equal thickness and halved together. As neither of the pieces is either ])arallel or perpendicular to the vertical plane, we do not know, in advance, the direction of their shadows. It will therefore be necessary to find the shadows of two points of one edge, and one, of the diago- nally opposite edge, of each timber. The edges which cast shadows are ac — a'c' and ht — e'h\ of one timber, and ed — e'k and niv—a'm" of the other. All this being understood, it will be enough to point out the shadows of the required points, without describing their construction. (See hb'. Fig. 81.) bb' is the shadow of aa', and dd^ is the shadow of cc' y' hence the shadow line b^ — d is determined. So uu' is the shadow of hh' ; hence the shadow line u'w' may be drawn parallel to b'd'. Similarly, for the other timber, ff is the shadow of ee', and oo'of mm\ One other point is necessary, which the student can construct. The process might be shortened some- what by finding the shadows of the points of intersection,/? and r, which would have been the points/*" and /' of the intersection of the shadows, and thus, points common to both shadows. i79. Prob. 11. To find the shadow of a pair of horizontal tini' Ocrs, which intersect as in the last problem, up>on the inclined fact of an oblique abutment. PI. IX., Fig. 90. Tiiis i:>roblem is so similar to Prob. 9, that we only note, as a key to it, that sd, corres- ponding to 2)q, Fig. 88, is the auxiliary shadow of the edge, Ab — n'b', upon a horizontal plane s'd', cutting the line sb from the face C of the abutment, and thence giving a point ss', intersection of sd, parallel to Ab, with sb of the shadow of A6 — n'b' on C. The folloAving examples of the very useful method of auxiliary elutdoivs (162, 6th), and Prob. 9, are here briefly added: Examples. -1st. To find the shadow of an abacus of any form, upon a conical column. PL IX., Fig. 88a. Circles OA and OB, with A'B'G' represent the conical pillar; and circle OC, with C'D'B', its cylindrical cap, or abacus. Then P'Q' is a horizontal i)lane, cutting from the pillar the circle P'Q', of radius OB=:nP; and pierced by the ray Oa — O'a' at a'a y centre of the cii'cle, of radius Od=^0"D' . Then (/, projected ate/', and intersection of the circle Od with the circle OQ, is a point of the shadow of circle OC — O'D' of the abacus, upon the pillar ; since the circle Oc£ is the auxiliary «" SHADOWS. Bhadow of the circle OC — (YD' upon the plane P'Q' (IG'2, Sff). OtKei points may be likewise found, 2d. 7h find the shadow of the front circle of a niche, upon its own sjjherical surface. PL IX., Fig. 88b. ABC — A'D'C is the quarter sphere,- which surmonts the vertical cylindrical part of the niche below the line ABC — A'O'C. Then the ray Oa — O'a' meets any plane E.S, parallel to the face, AOC — A'D'C, of the niche, at aa' y giving circle a',7iin'= circle O'A', for the auxiliary shadow of circle O'A' upon the plane KS (102, 3d), circle a', m'n' then cuts circle RS — E'ci'S', cut from the spherical part of the niche by the plane RS, at d' d' , a required point of shadow. Find other j)ointa likewise, and join them. The tangent ray at i gives that point. 3d. To firul the shculov} of a verticcd staff, upon a hemi spherical dome. PI. IX., Fig. 88c. Circle OC, with C'P'D' is the hemi- s])here, and A— A'B' the staff. By (162, ith) Ae the horizontal projection of a ray, is also the shadow of A — A'B' upon the assumed horizontal plane P'Q', and cutting from the hemisphere the circle P'Q'— PQ. Then dd', intersection of Ae with cii-cle PQ — P'Q', is one i^OLiit of the required shadow of A — A'B' upon the dome. 180. PnoB. 12. To find the shadoio of the floor of a bridge upon a verticcd cylindrical ahutment. PI. IX., Fig. 91. The line aq — a'g' is the edge of the floor which casts the shadow. hdg—h'e'g'n is the concave vertical abutment receiving the shadow. gg\ where the edge ag — g'a' of the floor meets this curved wall, is one point of the shadow, /'is thehcrizont:d projection of the point where the ray, ef — ef\ meets the al)utment ; its vertical projection is in ef, tlie vertical projection of the ray, ami in a peii)ondicular to the ground line, thi'ough f hence at/"'. Similnrly we find the points of shadoA\', errand bb', and joining them with /"'and g', have the boundary of the required shadow. Observe, that to find the shadow on any ])articular vertical line, as b — I/'b', we draw the ray in the direction b — a ; then project a at a', &c. Hemark. — The student may profitably exercise himself in chang- ing the ])Ositions of the given j^arts, while ret.niiiing the methods cf solution now given. For example, let the parts of the last problem be placed side by side, as two elevations, giving the shadow of a vertical wall on a horizontal concave cylindrical surface; or, let tiie tinilicrs, Fig. 89, be in vertical planes, and let their shadows then be found on a horizontal surface. i A CHAPTER II. SHADING. 131. The distinction between a shade and a shadotu is this. A shadow, as indicated by the preceding problems, is the portion of a body from which light is excluded by some other 'body. A shade, is that portion of the surface of a body from Avhich light is excluded by the body itself (158, 159). The accurate representation of shades assists in judging of the forms of bodies ; that of shadoios is similarly useful, besides aid- ing in showing their relations io surrounding bodies. In either case, a, flat tint mainly shows only ivhere the shade, or shadow, is ; while finished shading helps to show the forin and position of the body. 182. Example 1°. To shade the elevation of a vertical right hex^ agonal prism, and its shadow on the horizontal plane. PI. X., Figs. '.)2 and A. Let the prism be placed as represented, at some dis- tance from the vertical plane, and with none of its vertical fac-ea parallel to the vertical plane. The face, A, of the prism is in tho light ; in fact, the light strikes it nearly perpendicularly, as may be seen by reference to the plan ; hence it should receive a very light tint of Indian ink. The left hand portion of the face, A, is made slightly darker than the right hand part, it being more distant; for the reflected rays, which reach us from the left hand portion, have to traverse a greater extent of air than those fiom the neigh- borhood of the line tt', and hence are more absorbed or retarded; since the atmosphere is not perfectly transparent. That is, these rays make a weaker impression on the eye, causing the left hand portion of A, from which they come, to appear darker than at tt' Remarks. — a. It should be remembered that the whole of face A is very light, and the difference in tint between its opposite sides very slight. b. As coi'ollai'ies from the preceding, it appears : Jirst^ that a sur- face parallel to the vertical plane would receive a uniform tint throughout ; and, second^ that of a series of such surfaces, all of 78 SHADING which are in the light, tlie one nearest the eye would be Hghtest, and tlie one fuithest from the eye, darkest. c. It is only for great diiferences in distance that the above effects are manifest in nature; but drawing by j)rojections being artificial, bi)th in respect to the shapes which it gives in the drawings, and in the abscnc • of snrioiindlng objects which it allows, we are obliged to exaggerate natural appearances in some respects, in order to convey a clearei- idea of the forms of bodies. d. The mere manual process of shading small surfaces is hero briefly described. With a sharp-pointed camel's-hair brush, wet with a very light tint of Indian ink, make a narrow strip against the left hand line of A, and soften off its edge with another brush slightly wet with clear water. "When all this is dry, commence at the same line, and make a similar but wider strip, and so proceed till the -whole of face A is completed, when any little irregularities in the gradation of shade can be evened up with a delicate sable brush, dam2) only with very light ink. 183. Passing now 1o face B, it is, as a whole, a little darker than A, because, as may be seen by reference to the ])]an, while a beam of rays of the thickness njy strikes face A, a beam having only a thick nesspr, stiikes liice B ; i. e., we assume, _/?rs?, tliat the actual hrhjht- ness of a flat surface is proportioned to the number of rays of light which it receives; and, second^ that its apparent brightness is, other things being the same, proportioned to its actual brightness. Also the part at a — a' being a little more remote than the Hue t — 1\ the part at a — a' is made a very little darker. 184. The face C is very dark, as it receives no light, except a small amount by reflection from surrounding objects. Tliis side, C, is darkest at the edge a — a' which is nearest to the eye. This agrees with experience; for while the shady side of a house near to us appears in strong contrast with the ilhuninated side, the shady side of a remote building appears Bcarcely darker than the illuminated side. This fact is ex])lained as follows: The air, and particles floating in it, between tlie eye and the dark surface, C, are in the light, and reflect some light in a direction fiom the dark surface C to the eye; and as the air is invisible, these reflections appear to come from that dark surface. Now the more distant that surface io, the gieater will be tlie body of illuminated air between it and the eye, and therefore the greater will be the amount of light, appa- rently ))rocee;Jing from the surface, and its consequent apparent brightness. That is, the more distant a surface in the dark is, the SHADING. 79 lit,'hter it will appear. It may be objected that this would make o:it the remoter parts of illuminated surfaces as the lighter parts. IJut noi so ; for the air is a nearly perfect transparent medium, and hence reflects but little, compared with what it transmits to the opaque body ; but being not quite transparent, it absorbs the reflected rays from the distant body, in proportion to the distance of that body, making therefore the remoter portions darker; while t\\Q very tceak reflections from the shady side are reinforced, or replaced, by more of the comparatively stronger atmospheric reflections, in case of the remote, than in case of the near i3art of that shady side. Thus is made out a consistent theory. In relation, now, to the shadow, it will be lightest where furthest from the prism, since the atmospheric reflections evidently have to traverse a less depth of darkened air in the vicinity of de — d' than near the loiccr base of the prism at ahc. 185. Ex. 2". To shade the elevation of a vertical cylinder. PI. X., Fig. 93. Let the cylinder stand on the horizontal plane. The figures on the elevation suggest the comparative depth of color between the lines adjacent to the figures. The reasons for so distributing the tints will now be given. See also Fig. B. The darkest part of the figure may properly be assumed to be that to which the rays of light are tangent; viz., the vertical line at tt\ from which the tint becomes lighter in both directions. The lightest line is that which reflects most light to the eye. Now it is a principle of optics that the incident and the reflected i*ay make equal angles with a perpendicular to a surface. But nQ is the incident ray to the centre, and Qe the reflected ray from C to the eye (12). Hence d., which bisects ne, shows where the incident ray, ed, and reflected ray, dq^ make equal angles with the per})endicular (normal) c?C, to the surface. Hence d — d' is the lightest element. 186. Remark. — The question may here arise, "If all the light that is reflected towards the eye is reflected from d — as it appears to be — how can any other point of the body be seen ?" To answer this question requires a notice— ^rs^;, of the difierence between polished and dull surfaces ; and, second., between the case of light coming xohollyxw one direction, ox xyrincipally in one direction. If the cylinder CC, considered as perfectly polished, were deprived of all reflected light from the air and surrounding objects, the line at dr—d' would reflect to the eye all the light that the body would remit towards the eye, and would appear as a line of brilliant light, while other parts, remitting no light whatever, would be totally 6U SHADING. invisible. Let iis now suppose a reflecting inodium, tliough ua imperfect one, as the atmosphere, to be thrown arouml tlie body. By reflection, every part of the body would receive some liglit from all direction?, and so would remit some light to the eye, making the body visible, though faintly so. But no body has a polish that i? absolutely perfect ; rntlier, the great majority of those met with in engineering art have entirely dull sui'faces. Now a dull surface, greatly magnified, may be supposed to have a structure like that shown in PI. X., Fig. 97, in which many of the nsperities may be supposed to have one little facet each, so situated as to remit to the eye a ray received by the body directly from the pnncipa) source of light. 187. Having thus shown how any object placed before our eyes can be seen, we may proceed with an explanation of the distribu- tion of tints, b is midway between d and t. At A, the ink may be diluted, and at e — e\ much more diluted, as the gradation from a faint tint at e to absolute whiteness at d should be without any abrupt transition anywhere. The beam of incident rays which falls on the segment dn^ is broader than that wdiich strikes the equal segment de ,' hence the segment nd is, on the elevation, marked 5, as being the lightest band which is thited at all. The segment w, behig a little more obliquely illuminated, is less bright, and in elevation is marked 3, and may be made darkest at the left hand limit. Finally, the segment ru receives- about as much light as r?^, but reflects it within the very narrow limits, 5, hence appears brighter. This condensed beam, s, of reflected rays would make rytlie lightest band on the cylinder, but for two reasons; Jirst., on account of the exaggerated eflfect allowed to increase of distance from the eye; and, second^ because some of the asperities, Fig. 97, would obscui'e some of the reflected I'ays from asperities still more remote; hence rv is, in elevation, marked 4, and should be darkest at its right hand limit. 188. The process of shading is the same as in the last exercise. Each stripe of the preliminary ])rocess may extend past the i)reccd- ing one, a distance equal to that indicated by the short dashes at the top of Fig. 94. When the whole is finished, there should be a uniform gradation of shade froni the darkest to the lightest line, free from all sudden transitions and minor irregularities. ISO. Ex. 3°. To shade a rUjlit cone standing upon the horizon- lal plane, together with its shadow. Pi. X., Fig. 95. The sha- dow of the cone on the horizontal plane, will evidently be bounded SHADING. 81 by the shiidows of those Vines of the convex surface, at which the li<:;ht is tangent. The vertex is coninion to both these lines, and casts a shadow, y'". The sliadows being cast by strnight lines of the conic surface, are straiglit, and their extremities must be in the base, being cast by lines of the cone, which meet the horizontal plane in the cone's base; hence the tangents v"'t and v"'t" are tlu boundaries of the cone's shadow on the horizontal plane, and tlie lines joining i' and i!" with the vertex aie the lines to whicli th( rays of light are tangent; i.e., they are the darkest lines of the sliading ; hence tv, the visible one in elevation, is to be vertically projected at t'v'. The lightest line passes from vv' to the middle point, y, between ti and p in the base. At q and at/) a change in the darkness of the tint is made, as indicated by the figures seen in the elevation. In the case of the cone, it will be observed that the various bands of color are triangular rather than rectangular, as in tlie cylinder ; so that great care must be taken to avoid tilling up the whole of the upper part of the elevation with a dark shade. See PI. X., Fig. C. 190. Ex. 4°. To shade f/te elevation of a sphere. PI. X., Fig. 96. It is evident that, if there be a system of pai-allel rays, tan- gent to a s])here, their points of contact will form a great circle, perpendicular to these tangents; and which will divide the light from the shade of the sphere. That is, it will be its circle of shade. Each point of this circle is thus the point of contact of one tangent ray of light. If now, parallel planes of rays, that is, planes l)arallcl to the light, be passed thi'ough the sphere, each of thera will cut a circle, great or small, from the sphere, and there will be two rays tangent to it on opposite sides, whose jwiiUs of contact will be points of the circle of shade. In the construction, these parallel planes of rays will be taken perpendicular to the vertical plane of projection. Next, let us recolleci, that always, when a line is parallel to a jjlane, rts projection on that plane will be seen in its true direction. Now ]^Y)' being the direction of the liglit, as seen in elevation, let BD' bo the trace, on the vertical plane of projection — taken through the centre of the sphere — of a new plane perpendicular to the vertica. plane, and therefore parallel to the rays of light. The projection of a ray of light on this plane, BD', will be pai-allel to the ray itself, and therefore the angle made by this projection with the tiace BD will be equal to the angle made by the ray with the vertical plane But, referring to PI. IX., Fig. 80, we see that in the triangle 82 sriADixG. LL'L,, containing lli j angle LLiL' made by the ray LL, with tlie ver- tical plane, tlie side L'L, is the hypotliennse of tlic triangle L'i li,, each of A\hose other sides is equal to LL'. Hence in PI. IX., Fig. 96, take any distance, Be, make AB peri)endicular to BD', arid on it lay off BDrrBc, tlien make BD'i^Dc, join D and D', and DD' will be tlie projection of a ray upon the plane BD', and BD'D will be the true size of the ajigle made by the light with tlie vertical plane; it being understood that tlie plane BD', though in space perpendiculai- to tlic vertical iilaue, is, in the figure, rejn'esented as revolved over towards the right till it coincides with the vertical plane of projection, and with the paper. 191. We are now ready to find points in the curve of shade, oo' is the verticnl projection of a small circle parallel to the plane BD', and aLso of its tangent rays. The circle og'o\ on oo' as a diameter, represents the same circle revolved about oo' as an axis and into the vertical plane of projection. Drawing a tangent to og'o' ^ l)arallul to DD', we find g\ a visible point of the curve of shade, whicli, when the circle revolves back to the position oo' ^ appears at J/, since, as the axis oo' is in the vertical plane, an arc, g'g^ described about that axis, must be ye?*^«ea//y projected as a straight line. (See An. yi.) In a precisely similar manner ai"e found h.^ Jc, w?, and f. At A and B, rays are also evidently tangent to the sphere. Through A, y, «S:c., to B the visible i)ortiou of the curve of shade may now be sketch''d. 192. The niost highly illuminated point is 90° distant fioin tli< grcai circle of shade; Tience, on ABQ, the revolved position of a great circle which is perpendicular to the circle of shade, lay oft* ]c'Ql=^qV> = \\ic chord of 90°, and revolve this ]jerpendicular circle back to the })osi»ion qq', when Q will be found ^t, r'. But th«? brilliant point, as it appears to the eye, is not the one which receivei most light, but the one that reflects most, and this point is midway, iu space, between r' and r, i.e. at P, found by bisecting QB, and drawing RP; for at K the incident ray whose revolved position is parallel to Qr or DD', and the reflected ray whose revolved posi- tion is parallel to rB, make equal angles with R'R, the perpen- dicular (normal) to the surface of the sphere. See PL X., Fig. D. 193. In regard now to the second general division of the problem — the distribution of tints ; a small oval space around P should be It It blank. The first stripe of dark tint reaches from A to B, along tlie curve of shade, and the successive stripes of the same tint extend to B<7A on one side of BAvV, and to octjo on the other. Then take SHADING. 83 ca lighter tint on the lower half of the next zone, and a still lighter one on its upper half (2) and (3). In shading the next zone, use an intermediate tint (3-4), and in the zone next to P a very light tint on the lower side (4), and the lightest of all on the- upper side (5). After laying on these preliminary tints, even up all sudden transitions and minor irregularities as in other cases, 194. Ex. 5°. Shades and Shadows on a Model. PL XL, Figs. 1 and 2. General Description. — This plate contains two elevations of an architectural Model. It is introduced as affording excellent practice in tinting and shading large surfaces, and useful elementary studies of shadows. The construction of these elevations from given measurements is so simple, that only the base and several centre lines need be pointed out. QR is the ground line. ST is.a centre line for the flat topped tower in Fig. 1. UVis a centre line for the whole of Fig. 2, ex- cept the left-hand tower and its pedestal. WX is a centre line for the tower through which it passes. YZ is the centre line for the roofed tower in Fig. 1. The measurements are recorded in full, referred to the centre lines, base line, and bases of the towers, \vhich are the parts to be first drawn. Graphical construction of the shadows. 1". The roof, D — D'D", casts a shadow on its tower. The point, EE', casts a shadow where the ray, Ee, pierces the side of ^le tower. e is one projection of this point ; e', the other projection of the same point, is at the intei'section of the line ee"e' with the other projection, E'd', of the ray. Tlie shadow of a line on a parallel plane (162) is parallel to itself, hence e'f\ parallel to E'F', is the shadow of E'F'. The shadow of DE — D'E' joins e' with the shadow of D — ^D'. The point c?, determined by the ray Dc?, is one projection of the latter shadow ; the other projection, cl\ is at the intersection of dcl'd' witli tlic other projection, T>'cl\ of the ray, d' is on the side of the tower, produced, hence e'd' is only a real shadow line from e till it intersects the edge of the tower. Remembering that the direction of the light is supposed to change with each position of the observer, so that as he faces each side of the model, in succession, the light comes from left to right and from behind his left shoulder, it appears that the point, DD", casts a shadow on the fece of the tower, seen in Fig. 2, and that \y"d"" will be the position of the ray, through this point, on F'ig, 84" SHADIXG. 1. The point tV" is therefore one projection of the shadow of I)D". The other is at d"'\ the intersection of the lines d"'d"" with Y)d"'\ the other projection of the ray. Likewise EE" casts a shadow, e"'e"'\ on the same face of tlie tower, produced. DD'", being parallel to the face of the tower now being considered, its shadow, d""q^ is parallel to it. The line from d"" towards e"", till the edge of the tower, is the real i)ortiou of the shadow of ])K_D"E". From the foregoing it will be seen how most r)f these sha- dows are found, so that each step in the process of finding similar hadows will not be repeated. 2°. The body of the building — or model — casts a shadow on the roofed tower, beginning at AA' (lOl). The shadow of BB' on tlie side of this tower is bb\ ibund as in previous cases, and A'b' is the shadow of AB — A'B'. From b' downwards, a vertical line is the shadow of the verti(;al corner edge of tlie body of the model upon the parallel face of the tower. 3'. The line CC" — C, which is perpendicular to the side of the roofed tower, casts a shadow, C'c', in the direction of the projection of a ray of light on the side of the tower. 4°. In Fig. 1, a similar shadow, »Y, is cast by the edge s' — sa' or" the smaller pedestal. 5°. In Fig. 2, is visible the curved shadow, c''rg^ cast by the vertical edge, at c", of the tower, on the curved part of tlie pedestal of the to\i'er. The point r/ is found by drawing a ray, G'C — G^, which meets the upper edge of the pedestal at r/C. The point c'l the intersection of the edge of the tower with the curved part of the pedestal, is another point. Any intermediate point, as r, is found by drawing the ray R'r'/ r' is then one projection of the sh.idow of li'K, and the other is at the intersection of the line r'r with the other projection llrof the ray. These are all the shadows which are very near to the objects casting them. 0". The flat topped tower casts a shadow on the roof of the body. The upper back corner, IIII', casts a shadow on the roof, of which h is one ])rojection aii.l '' the other. The back upper edge II — IIT' oeing parallel to the roof, the short shadow /i'A", leaving the roof at A", is pai-allel to HI'. The left hand back edge IIJ — IIM' casts a BhaV' ^M \ ^; «"(/' — A'D, Fig. A, become parallel to the plane of i^rojection, XY. All other lines of Fig. 100 appear less than their real size. 4°. By 3°, e"f"n is evidently an equilateral triangle ; and, by 1° and 2°, all the other lines make equal angles of 30'' with these. Hence the perimeter Df'onpe" is a regular hexagon, and the projections of the visible foremost, and invisible hind- most corners of the cube coincide at C. Hence the diagonal of the cube, which joins these points, is perpendicular to the plane of projection. 201. Definitions. Fig. 100: C is the isometric centre, Ce", Qf", C'n, are the isometric axes. Lines parallel to these axes are isometric lines ; others are non-isometric lines. Planes par- allel to the faces of the cube arc isometric planes. The edges of a cube, or similar rectangular body, are the lines FIRST PRINCIPLES OF ISOMETRICAL DRAWING. 89 whose dimensions would naturally be desired. Hence it is usual to make Ce", Cf, He", etc.. Fig. 100, equal to the edges of the cube itself ; as in Fig. 101, Such a figure is, for distinction, called the Isometrical Drawing of the cube, and it is the isomet- rical projection of an imaginary larger cube, which is to Fig. 101 as Fig. 99 is to Fig. 100. 202. Another demonstration* The first principles of iso- metrical projection may be developed from the following Proposition. The plane luhich is perpendicular to the body diagonal of a cube, is equally inclined to its faces and edges. Let abcd—a'b'c'd'a"b"c"d", PI. XXL, Fig. A, be the plan and elevation of a cube, shown as in PI. XII., Fig. 99, only that the ground line, GL, is inclined, to permit the construction of the isometrical figure in an upright position, as in Fig. 100, by direct projection from the given elevation. The body diagonal, ac — a'c" , parallel to V, is the common hypothenuse of three equal right-angled triangles, similarly situ- ated relative to the cube. One base of each of these triangles is an edge, beginning at cc" ; the other is a face diagonal, begin- ning at aa' . Thus, one of these three triangles is ac — a'c'c" , which, being parallel to V, shows its real size on V. Another has for its bases the edge be — b"c" and the face diagonal ab — a'b" ; and the third has for its bases the edge dc — d"c" , and the face diagonal ad~a'd". !N^ow since these triangles arc tbus equal, and similarly i)laced on the cube, their common hypothenuse, the body diagonal a'c" , making equal angles with the three face diagonals which meet at aa' , also makes equal angles with the faces containing these diagonals. Hence a plane of projection XY, or Ti, perpendicular to this body diagonal, is equally inclined to the three faces of the cube "which meet at aa' , or at cc" ; which agrees with the enunciation. From this conclusion follow all the other particulars in the preceding articles. In making the figure, H and the plane XY (Vi) are both per- pendicular to Y. Hence Cn=cc" ; 'Do=dd"j etc. * This may be omitted at discretion, but may be preferred by Teachers and others, as fresher, and more concise and strictly geometrical. CHAPTER II. PROBLEMS INVOLVIXG OKLY ISOMETRIC LINES. 203. Prob. 1. To construct the isomctrical jJrojections, and draioings of cubes, and a rectangular hloch cut from a corner of one of them. The principles of the last chapter yield several simple con- structions, as follows: 204. First Method. Draw an equilateral triangle, as e"f"n. Fig. 100, each of whose sides shall be equal to a face diagonal of the cube. Then (200, 4°) draw two lines, as e"D and e"Q, with the 30° angle of the 30° and 60° triangle, making angles of 30° with each side of the triangle e"f"n, at each extremity. These lines will intersect as at C, D, o, etc., forming the isometric j9ro/ec^«ow of the cube. By extending Q,e" , C/", Qn, till equal to the edge of the cube, and joining their new outer extremities, the projec- tion will be converted into the isometric drawing. 205. Second 3Iethod. Fig. 100. Draw the isometric axes indefi- nitely, Ce" and Cf" each making angles of 30° with a horizontal EF, on which lay off half of a face diagonal e/. Fig. 99, each way from C, giving E and F. Then perpendiculars at E and F will limit the right and left axes at e" and f". Cn is then made equal to Ce" or C/", and the remaining edges are drawn parallel to these three. From this ^Jrojeciion, make the draiviyig as in the first method. 206. Third Method. Drawing tlie axes as before, lay off on each the true length of an edge of the cube, and complete the figure as just described. This at once makes the draining. 207. Fourth Method. Fig. 101. With centre C and radius equal to an edge of the cube, draw a circle, and in it inscribe a regu- lar hexagon in the position shown, adding the alternate radii Ca, Ch, Cc, which again gives the draiving of the cube. Let a prismatic block be cut from the front corner of this cube. Suppose the edge of the cube to be five inches long, drawn PROBLEMS rSTVOLVING OXLY ISOMETRIC LINES. 91 to a scale of one-fifth. Let Ca'=2 inches; CJ'^S inches; Cc' = 1 inch. Lay off these distances upon the axes, and at a', V , c' , draw isometric lines which will be the remaining visible edges of the block. Examples. — \8t. Draw a square panel in each visible face. '^d. Draw a square tablet in each visible face. od. Let a cube l^- inches on each edge be cut from every visi- ble corner of the cube. 208. Prob. 2. To find the shade lines on a cube, and the shadows of isometric lines upon isometric pkmes. Three faces of the cube, PL XII., Fig. 102, will be illuminated by light passing in the direction LL', from a to p. Two of these faces, ao7iC, and aCqD are visible. The under and right-hand faces, ojipC, npqC, and pqDC, are in the dark, hence by the rule (18) the edges on, nC, Cq and qD are to be inked heavy. The shadoivs of al)=-oa produced ; and of ad=J)a produced. ai is perpendicular to aCqD, hence (162, Ath) its shadow will be in the direction of the projection of the light upon that face. But aq is evidently the projection of the ra}', LL' upon aCqD', and as a is where ab meets aCqJ) (1G2, btJi), ah' is the shadow of ab on aQqT). In like manner, ad' is the shadow of ad upon aonO. Otherwise. A plane containing oa and ap, will contain rays through points of oab, and will also contain jog and W'ill therefore cut aQqJ) in the line aq. Hence rays from all points of ab will pierce aCqJ} in aq. Hence ab' is the shadow of ab. Again ; Idap is the plane of rays, containing DcZ and pn, and cutting the face aonQ in the line an. Hence d' is the shadow of d, and ad', that of ad. Similar constructions apply to the isometric projections and drawings of all rectangular bodies. Examples. — 1st. Find the shadow of the cube on the plane of its lower base. 2t?. Find the shadow of aL (considered as Ca produced) upon the rear face Dao. Zd. Find the shadow of a line parallel in space to Ca, upon the face Gaon, 92 PROBLEMS IXVOLTIXG OXLT ISOMETEIC LIN"ES. Wi. Find the shadow of Dq produced, on tlie plane of the lower base of the cube. 209. Pkob. 3. To construct the isometrical drawing of a car- penter's oil-stone lox. PL XIIL, Fig. 103. This problem involves the finding of points which are in the planes of the isometric axes. Let the box containing the stone be 10 inches long, 4 inches wide, and 3 inches high, and let it be drawn on a scale of \. Assume C then, and make Ca=4: inches, C5=10 inches, and Cc=3^ inches, and by other lines «D, Db, &c., equal and parallel to these, complete the outline of the box. Eepresent the joint between the cover and the box as being 1 inch below the top, aCh. Do this by making Cj) = l inch, and through p drawing the isometric lines which rejiresent the joint. Sujipose, now, a piece of ivory 5 inches long and 1 inch wide to be inlaid in the longer side of the box. Bisect the lower edge at (I, make de=l inch, and ee=l inch. Through e and e', draw the top and bottom lines of the ivory, making them 2^ inches long on each side of de', as at e't, e't'. Draw the vertical lines at t and i', which will complete the ivory. 210. To show another way of representing a similar inlaid piece, let us suppose one to be in tlie top of the cover, 5 inches long and 1\ inches wide. Draw the diagonals ab and CD, and through their intersection, o, draw isometric lines ; lay off oo'= 2^ inches, and cio" = | of an inch, and lay off equal distances in the opposite directions on these centre lines. Through o', o", &c., draw isometric lines to complete the representation of the inlaid jiiece. PL XII., Fig, B, suggests other exercises, and illustrates the gaining of room by making the longest lines of the figure hori- zontal, as is often done in practice, when the longest lines of the original object are horizontal, since it is not tlie position, but the /orm of the figure wliicli makes it isometrical. Examples. — 1st. Reconstruct, isometricalh', two courses of any of the examples in ])rick-W()rk on PL V. '2d. Do likewise witli any of the figures (4G-49) of PL VII., assuming any convenient width for the timbers in each case. PL.Xll. r r PROBLEMS INVOLVING ONLY ISOMETRIC LINES. 93 211. Prob. 4. To represent the same hox {Frob. 3) tvith the cover reiiiovcd. PL XIIL, Fig. 104. This j^roblem involves the finding of the positions of jooints not in the given isometric planes. Suijposing the edges of the box to be indicated by the same let- ters as are seen in Fig. 103, and suj)posing the body of the box to be drawn, 10 inches long, 4 inches wide, and 2i} inches high; then, to find the nearest upper corner of the oil stone, lay oif on Cb li- inches, and through the point /, thus found, draw a line, ff, parallel to Ca. Oiiff, lay oif 1 inch at each end, and from the points Jih', thus found, erect perpendiculars, as hn, each f of an inch long. Make the further end, x, of the oil stone li- inches from the further end of the box, and then complete the oil stone as shown. To find the panel in the side of the box, lay off 2 inches from each end of the box, on its lower edge; at the points thus found, erect perpendiculars, of half an inch in length, to the lower corners, as p, of the panel ; make the 2:>anels 1^ inches wide, as at ^t/;', and -^ an inch deep, as atpr, Avhich last line is parallel to Cd. ; and, Avith the isometric lines through r, p, p' , and ;*, completes the panel. 212. The manner of shadimj tliis figure Avill now be explained. The top of the box and stone is lightest. Their euds are a trifle darker, since they receive less of the light which is diffused through the atmosphere. The shadoAv of the oil stone on the top of the box is much darker than the surfaces just mentioned. The shadow of the foremost vertical edge of the stone is found in precisely the same Avay as was the shadow of the Avire iTpon the top of the cube, PI. XII., Fig. 102. The sides of the oil stone and of the box, Avhich are in the dark, are a little darker than the shadoAv, and all the surfaces of the panel are of equal darkness and a trifle darker than the other dark surfaces. In order to distinguish the separate faces of the panel, Avhen they are of the same darkness, leave their edges very light. The little light Avliich those edges receive is mostly perpendicular to them, re- garding them as rounded and polished by use. These light lines are left by tinting each surface of the panel, separately, with a small brush, leaving the blank edges, Avhich may, if necessary, be afterAvards made jierfectly straight hx inking them Avith a di PKOBLEMS IXVOLVI^'G OXLT ISOMETRIC LINES. light tint. The upper and left-hand edges of the panel, and all the lines corresijonding to those which are heavy in the previous figures, may be ruled with a dark tint. In the absence of an engraved copy, the figures will indicate tolerably the relative darkness of the different surfaces ; 1 being the lightest, and the numbers not being consecutive, so that they may assist in denot- ing relative differences of tint. When this figure is thus shaded, its edges should not be inked with ruled lines in black ink, but should be inked with pale ink. Examples. — 1st. Eeconstruct, isometrically, any of the figures 53, 54, 55, 58, on PL VIL; or figures 63, 64, 65, 67, of PI. VIII., omitting the bolts, [The few non-isometric lines that occur, being in given isometric planes, can be located by a litle care.] 2d. Construct the isometrical drawing of a low stout square- legged bench, and add the shadows. 3c?. Construct the isometrical drawing of a cattle yard -with pens of various sizes and heights. 4:th. Construct the isometrical drawing of a shallow partitioned drawer, or box, with the shadoAvs. 5th. Make the isometrical drawing of a cellar having a wall of irregular outline, and showing the chimneys, partitions, bins, etc. CHAPTER III. PnOBLEMS IXVOLVING NOX-ISOMETRICAL LINKS. 213. SixcE non-isometiical lines do not appear in their true size, each point in any of them, when it is in an isometric plane, must be ocated by two isometric lines, which, on the object itself, are at light angles to each other. Points, not in any known isometrical plane must be located by three such co-ordinates, as they are called, from a known point. 214. Prob. 5. To construct the isometrical draining of the scarfed splice, shoivn at PL VIII., Fig. 66. Let the scale be ^, or three-fourths of an inch to a foot. In this case, PI. XIII., Fig. 105, it will be necessary to reconstruct a portion of the elevation to the new scale (see PI. XIII., Fig. 106), where A;; = l|- feet, 7jA" = 6 inches, and the proportions and arrangement of parts are like PL VIII. , Fig. OG. In Fig. 105, draw AD, 3 feet; make DB and AA' each one foot, and draAv through A' and B isometric lines parallel to AD Join A — B, and from A lay off distances to 1, 2, 3, 4, equal to the corresponding distances on Fig. 106. Also lay off, on BK, the same distances from B, and at the points thus found on the edges of the timber, draw vertical isometric lines equal in length to those which locate the corners of the key in Fig. 106. Notice that opposite sides of the keys are parallel, and that AV, and its jiarallel at B, are both parallel to those sides of the keys which are in space per- pendicular to AB. To represent the obtuse end of the upjjcr tim berof the splice, bisect AA', and make va=Aa, Fig. 106, and draw Aa and A'a. Locate m by va produced, as at ar, Fig. 106, and a short perpendicular=rm. Fig. 106, and draw mV and mV ; W being parallel to AA', and am being parallel to AV. To represent the washer, nut, and bolt, draw a centre line, vv', and at t, the mid- dle point of Vw, draw the isometric lines tti and tie, which will givi c, the centre of the bolt hole or of the bottom of the washer. A point — coinciding in the drawing with the upper front comer of the nut — is the centre of the top of the washer, which may be made } of iui inch thick. 9G rU015LKSIS INVOLVIXG NO.\-ISO-METKICAL LINES. Tlnoup', and lay oli' on them, Ironi the same point, the radius of the washer, say 2^ inches, giving four points, as o, through Mhich, if an isometric Bquare be drawn, the top circle of the washer can be sketched in it^ being tangent to the sides of this square at the points, as o, and elliptical (-oval) in shape. The bottom circle of the washer is seen throughout a semi-circumference, i. e. till limited by vertical tan- gents to the upper curve. On the same centre lines, ^ay off from their intersection, the half side of the nut, 1 \ inches, and from the three corners which will be visible when the nut is drawn, lay off on vertical lines its thickness, 1;! inches, giving the upper corners, of which c is one. So much being done, the nut is easily finished, and the little fragment of bolt projecting through it can be sketched in. Example. — Construction of the nut when set ohliqudy. The not is here constructed in the simplest position, i. e. with its sides in the direction of isometric lines. If it had been determined to construct it in any oblique position, it would have been necessary to have constructed a portion of the plan of the timber with a plan of the nut — then to have circumscribed the plan of the nut by a square, parallel to the sides of the timber — then to have located the cor- ners of the nut in the sides of the isometric drawing of the circum- scribed square. Let the student draw the other nut on a large scale and in some such irregular position. See Fig. 107, Mhere the upper 6gure is the isometrical drawing of a squaie, as the top of a nut; this nut having its sides oblique to the edges of the timber, which are supposed to be parallel to ca. 215. Proi;. G. 7b make an isometrical drawing of an oblique timber framed into a horizontal one. PI. XIII., Fig.108. Let the ( riginal be a model in which the ho'"izontal piece is one inch square, and lot the scale be \. Make ag and aA, each one inch, complete the isometric end of the horiziint.d piece, :ind draw ad^hn ^wdi gk. Letf/J — 2 inches, ic = 2J inches, and draw bm. Let the slant of the oblique timber be such that if «7=2 inches, de shnil be | of an inch. Then by (21.3) de^ piirallel to ag, will give ce, whi(;h docs not show its true size. Draw edges parallel lo ce through b and m. In like mannerycan be found, by distances taken from a side elevation, like PI. VIII. , Fig. 59. 21 (J. Pro«. 7. To make an isometrical drawing of a pyramid ttandbaj upon a recessed pcdcaUd. PL XIII., Fig. 109. (From a k PltOllLESIS INVOLVING NON-ISOAIETKICAL LINKS. 97 Model.) Let the scale be ^. Assuming C, construct the isometrio square, CABD, of which each side is 4i inches. From each of its corners lay off on each adjacent side 1^ inches, giving points, as e and/'/ from all of these points lay off, on isometric lines, distances of I of an inch, giving points, as is one point of this shadoAV, hence pq is the direction of this shadow. Now n, where mn pierces the actual face of the wall, is one point of its shadow on that face, and q, where its shadow on the hoiizontal i)Iaue pierces the same face, is another point, hence 9tQ is the general direction of the shadow of tnn on the front of the wall, and the actual extent of this shadow is 7U\ r being where the ray ttir pierces the front of the wall. From r, the real shadow is cast by the edge, mu, of the counter- fort, r, the shadow of m, is one point of this shadow, and s, where um produced, meets yn i)roduced, is another (in the shadow pro- duced), since s is, by this construction, the point where Km, the line casting the shadow, pierces the surface receiving the shadow. Hence draw srt, and m't is the complete boundary of the shadow 6ou<;ht. The shadow of the hither counterfort is similar, so far aa it i'alls on the face of the wall. Otiier methods of constructing this shadow may be devised by the student. Let t be found by means of an auxiliary shadow of um on the plane of the ba8« of the walL liemark. — In case an object has but few isometrical lines, it ii most convenient to inscribe it in a right pri>ni, so that as many ot' its edges, as pos>il(I(', shrtll lie in the faces of the prism. CHAPTER IV. PBOHi.E.VS IHVOLTTNG THE COXSTRUCTION AND EQU^i DIVIKIOS Ot CIRCLES IX ISOMETRICAi DRAWING. 220. Prob. 9. To make an exact constructiofi of the isometrical drawing of a circle. PL XIII., Figs. 111-112. This coustruction is only a special application of the general problem requiring the con- struction of ]>oints in the isometric planes. Let PI. XIII., Fig. Ill, be a square by which a circle is circum- scribed. The rhombus — Fig. 112 — is the isometrical di-awing of the; same square, CA being equal to C'A'. The diameters g'h' and ef are those which are shown in their real size at gh and ef giving .V, /?, e, and ^/" as four pohits of the isometrical drawing of the circle. In Fig. Ill, draw h'a\ from the intersection, h\ of the circle with A'D' and parallel to C'A'. As a line equal to h'a\ and a distance equal to K'a' can be found at each corner of Fig. Ill, lay off each way from each corner of Fig. 112, a distance, as Aa, equal to A'a', and draw a line ah parallel to CA and note the point 5, where it meets AD. Similarly the points ?i, o, and r may be found. Having now eight points of the ellipse which will be the isometrical draw- ing of the circle, and knowing as further guides, that the curve is tangent to the circumscribing rhombus at 7=one inch, and draw eg and hg. Through A and b draw isometric lines which will meet, as at a'. On JE make bJc, hn and wE each equal to 3 inches, and let kl and mn each be 3j inches. At I and ?^ draw lines, as ?y, parallel to CB, and one inch long, and at their inner extremities erect perpendicu- lars, each 3^ inches long. Also at A, ?, m and n^ draw vertical iso- metrical Hues, as kt^ 3|- inches long. The rectangular openings thus formed are to be completed with semicircles whose real radius is If inches, hence produce the lines, as kt — on both Avindows — mak- ing lines, as /cG, 5\ inches long, and join their upper extremities a3 at GI. The horizontal lines, as ts, give a centre, as 5, for a larger aic, as tu. The intersection of Go with Jz — see the same letters on E'ig. 115 — gives the centre, jt?, of the small arc, uo. The same ope- rations on both openings make their front edges complete. Make oq and j>r parallel, and each, one inch long, and r will be the centre of a small arc from q Avhich forms the visible part of the inner edge of the window. Suppose the corners of the platform to be rounded by quadrants whose real radius is 14- inches. The lines a'b and bk each being 3 inches, k is the centre for the arc which repre- sents the isometric drawing of this quadrant, whose real centre on the object, is indicated on the drawing at y. So, near A, w ia the centre used in drawing an arc, which represents a quadrant vhose centre is x. — See the same letters on Fig 115. 102 PROBLEMS IXVOLVIXG THE ISOMETRICAL PRAWING OF CIRCLES. Of the Isometrical Draicing of Circles which are divided in Equal Parts. 228. Prob. 14. PI. XIV., Fig. 117. First method.— l^ tlie semi ellipse, ADB, be revolved up into a vertical position about AB as an axis, it will appear as a semicircle AD'B of which ADB is the iso- metrical projection. Since AB, the axis, is parallel to the vertical plane, the arc in which any i)oint, as D, revolves, is in a plane per pendicular to the vertical plane, and is therefore projected in a straight line DD'. Hence to divide the semi-ellipse ADB into parts corresponding to the parts of the circle which it represents, divide AD'B into the required number of equal parts, and through the points thus found, draw lines parallel to D'D, and they will divide ADB in the manner required. The opposite half of the curve can of course be divided in a similar manner. 229. Second method. — CE is the true diameter of the circle ot which ADB is the isometrical drawing. Let it also represent the side of the square in which the original circle to be drawn is inscribed. The centre of this circle is in the centre of the square, hence at O, found by making eO equal to half of CE, and perpendicular to that line at its middle point e. With O as a centre, draw a quarter circle, limited by CO and EO, and divide it into the required number of parts. Through the points of division, draw radii and produce them till they meet CE. CE, considered as the side of the isometrical drawing of the square, is the drawing of the original side CE of the square itself with all its points 1, 2, C, 7, &c., and O' is the isometrical posi- tion of O. Hence connect the points on CE with the point 0' and the lines thus made will divide the quadrant BC in the manner required. Aj^lications of the preceding Problem,. 230. Prob. 15. To make an isometric drawing of a segment oj an Ionic Column. PI. XIV., Fig. 118. Let aD be a side of the cir cumscribing piism of the column. By the second method of Prob. 14, fmd O', the centre of a section of the column, and with O' as a centre, draw any arc, as a'q'. The curved recesses in the surface of a column are called flutes, or the colunm is said to be fluted. In an Ionic, and in some other styles of cohmms, the flutings are semi- circular with narrow flat, or strictly, cylindrical surfaces, as ef"/), between them. Hence, in Fig. 118, assume a'b\ equal to q'v\ aa half of a space between two flutes, divide h'v' into four equal parts, and make the points of division central points of the spaces as f'e' PROBLEMS INVOLVING THE ISOMETRICAL DRAWING OP CIRCLES. 103 between the flutes. Lei the flutes be drawn witli points, as c' a» centres and touching the points as b'cl ; then draw an arc tangent, as at r, to the flutes. To proceed now with the isometrical draw- ing, draw, in the usual way, the isoraetiical drawing of the outer circumferences of the column, tangent to aD and h"'¥ — assuming DF for tlie tliickness of the segment. Now a'q' being any arc, and not one tangent to oD so as to represent the true size of a quadrant of the outer circumference, the true radius of the circle tangent to the inner points of all the flutes will be a fourth proportional, O'y' to 0/', Oi (=^0'y), and O's. On O^, lay off OY=Oy', draw IJ to find a centre I, and similarly find the other centres of the larger area of the inner ellipse. The points ?i, h and n\ h' are the centres of the small arcs (222) for the two bases. Having gone thus far, produce 0'b\ O'c', &c. to dD ; at i, c, &c., erect vertical lines, hh"\ cc"\ &c., then from 6, c, &c. draw lines to O, and note their intersec- tions, h'\ c", &c. with the curves of the low'er base ; and from h"\ c"\ Sec. draw lines to O" and note tlieir intersections, h"'\ c"'\ &c. with the elHpses of the upper base. This process gives three points for each flute by which they can be accurately sketched in, remembering that they are tangent to tlie inner dotted ellipses, as at c"", o"\ &c. and to the radii, as c"0" — at e" . Parts beyond FO" are projected over from the parts this side, thus drawn. 231. Prob, 16. To construct the isometrical drawing of a seg- ment of a Doric Coliunn. PL XIV., Fig. 119. The flutes of a Doric column are shallow and have no flat space between them. Adopt- ing the first method of Prob. 14, let the centre, A, of the plan be in the vertical axis, GA', of the elevation, produced. Let Ac and Ab be the outer and inner radii containing points of the flutes. Make Ad=^ of Ac, for the radius of the circle which shall contain the centres of the flute arcs. Let thei'e be four flutes in the qua- drant, shown in the plan. Their centres will be .at h, &c., where radii A^, &c., bisecting the flutes, meet the outermost arc. In pro ceeding to construct the isometrical drawing, project b and c, at b and c' on the axis A'd'. Now, owing to the variation at b and c' between the true and the approximate ellipse, we cannot make use of the latter, if we retain b' and c in their proper places, as projected from b and c, hence through b' and c' draw isometric lines which locate the points N' and Q' (the points are between these letters) which are the true positions of N and Q respectively. Correspond- ing points, between N'" and v, are similarly found. By an irregular cu>"ve the semi-ellipses vb'Q,' and N"V''N' can he quite accurately 104peoblems in^"Oltixg the isometeical dkawixg of circles- drawn. Next, project upon these curves the points u^ e, &c., r, g^ tfcc. of the flutes — as at u\ e\ &c., r', have the corresponding points c'c\ \ Fig. 122, equal to eV, we have the plan of a circle in the section G'E' — A'. Now draw o''x and ju'e' — Fig. 122 — make A'e and G'e'" and e"'p and eo — Fig. 121 — equal to e'a;, Ge', e'p' and xo" — Fig. 122 — draw ji:>Y and oU; and w'lJ and b'Y to intersect them, and we shall have U and Y as the isometric positions in the plane G'E' — A' of the points o' and p' which, considered as points on the circle, are evidently enough extre- mities of its horizontal diameter, at which points, the circle is tan- gent to the vertical lines whose isometric positions in the piano G'E' — A' are7:)Y and oU. T and a' are other points. The finding of intermediate points, which is not difficult, is left ad an exercise for the student. CHAPTER V. OBLIQUE PROJECTIONS. A. There is a kind of projection, examples of which, in the draw ing of details, etc., are oftener seen in French works than isometri cal projection (an English invention) is. It has been variously named, " Military," " Cavalier," or " Mechanical " Perspective. It may be called " Cabinet Projection," it being especially applicable to objects no larger than those of cabinet work, and being actually used in representing such work. It is properly called oblique projection, because in it the projecting lines, which have been hitherto made perpendicular to the plane of projection are oblique to that plane; and ^lictorial projection^ on account of its pictorial effect, as seen in PI. I., Fiirs. 1, 2, 3, etc. 2. This new projection differs from isometrical, chiefly in show- ing two of the tliree dimensions of a cube, for example, in their true direction as well as size. Thus; Fig. 1 is the isometrical drawing of a cube, and Fig. 2 is / /\ /\ h[ c '"/{ / Fig. 2. an oljiifjue })rojection of the same cube ; all the edges being of the same length in both figures. Hence we see, as stated, that in the latter figure, the faces DEFG, and ABCII, and by consequence every line in them, are shown in tlieir true form, as well as sIm ^ which is not true of isometrical drawing. OBLIQUE PROJECTION'S. 107 3. Another advantage of oblique projection, already apparent, is, that the remote corner, H, which, in the isonietrical drawing of a cube, coincides with the foremost corner, G, is seen separately in the oblique piojection. Also, of the four body diagonals of the cube, one, GIT, appears as a point only, in i.sometrical projection, and the other three, as FC, are all partly confounded with the projections, as FG, of edges. But in the oblique projection, all these diagonals show as lines, and, except BE, separately from the edges of the cube. 4. In the projections hitherto considered, the projecting lines of a point have uniformly been taken perpendicular to the planes of projection. Sometimes, however, the projecting lines, or direction of vision (12) are oblique to the plane of projection. There are thus two systems of projection in which the eye is at an infinite distance, l^lrst, common or peiyendicular p7'oJections, in which the projecting lines (5) are perpendicular to the plane of projection. Second. Oblique projections^ in which those lines are oblique to the plane of projection, Isometrical, is a species of perpendicular projection. We shall now proceed to ex2Dlain the simple and useful form of projection which is called oblique. projection. 5. If a line, AB, Figs. 3, or 3a, be perpendicular to any plane PQ, its projection on that plane, in common projection, would be simply the point B. But if we suppose the projecting line, AC, of any point. A, to make an angle of 45° with the plane of projection PQ, it is evident that the piojection of A would be at C, and the projection of AB on PQ would be BC; also that BC=AB. That is, the projectioti, as BC, of a j^eipendicular to the plane of project tion is equal to that perpendicular itself. Any line through A and /)a?'a/^eZ' to the plane PQ would evidently be projected in its real size on PQ. Hence, finally, the system of oblique ])rojections here described, allows us to show the three dimensions of a solid in their real size^ on a single figure; but only parallels., and perpendicidars to the plane of projection, appear in their true size. 6. It is now evident from Fig. 3a, that there may be an infinite number of lines from A, each making an angle of 45° with the plane PQ. Tnese lines, taken together, would form a right cone with a circular base, whose axis would be AB, whose vertex would be A, and whose base would be a circle in the plane PQ, drawn with B as a centre, and BC as a radius. Each radius of this circle 108 OBLIQUE PROJECTIONS. would be an oblique projection of AB, corresponding willi the element as C'A, from its extremity, taken as the direction of the projecting lines. That id, the ohliqiie projection of AB may be drrfrin equal to AB and in any direction. 7 Thus Figs. 4, h, 5 and many more are all equally oblique projec- tions of the same c ibe. The paper represents the plane of projec- tion ; FA is perpt.iditmlar to the paper at A. The eye, relative tc Fig. 2, is looking /roy.'' 'n, infinite distance above and to the right of the body, and in a direction making an angle of 45° with the pa}>er. And, generally, in oblique projection, the direction of Xiinon^^the projecting lines^ may have any direction (the same for ail points in the same problem) making an angle of <^b'^ with the jilane of projection. Ilence FA may have any direction relative tc» FG and y(it be always equal to FG, that is to the original of FA .n space. Thus, CDK = 45°, in Fig. 4 ; 30°, in Fig. 5; and 60°, in Fig. 6. Also, DC, FA, etc., may incline to the left^ or downward. Fig. 6. OBLIQUE PROJECTION'S. 109 In Fig. G, CDK = 60°. Accordingly DK==iDC, from wliich, having ibund K, the perpendicular KC can be drawn to limit DC. Or, as bef«»re, CK=i Vs^ DC being =1. That is, CK=r-iEC, since CDE=:12(>°. And for a square prism of any length, KC= half of the diagonal joining alternate vertices of a regular hexagon whose side equals the edge of the prism, lying in the direction of DC, an J whose length is supposed to be given. Fig. 6. With these illustrations, the student might proceed to investi- gate other relations between the parts of these, and still other oblique projections. But the above may suffice for now. We observe that, in Figs 5 and 6, none of the body diagonals are confounded with the edges ; and that each of the tJiree forma may be preferable for certain objects. 8. Points not on the axes EF, EH, and ED, or on parallels to them, are found by co-ordinates, as in isometrical drawing. Thus, if ED, Fig. 4, be 4 inches, and if we make Ea = 2 inches, ab paral- lel to EH, = 2|- inches, and be, parallel to EF,=::1|- inches, then c is the oblique projection of a point, 2 inches fi-oui the face FH ; 2^ inches from the face FEG, and l^ inches above the base EDC. This principle will enable the student to reconstruct any of the preceding isometrical examples of straight-edged objects, in ob- lique projection. 9. It only now remains to explain the oblique projections of cir- cles. Let Fig. 7 be the oblique projection of a cube, with circles Inscribed in its three visible faces. One of these circles, abed, will appear as a circle, and so would the invisible one on the parallel rear face. For the ellipse in BCDG, draw the diagonals, BD and CG, ol no OBLIQUE PROJECTIONS. that face. Then in the cube itself, horizontal lines joining corrO' epondin;^ points in the circles abcd^ and hpfu^ are parallel to the diagonal EC. Hence tu^ ef, mp^ and gh determine the points u^ftP Pig. 7. and A, by their intersections with the diagonals BD and CG. Tlie middle points of the sides of the fice BCDG, are also points of the ellipse, and are its points of contact with those sides. The ellipse also has tangents at h and/", paiallcl to BD, and at u and jy^ paral- lel to CG. Hence, having eight points, all of which are ])oints of contact of known tangents, the ellipse can be accurately sketched. 10. The ellipse in the upper face could be found in the same manner. But an approximate construction by circular arcs has been sliown, to test its accuiacy and appearance, as compared with the approximate isometrical ellipse. The ellipse being tangent at L and K, perpendiculars to FA and BA, at those points, will in- tersect at M, the centre of an arc tangent to FA and BA at those points. Then «N pei|)en(licidar to FG at a, and equal to ]\IK, gives N, the centre of the arc an. As the remaining arcs must bo tangent to those just drawn, their centres, r and «, must be the intersections of the radii of the large arcs, with the transverse axis, KM, of" the ellijise. The true extremities of the transverse axis are found by drawiny OBLIQUE PROJECTIONS. Ill pq parallel to AC, and a parallel to it from %c. Tlie error qq' al each end of the transverse axis, is thus seen to be considerable Also the jTieater difference between the radii, than occurs in niak- ins: the isonietrical ellipse, occasions a harsh change of curvature at K^ N, etc. ; so that the approximate construction of the oblique ellipse is of very little value. 11. It is found on trial, that the centre M falls both on BD and EC, so that neither KM nor LM really need be drawn. The reason of this property, Avhich so simplifies the construction, is evident. For BAmAF^^FE are in position as three sides of a regular octa- gon, so that the p .rpendiculars, as KM, from the middle points of those sides, will ueet at the same points with BD, AGM, EC, etc., which are obviously the bisecting lines of the angles of the octa gon, viz. at the centre, M, of the octagon. By varying the angle GFA, as in the previous figures, the stu- dent may discover similar coincidences, which he can explain for himself. Finally, it is to be noticed, that the pictorial diagrams of PI. I., Figs. 1, 2, 3, 5, etc., which are so effective a substitute for actual models, to most eyes, are merely oblique projections of models themselves. Practical Examples. 12. PI. XV. shows some further illustrations of oblique projec- tion in contrast with isometrical drawing. Fig. 1 is an isometi'ical drawing of a roller and axle, showing the parallel circumscribing squares of its several parallel circles; and the circumscribing prism, mnpo, of the roller, placed so as to show its lower base. The distances ah^ bd and de, between the ccntrea of the circles, are thus seen in their true size, in this, and on the next two figures. Also the several circles and their centres, have the same letters on the same figures. Fig. 2 shows an oblique projection of the same object, when its axis, «e, is made perpendicular to the paper. This is the simplest position to give to the object, since its several circles, being then parallel to the paper, will appear respectively as equal circles in the figure. And, generally, in making oblique projections of ob- jects having some circular outlines, the object should be so placed, that the majority of these outlines should be in planes parallel to Ihe plane of projection. Example. — Make ae in any other direction. Fig. 3 shows another oblique projection of the samo object, but 112 OBLIQUE PROJECTIOXS. witli its axis ae parallel to the paper, or plane of projection. Dif- fort-nt wheels and their axles in the same machine, might have the two positions indicated in Figs. 2 and 3. Hence it is necessary to understand both; thoiigh if drawing only a single object of this kind, we should for convenience make it as in Fig. 2, only remem- bering, as explained in previous principles, that ae may be drawn in any direction. Examples. — Is^. In Fig. 1, let the upper end of the axis be the visible one. 2d. In Fig. 3, let ae be horizo)ital and parallel to the paper and let the left hand end of the body be seen. PI. XV., Figs. 4, 5, and G are a plan and two isonietrical draw- ings, in full size, of a hexagonal nut. Fig. 4 is the plan of the nut with the circumscribing rectangle, ninop, containing two of its sides, CD and AF. Fig. 5 is the isonietrical drawing of the same, and thus shows the face CD/i, in its true size. The edges B^i, Cc, etc., and centre heights, IIA, of the faces, also show in their real size, as does the height Oo of the nut. BC is greater than its real size, being more nearly parallel to pn than pm is. AB is less than its true size, AB, Fig. 4; being nearer perpendicular to pn than pm is. Fig. 6 is, peihaps, a more agreeable looking isometrieal figure of the nut, but it shows only the heights in their true sizes, except an jyq equals the diameter of the circumscribing circle MNL of the hexagonal base of the nut, so that half of ^:>2' equals the true width of ilie faces. This figure makes an application of (Frob. XIV., J^irst Method). Thus, having made the isometrieal circle MNKL, in the usual way, describe the semicircle MN'K'L on ML as a diameter, and inscribe the semi-hexagon in it with vertices as M, N', K' and L. Then by revolving the semicircle back to its isometrieal position, N' and K' will fall at N and K. The surface Q, in Figs. 5 and G, represents a spherically rounded surface of the nut, while the surface, R, is plane. By finding three points as c, A and D, Fig. 5, in each upper edge of a face, those edges can be drawn as circular arcs; and the visible boundaries, grj, of the rounded surface Q can be sketched, as indicated. Examples. — 1*^^. Make oblique p)'>'oJections corresponding to Figs. 5 and G. 2d. Also with the top>, R, of the uwt 2yarallel to the plane of pro- jection, and either in isometrieal or oblique projection. Sd. Also as if Fig. were turned 90° about Oo, 8(t as to show only two faces of the nut. OBLIQUE PROJECTIONS. 113 Figs. 7, 8, and 9 show a plan and oblique projection of a ra(;de) of an oblique joint. Fig. V shows, once for all, that in every case of ohlique., as wel) as of isometrical drawing, where the lines as dg and gp^ of the ob- ject, are oblique to each other, the body must be conceived to be inclosed in a circumscribing rectangular prism, whose sides shall contain its points, or from which they can be laid off by ordinatea as mo, parallel or perpendicular to those sides. Fig. 7 is on a scale of one half, and Fig. 9 is in full size. Then, supposing the scale to be the same Cw, Fig. 9, = en Fig. 7, mo. ?iP, ihy etc., in Fig. 9 = mo, np^ ih, etc., in Fig. 7. So fe and /a Fig. 9 = the same in Fig. 7. Thus the edges of timber A are shown in their real size, but those of B are distorted by their position. B is separately shown in its true proportions in Fig. 8, that is so far as its Hoes arc paral- lel to ^0, oO or op. Of the heavy lines in Ohlique Projection. These simply follow the same rule, relative to the given object that is applied in common, or perpendicular projections; (lG-20). Thus, in PL XV., Fig. 2, the semicircles of A, B and C below ae would be heavy, and the opposite parts of D, and E. Also if B, Fig. 8, represents a timber parallel to the ground line, the heavy lines would be as there shown. And likewise on Fig. 9, where these lines are indicated by double dashes across them. In short, conceive the common projections of an object to be given with the heavy lines drawn. The oblique projection of the same object, placed in the same position, would simply show the oblique projections of the same heavy lines. That is, the same lines would be heavy in both kinds of projection. DIVISION FIFTH. ELEMENTS OP MACHINES. CHAPTER I. PRINCIPLES. SUPPORTEllS AND CRANK MOTIONS. General Ideas. 1. Machines generally effect only jo/iyszcaZ changes. That is, they are designed to change either the form or the position of matter. They do this either directly, as in machines that ope- rate immediately on the raw material to be wrought, as looms, lathes, planers of wood or metal, etc., or indirectly, as in the machines called prime movers, like steam-engines and water- wheels, which actuate operating machines. We have, then. Prime movers and Operative machines. Also, of the latter, machines for changing the position of matter, as pumps, cranes, etc.; and machines for changing its form, as lathes, planers, etc. ; and each with many subdivisions. 2. In every machine there are to be distinguished the sup- porting parts, which are generally fixed and rigid, and the working parts, which are moving pieces. The supporting parts are general, supporting the entire ma- cliine; or local, supporting some one part, as the pillow -Mock, also called a plummer-block, or a pedestal, which supports a revolv- ing shaft ; or the guide bars, plainly seen in some form at the piston-rod end of the cylinder of any locomotive or other steam- engine, and which, by means of the stout block, called a cross- head, sliding between them, constrain the piston-rod, which is fastened to the cross-head, to move in a straight line. 3. 77ie working parts are connected together, forming a train, subject to this law, that a given position of any one piece deter- mines that of all the others. For the purpose of making the drawing of a machine, it is not enough, tliercfore, only to take PL. XIV. c SUPPOBTEES AND CRANK MOTIONS. 115 the measurements of its parts. This will suffice for the frame, but the motions of the train must be understood, so as to know what position to give to other parts, corresponding to a given position of some one part. 4. In some machines, however, there are subordinate trains, serving to adjust the position or speed of the principal trains, as in case of engine governors. Also some parts are adjustable by hand, as the position of the bed in a drilling machine, or of the tool and rest containing it, in a lathe. 5. The working parts of every machine consist of certain me- chanical elements or organs, which are comparatively few in number, and not always all present in any one machine. The principal of these are pistons, cross-heads, shafts, cranks, cams and eccentrics, toothed wheels, screws, band-pulleys, connecting- rods, bands or chains, sliding or lifting valves, grooved links, rocking arms and beams, flat or spiral springs, chambered parts and internal passages, as pump-barrels, steam-cylinders, valve- chests, etc. 6. These, considered separately, are of various degrees of com- plexity of design, many of them quite simple. By far the most imj^ortant, relative to the geometrical theory of their perfect action, are toothed wheels of various forms. These we shall therefore principally consider, together with a few other useful examples. Siipporters. Example 1. A Pillow-block. Pillow-blocks of various designs, adapted to horizontal, or vertical, or beam engines, are so common, and so generally represented in works on prac- tical mechanism, that the following figure, taken from a drawing to scale, is inserted here as a sufficient guide; the object, more- over, being symmetrical with respect to the centre line 00', and a little more than half shown. Descriptio7i. — BB', not definitely shown in plan, is a portion of the main bed of an engine. SS' is the sole of the pillow- block, DD' its body, CO' its cover, and dd — d'd' the brasses which immediately enclose the fly-wheel shaft of the engine. The holding-down bolts, as b — ¥b', pass through slotted holes pg, a little wider, that is, in the direction pq than the diameter SUPPORTERS AND CRANK MOTIONS. 117 of the bolt. This construction allows for adjustment of the position of the block by wedges, driven between the sole and stops I, solid with the bed B'. The cover CO' is held in place by bolts c — c'W, the heads h' being in recesses, sunk in the under side of the sole. The spaces at r and n between the cover and the body, allow for the wear of the brasses dd — d'd' against the shaft. To prevent lateral or rotary displacement of the brasses, ears ee — e'e' project from them into recesses in the cover and body of the block. The same end is often attained by making their outer or convex surfaces octagonal, and by providing them with flanges where they enter and leave the block. 00' is the oil cup, here solid with the cover, but oftener now a separate covered brass cup, contrived to supply oil gradually to the shaft. Construction. — The proportions of the figure being correct, assume ag, the half length of the sole, to be 12 inches, and meas- ure by a scale its actual length on the figure. A comparison of the two will indicate the corresponding scale of the figure.* Then, having determined the scale, all the other measure- ments can be determined by it to agree with each other, and the figure can be draiun on any scale desired, from ^ to ^ of the full size. The body being symmetrical, all the measurements to the left from 00' can be laid off to the right of it, and the complete projections thus constructed. The method of drawing hexagonal nuts has been shown in detail in Div. I., Problems 31, 32. Execution. — Note the heavy or shade lines as in previous ex- amples (Div. XL); but if the figure is to be shaded and tinted, ink it wholly in pale lines or none. Exercises. — 1. Construct from the two given projections an end eleva- tiou of the block. 2. Construct a vertical section on the centre line 05. 3. Construct a top view with the cover removed. (The dotted lines showing the internal construction will enable these sectional views to be made.) * The proportions adopted by different builders, and by the same builder for different cases, being not precisely alike, tl^ pupil is thus encouraged not to think any one set of given measurements indispensable. 118 SUPPORTEKS AXD CRANK MOTION'S. Ex. 2. A Standard for a Lathe. PI. XVI., Fig. 1. Description. — This example illustrates the application of tan- gent lines and circles to the designing of open frames, having outlines conveniently varied for use and economy of material. The figure shows half of the side view (the object being sym- metrical), also an edgewise view. The scale, -J, being given, and the operatioyis of construction being here more important than the precise measurements, only a few of the principal dimensions are given in this and in the following figures, leaving the rest to be assumed, or sufficiently determined by knowing the scale. The double lines on the edges indicate ribbed edges, so made to secure stiffness and strength. The central and triangular openings may also afford rests for long-handled tools or metal bars. The nut n secures the standard to the lathe-bed. Construction. — Having made the half widths 4:^" and 12^'' at top, and at AB, the outline BD may be drawn. This is com- posed of an arc of 60° with radius AB, a tangent to this, and a second arc of 60° tangent to the last line, and with its centre on a horizontal line through D. The outline of the central opening is partly concentric with the arc through D, partly circular with C as a centre, and partly circular as shown at the top, and there tangent to the side arcs. C is here taken on a horizontal line through the lower end of the arc from D. Execution. — The figure, both halves of which should be drawn, and on a little larger scale, as \ or -J-, gives occasion for the neat drawing of curved shade lines, and the neat connection of tangent outlines. Exercise. — 1. Vary the design by making the arcs from B and D each less than 60°, and so that C shall be on the radius through the lower limit of the arc througli D. Ex. 3. Section of an Engine-bed, Guides, and their Support. PL XVI., Fig. 2. Description. — ABCD is a cross-section of the bed of a hori- zontal engine, which is of uniform section throughout, hh is a vertical plate, bolted i»to the bed as shown at n. From this plate, and solid with it, project two or more arms HII, which c r^ SUPPORTERS AND CRAXK MOTIONS. 119 support the guide-bars GG, between which slides the cross-head, not shown, to which the outer end of the piston-rod is fastened, as may be understood from the equivalent parts of nearly any locomotive or stationary engine. Construction. — Only the principal measurements being given, the others can be assumed, or made out by the given scale. The left side of the bed having vertical faces, these may be used as lines of reference from which to lay off horizontal measurements. Vertical ones can be laid off from the base line AB; or, on the guide attachments, from the top of the guides downward. To give greater stiffness to the arm H, its lower principal curve is struck from a centre, h, 1^" to the right of a, the centre of its semicircular outlines. The curved outlines generally are com- posed of circular arcs tangent to each other. As a minute following of given copies is* not intended, these general explanations, measurements, and scale Avill sufficiently guide the learner in the construction of examples like the present. Execution. — The thin material of the bed gives occasion for section lines as fine and close together as can well be made. Exercises. — 1. Reverse the figure right for left. 2. Supposing the guides to be four feet long, make a side and a plan view, showing three arras to the supporter H. 7. Bearings. — This is a general term meaning any surface which immediately supports a moving piece. The bearings of a rotating piece are cylindrical and variously termed. Journals are formed in the frame of a machine and lined with brass or other anti-friction alloy. When detached, they are pillotu-Uochs, as already shown. Bushes are whole hollow cylindrical linings of journals, but being unadjustable to compensate for wear, separate brasses are better. Footsteps are the bearings at the base of vertical shafts, the lower end of which is a pivot. Axle boxes are the terminal supports of rail-car axles, and have a small vertical range of motion between the jaws of a stout iron frame. Cranhs and Eccentrics. 8. A crank, Fig. a, is an arm, CC, keyed at one end firmly to a revolving shaft SS' by a key hh', and hence revolving 120 SUPPOKTEES AKD CRAXK MOTIOlirS. with it ; and at the other end carrying a cranh-pin pp' , which is embraced loosely by a connecting rod. This connecting rod similarly embraces a parallel pin in the cross-head attached to a piston-rod, a pump-rod, or other piece having a reciprocating motion. Thus a rectilinear recijorocating motion, as of a piston, is conyerted into a rotary motion, as seen in any locomotire, or Ficj. a. a stationary engine of the iisual type, or vice versa, as in case of a pump. The length of the strolce of the piston must evidently be equal to the diameter of the circle described by the centre of the crank-pin ; that is, equal to twice Sp. See Fig. h, where the stroke qq^ of the forward end of the connecting rod mr is equal to pp^, the dotted circle being that described by the centre of the crank-pin. 9. Fig. l illustrates an important elementary point in crank motions. Remembering that any connecting rod is of invariable length, take the middle-point m- of the stroke qq^^% a centre, and tlio length ???S =z qp =^ q^p^ of the rod as a radius, and the arc SUPPORTEES Al^J) CRAXK MOTIOXS. 121 rSri thias described will intersect the crank-pin circle in the corresponding positions r and r^ of the crank-pin. Thns, while the cross-head j^in passes over mq and qm, the crank-pin describes the arc Tipr, greater than a semicircle ; but while the former is passing over mqi and qiin, the crank-pin proceeds over rp^r^ less than a semicircle. Conversely, while the crank-pin traverses the rear semicircle A/jB, the cross-head jjin only travels from n to q and back ; but when the crank-pin describes the semicircle BjOiA, the other pin travels from oi to q^ and back ; An being equal to Vim. With the use of the connecting rod, this inequality mn, be- tween the two partial double strokes (^nq and Qw^-i, would dis- appear only by using a rod of infinite length. But the em- ployment of a yoke with a slot, equal and parallel to AB, as in C -} Fig. c. Fig. c, produces the same result by finite means. Here a piston- rod, P, issuing from the steam-cylinder C, is rigidly attached to a yoke, AB, in which the crank-pin plays as it is driven by the yoke. In this case the piston is exactly at the middle point of its stroke when the crank-pin is at either end of the diameter AB. This movement is often seen in steam fire-engines. 10. Eccentrics. — The distance, Fig. a, from the centre of tlie shaft S to the centre of the crank-pin p, is called the a7'tn of the crank. When this arm is so short, as comj)ared with the diameter of the shaft, as to be entirely within the shaft, as at S^, Fig. cl, the crank-pin AB, whose centre is p, has to be 122 SUPPORTERS AND CRANK MOTIONS. made large enough to embrace the shaft. In this case the crank- pin is called an eccentric. That the eccentric is simply a short crank in principle and action, will be evident by substituting for the crank C, Fig. a, a circular plate with centre p and radius sufficient to include the shaft. In either form of Fig. a, and in Fig. d, a connecting rod attached to the crank-pin would actuate any piece at its opposite end through a stroke equal to twice S/). 11. A connecting rod is attached to a crank-pin by a method having many modifications in minor details. The general prin- ciple, alike for all, is shown in Fig. e. The object to be secured is an invariable distance between the centres of the crank-pin and the pin p, at which the rod R is attached to the cross-head. E IS the end of this rod, called the stub-end. ssss is the strap. 'b 'b ^ Fig.e. in one piece. B, shown sectionally, and B' in elevation, are the brasses, square outside and cyhndrical inside, which, together, embrace the shank of the crank-pin, and are kept from sliding off by the head of the crank-phi h, Fig. a. The whole is fas- tened by two bolts bb, bb. This arrangement being understood, sup^iose that by long wear the brasses play loosely upon the pin p. By driving in the slightly tapering key ^•^•, its side aa presses the brass B against the pin p, the width of the slots ac ac in the strap permitting this to be done. Then loosening the bolts b, the holes for which are oblong from right to left, as seen in a plan view, a further driving in of the kcj kic operates through the hooked piece ^, called a gib, to draw the strap s to the right, and thus draw up the brass B' against the pin p. Having understood one construction, the learner will be able SUPPORTERS AND CRANK MOTIONS. 123 to understand all the modifications which he may notice on loco- motive or other engines, such as the omission of the gib, which is unnecessary, with the bolts ; a separate key for each brass ; the stub-end extending to the left of kh, so as to wholly enclose it, when no bolts would be necessary ; a screw motion at the small end of h for drawing, instead of hammering in the key; etc. Ex. 4. A Crank. PL XVI., Fig. 3. This consists essen- tially of two collars connected by a tapering arm, the whole in one cast-iron piece. is the centre of the 8-inch shaft, and shaft collar of diameter ad, 19''. P is the centre of the crank- pin, of A!', and of its collar, of 9" diameter. The arm CO' is chambered as indicated by the dotted line around C. The linear arm OP is 24". The surfaces of the arm flow into those of the collars as indi- cated in line drawings by the curved ends of the upper edge of 0', lines whose geometrical con- struction is unnecessary in practice, but may be found as follows. Fig. /. In this figure the arc c'n' is that at en, Plate XVI., Fig. 3, enlarged, and the line On corresponds to PO. Then project points of c'n' upon en, as r' at r, and r7\ and r'r" are the tAvo projec- tions of the horizontal circle through rr', which cuts the edge 7i-iP of the crank at Vi, which, pro- jected upon the horizontal line r'r", srives r". Simi- larly, other points of the Fig.f required curve n'li" r"o' are found. Such curves are, however, after the full-sized construction of a few cases, to apprehend their general form, sketched by hand, as they are not essential to a working drawing. 124: SUPPORTERS AND CRAN'K MOTIOKS. Exercises — 1. Complete the crank, half of •u-liicli is shown in the figure. 2. Make a longitudinal section of the crank. 3. Draw from measurement any accessible ribbed, trussed, or cham- bered crank. 4. Draw a cranked axle (such as may be seen on old locomotives hav- ing "inside connections"). Ex. 5. A Ribbed Eccentric and Strap. PI. XVI., Figs. 4, 5. This may also be called an open or skeleton eccentric. is the centre, and Oh the radius of the shaft, S\" diameter, to which the eccentric is clamped by clamp-screws, one of which is n. The centre of the eccentric is a, which makes the crank- arm 0« of the eccentric d", and hence the stroke, called the throw, of the valve, or whatever piece is moved by the eccen- tric, Q". The width of the different parts of the eccentric is shown on the fragment of sectional view, as at o'o" the thickness of the flange, or feather, o, the width e'e" of the collar h and rim e, and the width of the rib c. Fig. 5 shows a little more than one-quarter of the strap which surrounds the eccentric, and a little more than half of one of its two halves, which are bolted together tlirough the ears, as cd. The arc ab is of the radius ae, Fig. 4 ; mn is of the radius ac, thus showing the groove in the strap wliich just fits the rib on the eccentric, and so i)rcvents the strap from slipping off. The portion of the strap shown carries the socket ne, in which is keyed or clamped the eccentric rod, corresponding to the con- necting rod of a crank. The opposite or left-hand half is unin- terrupted in outline. The figure r'g' shows the form of the section at rg. Execution. — The numerous tangent arcs and curved heavy lines tapering at their termination will afford occasion for special care. Exercises. — 1. Draw the whole of the eccentric and its strap. 2. Make a horizontal section of the eccentric. 3. Make an end elevation of the eccentric and strap. Ex. 6. A Grooved Eccentric. PL XVI., Figs. 6, 7. Tliis might also ])C called, by reason of its form, a chambered or box eccentric, since all of it between the solid collar Qa and the PL XVI r SUPPOKTERS AND CRANK MOTIONS. 125 rim cd consists essentially of two thin plates enclosing a hollow interior of width, '6%" , shown on the fragment of end elevation — the scale is jV? ^^ in Ex. 5. The shaft opening, of centre 0, is G" in diameter, surrounded by solid metal \" thick as indicated. The arm OQ being ^:\" , makes the throw of this eccentric 8|-". The opening, P, 5" diameter in the walls of the eccentric, gives access to the clamp- screw n by which it is fastened to the shaft. The strap. Fig. 7, a section of Avhich is shown at H, sets in the groove c'c" of the circumference of the eccentric. Its outer arc mn is drawn from a centre a little to the right of that of AB, so as to support the rod-socket BC. As in the last example, the strap is in two halves bolted together through ears as at D. Exercises. — 1, Draw the whole eccentric with the strap in place upon it, and an end view of both. 2. Make a horizontal and a vertical section (perpendicular to the paper) of the combined eccentric and strap. 12. Chech, loch, or janih nuts. — On parts of machinery which are exposed to a jarring motion at high speed, as in locomotive machinery, two nuts are commonly seen at the end of the bolts which secure such pieces. These serve to clamp each other against the screw- threads of the bolt, and thus hold each other from working off the bolt. Other contrivances for securing the same re- sult, are nuts with notched sides, into which a detent enters, as may be observed in winding up ^'^9- 9' a watch; or a forked key hh' through the bolt and outside of the nut, as in Fig. q. fc CHAPTER II. GEARING. 13. Gearing is the term applied to wheels or straight bars when they are armed with interlocking teeth enabling them to take a firmer hold of each other, for the jiurpose of communi- cating motion, than they could if they were smooth surfaces, tangent to each other and communicating motion only by means of the friction of their surfaces of contact. In order to a smooth and uniform motion, the teeth must be equal and equidistant, and those on each body adapted to the form of those on the other body. Also, in order that the toothed bodies, two cylinders tangent to each other, for exami)le, should preserve the distance between their centres, depressions heloto the original surface of each body must be made between the teeth, in order to receive the portions which project teyond that of the other body. A toothed bar is called a raclc; a toothed cylinder, a spur-ivlieel, or pinion if small ; a toothed cone, a conical or level wheel. 14. Forms of teeth.— 1°. When a circle C, PI. XVII., Fig. 1, rolls, without slipjjing, on a fixed straight line AB, any one i^oint of the circle describes the curve called a cycloid. Thus the point describes the cycloid, one-half of which is OE', while the circle C rolls to the position C. 2°. When, on the contrary, a straight line as 3,3', Fig. 2, rolls, without slipping, on a fixed circle, any point of the rolling line describes the curve called an involute of the circle. Thus 3' de- scribes the involute 3', 2', 0. 3°. Again: when one circle rolls on the exterior of another as the circle BB', Fig. 3, on the circumference BFA, any point on the rolling circumference traces the curve called an epicycloid. Thus the point P traces the half epicycloid EG, while the circle C rolls to the position AG. 4°. Finally: when BB', instead of rolling on the convex or ex- terior side of the circumference C, rolls upon its concave side. GEAEIXQ. 127 or within it, any point of the rolling circle generates the curve called a hypocycloid. In all these cases the fixed line is called the base line or circle. 15. Suppose now the circle BB', Fig. 3, to be revolved 180° about a tangent at B. It would then be tangent to the circle C interiorly at B, as it now is exteriorly at that point. If then, the diameter BB' being less than radius CB, the circle BB' rolls within C, on the arc BFA, the hypocycloid traced by B will be alove AB. But if a circle of diameter greater than CB were to roll within C on the arc BFA, the hypoc^'Cloid curve would be heloiu AB. It plainly follows that the hypocycloid traced by the point B, when the circle of diameter just equal to CB rolls within C on the arc BFA, would coincide with BO. That is: the hypocycloid traced hy any point of a circumference ivliich rolls on the inside of a base circle of twice its diameter, , is a straight line. 16. These four curves (14) are suitable forms for the teeth of wheels, for the simple reason that when two circles as C and C, PI. XVII. , Fig. 4, maintain rolling contact with each other, as at H, equal arcs of each come in contact in a given time. Hence the motion is the same in effect as if, separately, C rolled on C as a fixed circle, and then C rolled over an equal arc on C as a fixed circle, and therefore contact Avill be maintained by arming the wheels with teeth generated by the point H, for each case respectively. Similarly for a wheel C, and rack I'J'. The circle or line employed for generating the tooth curves is called their generating circle, or line. 17. Construction of tooth curves. — This is very simple, and follows directly from the definitions in (14). Cycloid. Thus PI. XVII., Fig. 1, the points 1, 2, 3, etc., on the circle C indicate the heights of above AB, corresponding to 1, 2, 3, etc., on AB as successive points of contact of C with AB. Then the inter- sections of the parallels to AB through 1, 2, 3, etc., on C, with the arcs of radii al, h2, c3, etc., will be points of the cycloid OE'. Both sets of spaces 01, 12, etc., are equal, since there is no slipping of the circle in rolling on AB. Epicycloid. Likewise in Fig. 3, arcs Fl, etc., on circle C, = arc Fl, etc., on circle C^ express 128 QEARIN-Q. the character of the motion; a, h, c, etc., are positions of the centre C corresponding to 5, 4, 3, etc., as points of contact of the circles; the arcs of radii Cl, C3, C3, show the radial dis- tances of F from BFA as C rolls on C; hence, finally, the iinter- sections, not lettered, of the arcs of radii e\ and Cl, d'Z and C2, c3 and C3 are points of the epicycloids FG and FE'. Tlie hypocycloid is constructed in a precisely similar manner. The involute is approximately rej)rcsented by tangent circular arcs as in Fig. 2. Here, 11', 22', 33', being positions of the roll- ing straight line at equidistant points of contact 1, 2, 3, we describe the arc 01' with radius 10 (taking the chord us approxi- mately equal to the arc), then an arc 1' 2' with radius 21', then the arc 2' 3' with radius 32', etc. The more numerous the points 1, 2, 3, etc., the closer Avill the compound curve thus found approximate to a true involute. These curves can le constructed mechanically on a large scale by means of a pin or pencil point inserted firmly in the edge of a wooden ruler or circle, cither of wliicli is made to roll without slipping on the other, or the circle on a fixed circle; or within a circular oj^cning in a thin board, in the case of the hyi^ocycloid. 18. Definitions. — Let the circles of radii CH and C'H, PI. XA^IL, Fig. 4, represent the original circumferences of two cylinders having IJ for a common tangent at H, and now pro- vided Avith interlocking teeth as shown, forming a pair of sp7ir- tvheels. These circles are called pitch-circles. The correspond- ing line I'J' of the racJc is called its pitch-line. The distance ab, or II'K on the rack, which includes a tooth and a space on the pitch-circle, is called the pitch. The circle of radius Cc is t\ic root-circle, and contains i\\Q7'oots of the teeth. The circle of radius Cd is the ])oi)it-circle, and coiitains the points of tlic teeth. The sui'faccs as hd are the faces of the teeth, and those as be are ihc'w flafiks. 10. Usual proportions. — Supposing the pitch divided into 15 equal parts, 7 of these are taken for the width, ah, of the tooth, leaving 8 of them for the width of the space, hb, to allow easy working of the teeth. Also 5 J of these spaces are taken for the GEAUIXG. 129 radial extent of the teeth beyond the pitch-circle and (>\ of them for their depth below the pitch-circle, to prevent the tootli points of one wheel from striking the rim of the other wheel. Application of Tooth Curves. 20. Designing of gearing. — Comparing (15) and (IG), the gen- erating circle of the tooth curves must be smaller than the pitch- circle in order to form the necessary flank surfaces (18). A common practice is, to employ for the flanks of each wheel a generating circle of diameter equal to the radius of the i^itch-cir- cle of that wheel; in order to produce radial Hanks, as most simple. Now, as seen by inspection of PI. XVIL, Fig. 4, the face of a tooth of each wheel is in contact with ihcjlanh of some tooth of the other wheel. Hence (IG) the same circle that gen- erates the flanhs of one wheel must generate the faces of the teeth of the other, since keeping the generating circle of diameter CH in contact with the pitch-circles at their point of contact II, requires in effect the equal rolling of that generating circle upon the exterior of the circumference of wheel C, and on the interior side of that wheel C. This can easily be seen experimentally by using three card-board circles. 21. Detailed description. — Ep icy cloided teeth. — The circle CH generates the radial flanks, as be, of the teeth of C (14, 4°), and by rolling on the exterior of C generates tlie face curves of the teeth of C. To avoid confusion, HK may represent one of these curves, though it is really an involute. Likewise, the circle CH generates the radial flanks of wheel C, and by rolling on the exterior of C will generate the epicycloidal faces of its teeth, found as in Fig. 3, but represented as before by the involute IIL. 22. Objections. — Each wheel having a separate generating circle, each will work correctly only with the other. But if one uniform generating circle be employed for the faces and Hanks of any number of different-sized wheels of the same pitch, any two of them will work together properly. This common gen- erator must not exceed half the size of the least wheel of the set, so as to avoid convex flanks (15). 23. Involute i'rr'/Z(.— Involute faces for both wheels can be 130 GEARING. formed as shown by the rolling of the common tangent IJ at H, first on C, giving the involute face curve HL (Fig, 4), and then on C, giving the face curve HK. Usually, however, when involute teeth are employed, they are not combined with radial flanks, since this violates the principle that the same generatrix should form the face and the flank which are to be m contact ; but IJ is made a common tangent through 11 to the root-circles of the two wheels, so that the in- volute teeth will be bounded by a single involute curve reaching to the root-circles, as they should, since a straight line cannot be rolled on the interior side of the pitch-circles to i)roduce sepa- rate flank curves. 24. The rach-generating circle continuhig to be half the size of its own pitch-circle, the generating circle for the rack flanks will be I'J', since a straight line is a circle of infinite radius and half of that radius is infinite still. This understood, the^^aw^-s of the rack are straight lines as AH' perpendicular to its pitch- line, and \X\Q facea of the teeth of C, being properly generated by the same line, are involutes as H'F'. Likewise \X\(iflanhs of the teeth of C are straight lines genera- ted by the circle of diameter ClI', while the faces of the rack teeth are cycloids as H'G' generated, as shown, by the rolling of the same circle on the pitch-line I'J'. But, as before, one generating circle can be used for faces and flanks of both wheels. Ex. 7. The Drawing of a Spur-vrheel. — It is convenient that the pitch should be some simple measure as 1", 1^", li", . . . V, . . . 2^-", . . . etc., and it is necessary that the pitch should be contained an exact number of times in the ])itch-circle. Hence the usual jn-oblcm is: Given the pitch and number of teeth of a pair of wheels, to find their radii. Let P = ])itch X, = number of teeth, R = radius, and C = circumference of pitch-circle. Then C = P X N = 3.141G X 2R, P X N whence 11 = i^ir ' °^ denoting as usual, 3.141G by it R = P X N 2 n ' GEAKING. 131 Suppose a wheel of 34 teeth and IV' pitch. Its circumference will thus be 30" and its radius very nearly 52". The four quarters of the wheel being alike, it is sufficient to draw one of them, with the lirst tooth on the adjacent quarters, and this can conveniently be done on a scale of half the full size. Three forms of Avheels are in use according to their size: solid wheels, as in PL XVII., Fig. 4; plale wheels, consisting of a cen- tral hub, or boss, keyed to the shaft, and connected by a thin plaie to the rim which carries the teeth; and armed Avheels, in Avhich the boss is connected with the 7'im by arms the perpen- dicular section of which is often an equally four-armed cross. Attendiug at first principally to the teeth, let the wheel now drawn be solid. Divide a quadrant of tlie pitch-circle carefully into six equal parts, one of which will be the pitch. Proportion the teeth by (19), giving the root and point circles. Lay off half the width of a tooth oa each side of each point of division of the pitch-circle, which will make the lines as CH' and CD, Fig. 4, centre lines of teeth instead of as shown in that figure. While teeth are shown in detailed working drawings, of full size and by the most accurate construction of their proper forms, they are approximately represented in general illustrative draw- ings, by various simple methods. Thus the faces may well be drawn by taking the pitch ab as a radius, with the centre, as at b, on the pitch-circle to draw the face aJ). A more summary process is shown in PI. XIX., Fig. 6. The flanks, if not radial, as shown in the figure, should be in reality hypocycloids (14, 15) which would diverge totoards the centre C, and which may suffi- ciently be represented by taking d, for example, for the centre of the flank beginning at n. Thus the elevation may be completed, placing the shade, or heavy lines, on each tooth by the usual rule, as shown. For the plan, draw two parallel lines at a distance apart equal to the luidth of the Avlieel; that is, the length of the teeth, Avhich may be twice the pitch. Then simply project down the point angles as d, and visible root angles as c, and the points of con- tact of the face curves with tangents parallel to IJ, as at a and 132 GEARIXG. near h. To become familiar with the subject, work out fully the following : Exercises. — 1. Construct the half, not shown, of tlie cycloid. PI. XVII., Fig. 1. 2. Complete both of the epicycloids half-shown in Fig. ?>, one with the diameter of C eijuul to the radius of C. Also one, given by making the circles C and C equal. 3. Construct the liypocycloid generated by the jioint II of circle C'll, Fig. 3, in rolling within circle C. 4. Construct the liypocycloid generated by the circle CII' rolling within tlic small circle C. 5. Construct an arc of the involute of circle C, generated by the point H' of the line I'J', and by dividing a quadrant of C into eight equal parts. G. Draw a spur-wheel and rack, the wheel having 33 teeth and 2" pitch. Make the drawing of full size, showing a quadrant only of the wheel, bisected at its point of contact with the rack, and let the faces and flanks of both pieces have one generating circle whose diameter shall be -J tiiat of the radius of the wheel. 7. In Ex. G, substitute for the rack a wheel of 20 teeth, and let the common generating circle of the teeth-profiles of both wheels be of less diameter than the radius of the smaller wheel. 8. Draw enough of a four-armed wheel of 30 teeth and 1\" pitch to show two arms fully, making the tliickuess of the rim^ and of the arms, and of the feather, and their width also (see o, PI. XVI., Fig. 4), which surrounds the openings between the arms, all equal -^'^ of the pitch, and the radial thickness of the hub ^ of the pitch. 9. Draw PI. XIX., Fig. G, twice its present size or larger, and first with involute teeth, and then with epicycloidal faces and hypocycloidal flanks, and after constructing carefully one tooth-profile, find by trial the centre and radius of the circular arc which will most nearly coincide with it, to use in drawing the other teeth.' 25. Velocities. — It is clear (13) that if one wheel has 30 teeth and another GO, the former must make two revolutions to one of the latter, also that the radius of the former is one half that of the latter. What is true for one such case is evidently true in principle for all cases. That is, the number of revolutions in a given time of each of a i)air of toothed Avheels is inversely as its number of teeth, or as its radius. The rircumference velocities, as at tlie point of contact II, PI. A'Vir., y\'y. 4, are necessarilv e(|u;il. luit the velocities at the GEARING. 133 same distance from the centre, as 1 foot, on both wlicels are as the numbers of revolutions, and hence inversely as their radii. The latter are termed angular velocities. Then denoting them by V and v for the wheels C and C respectively, and the radii CII by 11 and C'H by r, we have ■ V:v::r:Pt. Bevel and Mitre WJieels. 26. PI. XVIIL, Fig. 1, shows a pair of level wheels. These consist of a pair of frusta of cones, CAD and one of which CAB IS the half, provided with teeth which converge to the common vertex, 0, of the cones, whose axes, CB and CF, may make any angle with each other. When, as m Fig. 2, the axes are at right angles, the wheels are distinguished as mitre wheels. As C IS lowered nearer and nearer to AB, still continuing the common vertex of the cones, the wheel AB becomes flatter and flatter, and when finally C passes below AB, the wheel AB be- comes a hollow frustum toothed on its inner surface. On account of the intersection of the axes of bevel wheels, one or both of the axes terminate at the wheels, as m Figs. 1 and 2. 27. Velocities. — The principles of (25) apply to bevel wheels. Hence having given one wheel, as CAB, Fig. 1, and the ratio of the velocities, make a'd' and a'e' in this ratio, a'd' repre- senting the relative velocity of the required wheel, and Ce' will be the axis tangent from C to an arc of centre a' and radius a'e', and AD the diameter of th(j latter wheel. Or, having given the vertex C, and axes CB and CF, set off Cc' and C5' inversely as the two velocities (that is, set off on each axis a distance proportional to the velocity of the other axis), and complete the parallelogram Cl'a'c', and CA is the line which will divide the angle BCF included by the axes, so as to give the radii AB and iAD of the required wheels. When, as in Fig. 2, the axes are at right angles, the latter construction applies, but the parallelogram becomes the rect- angle CJaO. 134 GEARING. Ex. 8. To Draw a Pair of Bevel Wheels. PL XVIIL, Figs. 2-5. Let the cones CAB and CAD — called the pitch-cones, because they contain the pitch-circles — be given. At A, the point of contact of the pitch-circles, draw EAF perpendicular to CA, and draw EA, EB, FA, FD. Then EAB and FAD will be the cones containing the larger, or outer ends of the teeth. Next, laying off Al equal to the length of a tooth, and drawing IR par- allel to AD, IH parallel to AB, and GIJ parallel to EF, we have JIR and GIH, the cones containing the inner ends of the teeth. The wheels here shown have respectively 36 and 28 teeth. Then divide each quadrant of the semi-pitch-circle on A'B' into 9 nine equal parts, and each quadrant of the semi-pitch-circle on A"D' into 7 equal parts. Taking the proportions before used, make Be and Bb, each on BE, respectively 5^ and 6^ fiftecnilis of the pitch, to obtain the point and root circles parallel to AB through e and h, since the real height be of the teeth is shown in its real size on the extreme element EB of the cone EAB. The horizontal jorojections of these circles are those with radii C'e' and G'b'. The corresponding inner point and root circles are found by noting (jr, the intersection of eC and GH, and that of bG with GH. This last point is horizontally projected at n'. Having thus both projections of all the circles of construc- tion: 1°. Lay off y*^ of the pitch, that is, half the space between two teeth, on each side of A', B' and S, and from the points so found lay off the pitch, over and over, which will give all those points of the teeth which arc m tlip outer pitch-circle A'SB'; and project these points on AB. 2^. Througl) the points just found on A'SB' draw lines to C, limited by the circle C'w'; and through those on AB, lines to E limited by ah, for the outer ends of the flanks. y. From tlie points on ah draw lines to C, limited by the vertical ])rojoctiun of circle C'n', for the root lines of the teeth, and from \\w. ])()ints llius found, the inner ends of the Hanks radiating from G and limited by HL 4°. Make arcs, tangent to each other as at 0, Fig. 5, Avith radii EA and FA, which will be (l)iv. L, Prob. 28) arcs of PL:xvii. GEARING. 135 the developments of the outer pitch-circles of the two wheels. On these lay ofT the pitch, and proportion the teeth as already described, the flanks running to E (below the border) and F, and the faces drawn with convenient circular arcs to ]-e})lace the epicycloidal curves OP and OQ. Having thus found the width of the teeth at their outer points, lay off half this width on each side of the middle point of each tooth on the circle of radius C'e'. 5°. Through these points on circle C'e' draw lines to C, lim- ited by circle Cg' ; project the points of circle C'e' upon ec, vertical projection of circle Ce', and thence draw the point edges of the teeth towards C. 6°. Finally, the face curves at both ends of the teeth are sketched by hand, tangent to the flanks. By precisely similar operations, the two projections of the wheel AD — A"D' may be drawn. The hub and arms of both can be easily drawn, as shown. To become perfectly familiar with the operations here de- scribed, work out the following variations : Exercises. — 1. Changing the numbers of teeth, let the axis of the wheel AD be perpendicular to the paper at C, so as to appear as Fig. 4 now does. 2, Again changing the number of teeth, let the wheel AD be in gear with AB at BH, and then draw the figure as if PI. XVIII. were upside down, making C'A'SB' the elevation, instead of, as now, the plan. Screivs and terpentines. 28. Tri angular -tlireaded screws. — If the isosceles triangle cdl2, PL XIX., Fig. 1, whose base is in the vertical line Wi, be revolved, together with that line, uniforml}', around the vertical AB as an axis, having also a uniform vertical motion on E7i, it will generate the spiral solid called the thread of a triangular- threaded, also called a V-threaded screw. The surfaces gen- erated by (Z12 and cl2 are lielicoids, upper and lower. The lines generated by the points c, d and 12 are helices, inner and outer. E/i will generate a cylinder, called the core or neivel of the screw. 29. Square-threaded and other screws. — If, PI. XIX., Fig. 2, 136 GEARIXG, a square, EC6, be substituted for the triangle, the result will be a square-threaded screw. If, Fig. 3, a sphere whose centre describes a helix be the gen- eratrix, the resulting solid will be that called a serpentine. This is the form of a spiral spring formed of circular wire ; also of the hand-rail of circular stairs, when the rail has a circular section made by cutting it " square across." Again : if, Fig. 5, the profile of a tooth be taken as the gen- eratrix of the thread, there will be formed the kind of toothed wheel called an endless screw, since its constant rotation in one direction will actuate tlie wheel L. It is always the screw that is the '^driver''' and actuates the wheel, which is the ^^ follower,''^ and receives a very slow motion ; since the tooth G will be car- ried to the position of the next tooth above it, by one complete revolution of the screw. 30. JSfiwiber of threads. — In PI. XIX., Fig. 1, one helical revo- lution of the generating triangle brings the side 6-12 to the posi- tion di^, which allows no intermediate position of the triangle. The screw is therefore single-threaded. The like is true of the screws in Figs. 2 and 5. If, however, Fig. 1, one such revolution of cdl2, had, by means of a greater ascending motion, bror.ght cl2 to the position rs, the screw would have been tivo-tltreaded; and if to the posi- tion no, it would have been three-threaded. The like again is true of other screws, the number of threads being adapted to the advance parallel to AB, of any point of the screw in one revolution. This advance is called the pitch of the screw. Evidently the coils of a second spiral like Fig. 3 could be laid between those shown. It would then be two-threaded. Ex. 0. To Construct the Projections of a Triangu- lar-threaded Screw. V\. XIX., Fig. 1. The construction of the screw consists principally in that of its helices. Accordingly, let AE and AC be the radii of the circles which represent the circular motions of the points c and 12, and which are the horizontal projections of the inner and outer helices. And let 0,12 be the pitch of the screw. As both component motions, circular and rectilinear, of the F L xvm GEARING. 137 compound helical motion are uniform, divide the circles AE and AC and the pitch 0,12 all into the same number of equal j^arts, here 12, and draw horizontal lines through the points of division on 0,12. Then for an outer helix, project C at 0; 1 on the first horizontal above it ; 2 on the second horizontal, C, at 6 on the sixth horizontal, and so on till C is projected again at 12. Proceed in a precisely similar manner, beginning by jiroject- ing E at c, to find points of an inner helix. The lines as cl2 and dVl complete the figure. Each half, to the right and left of AB, of the visible front half, as 06, of an outer helix is like the other half reversed, both right for left and upside down. Hence, as all the outer helices are alike, the portion of an irregular curve which will fit one half of one, will serve in ruling them all. Similar remarks apply to the inner helices. Had the ascent been from D to the left on the front half of the screw instead of from O to the right, the screw would have been left-handed. Left-handed screws are only employed for special purposes, as when two rods, placed end to end, are to be separated or brought together by a screw link working on both, as seen in the truss-rods under rail-car bodies. In this case the screw-threads on one rod would be right-handed, and those on the other left-handed. Exercises. — 1. Construct the projections of a two-threaded and of a three-threaded trianguhir screw. 2. Construct the projections of a two-threaded and of a three-threaded Mt-handed screw. Ex. 10. To DraviT a Square-threaded Scre-w. PI. XIX., Fig. 2. The operations in this case are so similar to those of the last problem, as is evident from the figure, that they need no de- tailed description. The form of the thread renders the under outer helices of the left side, and the iipper outer helices of the right side, of the screw visible on the back half of the screw until they disappear behind the cylindrical core. Also, the inner helices are visible only on the under left-hand side and upper right-hand side of the thread. In the execution, it is very important to remember that an}/ 138 GEAEI]S"G. one helix is, on the screw itself, of uniform curvature through- out, hence though very sharply curved in projection at the extreme points, as 6 and 12, especially in a single-threaded screw, they are not there pointed, except in drawings on a small scale where they may be approximately represented by straight lines, as in Figs. 7, 9, and 10. Exercise. — Draw a square-threaded screw with three threads, and show all four helices of one thread throughout, but dotted where invisible. Ex. 11. To Draw the Interior of a Nut or Internal Screvr. PI. XIX., Fig. 8, shows the interior of one half of the nut for a square-threaded screw ; that is, of the hollow cylmder with a thread on its interior surface, adapted to work in the spaces between the threads of the screw. The figure representing the rear half of the nut, the threads must there ascend to the left, as they do on the rear half of the screw. Exercises. — 1. Draw the vertical section of the nut corresponding with Fig. 1. 2. Draw that of the nut of a square-threaded screw of two threads. Ex. 12. To Dravr the Endless Screw and Worm Wheel. PI. XIX., Figs. 4, 5. The profile of a tooth here becomes the generatrix of a screw- thread bounded by helices found as before. The pitch-line MIST is divided by the pitch as in the casQ of a rack, the pitch of the screw and wheel being the same. The wheel, having its axis in a direction perpendicular to tluit of the screw, is in reality a short piece of a screw having a very great pitch. That is, the angle made by the helices of the wheel-teeth with a i)lane perpendicular to the axis of the wheel, that is, with the plane of the paper, is the complement of the angle made by the screw helices with a plane perpendicular to its axis, that is, to a plane perpendicular to the paper on GD. The curves, as that to the left of N, which represent the further ends of the teeth, are assumed, unless the width of the wheel is shown by a plan view.. GEARING. 139 Ex. 13. To Draw a Serpentine. PI. XIX., Fig. 3. This surface is one which, like a thin helical tube, would in- close, tangentially, all the positions of a sphere, indicated by the dotted circles, whose centre should describe a helix, ACB — 2345. The contours, or apparent bounding lines, of the serpentine are not helices, though at a uniform perpendicular distance from the central helix, but are drawn tangent to the numerous equal dotted circles having their centres on the helix, and which repre- sent as many positions of the generating sphere. Surfaces which, like the sphere and serpentine, are nowhere straight, are call double-curved. Where partly convex, as on the outer circle, or in the circle OF, and partly concave, as on the inner side, or on the circle OD, the contour vanishes into the surfaces, at certain points, when shown by a line drawing, as is seen at the left of the under contours, and the right of the upper ones. The lower coil is shown approximately as straight, indicating jfhat would be permissible in rough drawings or on a small scale. DIVISION SIXTH. SIMPLE STRUCTURES AND MACHINES. '23 7. Note. The objects of this Division" are, to acquaint the student with a few things respecting the drawing of whole structures which are not met with in the drawing of mere details ; to serve as a sort of review of practice in certain processes of execution ; and to afford illustrations of }»arts of structures whose names have yet to be defined. Proceedino; with the same order as regards material that was observed in Division Second, we have : — CHAPTER I. STONE STRUCTUKES. 238. Example 1°. A brick segmental Arch. I'l. XX., Fig. 123. Description of the structure. — A segmental arch is one whose curved edges, as aCc, are less than semicircles. A brick segmental arch is usually built Avith tlie widths of the bricks placed radially, since, as the bricks are rectangular, the mortar is disposed between tliem in a wedge form in order that each brick with the mortar attached may act as a wedge ; while if the length of the bricks be radial, the mortar spaces will be inconveniently wide at their outer ends, unless the arch be a very wide one, or unless it have a very large radius. The 2:)ermanent sujjports of the arch, as nPT, are called abut ments^ and the radial surface, as nab^ against which the arch rests, is called a skew-hack. The temporary supports of an arch wliile it is being built are called centres or centrings., and vary fi-om a mere curved frame made of jiieces of board — as used in case of a small drain or round PL.XIX. STONE STKUCTUUES. J 4 1 topped window — to a heavy and complicated fianiini,', as used foi the temporary support of heavy stone bridejes. Note. The general designing of these massive centrings may rail for as much of scientific engineei'ing /;«o?cZec?(/e, and their details and management may call for as much practical engineering skill, as does the construction of the permanent works to which these centrings are auxiliary. In short, the detailed design and manage- ment of auxiliary constructions, in general, is no unimportant depart- ment of engineering study. The span is the distance, as ac, between the points of support, on the under surface of the arch. The stones over the arch and abut- ment, form the spandril^ or hacking^ Qc^P. 239. G-rapli'tcal construction. — Let the scale be one of four feet to the inch=48 inches to one inch=:j\. Draw RT to represent tlie horizontal surface on which the arch rests. Let the radius of the inner curve of the arch be V feet, the height of the line ac from the ground 2 feet 8 inches, and the span 7 feet. Then at some point of the ground line, draw a vertical line, OC, for a centre line ; then draw the abutments at equal distances on each side of the centre line, and 6 feet 8 inches aj)art. Let them be 2 feet 6 inches wide. Since the span and radius have been made equal, Oh and Oc? may be drawn, in this example, with the 60" triangle. Drawing these lines, and making Oa — 7 feet, make ab = one foot, draw the two curves at the end of the arch, and make b and d points in the top surfaces of the abutments. To locate the bricks, since the thickness of the mortar between the bricks, at the inner curve of the arch, would be very slight, lay off two inches on the arc aCc an exact number of times. The dis- tance taken in the compasses as two inches, may be so adapted aa to be contained an exact number of times in aCc, since the thick- ness of the mortar has been neglected, but would in practice be so adjusted, as to allow an exact number of whole bricks in each course. The arch being a foot thick, there will be three rows of bricka seen in its front. Draw therefore two arcs, dividing ab and cd into three spaces of four inches each, and repeat the process of division on both of them. Having all the above-named divisions complete, fasten a fina needle vertically at O, and, keeping the edge of the ruler against it, to keep that edge on the centre without difiiculty, draw the lines which represent the joints in each of the three courses of brick. 142 STON^E STRUCTURES. 240. Ex. 2®. A semi-cylindrical Culvert, having vertical quarter-cylindrical Wing Walls, truncated obliquely. PI XX., Fig. 124. Description of the structure, — A culvert is an arched passage, often tlat bottomed, constructed for the purpose of carrying water under a canal or other thoroughfare. Wing walls are curved con- tinuations of the vertical fiat Avail in which the end of the arch is seen. Their use is to support tlie embankment through which the culvert is made to pass, and to prevent loose materials I'rom the embankment fx'om working their way or being washed into the cul- vert. Partly, perhaps, for appearance's sake, the slope of the plane which truncates the flat arch-wall, called the spandrll wall, and the wing walls, is parallel to the slope of tlie embankment. The wing walls are often terminated by rectangular flat-topped posts — " piers" or "buttresses," AA', and the tops, both of these piers and of the walls, are covered with thin stones, abed — a"b"c"d\ broader than the wall is thick, and collectively called the coping. Since the parts of stone structures are not usually firmly bound or framed together, each course cannot be regarded as one solid piece, but rather each stone, in case, for instance, of the lowermost course, rests directly on the ground, independently of other stones of the same course, hence if the ground were softer in some spots, under such a course, than in others, the stone resting on that spot would settle more than others, causing, in time, a general disloca- tion of the structure. Hence it is important to have what are called continuous bearings, that is, virtually, a single solid piece of some material on which several stones may rest, and placed between the lowest course and the ground. Timbers buried away from the air are nearly imperishable ; hence, tiiiiljcrs laid upon the ground, if that be fii-m, and covered with a double floor of plank, form a good foundation for stone structures; and in the case of a culvert, if such a flooring is made continuous over the whole space covered by the arch, it will prevent the flow- ing water from washing out the earth under the sides of the arch. When the wing walls and spandril are built in courses of uni- form thickness, the arrangement of the stones forming the arch, so as to bond neatly with those of the walls, offers some difficul- ties, as several things are to be harmonized. Thus, the arch stones must be of equal thickness, at least all except tlie top one, and then, there must be but little difference between the widths of the top, or key stone, and the other stones ; the stones must not be di.s- projiortionately thin or very wide, they .should have no re-entranf STONE STnUCTURES. 143 angles, or very acute angles, and there must not be any groal extent of unbroken joint. 241. Grapliical constmiction. — Let the scale be that of five feet to an inch = 60 inches to an inchrz:^'^, a. Draw a centre line, 15B', for the plan, h. 8nj)posing the radius of the outer surface, or back, of the arch to be b\ feet, draw CC parallel to BB' and 5^ feet from it. c. Draw BE, and on CC produced, make EC = 9 feet 8 inches, CD, the thickness of the face wall of the arch = 2 feet 4 inches, and the radius, oD, of the lace of the wing wall=4 feet. d. With o as a centre, draw the quadrants CG, and DF, and Avith a radius of 3 feet 8 inches, draw the arc oA, the plan of the inner edge of the coping. Also draw at D and C, lines perpendicular to BB' to represent the face Avail of the arch. e. At G, draw G/i towards o, and =3 feet, for the length of the cap stone of the buttress, A A', and make its Avidth =2 feet 10 inches, tangent to CG at G. The top of this cap stone, being a flat quadrangular pyramid, draw diagonals through G and A, to represent its slanting edges. f. Supposing the arch to be \\ feet thick, make C'II=1^ feet, and at C and H, draw the irregular curved lines of the broken end of the arch, and the broken line near the centre line, also a fragment of the straight part of the coping. g. Let the horizontal course on Avhich the arch rests, be 2 feet 9 inches wide, i.e., make He = 3 inches, and CW=1 foot; and let the planking project 3 inches beyond the said course, making e;-=3 feet. Through e, n and r, draAV lines parallel to BB' and extending a little to the right of C'll. h. Proceeding to represent the parts of the arch substantially in the order of their distance from the eye, as seen in a plan A^iew, a portion of the planking may next be represented. The paira of broken edges, and the position of the j(jints, shoAv that there are two layers of plank and that they break joints. i. Under these planks, appear the foundation timbers, Avliich being laid transversely, and being one foot wide and one foot apart, are represented by parallels one foot apart, and perpendicular to BB'. Let the planking project 4 inches beyond the left hand tim ber. Observe that tAVO timbers touch each other under the arch front. j. The general arrangement of stones in the curved courses of the ^ing wall, in order that they may break joints, is, to have three and four stones, respectively, in the consecutive courses. To indicate J 44 STONE STliUCTUKES. this arrangement in tbe plan, hG,f//, (^(^and DC will represent tJQf joints of alternate courses, and the lines km, &g. midway betweec the former, will represent the joints of the remaining interraediato toursos. Tliis completes a partial and dissected plan which shows more of the construction than would a plan view of the finished culvert, and a= much, as if the j^arts on both sides of the centre line were shown. In fact, in drawings which, are strictly working drawings, each i)ro- jection should show as much as possible in regard to each distinct part of the object represented. 242. Passi7ig to tJie side elevation, which is a sectional one, show- ing parts in and beyond a vertical plane through the axis of the arch, we have : — a. The foundation timbers, as rn'q, &c., projected up from the plan ; or, one of them being so projected, the others may be ecu- Btructed, independently of the plan, by the given measurements. b. The double course of planking ojo, appears next with an occa- sional vertical joint, showing where a plank ends. '•. The buttress, A, and its cap stone Y, are projected up from tiie plan, and made 6 feet high, from the planking to G'. d. From G' and A', the slanting top of the wing walls are shown, as having a slope of 1^ to 1 — i. e. //,7t"=f lo'u — and the vertical lines at C, D' and D" are projected up from C, D and D'". The remaining lines of the side elevation are best projected back from the end elevation, when that shall have been drawn. 243. In the end or front elevation, we have: — a. At m''m"', a side view of one of the foundation timbers, broken at vi'", so as to show other timbers behind it. b. The planking o'o" in this view, shows the ends of the planks in both layers — breaking joints. c. Nt>' = Bo"', taken from the plan ; and in general, all the hori- zontal distances on this elevation, are taken from the plan, on linea perpendicular to BB'. d. Tlie vertical sides of the buttress, A', are thus found. Tho heights of its parts are projected over from the side elevation. e. The thickness of the foundation course, ts=\\ feet, and tr'—en, on the plan. /. The centre, O, of the face of the arch, is in the line r't pro- duced. The radius of the inner curve (intrados) of the arch is 4 feet »nd of the cylindrical back, behind the lace wall, 5} feet — shown by a dotted arc. In representing the stones forming the arch, it is to be remembered that they must be equal, except the " key stone," STONE STKUCTURE^>. I <5 ^, which may be a little thicker than the others; they must also be of agreeable proportions, free from very acute angles, or from re- entrant obtuse angles ; and must interfere as little as possible with the bond of the regular horizontal courses of the whig walls. There must also be an odd number of stones (ring stones) in the front of the ai'ch. On both elevations, draw the horizontal lines representing th «ing wall courses as one foot in thickness, and divide the inner curve of the arch into 15 equal parts. Draw radial lines tlirough the points of division. Their intersections with the horizontal lines are managed according to the principles just laid down. g. The points, as h and/", in the plan, arc then projected into the alternate courses of the side elevation, and into the line, Bo'", of the plan. From the latter line, the several distances, o"'h"\ &c., from o"\ thus found, are transferred to the line o'N, as at o'b"'\ (fee, and at these points the vertical joints of the front elevation are drawn in their proper position, as being the same actual joints, shown by the veitical lines of the side elevation. In the stones immediately under the coping, there must generally be some irregularity, in order to avoid triangular stones, or stones of inai>propriate size. //. To construct the front elevation of the coping. All points, aa a, a', a", in either the front or back, or upper or lower edges of the co])ing, are found in the same way, and as follows : a" is in a horizontal line through a' and in a line a"a"'\ whose distance from o' equals the distance o"'a"' on the plan. Construct- ing other points similarly, the edges of the coping may be drawn with an " irregular curve." The horizontal portion of the coping, over the arch, is projected over from C and from the two ends of the vertical line at D'. Execution. — In respect to this, the drawing explains itselC Example. Let this design, or any similar one, be drawn on a scale of four fest to the inch, on a larger plate; not forgetthig to place tlie three projections in their proper relative position, as shown (15) and (32). 1 CHAFrER II. WOODEN sxnucTunES. 244. Ex. 3». Elevation of a " King Post Truss." Mechanical constmction, <£c. — A Truss is an assemblage of pieces 90 fastened together as to be virtually a single piece, and therefore exerting only a vertical force, due to its weight, upon the support- ing walls. In PL XX., Fig. 125, A is a tie beam/ B is a. pri7icipal / C is a rafter / D is the kin// post/ E is a strut / F is a tcall plate / G is a purlin — running parallel to the ridge of the roof, from truss to truss, and supporting the rafters. H is the ridgepole; AV is the wall^ and ab is a strap by which the tic beam is suspended from the king post. 245. Grraphical constrnction. — In the figure, only half of the truss is shown, but the directions apply to the drawing of the whole. In these directions an accent, thus ' , indicates feet, and two accents, ' , inches. For ])ractice draw the whole Ugure, and on a larger scale. a. Draw the vertical centre line 6D. b. Draw the upper and lower edges of the tie beam, one foot apart, and 12' in length, on each side of the vertical line. c. On the centre line, lay off from the top of the tie beam, 5' — G' to locate the intersection of the tops of the principals ; and on the top of the tie beam, lay off II' on each side, to locate the intersec- tion of the upper faces of the principals with the top of the tie beam. d. Draw the line joining the two points just found, and on any perpendicular to it, as^]/, lay off its depth = 8", and draw its lower edge parallel to the ui)per edge. Make the shoulder at = 3' and parallel tofg. e. From tlie top of the beam, draw short indefinite lines, c, 0* each side of the centre line, and note the points, as c, where they would meet the upper sides of the principals. f. Draw vertical lines on each side of the centre line and 4' from it. g. From tlie points, as e, draw lines parallel to fg till they inter- «ect the last named vertical lines. •WOODEN STEUCTURKS. 147 h. Jviake ns = o' — 9". Make the short vertical distance at c-.=4' draw so, and make tlie upper side of the strut parallel to sc, and 4* from it. Note the intersection of this parallel with the line to the left of D, and comiect this point with the upper end of c, to com- plete the strut. i. Draw the edges of the rafter, parallel to those of the principal, 4' apart, and leaving 4" between the rafter and the principal. At Of draw a vertical line till it meets the lower edge of C, and from this intersection draw a horizontal line till it meets the upper edge of C ; Avhich gives proper dimensions to the wall plate. J. From the intersections of the upper edges of the rafters, lay off downwards on the centre line 12", and make the ridge pole, thus located, 3" Avide. k. In the middle of the upper edge of the principal, place the purlin 4" x 6", and setting 2" into the principal. I. Let the strap, ab, be 2" wide, and 2' — 6" long from the bottom of the tie beam. Let it be spiked to the king post and tie beam, and let it be half an inch thick, as shown below the beam. W, the supportmg wall, is made at pleasure. Execution. — This mainly explains itself. As working drawmga usually have the dimensions figured upon them, let the dimensions be recorded in small hair line figures, between arrow heads which denote what points the measurements refer to. 246. Ex. 4«. A "Queen Post Truss" Bridge. PI. XXL, Fig. 126. Mechcmical construction. — ^This is a bridge of 33 feet span, ovei a canal 20' — 6* wide between its banks at top, and 20' — 2" at the water line. It rests on stone abutments, R and P, )ne of \\hich ia represented as resting on a plank and timber foundation, the other on " piles." A is the tie beam ; B, B' the queen posts ; C, C the principals; D the collar beam., or straining sill ^ R, P, the abutments ; eQ,t the pavement of the tow path ; iK. the stone side walls of the canal; TP the opposite timber wall, held by timbers UU', N, dovetailed into the wall timbers ; E, S, the piles, iron shod at bottom. These are the principal parts. 247. Graphical construction. — Let the scale be one of five feet to the inch. a. All parts of the truss are laid off on, or from, the centre line AD. A is 14" deep; the dimensions of BB' are 12' x 6', except at top, where they are 10" wide for a vertical space of 16". C and D 148 WOODRX STRUCTURE8. are each 10' deep. BB' are 10' apart, and the feet ofC and C, 12* from the ends of the tie beam, which is 36' long. D is 6" below the top of the queen posts, rr are inch rods with five inch wasliers, \" thick, and nuts2^''xl'. 6i' is a f bolt; with washer 4"x?' and nuts, 2''xl'; and perpendicular to the joint, ad. b. From each end of the tie beam, lay oflf 1' — 9" each way for the width of the abutments, at the top. Make the right hand abutment rectangular in section and 11' high, of rectangular stones in irregu- lar bond (70). Let the left hand abutment have a batter of 1" in 1' on the side towards the canal, and let it be eleven feet high, in eleven equal courses. c. Make et^ the width of the paved tow path = 7' — 6', with a rise in the centre, at Q, of 6". d. The side wall is of rubble, 4' thick at bottom, and extenduig 18" below the water, with a batter of 1" in 1', and having its upper edge formed of a timber 12" square. e. The right hand abutment rests on a double cour.se of three-inch planks, qq\ 5^' broad, and resting on four rows of 10" pUes, ES. S is the sheet iron conical shoe at the lower end of one of these piles, the dots at the upper end of which rejiresent nails which fasten it to the pile. f. IT is a timber wall having a batter of l" to 1', and held in place by timbers, UU', N, dovetailed into it at its horizontal joints, hi various places. (J. The water line is 2' below T<, and the water is 4^ feet deep. 248. Execution. — It is intended that this plate should be tinted, though, on account of the difficulty of procuring adequate engraved fac-similes of tinted hand-made drawings, it is here shown only aa a finished line drawing, and as such, explains itself, after observing that as the left hand abutment is shown in elevation, it is dotted below the ground ; while, as the right hand abutment is shown in Bection, it is made wholly in full lines, and earth is shown only at each side of it. The usual conventional rule is, to fill the sectional elevation of a fctonc wall with wavy lines; but where other niaiks servo to distin- guish elevations from sections, as in the case just described, this labor is unnecessary. The following would be the general order of operations, in case this drawing weie shaded. a. Pencil all parts in fine faint lines. h. Ink all paits in fine lines. c. Grain the wood work with a very tine pen and light indi/xn ink, ^^tJ^^^^^^J^g i^ PL.XX-. WOODEX STRUCTURES. 149 the sides of timbers as seen on a ncAvly-planed board, the ends oi large timbers in rings and radial cracks, and the ends of planks in di3.gonal straight lines. See also the figures at y, where the lines of graining outside of the knots, are to extend throughout the tie beam. d. Tint the wood work — the sides with pale clear burnt sienna, the ends with a darker tint of burnt sienna and indian ink. e. Tint the abutments, and other stone work, with prussian blue mixed with a little carmine and indian ink, put on in a very light tint. /. Grain the abutments in waving rows of fine, pale, verti- tical lines of uniform thickness, about one sixteenth of an inch long, leaving the uj)per and left- hand edges of the stones blank, to represent the mortar. The part of the left-hand abutment which is under ground is dotted only, as in the plate. g. Grain the canal walls and paving, as shoAvn in the plate, to indicate boulder rubble. h. Shade the piles roughly, they being roughly cylindrical; tint them with pale burnt sien- na, and the shoe, S, with prus- sian blue, the conventional tint for iron. i. Eule the water in blue lines, distributed as in the figure. j. Tint the dirt in fine horizon- tal strokes of any dingy mixture. Note. — The above figure shows a little more than half of a queen-post ro^-truss of 43 feet span. Omitting the light upper pieces, it may serve in place of Fig. 126 as a longer bridge truss; and may be drawn on any convenient scale ivova.four to six feet to an inch. 150 WOODEX STKUCTUEES. in which burnt sienna prevails, in the parts above the water, and ink, in the muddy parts below the water, and then add, oi not, the pen strokes shown in the plate, to represent sand, gravel, &c. k. Place heavy lines on the right-hand and lower edges of all surfaces, except where such lines form dividing lines between two surfaces in the same plane. A heavy line on the under side of the floor planks, indicates that those planks project beyond the tie beam A. ~K. ■D PL.XXI, My<^'^y'sv.Tiy^-y„j,'. t/i.m.t. AiCTtig art' BIllUOlll oxivi oiuaix iiiuv vuv/ j7ic»v».t CHAPTER m. IRON CONSTEUCnONS. 262. Kx. 6". A Railway Track. PI. XXII., Figs. 129-13-i. Mechanical construction^ tbc. — It may be thought an oversight to style tliis plate the drawing of a railroad track ; but taking the track alone, or separate from its various special supports, as bridges, &c., its graphical lopresentation is mainly summed up in that of two parts ; first, the union of two rails at their joints; second, the inter- section of two rails at the crossmg of tracks, or at turn-outs. The fixture shown in Fig. 129, placed at the intersection of two rails tc allow the unobstructed passage of car wheels, in either direction on either rail, is called a " Frog." Let y and z be fragments of two rails of the same track, then the side H/' of the point of the frog, and the portion k k' of its side flange, B, are in a line with the edges, denoted by dots, of the rails y and s, so that as the wheel passes either way, its flange rolls through the groove, I, without obstruction. When the wheel passes from y towards z there is a possibility of the flange's being caught in the groove, J, by dodg- ing the point,/*. To guard against this, a guard rail, g g, is placed near to the inside of the other rail, supposed to be on the side of the frog towards Fig. 132, as shown in the small sketch, Fig. 132, which prevents the pair of wheels, or the car-truck, from working BO far towards the flange, B, as to allow the flange of the wheel to run into the groove, J, and so run off" the track. F/", and the por- tion, I /', of the flange. A, arc in a line with the inner edge of the rail of a turn-out, for instance, the opposite rail being on the side of the frog towards the upper border of the plate, as shown in Fig. 132. Hence the flange of a car wheel in passing in either direc- tion on the turn-out, passes through the groove, J, and is prevented from running into the groove, I, by a guard rail, near the inner edge of the opposite turn-out rail, as at U, Fig. 132. 253. Fig. 130 represents the under side of the right hand portion of the frog, and shu^v8 the nuts which secure one of the bolts which aecuve the steel plates, as D, E ; bolts whose heads, as at u and v, are smooth and sunk into the plates so that their upper surfaces ar« 153 ''^ON CONSTRUCTIONS. flush. It will be seen that there aie two imts on each bolt, aa at D', on the bolt u — DD', which appears below the elevation, since it occurs between two of the cross-ties (sleepers) of the track. The nuts, as L, belonging to the bolt, b", which are in the chaiis, q'p\ to', x', are sunk in cylindrical recesses in the bottom of the frog, so as not to interfere with the cross-tie on which the surliice, L, rests. The extra init is called a check or "jam" nut. When screwed on snugly it wedges the first nut and itself also against the threads of the screw, so that the violent tremulous motion to which the frog is subjected during the rapid passage of heavy trains cannot start either of them. Li the end elevation. Fig. 131, A is the recess in the chair x x\ fitted for the reception of the rail, and B is the end of a rail in its place, as shown at y in the plan. 254. Graphical Construction. — From the above description it follows that the whole length of the frog depends on the shape of the part 11/* F, and the distance between this part and the side rails, as c ?. In the piesent example a c = l' — 11" and cf =2 20". ed is 11" and nk is 2' from Ff. Having these relations given, and knowing that the lines at the extreme ends are perpendicular to the rails at those ends, the several figures of the frog can be constructed from the given measurements, without further explanation. 255. The construction of railway-track joints so as to secure as nearly as possible the uniform firmness of a continuous rail, has long exercised the minds of railway inventors. Cast-iron chairs, wrouglit-iron chairs, long chairs resting on ties each side of the joint, compound rails (Div. 11. , 140) solid -headed, or split through their entire height, and fish-joints have all been used ; several of them in various forms. Fig. 133 is an isometrical drawing — scale -^ — of a wooden fish-joint which allows great smoothness of "toiotion and freedom from the loud clack which accompanies the use of ordinary chairs. A, A, A, are the sleep- ers (cross-tics), D is a stout oak plank, perhaps six feet long, resting on three sleepers, and fitted to the curved side of the rail, as shown at d. This plank is on the outside of the track. On the inner side the rails are spiked in the usual way with hook- headed spikes s s s, of which those at the joint, r, pass through a flat wrought-iron plate, P, which gives a better bearing to the end of the rail, and prevents dislocation of parts. Each plank, as D, is bolted to the rail by four horizontal half-inch bolts, b, b, b, b, furnished with nuts and washers on the further side of D (not seen). IROX CONSTRUCTIONS. 152 A modification of the above construction consists in substi- tuting for the plate P, a short piece or strap of iron fitted to the surface of the inside of the rail, and through which the two bolts hh, next to the joint, pass. With the now extended use of steel rails, the fish-joint, also in very general use, consists of an iron fish-plate on each side of the, rail, with two bolts on each side of the joint. This makes a very- firm joint. The plan has also been sometimes adopted of having the track break joints. That is, a joint, as a, Pig. 132, on one rail of a track, is placed opposite the centre of the rail he of the other line of the same track. As a track always tends to settle at the joints, a jumping motion is induced in a passing train, which perhaps may be thought to be less violent if only on one rail at a time. 256. Graphical Construction. — Tlireo lines through X, making angles of 60° witli eacli otlier, will be llic isometric axes. Reincni bering that it is the relative position of the lines which distinguishes an isometrical drawing, we can place XX' parallel to the lower border, and thus fill out the plate to better advantage. The rail being 4" vdde at bottom, and 4" high, circumscribe it by a square "Kcan, from the sides of which, or from its vertical centre line, lay off, on isometric lines, the distances to the various points on the rail. Thus, let the widest part of the rail, near the top, be 3" across, and h an inch below the top ac. Let the width at the top be 2", and at the narrowest part 1"; and let the mean thickness of thf lower flange be f ". The sides of the rail are represented by the bottom lines at XX', and the tangents each side of R, to the curvea of the section. Let the plank D be 6" wide, and 4" high. All the lines of the spikes, ss, are isometrical lines except their top edges, as st. The curve at the joint r, and at X', are similar to the corresponding parts of the section at X. To secure case of graphical construction, let the bolt heads, A, fee , be placed so that their edges shall be isometric lines. Fig. 134, is a plan and end elevation of a heavy cast-iron chair designed as a partial equivalent for a continuous rail, by making the outside of the chair extend to the top of the rail. The fault in every such contrivance, the best of which at present seems to be the fish-joint, is that, as the joint cannot be made as solid as the unbroken rail, the wave of depression just in advance of the engine is more or less completely broken at every joint. 258. Ex. 7°. The Hydraulic Ram. In order to give an iron construction, from the department of machinery, so as to render this volume a more fit elementary course for the machinist as well 154 IRON COXSTKUCnONfe. as for the civil engineer, a simple and generally useful structure viz. a hydraulic ram, has been chosen, as a fit example for the last to bo described in detail. This machine is designed to employ the power of running watei to elevate water to any desired heiglit. PI. XXIII., Figs. 135-137, shows a hydraulic ram, of highly appioved construction, and of half the full size. 259. Mechanical conUructlon. — FF — F'F' are feet to support the machine. These are screwed to a floor or other firm support. Ar> — A'B'B' is theinlet pipe, openinginto the air chamber C, at a — a!V and ending at dd — d' d' — d"d' the opening in the top of the waste valve chamber, E — E' — E". At a — ah' is the opening as just noticed from the inlet pipe into the air chamber C (not seen in the plan). This opening is controlled by a leather valve ee', weighted A\ ith a bit of copper e''e"\ and is fastened by a screw A'A'", and an oblong washer g'g. At N and H arc the extremities of two outlet pipes leading from the air chamber at F"F"'. Either one, but not both of these outlet pipes together, may be used, as one of the exchangeable flanges, H' is solid, while the otlier is per- forated, as seen at M', Fig. 137. The air chamber is secured by bolts passing through its flange f'f\ through the pasteboard or leather packing, 2^P — p'l ^"cl the flange D — D'D' at cc. Tliis flange, and part of the inlet pipe are shown as broken in the elevation, so as to exi)ose the valve ee\ and the adjacent parts. LL' is a flange through which the inlet pipe passes, and this pipe is slit and bent over the inner edge of the aperture in LL', forming a flange, which presses against a leather packing, U\ and makes a tight joint. The outlet pipes are secured in tlie same way. At uu — u' are the square heads of bolts which fasten the flanges to the projections UU — U'. K — K' is a shelf bearing the waste valve chamber, E — E'E", and the adjacent parts. W — W is the flange of this valve chamber, secui-ed by two bolts at r^v" — v\ which ])ass through the leather packing y. Ji'h" is the waste valve, perforated with holes, ic, to allow water to flow through it. mm' is the valve stem, d'd'k'k' is a perforated standard serving as a guide to the valve stem, and also as a support to the hollow screw s. w is a rest, secured to the valve stem by a pin p". ^' is a nut, part of which, qq'^ is made hexagonal, r is a "jam'' nut (253). In the plan of this portion of the machine, the innermost circle IH the top of the valve stem; next is the body of the valve stem ; next, the top of the rest; next, the bottom of the same; next, the :pi..xxii c c inOX COXSTRtJCTlOXS. J 55 nut q" ; and outside of that, and resting on the toj) of tlie waste valve chamber, are the standards, dd. 960. Operation. — Principles ifivolved. — In the case of wliat miglit l)c called passive consti'UctionSy that is mere stationary sup- ports, like bridges, &c., a knowledge of the construction of tho parts enables one to proceed intelligently in making a drawing ; hut, in the case of what may, in opposition to the foregoing, be called active constructions^ or machines, a knowledge of their mode of operation is usually essential to the most expeditious and accu- rate graphical construction of them, because a machine consists of a train of connected pieces, so that a given position of any piece implies a corresponding position for every other pirt. Having, then, in a drawing, assumed a definite position for some important part, the remaining parts must be located from a knowledge of the machine, though drawn by measurements of the dimensions of that part. OnXj fixed bearings^ and centres of niotio7X, can properly be located by measurement, in machine drawing. The principles involved in the operation of the hydraulic ram may be summed up under three heads, as follows: 261. I. Work. a. When a certain iceight is moved through a certain space^ a certain amount of xoorh is expended. h. Thus ; when a quantity of water descends through a certain space, a certain amoimt of work is developed. c. As the idea of work involves the idea both of weight moved, and space traversed, it follows that tcorks may be equal, while the weights and spaces may be unequal. Thus the work developed by a certain quantity of water, while descending through a certain height, may be equal to that expended in raising a portion of that water to a greater height. 262. II. Equilibrium, a. Where forces are balanced, or mutu- ally neutralized, they are said to be in equilibrium. Now the usual fact is, that when such equilibrium is disturbed, it does not restore Itself at once, but gradually, by a scries of alternations about the ftate of equilibrium. Thus a stationary pendulum, being swung from its position of equilibrium, does not, at the first returning vibration, stop at the lowest point, but does so only after many vibrations. b. Theoretically, these vibrations, as in the case of the pendulum, would never stop, but in practice the resistance of the air, fription, &c., make a continual supply of a greater or less amount of forc^ necessary to perpetuate the alternations about the position oi state of equilibrium. 3.56 rnox constructions. 263. m. A physical fact taken account of in the hydraulic ram, 18, that water in contact with compressed air will absorb a certain portion of such air. 264. Passing now more particularly to a description of the ope ration of the hydraulic ram: 1°. Water from some elevated pond or reservoir flows into the machine, through the inlet pipe AA' and continues through the machine, and flows out through the hole in the waste valve A'A", pressing meanwhile against the solid parts of the roof of this valve, whose hollow form — open at the bottonj — is clearly shown in Fig, 136. 2°. Presently the water acquires such a velocity as to press so strongly against the roof of the Avaste valve, that this valve is lifted against the under side of the roof of its chamber which it fita accurately. 3*^. The water thus instantly checked, expends its acquired force in rushing through the valve e — e'e'" and in compressing the air in *.he air chamber C. 4°. The holes F" or F'" of the outlet pipe, leading to an unob- structed outlet, the compressed air immediately forces the water out through the outlet pipe until, after a number of repetitions of this chain of operations, the portion of the water thus expelled from the air chamber is i-aised to a considerable height. 5°. In accordance with the second principle, the flow of water from the air chamber does not cease at the moment when the confined air is restored to its natural density, but continues, so that — taking account also of the absorption of the air by the water at the time of compression — for a moment the air of the air chamber is moi-e rare than the external atmosphere. Hence to keep a constant supply of air to the air chamber, a fine hole called a snifting hole, is punc- tured, as with a needle, at ss\ i.e., just at the entrance of the inlet pi]ie into the machine. Through this hole air enters, with a snift- ing sound, when the flow of water recommences, so as to supply the air chamber with a constant quantity of air. When the waste valve is at the bottom of the chamber EE', the nut and "jam" are together at the bottom of the screw s\ and the valve is at liberty to make a full stroke. By raising the valve to its highest point and turning the nut and "jam'' to some position as shown in the figure, the stroke of the valve can be shortened at pleasure, and, at its lowest point, will be as far from the bottom of the chamber as the "jam," q'\ is above its lowest position. 266. In practice, it is found that the strokes of the waste valvfl ihortly become regular ; their frequency depending in any given IRON CONSTRUCTIONS. 157 case on the height of the supply reservoir, the height of the ejected column, the size of the machine, the length of the stroke of the valve, &c. 267. The proportion of water discharged into the receiving eservoir will also depend on the above named circumstances, oeing more or less than one third of the quantity entering the machine at AA'. In a machine by M. Montgolfier of France, said to be the original inventor — water falling 7tV feet, raised ^j of itself to a height of 50 feet. 2G8. Graphical Construction. — Scale; half the full size. u. Hav- ing the extreme dimensions ol the plan, in round numbers 9" and 12", proceed to arrange the ground line, leaving room for the plan below it. b. Draw a centre line, NC, for plan and elevation, about in the middle of the width of the plate. c. Draw a centre line, AK, for the plan, parallel to the ground line. d. Exactly 4^" from the centre line NC, draw the centre line vu" — K'm' for the waste valve chamber and parts adjacent. e. With the intersection, *, of the centre lines of the plan, as a centre, draw circles having radii of \%" and 3Jj" respectively, and through the same centre, draw diagonals, as cc. f. On the centre line, NC, are the centres of the circles, F"F"', 'a'hose circumferences come within p'^ of an inch of the inner onw of the two circles just drawn. g. Draw the valve, e, the copper weight e", the screw end, A, and the nut and oblong washer, It," and g. h. Locate, at once, the centres of all the small circles, cc, cfec, by the intersection of arcs, \" from the circle /»jo having * for a centre, with the diagonals ; then proceed to draw tliese circles. i. Draw the projections, as U, drawing the opposite ones simul- taneously, and using an auxiliary end view of the nuts m, as ollen explained before. J. Draw the feet, F, with their grooves, F, and bevel edged screw holes, L". k. In drawing the shelf, K, and flange W, the intersection of the centre lines BK and m'm, is the centre for the curves which inter- sect the centre line AK; the corners, 1 1, of the nuts, v,v'\ are the centres for the curves that cross the centre line, vv" ; and the remaining outlines of the shelf are tangents to the arcs thus drawn, and those of the flange are lines sketched in so as to give curves tangent to the arcs already drawn, and short straight lines parallel to w" 158 IRON COXSTRUCTIONS. /, The reiiKiining cii-cles and larger hexagon, u\ of this portiorj of the plan, have the intersection of the centre lines for a centre ; and inav be drawn by nieasuiements independently of the elevation, or by projection from the elevation, after that shall have been finished. 269. Passing to the elevation; — a. Constrnct, at one position of the T square, the horizontal lines of both feet; then the horizontal lines of the nuts w', and flange L', and projection U' ; with the horizontal lines of the floor of the air chamber and adjacent parts. h. Project up from the plan the vertical edges of the feet, F'F', the flange, nut, and projection L', u' and TJ', the valv? e\ the cop- per e", the screw A", the washer ^, the air chamber flange/"/'', and screw z. Break away the portion D — see plan — of the body of the machine, and the near wall of the water channel A'B', Break away also the further wall of the water channel so as to show 3 Bection, ir, of the further outlet pipe, H — see plan. Q 's the centre of the spherical part of the air chamber to which the conical part is tangent. c. Draw all the horizontal lines of the waste valve chamber and parts adjacent. Make the edges of the threads of the screw straight ami slightly inclined upwards toward the right. d. Project up from the plan, or lay ofl", by measurement, the widths of various parts through which the valve stem passes, and draw their vertical edges. Fig. 136 is a section of the waste valve chamber, showing part both of the interior and exterior of the waste valve. The dotted circles form an auxiliary plan of this valve, in which the holes have two radial sides, and two circular sides with x" as a centre. The top of the valve is conical, so that in the detail below, two of the Bides of the hole /i, tend towards the vertex, x. At ii'^ one of these holes, of which there are supposed to be five, is shown in section. Fig. 137. The outlines of M, one of the outlet i)i])e flanges, are drawn by processes similar to those employed in drawing the shelf, K, in plan. 270. Kxecution. As a line drawing, the plate explains itself. It would make a very beautiful shaded drawing and one that the oarelul student of the chapter on shading and shadows, would be able to execute with substantial accuracy, without further izi3truo Lion. We conclude this division with the following additional excr- •isee as examples of iron constructions — one from civil engineer- IRON CONSTRUCTIONS. 159 ing practice, the other three from mechanical engineering; styl- ing them exercises, since, being partially shown (yet, with the description, sufficiently so for their purpose), they leave some- thing to be supplied by the student from the general insight gained from previous practice. Exercise 1. A Stop-valve. — The following figure shows one of many forms of valve differing more or less in detail, and made for the purpose of shutting off the passage of steam, water, etc., through pipes. Such halves either lift 'rom their seats, as in the example shown, or slide off them, in which case they are sometimes called gates. The figure represents what is called a ghhc-vahe, from the general external form of its valve-chamber NCCL. In this chamber is a bent par- tition, or diaphram, CEC, containing the seat, E, of the valve D. This valve is raised or lowered by means of the hand-wheel K, and screw valve-stem F working in the collar G, which is screwed into the top, NC, of the chamber. The head at the bottom of the valve-stem, working loosely in the hol- low head of the valve, raises tl>e latter vertically without turning it. The cap H secures the necessary packing. Opening the valve then allows of the passage of any fluid through AB and the pipes which may 160 mO'S CONSTRUCTIONS. be attached at A and B. Thesn openings are from J" to 2" diameter. The measurements and scale may thererore be assumed, and plan and end elevation added. Exercise 2. An iron truss bridge. Pi XXIV., Figs. 1-7. — This bridge is partly of wrought, and partly of cast iron, and known from its form and its inventor as Whipple's trapezoidal-trtiss bridge. The ujiper chords, a — a'a\ are lioUow cast-iron cylinders 7^" diameter, and I" to %" thickness of metal. The j)o^ts, p', and struts, 3, S'S", arq also of cast-iron, the latter, double, as seen in the fragment of end ele- vation. Fig. 2, and fragment of plan. Fig. 3. The posts extend through the flooring, where they are 5" in diameter, and rest on seats on the tops of the cast-iron coupling blocks, n'n', as shown in the plan. Fig. 5, and end elevation, Fig. 6, of one of these blocks. The loicer chord, bl> — b'V, is composed of heavy wrought-iron rods made in links embracing two successive coupling-blocks, in the manner shown in Figs. 4-6. The two end lengths, however, are single, as shown, and are secured by nuts, g', at the outer end of the shoes s, s", which holds the feet of the struts SS'S". The structure is further held in shape, and the forces acting in it suit- ably sustained and distributed by the diagonal and vertical rods r'r', each of which, after the first two from the end, crosses two panels of the bridge, as the spaces between the posts are called. The horizontal diagonal rods, y, under the floor, tightened by links I woiking on right- and left-handed screws (Div. V., Ex. 9) in the adja- cent rod ends, provide against the horizontal force of winds. The light transverse flanged beams k — k", overhead, also help to stiffen th6 struc- ture laterally. The main transverse beams c — c' — c" rest on the coupling-blocks, and support the floor joists dd'd", on which the floor planks gg'g" rest. CCC" is the coping, serving to cover the irregular ends of the floor planks, and as a guard to prevent vehicles from striking the truss. Fig. 7 is an enlarged view of the centre joint where the two halves of the posts meet. Other useful details would be vertical longitudinal sections of the joints as ee" and e', which Avould show an opening in the under side of the upper chord, sufficient for the entrance of the diagonal rods, and tliese rods forged into rings clasping the stout wroiiglit-iron pins e, e'c" \ also the level bearing for the head of the post, except at the joint at the head of the strut. Tlie three top cross-beams indicated at Teh, Fig. 3, show that a slcew- bridge is represented, that is, one which crosses the stream obliquely, the extreme timber, h, being parallel to the length of the stream. jL'j^JCxar IROX COXSTRUCTIOXS. 161 The span of the bridge is 114 leet, in 12 equal panels of 9^ feet eacn; the roadway is 19 feet wide from centre to centre of the trusses, which are 15 feet 9 inches in height from the centre of the coupling-blocks, n', to that of the upper-chord pins as at e'. Suitable scales are 3 to 5 feet to 1 inch for the general views, and from 6 inches to 1 foot to one inch for tht; details. Exercise 3. A vertical loiler. PI. XXIV., Fig. 8. — This figure, being given partly as an excellent example in shading, and of certain flame effects instructive to the draftsman, is described without letters of ref- erence. The figure represents a vertical section of what is known as the Shap- ley patent boiler, differing from the ordinary tubular vertical boiler as appears from the figure and following description. The central combustion chamber, being tall, is designed to effect three results ; viz., to raise its top, called tlie crown sheet, so far above the fire as to retard burning out; to afford abundant room for perfect combustion, thereby generating more heat; and to effectually convey this heat to the water which surrounds the fire-box in a thin sheet. Heat is further conveyed to the water by passing, as shown by the arrows, through short transverse tubes, two of them seen in section, and vertical tubes between the fire-box and the outer shell. These open into the annular base flue (interrupted by the ash-pit door), which leads to the smoke-pipe (sometimes called the uptake). The upper section, or steam-dome, is mostly occupied by steam, and is stayed by bolts to the crown sheet. Since the tubes, when sooty, lose much of their heat-conducting power, they are, in this boiler, made very easily accessible for frequent clean- ing by connecting the two sections of the boiler by a double annular jacket which contains no steam or water and sustains no pressure. It is made in sections for easy removal, and thus allows ready access to the tubes. Exercise 4. A direct- acting/ steam-pump. PI. XXIV., Fig. 9. — The mag- nitude and variety of pumping requirements for water, oil, and various other liquids, hot or cold, thin or viscid, pure or gritty, and for drain- age, mining, city, hotel, railroad - station, and other purposes, have called forth a large amount of inventive talent and many ingenious and effective pumping engines. The figure represents a vertical longitudinal section of the Knowles steam-pump, affording a useful study and guide in making a finished drawing. BB is the pump barrel or water cylinder — lined, when the character of the fluid to be pumped requires it, with composition linings, shown at XX and similarly in section on the upper side of the barrel. P is the IQ2 IROIS" COXSTRUCTIOIirS. •water piston with its packing p, and secured to the piston-rod a by nut and lock-nut (Div. V., 12) seen at the left. JEF is the steam cylinder with its piston on the same piston-rod, a, with the water piston, thus forming what is called a direct-acting pump. Both cylinders are provided with stuffing-boxes hg and K. The pump valves under the letter b are here shown as lifting disk valves circular in plan, but may be cage, or hinge, or any other valves. In the posiiion shown, and the piston still moving towards the left, water is entering through the lower or suction circular inlet and the lower right-hand valve, and is discharging by the upper or discharge pipe, which is smaller than the suction pipe. By raising the upper left- hand valve the discharge water also partly enters the air-chamber A, where, by compressing the confined air, a steady discharge is obtained. The valves rise and fall, each working on a short spindle, and are quickly closed by the aid of sjiiral springs above them; seen on the two closed valves. The steam and exhaust ports and passages to the steam cylinder are of the usual form ; n, the orifice for the admission of steam from the boiler, and the central orifice is the exhaust. The steam-valve is a double D valve. A stroke to the right being about to begin, a roller on the opposite side of the tappet arm CC, carried by the piston-rod a, raises the left end of the rocker DD. This, by means of the link s, slightly rotates the valve-rod Z and its "chest-piston," Fig. 11, so as to bring it into a posi- tion to take steam through the small passage at the lower right-hand corner of the steam-chest G, which throws the piston to the opposite end of its stroke, carrying the valve by means of its stem T, Fig. 10. Steam can then enter the left-hand end of the cylinder through the left-hand chamber of the D valve, while exhaust steam escapes through the pas- sage y and the right-hand chamber into the central, or exhaust passage. At i is the valve-rod guide. j\a a collar on the valve-rod. u clamps the rocker connection to the valve-rod. t adjusts the link s. M is the oil-cup, and 7i a stud to attach a hand lever. These pumps are made of a large range of sizes, from water cylinders of 2", and steam cylinders of 3J" diameter, and 4" stroke; to water cylinders of 20", and steam cylinders of 28" diameter, and 12" stroke. Fig. 9 may be regarded for drawing purposes as a sketcii (from a scale drawing and in true proportion, however) of a pump having a water cylinder of 7", and a steam cylinder of 12" diameter, with a 12" stroke. For variety of practice in the use of scales, the pump may then be drawn on a scale of i or \ the full size, with details on scales of 3" to a foot, or of full size. THE END. PL.xxrt; ^^W ^^ ^•^^>^3^i^'C:. l^w^ 1. '^^y^'^' ^M"' IBii