UC-NRLF ll il 1 ; '1 j 1 III i ■ , ■,■,,■ C 2 775 flflS TM' \ m'^ ^^\ i ^ i^^.4^^' i%: K §1 THEORETICAL AND PRACTICAL GRAPHICS AN EDUCATIONAL COURSE THEORY AND PRACTICAL APPLICATIONS DESCRIPTIVE GEOMETRY AND MECHANICAL DRAWING PREPARED FOR STUDENTS IN GENERAL SCIENCE, ENGINEERING OR ARCHITECTURE FREDERICK NEWTON WILLSON C. E. (rensselaer); A.M. (phinceton) Professor of Descriptive Oeotnetry, Stereolomy and l^echnical Drawing in the John C, Green School of Science, Princeton University ; Member Am. Sac. Mechanical Engineers; Member Am. Mathematical Society; Associate Am, Soc. Civil Engineers; Fellow American Association for the Advancement of Science. NEW YORK THE MACMILLAN COMPANY I/ONDON : MACMILLAN & CO., Ltd. 1898 ALL RIGHTS RESERVED JrtiH j^Q.B^'^J VVi' GROUPINGS OF CHAPTERS FOR INDEPENDENT COURSES. ?^79-^ I . Course in Free - Hand Sketching and Lettering, Note - Taking, Dimensioning and the Conventional Representation of Materials. Chapters II and VII. II. The Choice and Use of Instruments; Line and Brush Work; Plane Problems of the Line and Circle; Projections (Third Angle Method); Development of Surfaces for Sheet Metal Constructions; Intersections; Working Drawings of Rail Sections, Bridge Post Details, Gearing, Springs, Screws, Bolts, Slide Valve, etc. Chapters III, IV, VI, X (to Art. 445), XVII and Appendix. III. Course on the Helix, Conic Sections, Trochoidal Curves, Link -Motion Curves, Cen- troids, Spirals, etc. Chapter V and Appendix. IV. Working Drawings by the Third Angle Method; Intersections and the Development of Surfaces. Chapter X to Art. 445. V. Descriptive Geometry (Monge's), First Angle Method. Chapters IX, X (Arts. 445-522.) VI. Shades, Shadows and Perspective, with especial reference to Architectural Applications. Chapters XIII and XIV. VII. Axonometric Projection, Isometric Projection, One-plane Descriptive, Oblique (CUno- graphic) Projection, Cavalier Perspective, Chapters XV and XVI. V III . Broad Course in Descriptive Geometry, and its applications in Trihedrals, Spherical Projections, Shadows, Perspective, Axonometric and Oblique Projections. Chapters I, IX-XVI. COPrRIGHTEO IN PARTS, IS90, 1892, 1B96, BY FRECfK N. WILLSON. ALL RIGHTS RESERVED. COMPOSITION, UNIVERSITY PRESS, PRINCETON, N. J. CEROGRAPHIC PLATES, BRADLEY d POATES, N. X. PRESS-WORK, M. W. A C. PENNVPACKER, ASSURY PARK, N. J. PMOTO-ENQRAVINQ AND HALF-TONES, NEW YORK ENGRAVING ANO WOOD-ENGRAVINGS, A, P. NORMAN, ANO BARTLETT A CO., N. t. PRINTING CO., AND THE ELECTRO-LIQHT ENGHAVINQ CO., N, Y. PHOTOGRAVURES, THE PHOTOGRAVURC COMPANY, N. Y. * ELECTROTYPING, J. P. FELT A CO., N. Y., F. A. RINGLER A CO., N. PREFACE fHE preparation of this work was not undertaken until the author had felt the need of such a V)ook for his own classes, and a careful examination of the literature of graphical science had led to the conviction that it would occupy a distinct field. So great had been the cost and so highly specialized the nature of the finer text -books on the topics here treated, that to give a broad, educational course, by using the best work available on each branch, involved a far greater outlay than the average student could well afford, or a teacher would feel justified in requiring him to make. P^rt of the self-imposed task, therefore, was to endeavor to compress between the covers of a book not larger than the average more specialized work, and at no greater cost to the student, not only all the usual matter found in treatises on mechanical drawing and orthographic projection, but also much which should — but too often does not — form a part of a draughtsman's education. Of scarcely less importance than the proposed extended range of content was the method of presentation, the desire being not only to lay a broad and thorough foundation for advanced work along mathematico- graphical lines, but also in so doing to have every 'feature — illustrations, typog- raphy and even the quality of the paper- — contribute as much as possible to the creation and increase of an interest in some of the tojjics for their own sake, and to a desire to continue to work in some of the fields into which the student would be here introduced. While aiming to include nothing which might not reasonably be required of every candidate for a scientific degree, it was felt that it would increase the serviceability of the book, alike to teachers and to those dependent uj)on self- instruction, if it were so arranged that by taking its chapters in certain indicated groupings,* either elementary or advanced graphical courses could be taken from it with equal facility. On its practical side it will be found in fullest accord with the modern methods of the leading engineering and architectural draughting offices. The Third Angle Method for making machine-shop drawings receives special consideration, independently of the earlier system ; the latter, however, is of too great convenience for pure mathematical work and for stereotomy to ever become obsolete, and is therefore fully treated by itself. Since but little new matter is presented, whatever especial value the book may be found to possess must in chief measure depend upon the way in which old facts are here stated, illustrated and correlated; but the following may, however, be mentioned as original, although previously issued either in pamphlet form or in the advance sheets which have for some time been in use with the author's classes: A method for drawing a tangent to a Spiral of Archimedes at a given point, when the pole and a portion only of the arc are given; a demonstration of the property of double gene- ration of trochoidal curves when the tracing point is not on the circumference of the generator, with new terms completing a nomenclature of trochoids based on the property just mentioned; a simple *See opposite page. Some of these groupings are also to be separately issued as "parts." iv PREFACE. method for projecting the Pliicker conoid; and a few new terms in Chapter IX, suggested in the intei-est of brevity. The conchoidal hyperboloid of Catalan is probably treated in English for the first time, in this work; while such topics as the preparation of drawings for illustration, projective conies, relief per- spective, the theory of centroids and certain of the higher plane curves and algebraic surfaces, are among the features which will be noted as unusual in an elementary treatise. The Title. The comprehensive term Graphics was selected in the interest of brevity as well as appropriateness, as permitting the introduction of any science based upon the exact delineation of relations on paper, usually by the application of geometrical — and, in particular, of projective — properties by means of draughting instruments. No rigid line can be drawn between the theoretical and the practical part, except as the group- ing of the chapters, already alluded to, separates the elementary — and usually called "practical" — portions /rotn the advanced; but a knowledge of the mathematical properties of the h3'perbolic paraboloid, and the ability to make the drawings for a bridge portal of that form* when occasion requires, is obviously as "practical" as the drawing of an elbow joint; the classes these constructions represent therefore receive equal treatment, as this book is partly intended to be a concrete protest both against that spirit which regards a mathematical abstraction as degraded if some commercial application of it can be found, and against the disparagement of theory, as worthless for the "prac- tical man." Chapter I. A broad and comprehensive survey of the fields the student is about to enter seems the natural preliminary to intelligent work therein; the first chapter is, therefore, devoted to rigid definition and differentiation of the graphical sciences, and the arts in which they are applied. Some remarks on the nomenclature of geometries are also included, as further extending the draughtsman's usually too limited horizon. This would naturally be followed by the ninth and succeeding chapters in a course arranged more for educational than commercial purposes. Chapter II. As free-hand sketches are rightly made the basis of much of the practical draught- ing of the embryo engineer or architect, and as the graduate has frequent occasion, either as inspector or designer, to make clear and intelligible drawings without instruments, full instructions are given in this section as to what may be called technical, as distinguished from artistic, free-hand work, covering the following points: Sketching either in pictorial or orthographic view, dimensioning, free-hand lettering, conventional representation of materials, and note -taking on bridges and other trussed work, pins, bolts, screws, nuts and gearing. Chapters III and IV are devoted to the description of the draughtsman's equipment, and to pre- liminary practice in its use, during which the student is familiarized with the methods of represen- tation most employed, and with the solutions of the usual problems of the straight line and circle. The hyperboloid and anchor ring are also given as good tests of the beginner's skill in execution, but are so presented as to afford, with the other problems, material for recitation. Since these chapters were electrotyped an instrument of exceptional value has been placed on the market, a compass whose legs remain parallel as the instrument opens. This is a novelty of such merit as to justify a notice here, since it cannot be incorporated in the body of the work. Chapiter F, although appearing at that stage of a beginner's work when he will presumably be learning the use of the irregular curve and being ostensibly to furnish exercises therefor, is in * Although an unusual design, one is in process of erection at present writing. PREFACE. V, reality a treatise on the more important higher plane curves, and on the helix. It afforded an opportunity, in connection with the conic sections, to introduce the student to the beauties of the projective method, and give him his first notions of perspective. The close analogy between homological plane and space figures made it seem advisable to intro- duce the latter, if at all, immediately after the former; so that relief- perspective appears somewhat out of its logical mathematical setting. While employing Cremona's notation, the works of Burmester, Wiener and Peschka have been otherwise followed on projective geometry. The prominence given to the trochoidal curves, both in the main text and the Appendix, while primarily due to the interest in them which a reading of Proctor's Geometry of Cycloids aroused, is justified both by their intrinsic value, mathematically, and their important practical applications. Their tabular classification — an extension of Kennedy's scheme — contains distinctions among the hypo- curves whose acceptance by both Reuleaux and Proctor would seem to assure their permanence; while the reciprocal terms Ortho- cycloid and Cyclo - orthoid, incorporated at the suggestion of Professor Reuleaux, completed the system in a symmetrical manner. The remaining plane curves are treated with varying degrees of fullness, according to the impor- tance of their properties and applications; while throughout the chapter, as in other portions of the work, historical or descriptive matter has been introduced in order to enliven as far as possible what would otherwise have been a bare statement of mathematical fact. Salmon, I.eslie, Eagles and Proctor were the authorities of most service in this connection. Chapters VI and VII. Proficiency with brush and colors is an indispensable qualification for suc- cess either as artist or architect. It is customary, however, in some quarters, to disparage such attainments in the engineer, as likely to be so infrequently in demand as to make the time spent in their acquisition a practical loss. If it is assumed that every student of engineering is to enter the draughting office of some bridge company, on the lowest round of the professional ladder, there to remain, ambitionless, then let him by all means learn only tracing and copying; but the instances of improved conditions, due to manual skill, are too numerous to justify any lowering of the standard for the embryo engineer, especially as he might otherwise find in later life, as has many another, a design that was inferior to his own accepted because more handsomely worked up. It is also well to remember, that in times of depression in the engineering world his abilities in this line and in lettering would aid him in other fields, and that superior skill in both, combined with originality, often commands the same rate per week in illustrating establishments as is paid per month for shop drawing. Chapters on the methods of obtaining varied effects, and on lettering, are therefore among the most important relating to the less theoretical part of the student's preparatory work. The full instructions given in Chapter VII on spacing and proportioning, mechanical short-cuts, ornamentation, etc., will, it is believed, make this portion unusually serviceable to those who have felt the lack of such features in many otherwise most valuable works. In the Appendix a large number of complete alphabets affords a considerable range of choice, among forms which are of special service to engineers, architects and others. Chapter VIII. In addition to acquaintance with the blue -print process, whose use is at present so well-nigh universal, some familiarity with other modern methods of graphic reproduction ma}"- well form a part of the education of a scientific man, both as a means of enhancing his interest in the work of others, and of enabling him, with the least expenditure of time, to prepare the draw- ings for the illustration of his own researches or original designs. Full information is therefore given in this chapter on all the technicalities with which it is requisite that the amateur illustrator should^ be familiar, and a list of reference works is furnished the intending specialist. vi PREFACE. Chapters IX and X. In these chapters, covering an even hundred of pages, the Descriptive Geometry of Monge is treated in a manner intended not only to reduce to a minimum the difficul- ties ordinarily encountered in its study by students who are deficient in the imaginative faculty, but also at the same time to arouse an interest in this fundamental science of the constructive arts. Considerable reliance is placed, for the attainment of these ends, upon the use of pictorial views; and for the surfaces involved a series of wood -cuts are presented, which ought to prove a fair equivalent for a collection of models to those who unfortunately have not access to the latter. Believing with Cremona that the association of the names of illustrious investigators with the products of their labors is "not without advantage in assisting the mind to retain the results them- selves, and in exciting that scientific curiosity which so often contributes to enlarge our knowledge," the author has given both as to curves and surfaces, the commonly accredited source, although without undertaking verification. The Idea of defining a straight line as determined by two points (footnote to Art. 336) is due to Halsted (Appendix to translation of Bolyai), but since it was electrotyped it would seem to bo an improvement to have it read "the line that is completely determined by any two of its points." In Chapter X the choice is offered of dealing with figures by either the First Angle or Third Angle Methods. The latter is given first, being usually applied to more elementary surfaces than the other; and in connection with it the development of surfaces receives full treatment, followed by a large number of problems on the intersection of developable surfaces, which it is assumed will be worked out, like those in the section preceding them, to their logical conclusion — a finished model in Bristol -board. Variations of the problems on projection, sections, etc., can be readily made by employing the designs given in the Appendix. The portion of Chapter X which is devoted to the First Angle Method is supposed to be taken in close connection with Chapter IX, and may, if preferred, follow directly after a reading of pages 105-119, in order to model the course more closely along Continental lines. Chapter XI is on Trihedrals, which are treated in the usual way, except that in several cases solutions are given by both the one -plane and two -plane methods. Chapter- XII, on Spherical Projections, differs from the usual treatment of the topic considerably, the scientific classification of Craig having been adopted, much of the space usually devoted to orthographic projection having been transferred to stereographic, and a larger number of methods described than in other elementary treatises on this topic. Chapters XIII and XIV, on Shadows and Perspective, have been written with especial reference to the needs of architectural draughtsmen, and, though brief, cover all necessary principles, and the methods of best American practice. Chapters XV and XVI give not only the theory of axonometric and oblique projections, but also their api>lications in shadows, timber framings and stone cutting; and the contrast between the two systems is shown more clearly by applying them to the same arch voussoirs and structural articula- tions. The method of drawing crystals in oblique projection is also illustrated. One -plane Descriptive Geometry receives brief treatment, as being in theory so simple and in application so limited as to warrant the devotion to it of but little space. Chapter XVII, on bridge details, gearing, screws, springs, etc., might more logically have followed the theory of the Third Angle Method in Chapter X, but would there have interrupted the con- tinuity of that portion, and was therefore relegated to its present position. It is supplemented by working drawings in the Appendix. PREFACE. vii The Illustratims. Believing that a good illustration reduces very materially the number of words necessary to a demonstration, the author has taken especial pains in designing and drawing the figures, so as to have them, in as large degree as possible, self-explanatory; and for their repro- duction the five modern illustrative processes have been employed which seemed best adapted to the purpose, viz., cerography, photo - engraving, "half-tone," photo-gravure and wood - engraving. With regard to some of the figures the following acknowledgments are due: The wood -cut of the Pliicker conoid was made, by kind permission of Sir Robert Ball, from his illustration of that surface in his Theory of Screws, and is an exact reduction thereof, to scale. Figures 90 and 91 are slight modifications of designs by Adhemar. Figure 95 is from a photograph of a model by Burinester. Figure 99 is in its essential features a combination of two illustrations in Reuleaux' Kinematics. For the adaptation of the principle of the wedge to the tractrix (Fig. 115) indebtedness must be expressed to Halliday's Mechanical Graphics. It is impossible to give credit for Figures 138 and 141, as their origin is unknown. Figures 208, 211, 212 and 224-227 are from surfaces in Princeton's mathematical collection. Figures 345, 346, 370 and 371 are half-tone reproductions of photographs taken at the Paris Conservatoire for Columbia University, a duplicate set of which were made for Princeton from the original negatives, which were kindly loaned the author for that purpose by the late Dr. F. A. P. Barnatd, then president of Columbia. Reference Literature. The more important treatises consulted are mentioned at the end of the book, as constituting a valuable reference library for the si)ecialist in any of the lines named. The list includes some works already referred to in the text, as also those mentioned under some of the previous topical headings. There is so much in common in tliom that it has been impossible in many cases to say which has been an "original" source; but credit has been given whenever it could be with definiteness. Being the fortunate possessor of a cojjy of the first edition of Monge's Descriptive Geometry, there was at hand one authority, at least, whose originality was beyond doubt. With the following concluding remarks a long and frequently interrupted undertaking is completed, and a foundation course in graphical science presented on a University plane, in such shape, it is hoped, as to be almost as serviceable to those who cannot use it amid University surroundings, as to the more fortunate ones who can. These remarks would include the conventional acknowledg- ments to advisers, proof-readers and publishers had not the original plan been adhered to, of having ' the work represent only so near an approach to an ideal then in mind as could be secured by carrying it through to a finished edition under the author's personal supervision of every feature. Having purchased new type in order to have the plates fiawless, and the final type -proofs hav- ing practically been such, it is a disappointment to find that standard unattained in the end; equally so to have a few of the later illustrations fall below the general average. Others represent the second or even third attempt of the plate -maker, notably Fig. 228 (b), which, however, as finally accepted, is a triumph of the engraver's art. Previous editions of some of the earlier pages were printed from the type, for their care with which acknowledgment is due to the press -men, Messrs. J. P. Leigh and P. Bennett, of Princeton. With the exception of a page of designs in the Appendix, material for the variation of problems is left for separate issue; as also chapters on valve motion, stereotomy and perspective of reflections. F. N. W. Princkton, N. J., July, 1897. TABLE OF CONTENTS CHAPTER I. Fundamental principles of Graphic Science. — Di- visions of Projections. — Definitions and Appli- cations of the Sciences Based on Central Pro- jection, as Projective Geometry, Perspective, Relief - Perspective, Sciography, Photogram- metry. — Df^nitions and Applications of the Sciences based on Parallel Projection, as Clino- graphic Projection, Cavalier Perspective, One- plane Descriptive Geometry, Axonometric Pro- jection and the Descriptive Geometry of Monge. — Remarks on the Nomenclature and Differ- entiation of Geometries. Pages 1-41 CHAPTER U. Technical Free - Hand Sketching and Lettering. — Note -Taking from Measurement. — Dimension- ing. — Conventional Representations. Pages 5-10. CHAPTER til. The Choice and Use of Drawing Instruments and the Various Elements of the Draughtsman's Equipment. — General remarks preliminary to instrumental work. Pages 11-20. CHAPTER IV. Kinds and Signification of Lines. — Designs for Elementary Practice with the Right Line Pen. — Standard Methods of Representing Materials. — Line Shading. — Plane Problems of the Right Line and Circle, including Rankine's and Kochansky's approximations. — Exercises for the Compass and Bow -pen, including uniform and tapered curves. — The Anchor Ring. ^ The Hy- perboloid. — A Standard Rail Section. Pages 21 -38. CHAPTER V. ,,^^ Regarding the Irregular Curve. — The Helix. — The Ellipse, Hyperbola and Parabola, by various methods of construction. — Homological Plane Curves. — Relief- Perspective. — Link- Motion Curves, — Centroids. — The Cycloid. — The Companion to the Cycloid. — The Curtate and Prolate Trochoids. — Hypo-, Epi-, and Peri- Trochoids. — Special Trochoids, as the Ellipse, Straight Line, Limagon, Cardioid, Trisectrix, Involute and Spiral of Archimedes. — Parallel Curves. — Conchoid. — Quadratrix. — Cissoid. — Tractrix. — Witch of Agnesi. — Cartesian Ovals — Cassian Ovals. — Catenary. — Logarith- mic Spiral. — Hyperbolic Spiral. — Lituus. — Ionic Volute. Pages 39-78. CHAPTER VI. Brush Tinting, Flat and Graduated. — Masonry, Tiling, Wood Graining, River -Beds, etc., with brush alone, or in combined brush and line work. Pages 79-87- CHAPTER VU, Free - Hand Lettering. — Mechanical Expedients. — Proportioning of Titles. — Discussion of Fonns. — Half- Block, Full Block and Railroad Types. — Borders and how to draw them, (Alphabets in Appendix). Pages 88-96. CHAPTER VIII. The Blue -print Process. — Photo-, and other Re- productive Graphic Processes, how to Prepare Drawings for Illustration by them ; and including Wood Engraving, Cerography, Lithography, Photo - lithography, Chromo - lithography, Pho- to-engraving, " Half-Tones," Photo-gravure and allied processes. Pages 97- 103. CHAPTER IX. Orthographic Projection upon Mutually Perpen- dicular Planes, or the Descriptive Geometry of Monge. — Fundamental Principles and Problems (Arts. 283 - 330). — Definitions and Various Classi- fications of Lines and Surfaces, and suiftmation of the principles on which later problems relating to them are solved. (Arts. 331-382). Pages 105- 130. CHAPTER X. Monge's Descriptive Geometry, (continued). — Working Drawings by the Third Angle Meth- od. — The Development of Surfaces, for Sheet Metal or Arch Constructions. — Intersecting Sur- faces. — Projections, Intersections and Tangen- cies of Developable, Warped and Double - Curved Surfaces, by the First Angle Method. Pages 131-205. CHAPTER XI. Trihedrals, or the Solution of Spherical Triangles by Projection. Pages 206-210. CHAPTER Xtl. Map Projection.— Orthographic Projection of the Sphere. — Stereographic. — Gnomonic. — Nico- lisi's Globular. -De la Hire's Method. — Sir Henry James' Method. — Mercator's Chart. — Conic Projection, — Bonne's Method. — Rectan- gular Polyconic. —Equidistant Polyconic. —Or- dinary Polyconic Projection. Pages 211- 218. CHAPTER XIII. Shades and Shadows, Fundamental Principles and Definitions. — Shadows of Plane Sided Surfaces, as the Cube, Pyramid, Steps and Pier. — Col- umns and Abaci. — Hollow Cone, inverted. — Brilliant Points in general, and on given sur- faces, — Shade Line on Torus. — Shadow of Niche. — Shades and Shadows of Triangular- threaded Screw. Pages 319-227. CHAPTER XIV. Linear Perspective, Definitions and Illustration of Methods, — Perspective of Cube by various meth- ods. — Perspective of Cur\'es, — Method by Trace and Vanishing Point, as used in Architec- tural Work. — Perspective of Shadows, two methods. —Method of Scales, applied to Inter- iors. — Right Lunette. — Groined Arch. Pages 228-240, CHAPTER XV. Orthographic Projection upon a Single Plane, — Axonometric Projection. — General Fundamental Problem, inclinations known for two of the three axes, — Isometric Projection vs. Isometric Draw- ing. — Shadows on Isometric Drawings. — Tim- ber Framings and Arch Voussoirs in Isometric View, — One -Plane Descriptive Geometry. Pages 241-247. CHAPTER XVI. Oblique or Clinographic Projection, Cavalier Per- spective, Cabinet Projection, Military Perspec- tive. — Applications to Timber Framings, Arch Voussoirs and Drawing of Crystals, Pages 248-250. CHAPTER XVII. Working Drawings of Bridge Post Connection. — Structural Iron. — Spur Gearing, (Approxi- mate Involute Outlines). — Helical Springs, Rectangular and Circular Section. — Screws and Bolts (U. S. Standard), and Table of Propor- tions. Pages 251-258. APPENDIX. Working Drawings of Standard 100 - lb. Rail and of Allen- Richardson Slide Valve. — Designs for Variation of Problems in Chapters X, XIII, XIV, XV and XVI, — Notes to Arts. 113 and 131, on Properties of Torus and Ellipse. — Article on the Nomenclature and Double Gen- eration of Trochoidal Curves. — Alphabets. — Index. — List of Reference Works. Pages 259-293. THEORETICAL AND PRACTICAL GRAPHICS. CHAPTER I. FIRST PRINCIPLES, WITH GENERAL SURVEY OF THE FIELD OF GBAPHIC SCIENCE. Fig-. 1. 1. Geometrically considered, any combination of points, lines and surfaces is called a figure. A figure lying wholly in one plane is called a "pla^u figure; otherwise a space figure. 2. Among the methods of investigating and demonstrating the mathematical properties of figures, and of solving problems relating to them, that called projection is at once one of the most valuable and interesting, constituting, as it does, the common basis of nearly all graphic representations, whether of artist, architect or engineer. When using this method figures are always considered in connection with a certain point called a ceiitre of projection. In Fig. 1 let S be an assumed centre of projection and A any point in space. The straight line SA, joining S with A, is called a projecting Ihie or ray, or simply a projector, and its intersection, a, with any line CD, is its projection upon that lino. It is otherwise expressed by saying that A is pro- jected upon CD at a. In the same way the point B is projected ' from S upon the plane M N at b ; or, in other words, b is — for the assumed position of H — the projection of B upon the plane. It is with projection upon a plane that we are principally concerned. The word "projection" is used not only to indicate the method of representation but also the representation itself. In certain other branches of mathematics it has a yet more extended significance, being employed to denote the represen- tation of any curve or surface upon any other. 3. A figure, as ABC (Fig. 2), is projected upon a plane, M N, by drawing projectors, S A, SB, S C, through its vertices and prolonging them, if necessary^, to meet the plane. The figure abc, formed by joining the points in which the projectors intersect the plane, is then the projection of the first, or original figure. The plane upon which the projection is made is called the plane of prcgection. S A 4. Were abc (Fig. 3) the original figure and MN the plane of projection, then would ABC he the projection desired. Each figure may thus be considered a projection of the other for a given position of S, and when so related figures are said to correspond to each other. Points that are colUnear (or in line) with the centre of projection, as a and A, are called corresponding points. I'ig:. 3. 1 Were S the muzzle of a gun, and B a bullet speeding from it toward the plane, it would he projected against or through the plane at 6. The jipproprialeness of the term "projection" is obvious. 2 In Fig. 3 the projectors meet the plane between the centre S and the given figure. 2 THEORETICAL AND PRACTICAL GRAPHICS. 6. Ha\dng indicated what projections are and how obtained, it will be well, before giving their grand divisions and sub -divisions, to state the nature and extent of the field in which they may be employed. The mathematical properties of geometrical figures, as also the propositions and problems involving them, are divided into two classes, metiical and desaiptive. In the first class the idea of quantity necessarily enters, either directly — as in measurement, or indirectly — as in ratio." In the second or descriptive class, however, we find involved only those properties dependent upon relative position.^ Descriptive properties are unaltered by projection, while, as ordinarily regarded, but few metrical properties are projective.' The main pro\dnce of projection is obvious. 6. Descriptive Geometry is that branch of mathematics in which figures are represented and their descriptive properties investigated and demonstrated by means of projection. , DIVISIONS OF PROJECTION. 7. All projections may be divided into two general classes, Central and Parallel. If the centre of projection be at a finite distance, as in Figs. 2 and 3, the projection obtained is called a central projection; but if we suppose it to be at, infinity, as in Fig. 4, projectors from it will then evidently be parallel, and the resulting figure is called a parallel projection of the original figure. Parallel projection is thus seen to be merely a special case of central projection, yet each has been independently developed to a high degree and has an extensive literature. 8. The terms Conical and Cylindrical are employed by many writers synonymously with central and parallel respectively. Central projections are also occasionally called Radial or Polar. Remark. — ^A straight line is said to generate a conical surface (see Fig. 5) when it constantly passes through a fixed point (the vertex), and is guided in its motion hy a given fixed curve (the directrix). The moving straight line is a generatrix of the surface, and its various positions are called elements of the surface. If the vertex of a conical surface he removed to infinity the elements will become parallel, and we shall have a cylindrical surface, which may he also defined as the surface (see Fig. 6) generated by a straight line that is guided in its motion by a given fixed curve, and is in any position parallel to a given, fixed, straight line. The origin of the terms conical and cylindrical as applied to projection is obvious. We have now to mention the more important sub - divisions of projections, with the sciences based upon them. The names depend in certain cases upon the nature of the centre of projection, while in others they are due to some particular application. ^toS~ f Under Central (or Conical) jrrqjection we have: — 9. Projective Geometry (Geometry of Position). While in its most general sense this science includes all projections, yet in its ordinary acceptation it may be defined as that branch of mathematics in which — with the centre of projection considered as a mathematical point at a finite distance from the line or plane of projection — the projective properties of figures are investigated and established. 1 The following are metrical relations : (a) The lateral area of a cylinder is equal to the product of the perimeter of its right section by an element of the surface. (b) Two tetrahedrons which have a trihedral angle of the one equal to a trihedral angle of the other, are to each other as the products of the three edges of the equal trihedral angles. 2 lUusti'ating descriptive or positional properties : (a) If a-llne is perpendicular to a plane, any plane containing the line will also be perpendicular to the plane. fb) Planes that are perpendicular to the same straight line are parallel to each other. 3 See Klein's Eevieiv of liece7U Researches in Geometry regarding the point of view which enables the projective method to Include the whole of geometry. DEFINITIONS. 3 Its chief practical application is in Graphical Statics, in which the stresses in bridge and roof trusses or other engineering constructions are determined graphically, by means of diagrams. 10. Perspective. — If the centre of projection is the eye of the observer the projection is called a perspective or scenographic projection, or — more commonly — simply a perspective. The plane of projection is then called the perspective plane or picture plane, and is always vertical. The position of the eye is called the point of sight or station point, and the projectors are termed visual rays. Applied in the graphical construction usually preliminary to art work in water colors or oil ; also in architectural perspectives and in scientific illustrations of machinery, etc. It may be remarked that any projection, central or parallel, presents to the eye the same appearance as the figure projected would if viewed from the centre of projection. 11. Relief -perspective. — This differs from the perspective just defined in requiring, in addition to the usual perspective plane, a second plane parallel to it called a vanishing plane, the required repre- eentation appearing in relief between the two planes — a solid perspective, so to speak. Employed chiefly in the construction of bas-reliefs and theatre decorations. 12. Sciography or Shadows (artificial light). — If the centre of projection is an artificial light, as the electric or that of a candle — either of which may, without appreciable error, be treated in graphical constructions as a mere point — the projectors will be rays of light and the projection will be the shadow of the figure projected. Employed in obtaining shadow effects in paintings or architectural drawings. 13. Photogrammetry or Photometrography, the application of photography to surveying, the optical centre of the lens being the centre of projection. 14. Under Parallel (or Cylindrical) projection we have: — (a) Oblique or Clinographic, and (b) Perpendicidar or Orthographic, also called Orthogonal or Rectangular. These divisions are based upon the direction of the projectors with respect to the plane of pro- jection, they being — as the names imply — inclined to it in oblique projection and perpendicular to it in orthographic. OBLIQUE PROJECTION. 15. The shadow of an object in the sunlight would be its oblique projection, the sun's rays being practically parallel. 16. Oblique projection is usually called Clinographic when employed in Crystallography. 17. In its other applications, when not simply called oblique, this projection is variously termed Cavalier Perspective, Cabinet Projection and Military Perspective, the plane of projection being vertical in the first and second, while in the last it is horizontal. Oblique projection gives a pictorial effect closely analogous to a true perspective, yet is far more simple in its con- struction, and is much used for showing the form or method of assemblage of parts, or details, of machinery and archi- tectural work ; aiso in the representation of crystals. , ORTHOGRAPHIC PROJECTION UPON A SINGLE PLANE. 18. When but one plane of projection is employed the only important applications of ortho- graphic projection having special names are — (a) One- Plane Descriptive, otherwise called Horizontal Projection. Employed chiefly in fortification and general topographical work, in which the lines and surfaces represented are mainly horizontal. (b) Axonometric (including Isometric) Projection. Has the same range of application as oblique projection, viz., to objects whose lines lie mainly in directions mutually perpendicular to each other, or having axes so related. THEORETICAL AND PRACTICAL GRAPHICS. ORTHOGRAPHIC PROJECTION UPON MUTUALLY PERPENDICULAR PLANES. 19. When upon two (or more) mutually perpendicular planes orthographic projection becomes the Geometrie Descriptive of Gaspare! Monge, who reduced its principles to scientific form in the latter part of the eighteenth century. The tendency — a logical one — toward the general adoption of the title "Descriptive Geometry" in the broad sense of Art. 6 would make it seem advisable to appropriate the name Mongers Descriptive to this — the most important division of graphic science, that we may not only find in it a hint as to its source but at the same time also pay to its inventor the honor of perpetual association of his name with his creation. As originally defined by Monge it is the application of orthographic projection (a) to the exact representation upon a plane surface, as that of a drawing-board, of all objects capable of rigorous definition, and (b) to the solution of problems relating to these objects in space and involving only their projoerties of form and position. It inight with propriety be divided into pm-e and applied, the former being the abstract science in which the mathematical relations existing between figures and their projections are examined and applied in the solution of certain fundamental problems of the point, line and plane; while the latter division would naturally include the application of these principles and methods to the solution of problems relating to the various elementary and higher mathematical surfaces, and to machine drawing and design, shades, shadows, perspective, stone - cutting, spherical projections, crystallography, pattern - making, carpentry, etc. ADDITIONAL REMARKS ON NOMENCLATVRE. The student may And the following serviceable by way of enabling him to get clear ideas of the distinctions between certain divisions of geometrical science. The term Geometry, unqualified, Is usually understood to refer to the synthetic method of investigation of the foi-m, position, ratio and measurement of geometrical figures, the reasoning being from particular to general truths by the aid of diagrams. In contra-distinction to other geometries it Is frequently called Euclidean, after the celebrated Greek geometer, Euclid, (about 330-275 B. C. ) who organized its theorems and problems into a science. In Coordinate (or Analytical) Geometry the figure considei'cd is referred to a system of coordinates, the relation between which, for every point of the figure, is expressed by means of an equation in which the coordinates are represented by alge- braic symbols. The operations performed are algebraic, and the method of reasoning is from general to particular truths. Although the invention of Analytical Geonietiy has been attributed to Descartes •it is now recognized that he neither originated the use of coordinates nor the representation of curves and surfaces by means of equations. As the first to give, complete scientific form to the analytic method his name has justly been given, however, to the most important division of coor- dinate geometry, Cartesian. But the writers of the present day under that head do not by any means confine themselves to the system of coordinates employed by him, which consisted of intersecting straight lines, usual!}' perpendicular to each other. In selecting the title "Projective Geometry" for the science defined in Art. 9 the eminent Cremona says, "I prefer not to adopt that of Higher Geometry (G^om^trie sup&rieure, hbhere Geometrie) because that to which the title 'higher' at one time seemed appropriate, may to-day have become very elementary; nor that of Modem Geometry {neuere Geometrie) which in like manner ex- presses a merely relative idea, and is moreover open to the objection that althotigh the methods may be regarded as modern, yet the matter is to a great extent old. Nor does the title Geometry of Position {Geometrie der Lage) as used by Staudt seem to me a suitable one, since it excludes the consideration of the metrical properties of figures. I have chosen the name of Projective Geometry as expressing the true nature of the methods, which are based essentially upon central projection or perspective. And one reason which has detertnined this choice is that the great Poncelet, the chief creator of the modern methods, gave to his immortal book the title of Traits des propriSfSs projectives des figures/^ Cremona further states that "there is one Important class of metrical properties (anharmonic properties) which are pro- jective, and the discussion of which therefore finds a place in the Projective Geometry." But the positional definition given by Staudt for the auharraonie ratio of four points, which removes these properties from the class metrical to the class descrip- tive (which last are always projective), to that extent justifies the title employed by him, while making Cremona's choice none the less a fortunate one. Among other geometries some belong to what may be called speculative mathematics, based upon "quasi -geometrical notions, those of more -than -three -dimensional space, and of non-Euclidean two -and -three -dimensional space, and also of the generalized notion of distance."* The following will illustrate a method of arriviug'at a conception of non -Euclidean two-dimensional geometry. "Imagine the earth a perfectly smooth sphere. Understand by a plane the surface of the earth, and by a line the apparently straight line (in fact an arc of a great circle) drawn on the surface. What experience would in the first instance teach would be Euclidean two-dimensional geometry; there would be intersecting lines, which, produced a few miles or so, would seem to go on diverging, and apparently parallel lines which would exhibit no tendency to approach each other; and the Inhabitants might very well conceive that they had by experience established the axiom that two straight lines cannot enclose a space, and also the axiom as to parallel lines. A more extended experience and more accurate measurements would teach them that the axioms were each of them false; and that any two lines, if produced far enough each way, would meet in two points; they would, in fact, arrive at a spherical geometry accurately representing the properties of the two-dimensional space of their experience. But their original Euclidean geometry would not the less be a true system; only it would apply to an ideal space, not the space of their experience."* * Cay ley. FREE-HAND DRAWING. 6 CHAPTER II. AKTISTIC AND TECHNICAL FEEE-HAND DRAWING. — SKETCHING PROM MEASUREMENT. — FREE- HAND LETTERING.— CONVENTIONAL REPRESENTATIONS. 20. Drawings, if classified as to the method of their production, are either free-hand or mechanical; while as to purpose they may be working drawings, so fully dimensioned that they can be worked from and what they represent may be manufactured; or finished drawings, illustrative or artistic in character and therefore shaded either with pen or brush, and having no hidden parts indicated by dotted lines as in the preceding division. Finished drawings also lack figured dimensions. Working drawings of parts or " details " of a structure are called detail drawings; while the representation of a structure as a whole, with all its details in their proper relative position, hidden parts indicated by dotted lines, etc., is termed a general or assembly drawing. 21. While mechanical drawing is involved in making the various essential views — plans, eleva- tions and sections — of all engineering and architectural constructions, and in solving the problems of form and relative position arising in their design, yet, to the engineer, the ability to sketch effectively and rapidly, free-hand, is of scarcely less importance than to handle the drawing instruments skill- fully ; while the success of an architect depends in still greater measure upon it. We must distinguish, however, between artistic and technical free-hand work. The architect must be master of both; the engineer necessarily only of the latter. To secure the adoption of his designs the architect relies largely upon the effective way in which he can finish, either with pen and ink or in water- colors, the perspectives of exterior and interior views; and such drawings are judged mainly from the artistic standpoint. While it is not the province of this treatise to instruct in such work a word of suggestion may properly be introduced for the student looking forward to architecture as a profession. He should procure Linfoot's Picture Making in Pen and Ink, Miller's Essentials of Perspective and Delamotte's Art of Sketching from Nature; ,and with an experienced architect or artist, if possible, but otherwise by himself, master the prin- ciples and act on the instructions of these writers. 22. Since the camera makes it, fortunately, no longer essential that a civil engineer should be a landscape artist as well, his free-hand work has become more restricted in its scope and more rigid in its character, and like that of the machine designer it may properly be called technical, from its object. Yet to attain a sufficient degree of skill in it for all practical and commercial purposes is possible to all, and among them many who could never hope to produce artistic results. It is con- fined mainly to the making of working sketches, conventional representations and free-hand lettering, and the equipment therefor consists of a pencil of medium grade as to hardness; lettering pens — Falcon or Gillott's 303, with Miller Bros. "Carbon" pen No. 4; either a note -book or a sketch -block or pad ; also the following for sketching from measurement : a two - foot pocket - rule ; calipers, both external and internal, for taking outside and inside diameters; a pair of pencil compasses for making an occasional circle too large to be drawn absolutely free - hand ; and a steel tape - measure for large work, if one can have assistance in taking notes, but otherwise a long rod graduated to eighths. THEORETICAL AND PRACTICAL GRAPHICS. 23. In the evolution of a machine or other engineering project the designer places his ideas on paper in the form of rough and mainly free-hand sketches, beginning with a general outline, or "skeleton" drawing of the whole, on as large a scale as possible, then filling in the details, separate — and larger — drawings of which are later made to exact scale. While such preliminary sketches are not drawn literally "to scale" it is obviously desirable that something like the relative proportions should be preserved and that the closer the approximation thereto the clearer the idea they will give to the draughtsman or workman who has to work from them. A habit of close observation must therefore be cultivated, of analysis of form and of relative direction and proportion, by all who would succeed in draughting, whether as designers or merely as copyists of existing construc- tions. While the beginner belongs necessarily in the latter category he must not forget that his aim should be to place himself in the ranks of the former, both by a thorough mastery of the funda- mental theory that lies back of all correct design and by such training of the hand as shall facilitate the graphic expression of his ideas. To that end he should improve every opportunity to i)ut in practice the following instructions as to SKETCHING FROM MEASUREMENT, as each structure sketched and measured will not only give exercise to the hand but also prove a valuable object lesson in the proportioning of parts and the modes of their assemblage. A free-hand sketch may be as good a working drawing as the exactly scaled — and usually inked — drawing that is generally made from it to be sent to the shop. While several views are usually required, yet for objects of not too compUcated form, and whose lines lie mainly in mutually perpendicular directions, the method of representation illustrated by Fig. 7, is admirably adapted,* and ob\aates all necessity for additional sketches. It is an oblique projection Fler. 7. ^ J/k A- FREE-HAND SKETCH OF TIMBER FRAMINQ. (Art. 17) the theory of whose construction will be found in a subsequent chapter, but with regard to which it is sufficient at this point to say that the right angles of the front face are seen in their true fonn, while the other right angles are shown either of 30°, 60°, or 120°; although almost any oblique angle Avill give the same general effect and may be adopted. Lines parallel to each other on the object are also parallel in the drawing. Draw first the front face, whose angles are seen in their true form; then run the oblique lines ofi" in the direction which will give the best view. (Refer to Figs. 42, 44, 45 and 46.) 24. While Fig. 7 gives almost the pictorial effect of a true perspective and the object requires no other description, yet for complicated and irregular forms it gives place to the plan - and - eleva - tion mode of representation, the plan being a top and the elevation a front view of the object. And * The figures In this chapter are photo -reproductions of free-hand work and are intended not only to Illustrate the text but also to set a reasonable standard for sketch - notes. SKETCHING FROM MEASUREMENT. if two views are not enough for clearness as many more should be added as seem necessary, includ- ing what are called sections, which represent the object as if cut apart by a plane, separated and a view obtained {perpendicular to the cutting plane, showing the internal arrangement and shape of parts. In Fig. 8 we have the same object as in Fig. 7, but represented by the method just mentioned. The front view (elevation) is evidently the same in both Figs. 7 and 8, except that in the latter we indicate by dotted lines the hidden recess which is in full sight in Fig. 7. The view of the top js placed at the top in conformity to the now quite general practice as to location, viz., grouping the various sketches about the elevation, so that the view of the left end is at the left, of the right at the right, etc. .&. •a* '4 Jo N i. —PLAN- J^ 2 "Tnt: 1 L /r ■ t" 1 1 **• f yC 0' Y y^'Li H e %' •^- fr— -fro -£L£VAT/OAf- FREE'HANO SKETCH OF TIMBER FRAMING, IN PLAN AND ELEVATION In these views, which fall under Art. 19 as to theoretical construction, entire surfaces are pra- 'iected as straight lines, as G B C H in the straight line H' C. Were this a metallic surface and "finished" or "machined" to smoothness, as distinguished from the surface of a rough casting, that fact would be denoted by an "/" on the line H' C which represents the entire surface, the cross- line of the "/" cutting the line obliquely, as shown. CENTRE-LINES. — DIMENSIONING. 25. Dimensioning. In sketching, centre-lines and all important centres should be located first, and measurements taken from them or from finished surfaces. Feet and inches are abbreviated to "Ft.," and "In.," as 4 Ft. 6f In.: also written 4' 6f", and occasionally 4 Ft. 6f". A dimension should not be written as an improper fraction, ^" for ex- ample, but as a mixed number, If". Fractions should have horizontal dividing lines. Not only should, dimensions of successive parts be given but an "over-all" dimension, which, it need hardly be said, should sustain the axiom regarding the whole and the sum of its parts. Dimensions should read in line with the line they are on, and either from the bottom or the right hand. The arrow tips should touch the lines between which a distance is given. THEORETICAL AND PRACTICAL GRAPHICS. Extension lines should be drawn and the dimension given outside the drawing whenever such course will add to the clearness. (See D' F', Fig. 8.) An opening should always be left in the dimension line for the figures. In case of very small dimensions the arrow tips may be located outside the lines, as in Fig. 9, and the dimension indicated by an arrow, as at A, or inserted as at B if there is room. Should a piece of uniform cross-section (as, for example, a rail, angle -iron, channel bar, Phoenix column or other form of structural iron) be too long to be represented in. its proper relative length on the sketch it may be broken as in Fig. 9, and the form of the section (which in the case sup- posed will be the same as an end view) may be inserted with its dimensions, as in the shaded figure. If the kind of bar and the number of pounds per yard are known the dimensions can be obtained by reference to the handbook issued by the manufacturers. ffy/niol- /■»>}■ ; 2.00 l-as. i»/c. >"£> f9% w? r^A FREE-HAND SKETCH OF A CHANNEL BAR. The same dimension should not appear on each view, but each dimension must be given at least once on some view. Notes on Riveted Work, Pins, Bolts, Screws and Nuts. In riveted work the " pitch " of the rivets, i. e., their distance from centre to centre (" c. to c") should be noted, as also that between centre lines or rows, and of the latter from main centre lines. Similarly for bolts and holes. If the latter are located in a circle note the diameter of the circle containing their centres. Note that a hole for a rivet is usually about one -half the diameter of the forged head. In measuring nuts take the width between parallel sides ("width across the flats") and abbreviate for the shape, as "sq.," "hex.," "oct." For a piece of cylindrical shape a frequently used symbol is the circle, as 4" O (read "four inches, round," not around,) for 4" diameter; but it is even clearer to use the abbreviation of the latter word, viz., "diam." In taking notes on bolts and screws the outside diameter is sufficient if they are " standard," that is, proportioned after either the Sellers (U. S. Standard) or Whitworth (English Standard) sys- tems, as the proportions of heads and nuts, number of threads to the inch, etc., can be obtained from the tables in the Appendix. If not "standard" note the number of threads to the inch. Record whether a screw is right- or left-handed. If right-handed it will advance if turned clock -wise. The shape of thread, whether triangular or square, would also be noted. Notes on Gearing. On cog, or "gear," wheels obtain the distance between centres and the number of teeth on each wheel. The remaining data are then obtained by calculation. Bridge Notes. In taking bridge notes there would be required general sketches of front and end view; of the flooring system, showing arrangement of tracks, ties, guard -beam and side -walk; a cross -section; also detail drawings of the top and foot of each post -connection in one longitudinal line from one end to the middle of the structure. In case of a double -track bridge the outside rows of posts are alike but differ fi-om those of the middle truss. 10 THEORETICAL AND PRACTICAL GRAPHICS. sections they may employ burnt umber undertone for the earthy bed, jpale blue or india ink tint for the rock, and prussian blue for the water lines. FREE-HAND LETTERING. 27. Although later on in this work an entire chapter is devoted to the subject of lettering, yet at this point a word should be said regarding those forms of letters which ought to be mastered, early in a draughting course, as the most ser\'iceable to the practical worker. Fi-s- 3.2. ABCDEFGHIJKLMNOPQRSTUVWXYZ& 1234567890 ABCDEFGHIJKLMNOPQRSTUVWXYZ<& I234567890 The first, known as the Gothic, is the simplest form of letter, and is illustrated in both its vertical and inclined (or Italic) forms in Fig. 12. It is much used in dimensioning, as well as for FLs X2 (a). titles. The lettering and numerals are Gothic in Figs. 7 and 8, with the exception , _ — . of the 1 and 4, which, by the addition of feet, are no longer a pure form although C^v^w ^ enhanced in appearance. In Fig. 12 (a) some modifications of the forms of certain numerals are shown; also the omission of the dividing line in a mixed number, as is customary in some offices. For Gothic letters and for all others in which there is to be no shading it is well to use a pen with a blunt end, preferably "ball -pointed," but otherwise a medium stub, like Miller Bros'. "Carbon" No. 4, which gives the desired result when used on a smooth surface and without undue pressure. Fig. 13 illustrates the Italic (or inclined) form of a letter which when vertical is known as the Roman. The Roman and Italic Roman are much used on Government and other map work, and in ng-. 1.3. ABCDUFGHTJKLMWOPQRS 12 3 4 3 T IT Y W X Y Z 6 7 8 9 al) c d ef a hiyj k Ivvnojf q^r s Itivw xy z the draughting offices of many prominent mechanical engineers. Regarding them the student may profitably read Arts. 260-262. Make the spaces between letters as nearly uniform as possible, and the small letters usually about three -fifths the height of the capitals in the same line. For Roman and other forms of letter requiring shading use a fine pen; Gillott's No. 303 for small work, and a " Falcon " pen for larger. A form of letter much used in Europe and growing in favor here is the Soennecken Round Writin;/, referred to more particularly in Art. 265 and illustrated by a complete alphabet in the Appendix. The text -book and special pens required for it can be ordered through any dealer in draughtsmen's supplies. CONVENTIONAL REP RE SENTATIO N S. — FREE-HAND LETTERING. 9 All notes should be taken on as large a scale as possible, and so indexed that drawings of parts may readily be understood in their relation to the whole. The foregoing hints might be considerably extended to embrace other and special cases, but experience will prove a sufficient teacher if the student will act on the suggestions given, and will remember that to get an excess of data is to err on the side of safety. It need hardly be added that what has preceded is intended to be merely a partial summary of the instructions which would be given in the more or less brief practice in technical sketching which, presumably, constitutes a part of every course in Graphics; and that unless the draughtsman can be under the direction of a teacher he will be able to sketch much more intelligently after studying more of the theory involved in Mechanical Drawing and given in the later pages of this work. CONVENTIONAL REPRESENTATIONS. 26. Conventional representations of the natural leatures of the country or of the materials of construction are so called on the assumption, none too well founded, that the engineering profession has agreed in convention that they shall indicate that which they also more or less resemble. While there is no universal agreement in this matter there is usually but little ambiguity in their use, especially in those that are drawn free-hand, since in them there can be a nearer approach to the natural appearance. This is well illustrated by Figs. 10 and 11. ■Fi-s- il- In addition to a rock section Fig. 11 (a) shows the method of indicating a mud or sand bed with small random boulders. Water either in section or as a receding surface may be shown by parallel lines, the spaces between them increasing gradually. Conventional representations of wood, masonry and the metals will be found in Chapter VI, after hints on coloring have been given, the foregoing figures appearing at this point merely to illustrate, in black and white, one of the important divisions of technical free-hand work. Those, however, who have already had some practice in drawing may undertake them either with pen and ink or in colors, in the latter case observing the instructions of Arts. 237-241 for wood, while for the river THE DRAUGHTSMAN'S EQUIPMENT. 11 CHAPTER III. DEAWING INSTEUMENTS AND MATERIALS. -INSTRUCTIONS AS TO USE. — GENERAL PRELIMINARIES AND TECHNICALITIES. x-igr- is- 28. The draughtsman's equipment for graphical work should be the best con- ^^s- ^^- ^^^- ^^• sistent with his means. It is mistaken economy to buy inferior instruments. The best obtainable will be found in the end to have been the cheapest. The set of instruments illustrated in the following figures contains only those which may be considered absolutely essential for the beginner. THE DRAWING PEN. The right line pen (Fig. 14) is ordinarily used for drawing straight lines, with either a rule or triangle to guide it; but it is also employed for the draw- ing of curves when directed in its motion by curves of wood or hard rubber. For average work a pen about five inches long is best. The figure illustrates the most approved type, i. e., made from a single piece of steel. The distance between its points, or "nibs," is adjustable by means of the screw H. An older form of pen has the outer blade connected with the inner by a hinge. The convenience with which such a pen may be cleaned is more than offset by the certainty that it will not do satisfactory work after the joint has become in the slightest degree loose and inaccurate through wear. 29. If the points wear unequally or become blunt the draughtsman may sharpen them readily himself upon a fine oil-stone. The process is as follows: Screw up the blades till they nearly touch. Inclinis the pen at a small angle to the surface of the stone and draw it lightly from left to right (supposing the initial position as in Fig. 16). Before reaching the right | I H end of the stone begin turning the pen in a plane perpendic- ular to the surface, and draw in the opposite direction at the same angle. After frequent examination and trial, to see that the blades have become equal in length and similarly rounded, the process is completed by lightly dressing the outside of each blade separately upon the stone. No grinding should be done on the inside of the blade. Any " burr " or rough edge resulting from the operation may be removed with fine emery paper. For the best results, obtained in the shortest possible time, a magnifying glass should be used. The student should take particular notice of the shape of the pen when new, as a standard to be aimed at when compelled to act on the above suggestions. 30. The pen may be supplied with ink by means of an ordinary writing pen dipped in the ink and then passed between the blades; or by using in the same manner a strip of Bristol board about a quarter of an inch in width. Should any fresh ink get on the outside of the pen it must 12 THEORETICAL AND PRACTICAL GRAPHICS. x'ler- IT. be removed; otherwise it will be transferred to the edge of the rule and thence to the paper, caus- ing a blot. 31. As with the pencil, so with the pen, horizontal lines are to be drawn from left to right, while vertical or inclined lines are drawn either from or toward the worker, according to the position of the guiding edge with respect to the line to be drawn. If the line were m n, Fig. 17, the motion would be away from the draughtsman, i. e., from n toward m; while o jp would be drawn toward the worker, being on the right of the triangle. 32. To make a sharply defined, clean-cut line — the only kind allowable — the pen should be held lightly but firmly with one blade resting against the guiding edge, and with both points resting equally upon the paper so that they may wear at the same rate. 33. The inclination of the pen to the paper may best be about 70°. When properly held the pen will make a line about a fortieth of an inch from the edge of the rule or triangle, leaving visible a white hne of the paper of that width. If, then, we wish to connect two points by an inked straight line, the rule must be so placed that its edge will be from them the distance indicated. It need hardly be said that a drawing -'p&a. should not be "pushed. The more frequently the draughtsman will take the trouble to clean out the point of the pen and supply fresh ink the more satisfactory results will he obtain. When through with the pen clean it carefully, and lay it away with the points not in contact. Equal care should be taken of all the instruments, and for cleaning them nothing is superior to chamois skin. DIVIDERS. 34. The hair -spring dividers (Fig. 15) are employed in dividing lines and spacing off distances, and are capable of the most delicate adjustment by means of the screw G and spring in one of the legs. When but one pair of dividers is purchased the kind illustrated should have the preference over plain dividers, which lack the spring. It will, however, be frequently found convenient to have at hand a pair of each. Should the joint at F become loose through wear it can be tightened by means of a key having two projections which fit into the holes shown in the joint. 35. In spacing off distances the pressure exerted should be the slightest consistent with the loca- tion of a point, the puncture to be merely in the surface of the paper and the points determined by lightly pencilled circles about them, thus q © . In laying off several equal distances along a line all the arcs described by one x'igr- le. jgg Qf ^j^g dividers should be on the same side of the line. Thus, in Fig. 19, with b the first centre of turning, the leg x describes the Fig:. i9. arc R, then rests and pivots on c while the leg y describes the arc S; x then traces arc T, etc. THE COMPASSES.— BOW-PENCIL AND PEN. 18 COMPASS SET. Fig-. 20. Fi-S- 21. ^'igr. 22. 36. The compasses (Fig. 20) resemble the dividers in form and may he used to perform the same office, but are usually employed for the drawing of circles. Unlike the dividers one or both of the legs of compasses are detachable. Those illustrated have one perma- nent leg, with pivot or "needle-point" adjustable by means of screw R. The other leg is detachable by turning the screw 0, when the pen leg LM (Fig. 21) may be inserted for ink work; or, where large work is involved, the lengthening bar on the right (Fig. 22) may be first attached at O and the pencil or pen leg then inserted at /. The metallic point held by screw S is usually replaced by a hard lead, sharpened as indicated in Art 54. 37. When in use the legs should be bent at the joints P and L, so that they will be perpendicular to the paper when the compasses are held in a vertical plane. The turning may be in either direction, but is usually " clock - wise ; " and the compasses may be slightly inclined toward the direction of turning. When so used, and if no undue pressure be exerted on the pivot leg, there should be but the slightest puncture at the centre, while the pen points having rested equally upon the paper have sustained equal wear, and the resulting line has been sharply defined on both sides. Obviously the legs must be re -adjusted as to angle, for any material change in the size of the circles wanted. The compasses should be held and turned by the milled head which projects above the joint N. Dividers and compasses should open and shut with an absolutely uniform motion and somewhat stiffly. s 1 BOW -PENCIL AND PEN. ^-Ig-. 23. F5.gr. s-a. 38. For extremely accurate work, in diameters from one -sixteenth of an inch to about two inches, the bow -pencil (Fig. 23) and bow -pen (Fig. 24) are especially adapted. The pencil -bow has a needle-point, adjustable by means of screw E, which gives it a great advantage over the fixed pivot -point of the bow -pen, not alone in that it permits of more delicate adjustment for unusually small work but also because it can be easily replaced by a new one in case of damage; whereas an injury to the other ren- ders the whole instrument useless. For very small circles the needle-point should project very slightly beyond the pen- point; theoretically by only the extremely small distance the needle- point is expected to sink into the paper. The spring of either bow should be strong; otherwise an attempt at a circle will result in a spiral. It will save wear upon the threads of the milled heads A and C if the draughtsman will press the legs of the bow together with his left hand and run the head up loosely on the screw with his right. 14 THEORETICAL AND PRACTICAL GRAPHICS. 39. To the above described — which we may call the minimum set of instruments — might be advantageously added a pair of bow -spacers (small dividers shaped like Fig. 24); beam -compasses, for extra large circles; parallel -rule; proportional dividers, and an extra — and larger — right-line pen. 40. The remainder of the necessary equipment consists of paper; a drawing-board; T-rule; tri- angles or "set squares;" scales; pencils; India ink; water colors; saucers for mixing ink or colors; brushes; water-glass and sponge; irregular (or "French") curves; india rubber; erasing knife; pro- tractor; file for sharpening pencils, or a pad of fine emery or sand paper; thumb-tacks (or "drawing- pins"); horn centre, for making a large number of concentric circles. PAPER AND TKACING CLOTH. 41. Drawing paper may be purchased by the sheet or roll and either unmounted or mounted, i. e., "backed" by muslin or heavy card -board. Smooth or " hot- pressed " paper is best for drawings in line -work only; but the rougher surfaced, or "cold -pressed," should always be employed when brush-work in ink or colors is involved: in the latter case, also, either mounted paper should be used or the sheets " stretched " by the process described in Art. 44. 42. The names and sizes of sheets are : — Cap 13 X 17 Elephant 23 x 28 Demi 15 x 20 Atlas 26 x 34 Medium 17 X 22 Columbia 23 x 35 Royal 19 X 24 Double Elephant 27 x 40 Super Royal 19 x 27 Antiquarian 31 x 53 Imperial 22 x 30 43. There are many makes of first-class papers, but the best known and still probably the most used is Whatman's. The draughtsman's choice of paper must, however, be determined largely by the value of the drawing to be made upon it, and by the probable usage to which it will be subjected. Where several copies of one drawing were desired it has been a general practice to make the original, or "construction" draAving, with the pencil, on paper of medium grade, then to lay over it a sheet of tracing -cloth and copy upon it, in ink, the lines underneath. Upon placing the tracing cloth over a sheet of sensitized paper, exposing both to the light and then immersing the sensitive paper in water, a copy or print of the drawing was found upon the sheet, in white lines on a blue ground — the well-known blue-print. The time of the draughtsman may, however, be economized, as also his purse, by making the original drawing in ink upon Crane's Bond paper, which combines in a remarkable degree the qualities of transparency and toughness. About as clear blue -prints can be made with it as with tracing -cloth, yet it will stand severe usage in the shop or the drafting -room. Better papers may yet be manufactured for such purposes, and the progressive draughtsman will be on the alert to avail himself of these as of all genuine improvements upon the materials and instruments before employed. 44. To stretch paper tightly upon the board lay the sheet right side up,* place the long rule with its edge about one -half inch back from each edge of the paper in turn, and fold up against it a margin of that width. Then thoroughly dampen the back of the paper with a full sponge, except on the folded margins. Turning the paper again face up gum the margins with strong mucilage or glue, and quickly but firmly press opposite edges down simultaneously, long sides first, exerting at the same time a slight outward pressure with the hands to bring the paper down somewhat closer to • The "right side" of a sheet is, presumably, that toward one when — on holding it up to the light— the manufacturer's name, in water- mark, reads correctly. TRACING-CLOTH. — DRAWING BO ARD. — T-RULE. — TRIANGLES. 15 the board. Until the gum "sets," so that the paper adheres perfectly where it should, the latter should not shrink; hence the necessity for so completely soaking it at first. The sponge may be applied to the face of the paper provided it is not rubbed over the surface, so as to damage it. The stretch should be horizontal when drying, and no excess of water should be left standing on the surface; otherwise a water -mark will form at the edge of each pool. 45. When tracing -cfoiA is used it must be fastened smoothly, with thumb-tacks, over the drawing to be copied, and the ink lining done upon the glazed side, any brush work that may be required — either in ink or colors — being always done upon the dull side of the cloth after the outlining has been completed. If the glazed surface be first dusted with powdered pipe -clay applied with chamois skin it will take the ink much more readily. When erasure is necessary use the rubber, after which the surface may be restored for further pen -work by rubbing it with soapstone. Tracing- cloth, like drawing paper, is most convenient to work upon if perfectly flat. When either has been purchased by the roll it should therefore be cut in sheets and laid away for some time in drawers to become flat before needed for use. DRAWING BOARD. 46. The drawing board should be slightly larger than the paper for which it is designed and of the most thoroughly seasoned material, preferably some soft wood, as pine, to facilitate the use of the drawing-pins or thumb-tacks. To prevent warping it should have battens of hard wood dove- tailed into it across the back, transversely to its length. The back of the board should be grooved longitudinally to a depth equal to half the thickness of the wood, which weakens the board trans- versely and to that degree facilitates the stiffening action of the battens. For work of moderate size, on stretched paper, yet without the use of mucilage, the " panel " board is recommended, provided that both frame and panel are made of the best seasoned hard wood. It will be found convenient for each student in a technical school to possess two boards, one 20" X 28" for paper of Super Royal size, which is suitable for much of a beginner's work, and another 28" X 41" for Double Elephant sheets (about twice Super Royal size) which are well adapted to large drawings of machinery, bridges, etc. A large board may of course be used for small sheets, and the expense of getting a second board avoided; but it is often a great convenience to have a medium- sized board, especially in case the student desires to do some work outside the draughting -room. THE T-RULE. 47. The T-rule should be slightly shorter than the drawing board. Its head and blade must have absolutely straight edges, and be so rigidly combined as to admit of no lateral play of the latter in the former. The head should also be so fastened to the blade as to be level with the surface of the board. This permits the triangles to slide freely over the head, a great convenience when the lines of the drawing run close to the edge of the paper. (See Fig. 32.) The head of the T-rule should always be used along the left-hand edge of the drawing board. TRIANGLES. 48. Triangles, or "set -squares" as they are also called, can be obtained in various materials, as hard rubber, celluloid, pear- wood, mahogany and steel; and either solid (Fig. 25) or open (Fig. 26). The open triangles are preferable, and two are required, one with acute angles of 30° and 60°, the other with 45" angles. Hard rubber has an advantage over metal or wood, the latter being likely to warp and the former to rust, unless plated. Celluloid is transparent and the most cleanly of all. 16 THEORETICAL AND PRACTICAL GRAPHICS. E'lgr- ss- The most frequently recurring problems involving the use of the triangles are the following: — 49. To draw parallel lines place either of the edges s-ig-. ss. against another triangle or the T-rule. If then moved along, in either direction, each of the other edges will take a series of parallel positions. 50. To draw a line perpendicular to a given line place the hypothenuse of the triangle, o a, (Fig. 26), so as to coincide with or be parallel to the given ■ — -^ another triangle against the base. By then turning the triangle so that the the hypothenuse will be found Fig-- ST'. line; then a rule or other side, o c, of its right angle shall be against the rule, as at o , perpendicular to its first position and therefore to the given line. 51. To construct regular hexagons place the shortest side of the 60° triangle against the rule (Pig. 27) if two sides are to be horizontal, as fe and 6 c of hexagon H. For vertical sides, as in H ', the position of the triangle is evident. By making a h indefinite at first, and knowing b c — the length of a side, we may obtain a by an arc, centre b, radius b c. If the inscribed circles were given, the hexagons might also be obtained by drawing a series of tangents to the circles, with the rule and triangles in the positions indicated. THE SCALE. 52. But rarely can a drawing be made of the same size as the object, or "full-size," as it is called; the lines of the drawing, therefore, usually bear a certain ratio to those of the object. This ratio is called the scale and should invariably bo indicated. If six inches on the drawing represent one foot on the object the scale is one-half and might be variously indicated, thus: SCALE |; SCALE 1:2; Scale 6 In. ^ 1 Ft. Scale 6" = 1'. At one foot to the inch any line of the drawing would be one -twelfth the actual size, and the fact indicated in either of the ways just illustrated. Although it is a simple matter for the draughtsman to make a scale for himself for any par- ticular case yet scales can be purchased in great variety, the most serviceable of which for the usual range of work is of box -wood, 12" long, (or 18", if for large work) of the form illustrated by Fig. x-j-er. se. 28, and graduated -^^ : -f^ : \ : \ : \ : \ : \: \ : \\ : Z inches to the foot. This is known as the architect's scale in contradistinction to the engineer's, which is decimally graduated. It will, however, be frequently convenient to have at \\\\\\\\\\\\\\\\\\\\\\\\\'t hand the latter as well as the former. When in use it should be laid along the line to be spaced, and a light dot made upon the % 11 10 987654331b D 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i 1 j 1 1 1 1 1 1 1 1 1 j 1 1 1 1 1 1 / 1 III It 1 1 1 1 1 / 1 1 / 1 ^1 X 1 1 1 1 II 1 1 tl 1 1 1 1 1 1 1 1 1 1 1 1 !c Id lb If h Ih k k Ik ll Im ^™ 1-ff B a paper with the pencil, opposite the proper division on the graduated edge. A distance should rarely SCALES. — PENCILS. — INKS. 17 be transferred from the scale to the drawing by the dividers, as such procedure damages the scale if not the paper. 53. For special cases diagonal scales can readily be constructed. If, for example, a scale of 3 inches to the foot is needed and measuring to fortieths of inches, draw eleven equidistant, parallel lines, enclosing ten equal spaces, as in Fig. 29, and from the end A lay off AB, B C, etc., each 8 inches and representing a foot. Then twelve parallel diagonal lines in the first space intercept quarter -inch spaces on AB or ah, each representative of an inch. There being ten equal spaces between B and h, the distance s x, of the diagonal 6 m from the vertical h B, taken on any horizontal line 8 X, is as many tenths of the space m 5 as there are spaces between s x and b ; six, in this case. The principle of construction may be generalized as follows : — The distance apart of the vertical lines represents the units of the scale, whether inches, feet, rods or miles. Except for decimal graduation divide the left-hand space at top and bottom into as many spaces as there are units of the next lower denomination in one of the original units (feet, for yards as units; inches in case of feet, etc.). Join the points of division by diagonal lines; and, if — is the smallest fraction that the scale is designed to give, rule x+1 equidistant horizontal lines, giving % equal horizontal spaces. The scale will then read to -th of the intermediate denomination of the scale. When a scale is properly used, the spaces on it which represent feet and inches are treated as if they were such in fact. On a scale of one -eighth actual size the edge graduated 1^ inches to the foot would be employed; each 1| inch space on the scale would be read as if it were a foot; and ten inches, for example, would be ten of the eighth -inch spaces, each of which is to represent an inch of the original line being scaled. The usual error of beginners would be to divide each original dimension by eight and lay off the result, actual size. The former method is the more expeditious. THE PENCILS. 54. For construction lines afterward to be inked the pencils should be of hard lead, grade 6H if Fabers or VVH if Dixon's. The pencilling should be light. It is easy to make a groove in the paper by exerting too great pressure when using a hard lead. The hexagonal form of pencil is usually indicative of the finest quality, and has an advantage over the cylindrical in not rolling off when on a board that is slightly inclined. Somewhat softer pencils should be used for drawings afterward to be traced, and for the prelim- inary free-hand sketches from which exact drawings are to be made; also in free-hand lettering. Sharpen to a chisel edge for work along the edges of the T-rule or triangles, but use another pencil with coned point for marking off distances with a scale, locating centres and other isolated points, and for free-hand lettering; also sharpen the compass leads to a point. Use the knife for cutting the wood of the pencil, beginning at least an inch from the end. Leave the lead exposed for a quarter of an inch and shape it as desired, either with a knive or on a fine file, or a pad of emery paper. THE INK. 55. Although for many purposes some of the liquid drawing -inks now in the market, partic- ularly Higgins', answer admirably, yet for the best results, either with pen or brush, the draughtsman should mix the ink himself with a stick of India — or, more correctly, China ink, selecting one of the higher- priced cakes, of rectangular cross -section. The best will show a lustrous, almost iridescent fracture, and will have a smooth, as contrasted with a gritty fed when tested by rubbing the moist- ened finger on the end of the cake. TT"KTTTTT-i-f 18 THEORETICAL AND PRACTICAL GRAPHICS. Sets of saucers, called "nests," designed for the mixing of ink and colors, form an essential part of an equipment. There are usually six in a set and so made that each answers as a cover for the one below it. Placing from fifteen to twenty drops of water in one of these the stick of ink should be rubbed on the saucer with moderate pressure. To properly mix ink requires great patience, as with too great pressure a mixture results having flakes and sand -like particles of ink in it, whereas an absolutely smooth and rather thick, slow- flowing liquid is wanted, whose surface will reflect the face like a mirror. The final test as to sufficiency of grinding is to draw a broad line and let it dry. It should then be a rich jet black, with a slight lustre. The end of the cake must be carefully dried on removing it from the saucer to prevent its flaking, which it will otherwise invariably do. One may say, almost without qualification, and particularly when for use on tracing -cloth, the thicker the ink the better; but if it should require thinning, on saving it from one day to another — which is possible with the close-fitting saucers described — add a few drops of water, or of ox -gall if for use on a glazed surface. When the ink has once dried on the saucer no attempt should be made to work it up again into solution. Clean the saucer and start anew. WATER COLORS. 56. The ordinary colored writing inks should never be used by the draughtsman. They lack the requisite "body" and are corrosive to the pen. Very good colored drawing inks are now manu- factured for line work, but Winsor and Newton's water colors, in the form called "moist," and in "half- pans" are the best if not the most convenient, for color work either with pen or brush. Those most frequently employed in engineering and architectural drawing are Prussian Blue, Carmine, Light Red, Burnt Sienna, Burnt Umber, Vermilion, Gamboge, Yellow Ochre, Chrome Yellow, Payne's Gray and Sepia. For some of their special uses see Art. 73. Although hardly properly called a color Chinese White may be mentioned at this point as a requisite, and obtainable of the same form and make as the colors above. DRAWING-PINS. 57. Drawing-pins or thumb-tacks, for fastening paper upon the board, are of various grades, the best, and at present the cheapest, being made from a single disc of metal one -half inch in diameter, from which a section is partially cut, then bent at right angles to the surface, forming the point of the pin. IRREGULAR CURVES. 58. Irregular or French curves, also called sweeps, for drawing non- circular arcs, are of great variety, and the draughtsman can hardly have too many of them. They may be either of pear Fig- SO- wood or hard rubber. A thoroughly equipped draughting office will have a large stock of these curves, which may be obtained in sets, and are known as railroad curves, ship curves, spirals, ellipses, hyperbolas, parabolas and combination curves. Some very serviceable flexible curves are also in the market. If but two are obtained (which would be a minimum stock for a beginner) the forms shown in Fig. 30 will probably prove as serviceable as any. When employing them for inked work the pen should be so turned, as it advances, that its blades will maintain the same relation (parallelism) to the edge of the guiding curve as they ordinarily do to the edge of CURVES. — RUB BER. — ERASERS. — PROTRACTORS. — BRUSHES. 19 the rule. And the student must content himself with drawing slightly less of the curve than might apparently be made with one setting of the sweep, such course being safer in order to avoid too close an approximation to angles in what should be a smooth curve. For the same reason, when placed in a new position, a portion of the irregular curve must coincide with a part of that last inked. The pencilled curve is usually drawn free-hand, after a number of the points through which it should pass have been definitely located. In sketching a curve free-hand it is much more naturally and smoothly done if the hand is always kept on the concave side of the curve. INDIA EUBBER. 59. For erasing pencil -lines and cleaning the paper india rubber is required, that known as " velvet " being recommended for the former purpose, and either " natural " or " sponge " rubber for the latter. Stale bread crumbs are equally good for cleaning the surface of the paper after the lines have been inked, but will damage pencilling to some extent. One end of the velvet rubber may well be wedge-shaped in order to erase lines without damag- ing others near them. INK ERASER. 60. The double-edged erasing knife gives the quickest and best results when an inked line is to be removed. The point should rarely be employed. The use of the knife will damage the paper more or less, to partially obviate which rub the surface with the thumb-nail or an ivory knife handle. PROTRACTOR. 61. For laying out angles a graduated arc called a "protractor" is used. Various materials are employed in the manufacture of protractors, x-ig-. 3i. as metal, horn, celluloid, Bristol board and tracing paper. The two last are quite accu- rate enough for ordinary purposes, although where the utmost precision is required, one of German silver should be obtained, with a moveable arm and vernier attachment. The graduation may advantageously be to half degrees for average work. To lay out an angle (say 40°) with a protractor, the radius CH (Fig. 31) should be made to coincide with one side of the desired angle; the centre, C, with the desired vertex; and a dot made with the pencil opposite division numbered 40 on the graduated edge. The line MC, through this ])oint and C, completes the construction. BRUSHES. 62. Sable -hair brushes are the best for laying flat or graduated tints, with ink or colors, upon small surfaces; while those of camel's hair, large, with a brush at each end of the handle, are better adapted for tinting large surfaces. Reject any brush that does not come to a perfect point on being moistened. Five or six brushes of different sizes are needed. PRELIMINARIES TO PRACTICAL WORK. 63. The first work of a draughtsman, like most of his later productions, consists of line as distin- guished irom brmh work, and for it the paper may be fastened upon the board with thumb-tacks only. 20 THEORETICAL AND PRACTICAL GRAPHICS. There is no universal standard as to size of sheets for drawings. As a rule each draughting office has its own set of standard sizes, and system of preserving and indexing. The columns of the various engineering papers present frequent notes on these points, and the best system of pre- serving and recording drawings, tracings and corrections is apparently in process of evolution. For the student the best plan is to have all drawings of the same size bound in neat but permanent form at the end of the course. The title-pages, which presumably have also been drawn, will suf- ficiently distinguish the different sets. In his elementary work the student may to advantage adopt two sizes of sheets which are con- siderably employed, 9" x 13", and its double, 13" x 18"; sizes into which a "Super Royal" sheet naturally divides, leaving ample margins for the mucilage in case a " stretch " is to be made. A " Double Elephant " sheet being twice the size of a " Super Royal " divides equally well into plates of the above size, but is preferable on account of its better quality. To lay out four rectangles upon the paper locate first the centre (see Fig. 32) by intersecting diagonals, as at 0. These should not be drawn entirely across the sheet, but one of them will necessarily pass a short distance each side of the point where the centre lies — judging by the eye alone ; the second definitely determines Fig-, as. the point. If the T-rule will not iPii\MMiMlliMllM^^ reach diagonally from corner to corner of the paper (and it usually will not) the edge may be practically extended by placing a triangle against but pro- jecting beyond it, as in the upper left- hand portion of the figure. The T-rule being placed as shown, with its head at the left end of the board — the correct and usual position — draw a horizontal line X Y, through the centre just located. The vertical centre line is then to be drawn, with one of the triangles placed as shown in the figure, i. e., so that a side, as mn or tr, is perpendicular to the edge. It is true that as long as the edges of the board are exactly at right angles with each other we might use the T-rule altogether for drawing mutually perpendicular lines. This condition being, however, rarely realized for any length of time, it has become the custom — a safe one, as long as rule and triangle remain "true" — to use them as stated. The outer rectangles for the drawings (or "plates," in the language of the technical school) are completed by drawing parallels, as JN and Y N, to the centre lines, at distances from them of 9" and 13" respectively, laid off from the centre, 0. An inner rectangle, as abed, should be laid out on each plate, with proper margins ; usually at least an inch at the top, right and bottom, and an extra half inch on the left as an allowance for binding. These margins are indicated by x and z in the figure, as variables to which any con- venient values may be assigned. The broad margin x in the upper rectangle will be at the draughts- man's left hand if he turns the board entirely around — as would be natural and convenient — when ready to draw on the rectangle Q Y. EXERCISES- FOR PEN AND COMPASS. 21 CHAPTER ir. GRADES or LINES. — LINE TINTING. — LINE SHADING. — CONVENTIONAL SECTION -LINING.— PEEQUENTLY KECUERING PLANE PROBLEMS.— MISCELLANEOUS PEN AND COMPASS EXEECISES. 65. Several kinds of lines employed in mechanical drawing are indicated in the figure below. While getting his elementary practice with the ruling -pen the student may group them as shown, or in any other symmetrical arrangement, either original with himself or suggested by other designs. Fig-. 33. CENTRE IJ.INE, if red. CENTRE UINE, if blaol<. -H' FOR ORDINARY OUTLINES. ll'DDElJnjNE I MEDIUM, };ontinuous. ^"-_ "dotted LINE'l usually empi )^ed as a construction line. \ ^^>.^ A^^ SH kDE i^aS^^B u1 f^E .-^-' "DOTTED LINE" line of moti >n in Kinematic Geometry. ^^^ \ ^'' DIMENSION LINE, if red. I ' ^-^ DIMENSION LINE, black. .-^ When drawing on tracing cloth or tracing paper, for the purpose of making blue- prints, all the lines will preferably be hlack, and the centre and dimension lines distinguished from others as indi- cated above, as also by being somewhat finer than those employed for the light outlines of the object. Heavy, opaque, red lines may, however, be used, as they will blue -print, though faintly. There is at present no universal agreement among the members of the engineering profession as to standard dimension and centre lines. Not wishing to add another to the systems already at variance, but preferring to facilitate the securing of the uniformity so desirable, I have presented those for some time employed by the Pennsylvania Railroad and now taught at Cornell University. The lines of Fig. 33, aa also of nearly all the other figures of this work, having been printed from hloeks made hy the cerographlc process (Art. 277), are for the most part too light to serve as examples for machine-shop work. Fig. 80 Is a sample of P. B. R. drawing, and Is a fair model as to weight of line for working drawings. 22 THEORETICAL AND PRACTICAL GRAPHICS. A dash -and -three -dot Una (not shown in the figure) is considerably used in Descriptive Geometry, either to represent an auxiliary plane or an invisible trace of any plane. (See Fig. 238). The so-called "dotted" line is actually composed of short dashes. Its use as a "line of motion" was suggested at Cornell. When colors are used without intent to blue -print they may be drawn as light, continuous lines. Colors will further add to the intelligibility of a drawing if employed for construction lines. Even if red dimension lines are used the arrow heads should invariably be black. They should be drawn free-hand^ with a writing pen, and their points touch the lines between which they give the distance. 66. The utmost accuracy is requisite in pencilling, as the draughtsman should be merely a copyist when using the pen. On a complicated drawing even the kind of line should be indicated at the outset, so that no time will be wasted, when inking, in the making of distinctions to which thought has already been given during the process of construction. No unnecessary lines should be drawn, or any exceeding of the intended limit of a line if it can possibly be avoided. If the work is symmetrical, in whole or in part, draw centre lines first, then main outlines; and continue the work from large parts to small. The visible lines of an object are to be drawn first; afterward those to be indicated as concealed. All lines of the same quality may to advantage be drawn with one setting of the pen, to ensure uniformity ; and the light outlines before the shade lines. In drawing arcs and their tangents ink the former first, invariably. All the inking may best be done at once, although for the sake of clearness, in making a large and complicated drawing, a portion — -usually the nearest and visible parts — may be inked, the draw- ing cleaned, and the pencilling of the construction lines of the remainder continued from that point. The inking of the centre, dimension and construction lines naturally follows the completion of the main design. a - 6 1 2 3 4 5 6 7 8 9 n^ I 2 3 4 6 6 7 1 2 3 4 5 I 2 3 1 2 3 4 5 1 \ ' 2 3 4 M 67. In Fig. 34 we have a straight- line design usually called the "Greek Fret," and giving the student his first illustration of the use of the "shade line" to bring a drawing out "in relief." The law of the construction will be evident on examination of the numbered squares. Without entering into the theory of shadows at this point we may state briefly the "shop rule" for drawing shade lines, viz., right-hand and lower. «3 That is, of any pair of lines making the same turns together or representing the limit of the same flat surface, the right- hand line is the heavier if the pair is vertical, but the lower if they run horizontally; always subject, however, to the proviso that the line of inter- section of two illumin- ated planes is never a shade line. 68. The conic section called the parabola fur- nishes another interesting exercise in ruled lines, when it is represented by its tangents as in Fig. 35. The angle OA E may be assumed at pleasure, and on the finished drawing the numbers may SECTION- LINING. — LINE- SHADING. 23 X-igr- 3S. be omitted, being given here merely to show the law of construction. All the divisions are equal, and like numbers are joined. Some interesting mathematical properties of the curve will l)e found in ('hapter V. 69. A pleasing design that will test the beginner's skill is that of Fig. 86. It is suggestive of a cobweb, and a skillful free-hand draughtsman could make it more realistic by adding the spider. Use the 60° triangle for the heavy diagonals and parallels to them; the T-rule for the hor- izontals. Pencil the diagonals first but ink them last. 70. The even or flat effect of equidistant parallel lines is called line - tinting ; or, if repre- senting an object that has been cut by a plane, as in Fig. 37, it is called section -lining. The section., strictly speaking, is the part actually in contact with the cutting plane; while the drawing as a whole is a sectional view, as it also shows what is back of the plane of section, the latter being always as- sumed to be transparent. E-ig-. S7. Adjacent pieces have the lines drawn in diff"erent directions in order to distinguish sufficiently between them. The curved eff'ect on the semi -cylinder is evidently obtained by prop- erly varying both the strength of the line and the, spacing. 71. The difference between the shading on the exterior and interior of a cylinder is sharply contrasted in Fig. 38. On the concavity the darkest line is at the toj), while on the convex surface it is near the bot- tom, and below it the spaces remain unchanged while the lines diminish. A better effect would haA^e been obtained in the figure had the engraver begun to increase the lines with the first decrease in the space between them. r'lgr- 3e. The spacing of the lines, in section -Jining, depends upon the scale of the drawing. It may run down to a thirtieth of an inch or as high as one -eighth; but from a twentieth to a twelfth of an inch would be best adapted to the ordinary range of work. Equal spacing and not fine spacing 24 THEORETICAL AND PRACTICAL GRAPHICS. should be the object, and neither scale nor patent section -liner should be employed, but distances gauged by the eye alone. 72. A refinement in execution which adds considerably to the effect is to leave a white line between the top and left-hand outlines of each piece and the section lines. When purposing to produce this effect rule light pencil lines as limits for the line -tints. 73. If the various pieces shown in a section are of different materials there are four ways of denoting the difference between them : (a) By the use of the brush and certain water -colors, a method considerably employed in Europe, but not used to any great extent in this country, probably owing to the fact that it is not applic- able where blue -prints of the original are desired. The use of colors may, however, be advantageously adopted when making a highly finished, shaded drawing; the shading being done first, in India ink or sepia, and then overlaid with a flat tint of the conventional color. The colors ordinarily used for the metals are Payne's gray or India ink for Cast Iron. Gamboge " Brass (outside view). Carmine " Brass (in section). Prussian Blue " Wrought Iron. Prussian Blue with a tinge of Carmine " Steel. FlC- 39. Last Iran. 5 + EEl. Wr't. Irnn. Brass. PEITlTiL. % 5. %t. r^//^^////../;^ ■A ^. /////,///,//, ^///, /, BtnnE. y/yy^y WDDd. Capper. Brick. CONVENTIONAL SECTION-LINING. 25 More imtural effects can also be given by the use of colors, in representing the other materials of construction; and the more of an artist the draughtsman proves to be the closer can he approx- imate to nature. Pale blue may be used for water lines; Burnt Sienna, whether grained or not, suggests wood; Burnt Umber is ordinarily employed for earth; either Light Red or Venetian Red are well adapted for brick, and a wash of India ink having a. tinge of blue gives a fair suggestion of masonry; although the actual tint and surface of any rock can be exactly represented after a little practice with the brush and colors. These points will l)e enlarged upon later. (b) By section - lining with the drawing pen in the conventional colors just mentioned, a process giving very handsome and thoroughly intelligible results on the original drawing, but, as before, unadapted to blue-printing and therefore not as often used as either of the following methods. (c) By section -lining uniformly in ink throughout and printing the name of the material upon each piece. ^igr- ■^^■ (d) By alternating light and heavy, continuous and In-oken lines according to some law. Said " law " is, unfortunately, by no means universal, despite the attempt made at a recent con- vention of the American Society of Mechanical Engineers to secure uniformity. P^aoli draughting office seems at present to be a law unto itself in tliis matter. 74. As affording valuable examples for further exercise with the ruling pen the system of section -lines adopted by the Pennsylvania Railroad is presented on the opposite page. The wood section is an exception to tliu rule, being drawn free-hand, with a Falcon pen. Fig-, -io. — jjy ^^,,^y (,f contrasting free-hand with me- chanical work Fig. 40 is introduced, in which the rings showing annual growth are drawn as concentric circles with the compass. COURSED RUBBLE MASONRY RUBBLE MASONRY Light India ink. BRICK Venetian Red. CONCRETE Yellow Ochre. selected from the designs of M. N. Forney and F. Van Vleck, and which are fortunate arrangements. ^^ 75. Figs. 42 and 43 are i^rofiles or outlines of mouldings, such as are of frequent occurrence in architectural work. It is good practice to convert such views into oblique projections, giving the effect of solidity; and to further bring out their form by line shading. Figs. 44-46 are sucli representations, the front of each being of the same form as Fig. 42. The oblique lines are all parallel to each other, and — where X'J.gr- -3=2. Fig-, -is. Fig-- -4-a. visible throughout— of the same lengtli. Their direction should be chosen with reference to best ex- hibiting the peculiar features of the object. Obviously the view in Fig. 44 is the least adapted to the conveying of a clear idea of the moulding, while that of Fig. 46 is evidently the best. THEORETICAL AND PRACTICAL GRAPHICS. 76. The student may, to advantage, design profiles for mouldings and line -shade them, after converting them into oblique views. As hints for such work two figures are given (47-48), taken from actual construction in wood. By setting a moulding vertically, as in Fig. 49, and projecting horizontally from its points, a front view is obtained, as in Fig. 50. x-isr- ^s- ng-- so. Fig-. ■V7. Fig-. -^S. M n \ '•■ N 77. The reverse curves on the mouldings may be drawn with the irregular curve, (see Art 58) ; or, if composed of circular arcs to be tangent to vertical lines, by the follow- Fi.s- si. ing construction: — Let M and N be the points of tangency on the verticals Mm and Nn, and let the arcs be tangent to each other at the middle point of the line MX. Draw Mn and Nm perpendicular to the vertical lines. The centres, r and c,, of the desired arcs, are at the intersection of Mn and Nm by per- pendiculars to MN from x and y, the middle points of the segments of MN. 78. The light is to be assumed as coming in the usual direction, i. e., descending from left to right at such an angle that any ray would be projected on the paper at an angle of 45° to the horizontal. In Fig. 43 several rays are shown. At x, where the light strikes the cylindrical portion most directly — technically is normal to the surface — is actuallj' the brightest part. A tangent ray st gives. t, the darkest part of the cylinder. The concave portion beginning at o would be darkest at o and get lighter as it approaches y. Flat parts are either to be left white, if in the light, or have equidistant lines if in the shade, unless the most elegant finish is desired, in which case both change of space and gradation of line must be resorted to as in Fig. 52, which represents a front view of a hexagonal nut. The front face, being parallel to the paper, receives an even tint. An inclined face in the light, as abhf, is lightest toward the observer, while an unillumined face tkdg is exactly the reverse. Notice that to give a fid efl'ect on the inclined faces the spacing- out as also the change in the size of lines must be more gradual than when indicating curvature. (Compare with Figs. 46 and 50.) S'iS- S2. REMARKS ON SHADING . — PL A KE PROBLEMS. 27 If two or more illuminated flat surfaces are parallel to the paper (as t g b h, Fig. 52) but at different distances from the eye, the nearest is to be the lightest; if unilluminated, the reverse would be the case. 79. In treating of the theory of shadows distinctions have to be made, not necessary here, between real and apparent brilliant points and lines. We may also remark at this point that to an exi)eri- enced draughtsman some license is always accorded, and that he can not be expected to adhere rigidly to theory when it involves a sacrifice of effect. For example, in Fig. 46 we are unable to see to the left of the (theoretically) lightest part of the cylinder, and find it, therefore, advisable to move the darkest part past the point where, according to Fig. 43, we know it in reality to be. The professional draughtsmen who draw for the best scientific papers and to illustrate the circulars of the leading machine designers allow themselves the latitude mentioned, with most pleasing results. Yet until one may be justly called an expert he can depart but little from the narrow confines of theory without being in danger of producing decidedly peculiar effects. 80. As from this point the student will make considerable use of the compasses, a few of the more important and frequently recurring plane problems, nearly all of which involve their use, may well be introduced. The proofs of the geometrical constructions are in several cases omitted, but if desired the student can readily obtain them by reference to any synthetic geometry or work on plane problems- All the problems given (except No. 20) have proved of value in shop practice and architectural work. The student should again read Arts. 48-51 regarding special uses of the 30° and 45° triangles, which, with the T- rule, enable him to employ so many " draughtsman's " as distinguished from "geometrician's" methods; also Arts. 86 and 87. 81. Prob. 1. To draw a perpendicular to a given line at a given point, as A (Fig. 53), use the tri- angles, or triangle and rule as previously described ; or lay off equal distances A a, A b, and with Y rotating M N with the circle a b about the axis, it is evident that in one of the three systems of planes mentioned in the last article each plane must contain the axis. When a surface can be generated by revolution about an axis one of its characteristics is that any plane perpendicular to the axis will cut it in a circle. The circles of Fig. 73 may then be, for the moment, considered as parallel cuts by a series of planes perpendicular to the axis, a few of which * Olivier, Memoires de OSomStrie Descriptive. Paris, IS."!!. ANNULAR TORUS.— WARPED HYPERBOLOID. 35 may be shown in mn, op, &c. (Fig. X). Each of these planes cuts two circles from the surface; the plane o p, for example, giving circles of diameters c d and v w respectively. A plane, perpendicular to the paper on the line P Q, would be a bi-tangent plane, because tan- gent to the surface at two points, P and Q; and such plane would cut two over -lapping circles from the torus, each of them running partly on the inner and partly on the outer portion of the surface. These sections are seen as ellipses in Fig. 74. For the proof that such sections are circles the stu- dent is probably not prepared at this point, but is referred to Olmer's Seventh Memoir, or to the appendix.* 114. Another interesting fact with regard to the torus is that a series of planes parallel to, but not containing the axis, cut it in a set of curves called the Cassian ovals (see Art. 212) of which the Lemniscate of Art. 158 is a special case, and which would result from using a plane parallel to the axis and tangent to the surface at a point on the smallest circle at a, (Fig. 75.)+ 115. Fig. Y is given to illustrate the fact that from mere untapered outlines, such as compose the central figure, we cannot determine the form of the object. ^i-s- 'T's. By shading ehf and DNr we get Fig. Z, and the form shown in Fig. Y would be instantly recognized without the drawing of the latter. An angular object must therefore have shade lines, as also the end view of a round object; but a side view m m'/,'/ of a cyHndrical piece must either have uniform outlines or be shaded with several lines. Thus, in Fig. 76, A would represent an angular piece, while B would indicate a circular cylin- der; if elliptical its section would be drawn at one side as shown. 116. Before presenting the crucial test for the learner — the railroad rail — two additional practice exercises, mainly in ruling, are given in Figs. 77 and 78. The former shows that, like the parabola the circle and hyperbola can be represented by their enveloping tangents. The upper and lower figures are merely two views of the surface called the warped hyperboloid, from the hyperbolas which constitute the curved outlines seen in the upper figure. The student can make this surface in a few moments by stringing threads through equidistant holes arranged in a circle on two circular discs of the same or difl'erent sizes, but having the same number of holes in each disc. By attaching weights to the threads to keep them in tension at all times, and giving the upper disc a twist, the surface will change from cylindrical or conical to the hyperboloidal form shown. Gear wheels are occasionally constructed, having their teeth upon such a surface and in the direction of the lines or elements forming it; but the hyperboloid is of more interest mathematically than mechanically. Begin the drawing by pencilling the three concentric circles of the lower figure. When inking, omit the smaller circle. Draw a series of tangents to the inner circle, each one beginning on the middle circle and terminating on the outer. Assume any vertical height, ts', for the upper figure, and draw H' M' and P' R' as its upper and lower limits. H' M' is the vertical projection, or eleva~ tion, of the circle H K M N, and all points on the latter, as 1,2,3,4, are projected, by perpendiculars to H' M', at 1', 2', 3',4', etc. All points on the larger circle PQR are similarly projected to P' R'. The extremities of the same tangent are then joined in the upper view, as 1' with 1 (a). •An originttl deiiioiistrution by Mr. George E. Barton (rriuceton, 'IB,) when a Junior in the John C. Green School ol Science. t These curves can also be obtained by assuming two foci, as if for an ellipse, but taking the product of the focal radii as a constant quantity, some perfect square. If pp'= 36 then a point on the curve would be found at the intersection of arcs having the foci as centres, and for radii 2" and 18", or 4" and 9", etc. The Lemniscate results when the constant assumed Is the square of half the distance between the foci. THEORETICAL AND PRACTICAL GRAPHICS. 3|n)2(ii) Vff) 32 31 13 14 TAPERING LINES.— RAIL SECTIONS. 37 Part of each line is dotted to represent its disappearing upon an invisible portion of the sur- face. The law of such change on the lower figure is evident from inspection, while on the eleva- tion the point of division on each line is exactly above the point where the other view of the same line runs through H M in the lower figure. 117. To reproduce Fig. 78 draw first the circle ajh n, then two circular arcs which would con- tain a and b if extended, and whose greatest distance from the original circle is x, (arbitrary). Six- teen equidistant radii as at a, c, d, etc., are next in order, of which the rule and 45 ° triangle give those through a, d, f and h. At their extremities, as m and n, lay off" the desired width, y, and draw toward the points thus determined lines radiating from the centre. Terminate these last upon the inner arcs. Ink by drawing from the centre, not through or toward it. All construction lines should be erased before the tapering lines are filled in. The "filling in" may be done very rapidly by ruling the edges in fine lines at first, then opening the pen slightly and beginning again where the opening between the lines is apparent and ruling from there, adding thickness to each edge on its inner side. It will then be but a moment's work to fill in, free- hand, with the Falcon pen or a fine -pointed sable -brush, between the — now heavy — edge-lines of the taper. To have the pen make a coarse line when starting from the centre would destroy the effect desired. 118. The draughtsman's ability can scarcely be put to a severer test on mere outline work than in the drawing of a railroad rail, so many are the changes of radii involved. As previously stated, where tangencies to straight lines are required, the arcs are to be drawn first, then th% tangents. Figs. 79 and 80 are photo - engravings of rail sections, showing two kinds of "finish." Fig. 80 is a "working drawing" of a Penn- sylvania Railroad rail, full-size. This makes one of the handsomest plates that can be undertaken, if finished with shade lines, as in Fig. 79, section -lined with Prussian blue, and the dimension lines drawn in carmine. A still higher effect is shown in the wood- cut on page 85, the rail being represented in oblique projection and shaded. Begin Fig. 80 by drawing the vertical centre-line, it being an axis of symmetry. Upon it lay off 6" for the total height, and locate two points between the top and base, at distances from tliem of If" and ^" respectively; these to be the points of convergence of the lower lines of the head and sloping sides of the base. From these points draw lines, at first indefinite in length, and inclined 13° to the horizontal. The top of the head is an arc of 10" radius, subtended by an angle of 90°. This changes into an arc of ■^^" radius on the upper corner, with its centre on the side of said 9° angle. The sides of the head are straight lines,, drawn at 4° to the vertical, and tangent to the comer arcs. The thin vertical portion of the rail is called the web, and is ^|" wide at its centre. The outlines of the web are arcs of 8" radius, subtended by angles of 15°, centres on line marked "centre line of bolt holes." Fis. 7S. THEORETICAL AND PRACTICAL GRAPHICS. The weight per yard of the rail shown is given as eighty- five pounds,* from which we know the area of the cross -section to be eight and one -half square inches, since a bar of iron a yard long and one square inch in cross -section weighs, approx- imately, ten pounds. (10.2 lbs., average). The proportions given are slightly different from those recommended in the report'' of the committee appointed by the American Society of Civil Engineers to examine into the proper relations to each other of the sections of railway wheels and rails. There was quite general agreement as to the following recommendations : a top radius of twelve inches ; a quarter-inch comer radius; vertical sides to the web; a lower- comer of one-sixteenth inch, and a broad head relatively to the depth. E-ig-. eo- 'See the Appendix for dimensions of a 100-lb. rail. + Transactions A. S. C. E., .January, 1891. EKERV18ES FOE THE IRREGULAR CURVE. 89 CHAPTER V. THE HELIX.— CONIC SECTIONS.— H0M0L06ICAL PLANE CURVES AND SPACE-FIGURES. — LINK- MOTION CURVES. — CENTROIDS. — THE CYCLOID. — COMPANION TO THE CYCLOID. — THE CUR- TATE TROCHOID. — THE PROLATE TROCHOID. — HYPO-, EPI-, AND PERI-TROCHOIDS. — SPECIAL TROCHOIDS— ELLIPSE, STRAIGHT LINE, LIMAgoN, CARDIOID, TRISECTRIX, INVOLUTE, SPIRAL OF ARCHIMEDES. — PARALLEL CURVES. — CONCHOID. — QUADRATRIX. — CISSOID. — TRACTRIX. — WITCH OF AGNESL — CARTESIAN OVALS. — CASSIAN OVALS. -CATENARY. — LOGARITHMIC SPIRAL. — HYPERBOLIC SPIRAL. — THE LITUUS.— THE IONIC VOLUTE. 119. There are many curves which the draughtsman has frequent occasion to make whose con- struction involves the use of the irregular curve. The more important of these are the Helix; Conic Sections — Ellipse, Parabola and Hyperbola; Link-motion curves or point -paths; Centroids; Trochoids; the Involute and the Spiral of Archimedes. Of less practical importance, though equally interesting geometrically, are the other curves mentioned in the heading. The student should become thoroughly acquainted with the more important geometrical jjroperties of these curves, both to facilitate their construction under the varying conditions that may arise and also as a matter of education. Considerable space is therefore allotted to them here. At this point Art. 58 should be reviewed, and in addition to its suggestions the student is fur- ther advised to work, at first, on as large a scale as possible, not undertaking small curves of sharp curvature until after acquiring some facility with the curved ruler. THE HELIX. 120. The ordinary helix is a curve which cuts all the elements of a right cylinder at the same angle. Or we may define it as the curve which would be generated by a point having a uniform motion around a straight line combined with a uniform motion parallel to the line. E'igr- ei- Qmuuumuuuuuumuuuuuuu The student can readily make a model of the cylinder and helix by drawing on thick paper or Bristol-board a rectangle A" B" C" D" (Fig. 81) and its diagonal, D" B" ; also equidistant elements, as m"b", n"c", etc. Allow at the right and bottom about a quarter of an inch extra for over- lapping, as shown by the lines x y and s z. Cut out the rectangle zx; also cut a series of vertical slits between D" C" and zs; put mucilage between B" C" and xy; then roll the {)aper up into cylindrical form, bringing A" D" t" h" in front of and upon the gummed portion, so that A" D" 40 THEORETICAL AND PRACTICAL GRAPHICS. will coincide with B" C". The diagonal D" B" will then be a helix on the outside of the cylinder, but half of which is visible in front view, as D'l', (see right-hand figure); the other half, T A', being indicated as unseen. To give the cylinder permanent form it can then be pasted to a cardb(jard base by mucilage on the under side of the marginal fiai)s below D" C", turning them outward, not in toward the axis. The rectangle A" B" C" D" is called the devdopmeni of the cylinder; and any surface like a cylinder or cone, which can be rolled out on a plane surface and its equivalent area obtained by bringing consecutive elements into the same plane, is called a developable surface. The elements m"b", n"c", etc., of the development stand vertically at b, e, d . . . . g of the half plan, and are seen in the elevation at m' b',n' c', o' d', etc. The point 8', where any element, as c', cuts the helix, is evidently as high as 3", where the same point appears on the- development. We may therefore get the curve D' T A' by erecting verticals from b,c,d....c/, to meet horizontals from the points where the diago- nal D" B" crosses those elements on the development. ]}"('" obviously equals 2 ir r, where r^OD. The shortest method of dramng a helix is to divide its plan (a circle) and its pitch {D' A', the i~ise in one turn) into the same number of equal parts; then verticals 6m', en', etc., from the points of division on the plan, will meet the horizontals dividing the pitch, in points 2', 3', etc., of the desired curve. The construction of the helix is involved in the designing of screws and screw-propellers, and in the building of winding stairs and skew-arches. Mathematically, both the curve and its orthographic projection are well worth study, the latter being alwaj's a sinusoid, and becoming the companion to the cycloid for a 45°-helix. (Arts. 170 and 171). For the conical helix, seen in projection and development as a Spiral of Archimedes, see Art. 191. THE CONIC SECTIONS. 121. The ellijise, parabola and h^-perbola are called conic sections- or conies because they may be obtained by cutting a cone by a plane. We will, however, first" obtain them by other methods. According to the definiticm given by Boscovich, the ellipse, i)arabola and hyperbola are curves in which there is a constant ratio between the distances of points on the curve from a certain fixed point (the focus) and their distances from a fixed straight line (the directrix). Referring to the parabola. Fig. 82, if S and B are points of the curve, F the focus and X Y the directrix, then, if S F : S T : : B F : B X, we conclude that B and S are points of a conic section. 122. The actual value of such ratio (or eccentricity) may be 1 or either greater or less than unity. When SF equals ST the ratio equals 1, and the relation is that of equality, or parity, which suggests the parabola. "f 123. If it is farther from a point oi the conic to the focus than to the directrix the ratio is greater than 1, and the hyperbola is indicated. 124. The ellipse, of course, comes in for the third possibility as to ratio, viz., less than 1. Its construction by this principle is not shown in Fig. 82 but later, the (Art. 142) method of generation here given illustrating the practical way in which, in landscape gardening, an elliptical plat would be laid out; it is tlierefore called the construction as the "gardener's ellipse." Taking A C and D E as representing the extreme length and width, the points F and I\ (foci) would be found by cutting A C by an arc of radius equal to one-half A C, centre I). Pegs or pins at F and F, , and a string, of length AC, with ends fastened at the foci, complete the preliminaries. The curve is then traced on the ground by sliding a pointed stake against the string, as at P, so that at all times the parts F, P. F P, are kept straight. CONIC SECTIONS. 41 125. According to the foregoing construction the ellipse may be defined as a curve in which the xum of the distances from any point of the curve to two fixed points is constant. That constant is evidently the longer or transrerse (major) axis, A C. The shorter or conjugate (or minor) axis, D E, is perpen- dicular to the other. With the compasses we can determine P and other points of the ellipse by using F and F, as centres, and for radii any two segments of A C. Q, for example, gives A Q and C Q as segments. Then arcs from /' and I\ , with radius equal to Q C, would intersect arcs from the same centres, radius QA, in four points of the ellipse, one. of which is P. 126. ]5y tlie l^oscovicli definition we are also enabled to construct the parabola and hyperbola by continuous motion along a string. For the parabola jjlace a triangle as in Fig. 82, with its altitude GX toward the focus. If a string of length G X be fastened at G, stretched tight from G to any point B, by putting a pencil at B, then the remainder BX swung around and the end fostened at F, it is then, evidently, as far from B to F as it is from B to the directrix; and that relation will remain constant as the tri- angle is slid along the (Hrectrix, if the pencil point remains against the edge of the triangle so that the portion of tlie string from G to the pencil is kept straight. 127. For tlie hyperbola, (Fig. 82), the construction is identical with the preceding, except that the string fastened at J runs down the hypothemise, and equals it in length. 128. Referring back to Fig. 35, it will be noticed tliat the focus and directrix of the parabola are there omitted; but the former would be the jjoint of intersection of a [)erpendicular from A upon the line joining C with E. A line through A, parallel to C E, would be the directrix. 129. Like the ellipse, the hyperbola can he constructed by using two foci, l)ut whereas in the ellipse (Fig. 82) it was the sum of two focal radii that was constant, i.e., FP-{- F, P= FD-\- 42 THEORETICAL AND PRACTICAL GRAPHICS. F^D=A C (the transverse axis), it is the difference of the radii that is constant for the hyperbola. In Fig. 83 let A B be the transverse axis of the two arcs, or "branches,"' which make the complete hyperbola; then using p and p to represent any two focal radii, as i^Q and F^Q, or FR and F^R, we will have p — p'=AB, the constant quantity. To get a point of the curve in accordance with this prin- ciple we may lay oflF from either focus, as F, any distance greater than FB, as FJ, and with it as a radius, and F as a centre, describe the indefinite arc JR. Subtracting the con- stant, A B, from FJ, by making JE=AB, we use the remainder, ' /'''£', as a radius, and F, as a centre, to cut the first arc at R. The same radii will evidently determine three other points fulfilling the conditions. Figr- S3- 130. The tangent to an hyperbola at any point, as Q, bisect.s the angle FQF^, between the focal radii. In the ellipse, (Fig. 82), it is the external angle between the radii that is bisected by the tan- gent. In the parabola, (Fig. 82), the same principle applies, but as one focus is supposed to be at infinity, the focal radius, B G, toward the latter, from any point, as B, would be parallel to the axis. The tangent at B would therefore bisect the angle F B X. 131. The ellipse as a circle viewed obliquely. If ARMBF (Fig. 84) were a circular disc and we ^^- S"*- were to rotate it on the diameter A B, it would become narrower in the direction F E until, if sufficiently turned, only an edge view of the disc would be obtained. Tlie axis of rotation A B would, however, still appear of its original length. In the rotation supposed, all ])oints not on the axis would describe circles about it with their planes perpendicular to it. M, for example, would describe an arc, part of which is shown in MM,, which is straight, as the plane of the arc is seen "edge-wise." If instead of a circular disc we turn an elliptical one, A G B D, upon its shorter axis CD, it is obvious that B would apparently approach on one side while A advanced on the other, and that the disc could reach a position in which it would be projected in the small circle CkD. If, then,- the axes of an ellipse are given, as A B and CD, use them as diam- eters of concentric circles; from their centre, 0, draw random radii, as T, OK; then either, as T, will cut the circles in points, t and T, through which a parallel and perpendicular, respectively, to the longer axis will give a point T, of an ellipse. The relation just illustrated is established analytically in the Appendix. 132. If TS is a tangent at T to the large circle, then when T has rotated to T, we shall have T,S as a tangent to the ellipse at the point derived from T, the point S having remained con- stant, being on the axis of rotation. CONIC SECTIONS. 43 s-i-g. es. • Similarl}', if a tangent at R, were wanted, we would first find r, corresponding to ^,; draw the tangent rJ to the small circle; then join /?, to /, the latter on the axis and therefore constant. 133. Occasionally we have given the length and inclination of a pair of diameters of the ellipse making oblique angles with each other. Such diameters, are called conjugate, and the curve may be constructed upon them thus: Draw the axes TD and H K at the assigned angle D H; construct the parallelogram MX X Y; divide D M and D into the same number of equal parts; then from K draw lines through the points of division on D 0, to meet similar lines drawn through II and the divisions on D M. The intersection of like - num- bered lines will give points of the ellipse. 134. It is the law of expansion of a perfect gas that the volume is inversely as the pressure. That is, if the volume be doubled the pressure drops one -half; if trebled the pressure becomes one- third, etc. Steam not being a perfect gas departs somewhat from the above law, but the curve indi- cating the fall in pressure due to its expansion is compared with that for a perfect gas. To construct the curve for the latter let us suppose CLKG (Fig. 8(5) to be a cylinder witli a volume of gas C Ghc behind the piston. Let c h indicate the Figr. ss. pressure before expansion begins. If the piston be forced ahead by the expanding gas until the volume is doubled, the pressure will drop, by Boyle's law, to one -half, and will be indicated by td. For three volumes the pressure becomes v f, etc. The curve c'sx is an hyperbola, of the form called equilateral, or rectangular. Suppose the cylinder were infinite in length. Since we cannot conceive a volume so great that it could not be doubled, or a pressure so small that it could not be halved, it is evident that theoretically the curve c x and the line G K will forever approach each other yet never meet; that is, they will be tangent at infinity. In such a case the straight line is called an asymptote to the curve. 135. Although the right cone (i. e., one having its axis perpendicular to the plane of its base) is usually employed in obtaining the ellipse, hyperbola and parabola, yet the same kind of sections can be cut from an oblique or scalene cone of circular base, as V. A B, Fig. 87. Two sets of circular sections can also be cut from such a cone, one set, obviously, by planes i)arallel to the base, while the other would be by planes like CD, making the same angle with the lowest element, V B, that the highest element, V A, makes with the base. The latter sections are called sub -contrary. Their planes are perpendicular to the plane VA B containing the highest and lowest elements — principal plane, as it is termed. To prove that the sub -contrary section xy is a. circle we note that both it and the section m n — the latter known to be a circle because parallel to the base — intersect in a line perpendicular to the paper at o. This line pierces the front surface of the cone at a point we may call r. It would be seen as the ordinate o r (Fig. 88), were the front half of the circle m n rotated until parallel to the paper. Then or'' = omxon. But in Fig. 87 we have om: oy ::ox:o n, whence oy X ox = om x on^^^or'', proving the section x y circular. s V ) d \e i/ S, and «S'.^ it must contain the horizontal trace, 0, of the line joining .S'.^ with S,. But this puts A.^ and A^ into \\\e same relation with that A.^ and A^ sustain to S.^; or that of A^ and ^i to iS',. Again, A^ c„ is the trace, on the vertical plane, of any plane containing A ' li '. This plane cuts the "axis of liomology," t^in, in Cj. As A^Bi lies in the plane of )S\ and A^ B^, and in the horizontal plane as well, it can only meet the vertical jjlane in c„, the point of intersection of all these planes. Sim- ilarly we find that A' C" and Ai C, , if pro- longed, meet the axis at the point /%; cor- respondingly 5' C" and 7^1 Ci meet at a,. But yl'/i' and A.^B.^, Ijeing corresponding lines, lie in the plane with S.^, though belonging to figures in two other planes; they must, there- fore, meet also at the same point, c„; and similarly for the other lines in the figures \ised with S.^. 146. Were A^B^C^ a circle, and all its points joined witli S^, the figure A^ B^ C would obviously be an ellipse; equally so were A^B.^C^ a circle used in connection with S.^. We may, therefore, F^Af- I 1»^ ^ 48 THEORETICAL AND PRACTICAL GRAPHICS. substitute a circle for A^B^C^, and using on the same plane with it get an ellipse in place of the triangle A^B^C^. Before illustrating this it is necessary to show the relation of the axis to the other elements of the problem, and supply a test as to the nature of the conic. 147. First as to the axis, and employing again for a time a space figure (Fig. 93), it is evident that raising or lowering the horizontal ])lane cXY parallel to itself, and with it, necessarily, the axis, would not alter the hind of curve that it would cut from the cone H. II A B, were the elements of .the latter ])rolonged. But raising or lowering the centre S, while the base circle AHBt remained, as before, in the same place, would decidedly aff'ect the curve. Where it is, there are two elements of the cone, SA and SB, which would never meet the plane c X Y. The shaded plane containing those elements meets the vertical plane in " vanishing line (a)," parallel to the axis. This contains the projections, A and B, of the points at infinity where the lower plane may be considered as cutting the elements SA and SB. \\'ere S and the shaded plane raised to the level of H, making "vanishing' line (a)" tangent to the base, there would be one element, S H, of the cone, parallel to the lower plane, and the section of the cone by the latter would be the parabola; as it is, the hyperbola is indicated. Tlie former would have but one point at infinity; the latter, two. 148. Raise the centre S so that the vanishing line does not cut the base, and evidently no line from S to the base would be parallel to the lower plane; but the latter would cut all the elements on one side of the vertex, giving the ellii)se. 149. Bearing in mind that the projection of the circle AHBt is on the lower i)lane produced, if we wish to bring both these figures and the centre S into one plane without destroying the relation between them, we maj' imagine the end plane Q LX removed, the rotation of the remaining system Fig-, ©-i. VANISHING UNE ( a J occurring about cr^ in a manner exactly similar to that which would occur were iojc a .system of four pivoted links, and the point o jiressed toward c. The motion of (S would be parallel and equal CONIC SECTIONS AS HOMOLOGOUS FIG URES. — RELIEF-PERSPECTIVE. 49 to that of 0, and, like the latter, S would evidently maintain its distance from the vanishing line and describe a circular arc about it. The vanishing line would remain parallel to the axis. 150. From the foregoing we see that to obtain the hyperbola, by projection of a circle from a point in the plane of the latter, we would re()uire simply a secant vanishing line, MN (Fig. 94), and an axis of homology parallel to it. Take any point P on the vanishing line and join it with any point K of the circle. PK meets the axis at y; hence whatever line corresponds to P K must also meet the axis at y. P is analogous to SA of Fig. 93, in tliat it meets its corresponding line at infinity, i.e., is ])arallel to it. Therefore yk, parallel to OP, corresponds to Py, and meets the ray K at k, corresponding to K. Then K joined with any other point R gives K z. Join z with k and 2>r<>l"iig R to intersect k z, obtaining r, another point of the hyperbola. 151. In Fig. 98, were a tangent drawn to arc A H B at B it would meet the axis in a point which, like all points on the axis, ''corresponds to itself" From that point the projection of that tangent on the lower plane would be parallel to SB, since they are to meet at infinity. Or, if SJ is parallel to the tangent at B, then / will be the projection of /' at infinity, where >S'/ meets the tangent; ./ will be therefore one j)oint of the projection of said tangent on the lower plane: while another point would be, as previously stated, that in which the tangent at B meets the &xia, 152. Analogously in Fig. 94, the tangents at M and N meet the axis, as at F and E; but the projectors OM and ON go to jxnnts of tangency at infinity; M and N are on a "vanishing line"; hence OM is ]>arullel to tlie tangent at infinity, that is, to tlie asymptote (see Art. 134) through F; while the other asymptote is a parallel through E to ON. 153. As in Fig. 93 the projector.^ from .S' to all points of the arc above the level of ,S' could cut the lower j)lane only ])y being produced to the right, giving the right-hand branch of the hyj)er- bola; so, in Fig. 94, the arc M H N, above the vanishing line, gives the lower branch of the hyperbola. To get a point of the latter, as h, and having already obtained any point x of the other branch, join H with X (the original of ») and get its intersection, g, with the axis. Then xgh corresponds to g X H, and the ray H meets it at h, the projection of H. The cases should be worked out in which the vanishing line is tangent to the circle or exterior to it. 154. The homological figures with which wo have been dealing were plane figures. But it is possible to have space figures homological with each other. In homological space figures corresponding lines meet at the same point in a plane, instead of the same point on a line. A vanishing plane takes the place of a van- isliing line. The figure that is in homology with the original figure is called the relief- perspective of the latter. (See Art. 11.) Remarkably beautiful effects can lie ol)tained by the construction of homological space figures, as a glance at Fig. 95 will show. The figure represents a triple row of groined arches, and is from a photograph of a model designed by Prof L. Burmester. Although not always reciuiring the use of the irregular curve and therefore not strictly the material for a topic in this chapter, its close analogy to the foregoing matter may justify a few words at this point on the construction of a relief- perspective. 155. In Fig. 9G the ])lane PQ is called the plane of homology or picture -plane, and— adopting Cremona's notation— we will denote it by tt. The vanishing plane M N, or <^', is parallel to it. is Fig-- ©5. 50 THEORETICAL AND PRACTICAL GRAPHICS. the centre of homology or 'perspective -centre. All jjoints in the plane t are their own i^erspectives, or, in other words, coirespond to themselves. Therefore B" is one point of the projection or perspective of the line A B, being the intersection of A B with ir. The line r, parallel to A B, would meet the x-ig-. ©s. ''/I latter at infinhji ; therefore r, in the vanisliing ])lane <^', would be the projection upon it of the point at infinity. Joining r witli B", and cutting r B" by rays OA and OB, gives A' B' as the relief- perspective of A B. The j)lane through and A B cuts ir in B" n, which is an axis of homoloijy for AB and A' B', exactly as mn in Fig. 92 is for .1,7?, and A.,B.,. ■FLs- ST. ,--,-^''' -^C.5v\ ! \ je \'\ '"- a// -t :::E I / ~xh| '% As Z) C in Fig. 96 is parallel to A B, & parallel to it througli is again the line Ov. LINK-MOTION CURVES. 51 The trace of DC on tt is C". Joining v witli C" and cutting v C" by rays D, C, obtains D' C in tlie same manner as A' B' was derived. The originals of A' B' and CD' are parallel lines; but we see that their relief- perspectives meet at r. The vanishing plane is therefore the locus* of the vanishing points of lines that are f)arallel on the original object, while the plane of homology is the locus of the axes of homology of corresponding lines; or, differently stated, any line and its relief- perspective will, if produced, meet on the plane of homology. 156. Fig. 97 is inserted here for the sake of completeness, although its study may be reserved, if necessary, until the chapter on projections has been read. In it a solid object is represented at the left, in the usual views, plan and elevation; GL being the grorind line or axis of intersection of the planes on which the views are made. The planes tt and <^' are interchanged, as compared with their positions in Fig. 96, and they are seen as lines, being assumed as perpendicular to the paper. The relief- perspective appears between them, in plan and elevation. The lettering of A B and D C, and the lines emplo3-ed in getting their relief- i)erspectives, being identical with the same constructions in Fig. 96, ought to make the matter clear at a glance to all who have mastered what has preceded. Burmester's Gnmcktu/e der Relief - I'er»pective and Wiener's Dardellende Geometn'e are valuable reference works on this topic for those wishing to pursue its study further; but for special work in the line of homological plane figures the student is recommended to read Cremona's Projective Gemnetry and Graham's Geometry of Position, the latter of which is especially valuable to the engineer or architect, since it illustrates more fully the practical application of central projection to Graphical Statics. LINK - MOTION CURVES 157. Kinematics is the science which treats of pure motion, regardless of the cause or the results of the motion. It is a purely kinematic problem if we lay out on the drawing-board the path of a point on the connecting-rod of a locomotive, or of a point on the piston of an oscillating cylinder, or of any point on one of the moving pieces of a mechanism. Such problems often arise in machine design, especially in the invention or modification of valve -motions. Some of the motion-curves or point-paths that are discovered by a study of relative motion are without special name. Others, whose mathematical properties had already been investigated and the curves dignified with names, it was later found could be mechanically traced. Among these the most familiar examples are the Ellipse and the Lemniscate, the latter of which is employed here to illustrate the general problem. The moving pieces in a mechanism are rigid and inextensible, and are always under certain conditions of restraint. "Conditions of restraint" may be illustrated by the familiar case of the con- necting-rod of the locomotive, one end of which is always attached to the driving-wheel at the crank-i)in and is therefore constrained to describe a circle about the axle of that wheel, while the other end of the rod must move in a straight line, being fastened l)y the "wrist-pin" to the "cross- head," w;hich slides between straight "guides." The first step in tracing a point-path of any mechanism is therefore the detennination of the fixed points, and a general analysis of the motion. * Locus Is the Latin for place: and In rather nntechnlcal language, aUhough in the exact sense in which it is used mathe- matically, we may say that the torus of points or lines is the place where you may expect to find them xinder their coiiditious of rbstiiction. t'ov example, the surface of a sphere is the locus of all points equidistant from a fixed point (Its centre). The locus of a point moving in a plane so as to remain at a constant distance from a given fixed point, is a circle having the latter point as its centre. 52 THEORETICAL AND PRACTICAL GRAPHICS. 158. We have t;iven, in Fig. 98, two linka or bars, MN and S P, fastened at X and P by pivots to a third link, X P, while their other extremities are pivoted on dutionary axes at M and S. The only movement possible to the point X is therefore in a circle about M; while 7* is equally limited to circular motion about .S. The points on the link A'^7', with the exception of its Fig-. SS. 2 MN _2 MS THE LEMNISCATE AS A LINK-MOTION CURVE — r V^ ^U extremities, have a compound motion, in curves whose form it is not easy to predict and which differ most curiously from each other. The figure-of-eight curve shown, otherwise the "Lemniscate of Bernoulli," is the point-path of Z, tlie link XP being supposed prolonged by an amount, P Z, equal to NP. Since XP is constant in length, if X were moved along to F, the point P would have to be at a distance XP from F and also on the circle to which it is confined; therefore its new position /, is at the intersection of the circle P«r by an arc of radius P X, centre F. Then Ff prolonged ])y an amount equal to itself, gives /, , another point of the Lemniscate, and to which Z has then moved. All other positions are similarly found. If the motion of X is toward D it will soon reach a limit. A, to its further movement in that direction, arriving there at the instant that P reaches n, when NP and PS will be in one straight line, SA. In this position any movement of P either side of a will drag X back over its former path; and unless /' moves to the left, past a, it would also retrace its path. P reaches a similar "dead point" at v. To obtain a Lenniiscate the links XP and PS had to be equal, as also the distance MS to MX. By varying the proportions of the links, the point- paths would be correspondingly affected. INST A NT A NEO US CENTRES. — CENTR OlDS. 53 By tracing the i)ath of a point on PN produced, and as for from N as Z is from /', the student will obtain an interestinj; contrast to the Lemniscate. If M and S were joined by a link, and the latter held rigidly in position, it would have been called the fixed link; and although its use would not have altered the motions illustrated, and it is not essential that it should be drawn, yet in considering a mechanism as a whole, the line joining the fixed centres always exists, in the imagination, as a link of the complete system. INSTANTANEOUS CENTRES. — CENTROIDS. 159. Let us imagine a boy about to hiu'l a stone from a sling. Just before ho releases it he runs forward a few steps, as if to add a little extra impetus to the stone. While taking those few steps a peculiar shadow is cast on the road bj' the end of the sling, if the day is bright. The boy moves with respect to the earth; his hand moves in relation to himself, and the end of the sling describes a circle about his hand. The last is the only definite element of the three, yet it is sufficient to simplify otherwise difficult constructions relating to the complex curve which is described relatively to the earth. 54 THEORETICAL AND PRACTICAL GRAPHICS. A tangent and a normal to a circle are easily obtained, the former being, as need hardly be stated at this point, perpendicular to tlie radius at the point of tangency, while the normal simply coincides in direction with such radius. If the stone were released at any instant it W(juld fly off in a straiglit line, tangent to the circle it was describing ahout the hand as a centre; but such line would, at the instant of release, be tangent also to the compound curve. If, then, we wish a tangent at a given point of any curve generated by a point in motion, we have but to reduce that motion to circular motion about some moving centre; then, joining the point of desired tangency with the — at that instant — position of the moving centre, we have the norjnnl, a perpendicular to which gives the tangent desired. A centre which is thus used for an instant only is called an instantaneous centre. 160. In Fig. 99 a series of instantaneous centres are shown and an im])ortant as well as inter- esting fact illustrated, viz., that every moving piece in a mechanism might be rigidly attached to a certain curve, and by the. rolling of the latter upon another curve the link might be brought into all the positions which its visible modes of restraint compel it to take. IBl. In the "Fundamental"' ])art of Fig. 99 A li is assumed to l)e one position of a link. We next find it, let us suppose, at A' B', A having moved over A A', and B over B B'. Bisecting A A' and B B' by perpendiculars intersecting at 0, and drawing A, OA', OB and OB', we have A A' = 6^^:^ B O B', and evidently a point about which, as a centre, the turning of AB through the angle ^, would have brought it tw A' B'. Himilarl}', if the next position in which we find AB is A" B", we may find a point ■•i as the centre about which it might have turned to bring it there ; the angle being 0.^ , probably different from 0^ . X and m are analogous to and .s. If Os' be drawn equal to Os and making with the latter an angle 6,, equal to the angle A O A' , and if Os were rigidly attached to A B, the latter would be brought over to A' B' by bringing s into coincidence with s. In the same manner, if we bring / n' upon s n thrcjugh an angle d.^ about s, then the next i)osition. A" B", wojild be reached by A B. 0' s' n' m' is then part of a polygon whose rolling uj)on Osnm would bring AB into all the positions shown, provided the polygon and the line were so attached as to move as one piece. Polygons whose vertices are thus obtained are called central polygons. If consecutive centres were joined we would have curves, called centroids*, instead of polygons; the one corresponding to s n ni being called the fixed, the other the rolling centroid. The perpen- dicular from upon A A ' is a normal to that path. But were ..4 to move in a circle, the normal to its path at any instant would be simply the radius to the position of A at that instant. If, then, both A and B were moving in circular paths, wo would find the instantaneous centre at the intersection of the normals (radii) at the pt)ints A and B. 162. In Fig. 98 the instantaneous centre about which the whole link NP is turning is at the intersection of radii MX and SP (produced); and calling it X we would have XZ for the normal at Z to the Lenniiscate. 163. The shaded portions of Fig. 99 illustrate some of the forms of centroids. The mechanism is of four links, ojjposite links equal. Unlike the usual quadrilateral fulfilling this condition, the long sides cross, hence the name "anti- parallelogram." The "fixed link (a)" corresponds to MS of Fig. 98, and its extremities are the centres of rotation of the short links, whose ends, / and /,, describe the dotted circles. For the given position T is evidently the instantaneous centre. Were a bar pivoted at T and *KcnIeaux' nomenclature; also called cenlrodes by a number of wrltei-s on Kinematics. TROCHOIDS. 55 fastened at right angles to "moving link (a)," an hijinitedmal turning about T would move "link (a)" exact!}' as under the old conditions. B}' taking "link (a)" in all possible positions, and, for each, prolonging the radii through its extremities, the points of the fixed centroid are determined. Inverting the combination so that "moving link (a)" and its opposite are interchanged, and proceeding as before, gives the points of •'rolling centroid (a)." These centroids are branches of hyperbolas having the extremities of the long links as foci. By holding a short link stationary, as "fixed link (b)," an elliptical fixed centroid results; "rolling centroid (b)" being obtained, as before, by inversion. The foci are again the extremities of the fixed and moving links. Obviously the curved pieces represented as screwed to the links would not be employed in a practical construction, and they are only introduced to give a more realistic effect to the figure and possibly thereby conduce to a clearer understanding of the subject. 164. It is interesting to notice that the Lemniscate occurs here under new conditions, being traced by the middle point of "moving link (a)." The study of kinematics is both fascinating and profitable, and it is hoped that this brief glance at the subject may create a desire on the part of the student to pursue it further in such works as Reauleaux' Kinematics of Machinery and Bunnester's Lehrbuch der Kinematik. 165. Before leaving this topic the important fact should be stated, which now needs no argument to establish, that the instantaneous centre, for any position of a moving piece, is the point of contact of the rolling and fixed centroids. We shall have occasion to use this principle in drawing tangents and normals to the TROCHOIDS I which are the principal Roulettes, or roll-traced curves, and which may be defined as follows: — If, in the same plane, one of two circles roll upon the other without sliding, the path of any point on a radius of the rolling circle or on the radius produced is a trochoid. 166. The Cycloid. Since a straight line may be considered a circle of infinite radius, the above definition would include the curve traced by a point on the circumference of a locomotive wheel as it rolls along the rail, or of a carriage wheel on the road. This curve is known as a cycloid* and is shown in Tnahc, Fig. 100. It is the proper outline for a portion of each tooth in a certain case of gearing, viz., where one wheel has an infinite radius, that is, becomes a " rack." Were T^ a ceiling -corner of a room, and r,^ the diagonally opposite floor -corner, a weight would slide from T, to T,, more quickly on guides curved in cycloidal shajie than if shaped to any other curve, or if straight. If started at c, or any other point oi the curve, it would reach T^ as soon as if started at 7'^. 167. In beginning the construction of the cycloid we notice, first, that as T V D rolls on the straight line A B, the arrow D RT will be reversed in position (as at D^ T^) as soon as the semi- circumference T8D has had rolling contact with A B. The tracing point will then be at T^, its maximum distance from A B. When the wheel has rolled itself out once upon the rail, the point T will again come in contact with the rail, as at T,.^. ♦"Although the invention of the cycloid l8 attributed to Galileo, it is certain that the familj- of curves to which It belongs had been Icnown and some of the properties of such curves investigated, nearly two thousand years before Galileo's time, it not earlier. For ancient astronomers explained the motion of the planets by supposing that each planet travels uniformly round a circle whose centre travels uniformly around another circle."— Proctor, Oeometry of Cycloids. 56 THEORETICAL AND PRACTICAL GRAPHICS. The distance TT^^ evidently equals ^vr, when r^^TR. We also have TD^^=D^T^^^=^irr. If the semi-circumference TZD (equal to vr) be divided into any number of equal parts, and also the path of centres RR^ (again = 5rr) into the same number of equal parts, then as the points 1, 2, etc., come in contact with the rail, the centre R will take the positions R^,R^, etc., directly above the corresponding points of contact. A sufficient rolling of the wheel to bring point 2 upon A B would evidently raise T from its original position to the former level of 2. But as T must always be at a radius' distance from R, and the latter would by that time be at R,, we would find T located at the intersection (n) of the dotted line of level through 2 by an arc of radius R T, centre R,. Similarly for other points. The construction, summarized, involves the drawing of lines of level through equidistant points of division on a semi-circumference of the rolling circle, and their intersection by arcs of constant radius (that of the rolling circle) from centres which are the successive positions taken by the centre of the rolling circle. It is worth while calling attention to a point occasionally overlooked by the novice, .although almost self-evident, that, in the position illustrated in the figure, the point T drags behind the centre R until the latter reaches R^, when it passes and goes ahead of it. From i?, the line of level through 5 could be cut not alone at c by an arc of radius cR^ but also in a second point; evidently but one of these points belongs to the cycloid, and the choice depends upon the direction of turning, and upon the relative position of the rolling centre and the moving point. This matter requires more thought in drawing trochoidal curves in which both circles have finite radii, as will appear later. x-igr- loo. THE CYCLOID. 168. Were points T^ and T^, given, and the semi -cycloid Tg 7',.2 desired, we can readily ascertain the "base," A B, and generating circle, as follows: Join T", with T,./, at any ])oint of such line, as X, erect a perpendicular, xy; from the similar triangles xyT„ and T^D.^T,,, having angle <^ common and angles 6 equal, we see that xy:xT,,::T^D^:D^T„::2r:-!rr::2:-!r::l:^^; or, very nearly, as 14:22. If, then, we lay off xT^^ equal to twenty -two equal parts on any scale, and a perpendicular, xy, fourteen parts of the same scale, the line y T^^ will be the base of the desired curve ; while the diameter of the generating circle will be the perpendicular from T^ to y r,^ prolonged. 169. To draw the tangent to a cycloid at any point is a simple matter, if we see the analog}' between the point of contact of the wheel and rail at any instant, and the hand used in the former illustration (Art. 159). At any one moment each point on the entire wheel may be considered as describing an infinitesimal arc of a circle whose radius is the line joining the point with the point of contact on the rail. The tangent at N, for example, (Fig. 100), would be t X, perpendicular to the nonnal, No, joining N with o; the latter point being found by using TV as a centre and THE CYCLOID.— COMPANION TO THE CYCLOID. 57 cutting AB by an arc of radius equal to m I, in which m is a point at the level of N on any position of the rolling circle, while I is the corresponding point of contact. The point o might also have been located by the following method: Cut the line of centres by an arc, centre N, radius TR; would obviously be vertically below the position of the rolling centre thus determined. 170. The Companion to the Cycloid. The kinematic method of drawing tangents, just applied, was devised by Roberval, as also the curve named by him the "Companion to the Cycloid," to which allusion has already been made (Art. 120) and which was invented by him in 1634 for the purpose of solving a problem upon which he had spent six years without success, and which had foiled Galileo, viz., the calculating of the area between a cycloid and its base. Galileo was reduced to the expedient of comparing the area of the cycloid with that of the rolling circle by weighing paper models of the two figures. He concluded that the area in question was nearly but not exactly three times that of the rolling circle. That the latter would have been the correct solution may be readily shown by means of the "Companion," as will be found demonstrated in Art. 172. 171. Suppose two points coincident at T (Fig. 101) and starting simultaneously to generate curves, the first of these points to trace the cycloid during the rolling of circle TVD, while the second is to move independently of the circle and so as to be always at the level of the point tracing the cycloid, yet at the same time vertically above the point of contact of the circle and base. This makes the second point always as far from the initial vertical diameter, or axis, of the cycloid, as the length of the arc from T to whatever level the tracing point of the latter has then reached; that is, MA equals arc THa; RO equals quadrant Tsy. Adopting the method of Analytical Geometry, and using as the ongin, we may reach any point. A, on the curve, by co-ordinates, as Ox, xA, of which the horizontal is called an abscissa, the vertical an ordinate. By the preceding construction Ox equals arc sfy, while xA equals sw — the sine of the same arc. The "Companion" is therefore a curve of sines or sinusoid, since, starting from 0, the abscissas are equal to or proportional to the arc of a circle, while the ordinates are the sines of those arcs. It is also the orthographic projection of a 45° -helix. This curve is particularly interesting as "expressing the law of the vibration of perfectly elastic solids; of the vibratory movement of a particle acted upon by a force which varies directly as the distance from the origin; approximately, the vibratory movement of a pendulum; and exactly the law of vibration of the so-called mathematical pendulum."* (See also Art. 356). 172. From the symmetry of the sinusoid with respect to R R^ and to 0, we have area TAOR=ECOR^; adding area D E L R to both mem- bers we have the area between the sinusoid and TD and D E equal to the rectangle R E, or one-half the rect- angle D E K T; or to ^-n-r x 2r = ■trr*, the area of the rolling circle. As T AC E IB but half of the entire sinusoid it is evident that the total area below the curve is twice that of the generating circle. The area between the cycloid and its "companion" remains to be determined, but is readily ascertained by noting that as any point of the latter, as A, is on the vertical diameter of the circle * Wood, Elements of Co-ordinate Geometry^ p. 209. x-igr- ioi. 58 THEORETICAL AND PRACTICAL GRAPHICS. passing through the then position of the tracing point, as a, the distance, A a, between tlie two curves at any level, is merely the semi-chord of the rolling circle at that level. But this, evidently, equals Ms, the semi-chord at the same level on the equal circle. The equality of Ma and A a makes the elementary rectangles Mss^m^ and AA^a^a equal; and considering all the possible similarly - constructed rectangles of infinitesimal altitude, the sum of those on semi-chords of the rolling circle would equal the area of the semi-circle TDy, which is therefore the extent of the area between the two curves under consideration. The figure showing but half of a cycloid, the total area between it and its "companion" must be that of the rolling circle. Adding this to the area between the "companion" and the base makes the total area between cycloid and base equal to three times that of the rolling circle. 173. The paths of points carried by and in the plane of the rolling circle, though not on its circumference, are obtained in a manner closely analogous to that emj^loyed for the cycloid. In Fig: 102 the looped curve, traced hy the arrow-point while the circle CHM rolls on the base A B, is called the Curtate Trochoid. To obtain the various positions of the tracing point T describe a circle through it from centre R. On this circle lay off any even number of equal arcs, and draw radii from R to the points of division; also "lines of level" through the latter. The radii drawn intercept equal arcs on the rolling circle CHM, whose straight equivalents are next laid off on the path of centres, giving R^, R.^, etc. While the first of these arcs rolls upon AB the point T turns through the angle TR 1 about R, and reaches the line of level through point 1. But T is always at the distance R T (called the tracing radius) from R; and, as R has reached i?, in the rolling supposed, we will find J", — the new position of T — by an arc from R^, radius TR, cutting said line of level. After what has preceded, the figure may be assumed to be self-interpreting, each position of T having been joined with the position of R which determined it. 174. Were a tangent wanted at any point, as T^, we have, as before, to determine the point of contact of rolling circle and line when T reached T,, and use it as an instantaneous centre. 3', was obtained from R.; and the point of contact must have been vertically below the latter and on A B. Joining such point to IT, gives the normal, from which the tangent follows in the usual way. 17.5. The Prolate Trochoid. Had we taken a point inside of the circle CHM and constructed its path the only difference between it and the curve illustrated would have been in the name and the HYPO-, EPl- AND PERl-TROCHOIDS. 69 shape of the curve. An undulating, wavy jjath would have resulted, called the prolate trochoid; but, as before, we would liave described a circle through the* tracing point; divided it into equal parts; drawn lines of level, and cut them by arcs of constant radius, using as centres the successive positions of R. A bicycle pedal describes a prolate trochoid relatively to the earth. HYPO-, EPI- AND PERI - TROCHOIDS. 176. Circles of finite radius can evidently be tangent in but two ways — either externally, or internally; if the latter, the larger may roll on the one within it, or the smaller may roll inside the larger. When a small circle rolls within a larger the radius of the latter may be greater than the diameter of the rolling circle, or may equal it, or be smaller. On account of an interesting property of the curves traced by points in the planes of such rolling circles, viz., their capability of being generated, trochoidally, in two ways, a nomenclature was necessary' which would indicate how each curve was obtained. This is included in the tal)ular arrangement of names below and which was the outcome of an investigation* made by the writer in 1887 and presented before the American Association for the Advancement of Science. In accei)ting the new terms, advanced at that time. Prof Francis Reuleaux suggested the names Ortho-cycloids and Cyclo - orthoids for the classes of curves of which the cj'cloid and involute are respectively representative; orthoids being the paths of points in a fixed position with respect to a straight line rolling upon any curve, and cyclo - nrthoid therefore implying a circular director or base- curve. These approj)riate terms have been incorj)orated in the table. For the last colunni a point is considered as within tlie rolling circle of infinite radius when on the normal to its initial position and on the side toward the centre of the fixed circle. As will be seen by reference to the Appendix, the curves whose names are j^receded by the same letter may be identical. Hence the terms curtate, and prolate, while indicating whether the tracing jxnnt is beyond or within the circumference of the rolling circle, give no hint as to the actual j'orm of the curves. In the table, R re])resents the radius of the rolling circle, F that of the fixed circle. NOMENCLATURE OF TROCHOIDS. position of Circle rolling Circle rolling upon circle. Straight Line rolling upon Circle. R=oo Tracing upon Straight Line. F=oo External contact. Internal contact. Larger Circle rolling. Smaller circle rolling. describing Kpitrochoids. 2 R>F. 2 R, and 2 ((9 — <^)-|-a:-|- <^:=180°, which gives x=2 by substituting the value of 6 from the previous equation. 186. The Involute. As the opposite extreme of a circle rolling on a straight line we may have the latter rolling on a circle. In this case the rolling circle has an infinite radius. A point on the straight line describes a curve called the involute. This would be the path of the end of a thread if the latter were in tension while l)eing unwound from a spool. In Fig. 107 a rule is shown, tangent at m to a circle on which it is supposed to roll. Were a pencil -point inserted in the centre of the circle at j (which is on the line ux produced) it would trace the involute. When j reaches a the rule will have had rolling contact with the base circle over an arc uts---a whose length equals line uxj. Were a the initial point we would obtain, b, c, 64 THEORETICAL AND PRACTICAL GRAPHICS. etc., by making tangent mb=^arc ma; tangent nc=arc na. Each tangent thus equals the arc from the initial point to the point of tangency. 187. The circle from which the involute is derived or evolved is called the evolute. Were a hexagon or other figure to be taken as an evolute a corresponding involute could be derived; but the name "involute," unqualified, is understood to be that obtained from a circle. From the law of formation of the involute the rolling line is in all its positions a normal to the curve; the point of tangency on the evolute is an instantaneous centre, and a tangent at any point, as /, is a perpendicular to the tangent, fq, from / to the base circle. Like the cycloid, the involute is a correct working outline for the teeth of gear-wheels; and gears manufactured on the involute system are to a considerable degree supplanting other forms. A surface known as the developable helicoid (see Figs. 209 and 270) is formed bj' moving a line x-igr- X07. so as to be always tangent to a given helix. It is interesting in this connection to notice that any plane perpendicular to the axis of the helix would cut such a surface in a pair of involutes.* 188. The Spiral of Archimedes. This curve is generated by a point having a uniform motion around a fixed point — the pole — combined with uniform motion toward or from it. In Fig. 107, with as the pole, if the angles 6 are equal, and D, OE and Oy, are in arith- metical progression, then the points D, E and y^ are points of an Archimedean Spiral. This spiral can be trochoidally generated, simultaneously with the involute, by inserting a pencil point at y in a piece carried by — -and at right angles with — the rule, the point y being at a distance, •The day of writing the above article the foUowing item appeared in the New York Evening Post: "Visitors to the Royal Observatory, Greenwich, will hereafter miss the great cylindrical structure which has for a quarter century and more covered the largest telescope possessed by the Observatory. Notwithstanding its size the Astronomar Koyal has now procured through the Lords Commissioners a telescope more than twice as large as the old one.... The optical peculiarities embodied in the new Instrument will render it one of the three most powerful telescopes at present in existence.... The peculiar architectural feature of the building which is to shelter the new telescope is that its dome, of thirty-six feet diameter, will surmount a tower having a diameter of only thirty-one feet. Technically, the form adopted is the surface generated by the revolution of an involute of a circle." SPECIAL TROCHOIDS. 65 E'ig-. loe. xy, from the contact-edge of the rule, equal to the radius Os of the base circle of the involute; for after the rolling of ux over an arc ut we shall have tx, as the portion of the rolling line between X and the point of tangency, and xy will have reached x^y^. If the rolling be continued y will evidently reach 0. We see that Oy = ux, and Oyt = tx^; but the lengths ux and tx^ are propor- tional to the angular movement of the rolling line about 0, and as the spiral may be defined as that curve in which the length of a radius vector is directly proportional to the angle through which it has turned about the pole, the various positions of y are evidently points of such a curve. 189. A Tangent to the Spiral of Archimedes. Were the pole, 0, given, and a portion only of the spiral, we could draw a tangent at any point, y,, by determining the circle on which the spiral could be trochoidally generated, then the instantaneous centre for the given position of the tracing- point, whence the normal and tangent would be derived in the usual way. The radius 1 of the base circle would equal wy — the difference between two radii vectores Oy and Oz which include an angle of 57° 29+, (the angle which at the centre of a circle subtends an arc equal to the radius). The instantaneous centre, t, would be the extremity of that radius which was perpendicular to Oy,. The normal would be iy,, and the tangent TT^ perpendicular to it. 190. The spiral of Archimedes is the right section of an oblique helicoid. (Art. 357). It is also the proper outline for a cam to convert uni- form rotary into uniform rectilinear motion, and when combined with an equal and oppo- site spiral gives the well-known form called the heart- cam. As usually constructed the act- ing curve is not the true spiral, but a curve whose points are at a constant distance from the theoretical outline equal to the radius of the friction -roller which is on the end of the piece to be raised. Qs^ (Fig. 107) is a small portion of such a "parallel curve." 191. If a point travel on the surface of a cone so as to combine a uniform motion around the axis with a uniform motion toward the vertex it will trace a conical helix, whose orthographic projection on the plane of the base will be a spiral of Archimedes. In Fig. 108 a top and front view are given of a cone and helix. The shaded por- tion is the development of the cone, that is, the area equal to the convex surface, and which — if rolled up — would form the cone. To obtain the development draw an arc A'G"A" of radius equal to an element. The convex surface of the cone will then be repre- sented by the sector A'O'A", whose angle may be found by the proportion A iff A' :: 6:360°, since the arc A'G"A" must equal the entire circumference of the cone's base. The student can make a paper model of the cone and helix by cutting out a sector of a circle, 66 . THEORETICAL AND PRACTICAL GRAPHICS. making allowance for an overlap on which to put the mucilage, as shown by the dotted lines O'y and yvz in the figure. The development of a conical helix is the same kind of spiral as its orthographic projection. PARALLEL CURVES. 192. A parallel curve is one whose points are at a constant normal distance from some other curve. Parallel curves have not the same mathematical properties as those from wliich they are derived, except in the case of a circle; this can readily be seen from the cam figure under the last heading, in which a point, as (S, , of the true spiral, is located on a line from which is by no means in the direction of the normal to the curve at iS',, ui)on which lies tlie point -S'^ of the parallel curve. Instead of actually determining the normals to a curve and on each laying off a constant distance, we maj' draw many arcs of constant radius, having their centres on the original curve; the desired parallel Avill be tangent to all these arcs. In strictly mathematical language a parallel curve is the envelope of a circle of constant radius whose centre is on the original curve. We may also define it as the locus of consecutive inter- sections of a system of equal circles having their centres on the original curve. If on the convex side of the original the jmrallel will resemble it in form, but if within, the two may be totally dissimilar. This is well .illustrated in Fig. 109, in which the parallel to a Lemniscate is shown. The student will obtain some interesting resuHs by constructing the parallels to ellipses, parabolas and other plane curves. THE CONCHOID OF NICOMEDES. 193. The Conchoid, named after the Greek word for shell,' may be obtained by laying off a con- stant length on each side of a given line M N (the directrix) upon radials through a fixed point or pole, (Fig. 110). If mv^mn = sx then v, n and x are points of the curve. Denote by a the distance of from MN, and use c for the constant length to be laid off; then if a<.c there will be a loop in that branch of the curve which is nearest the pole, — the inferior branch. If a = c the curve has a point or cusp at the pole. When a>c the curve has an undulation or wave -form towards the pole. • A series of curves much more closely resembling those of a shell can be obtained by tracing the paths of points on the piston-rod of an oscillating cylinder. See Arts. 167 and 168 for the principles of their construction. THE CONCHOID.— THE QUADRATRIX. 67 Ov^c+ Om; On=c — Om; we maj' therefore express the relation to of points on the curve by the equation p = cdzO m^^c±asec. I^lgr- no. 194. Mention has ah-eady been made (Art. 184) of the fact that this was one of the curves invented in part for the purpose of solving the problem of the trisection of an angle. Were mOx (or <^) the angle to be trisected we would first draw pqr, the superior branch of a conchoid having the constant, c, equal to twice Om. A parallel from m to the axis will intersect the curve at q; the angle pOq will then be one -third of 4>: for since b q=^20m we have mq=20mcos P; also mq: Om: : sin6: gin ^; hence 20mcos p-.Om: : .nn6 : sin fi, whence ■•iin0^-^2ti!n ^ cos ^=^sin2 ^ (from known trigonometric relations). The angle <^ is therefore equal to twice /8, which makes the latter one -third of angle <(>■ 195. To draw a tangent and normal at any point r we find the instantaneous centre o on the principle that it is at the intersection of normals to the paths of two moving points of a line, the distance between said points remaining constant. In tracing the curve the motion of (on Ov) is — at the instant considered — in the direction Ov; Oo is therefore the normal. The point m of Ov is at the same moment moving along M N, for which mo is the normal. Their intersection o is then the instantaneous centre, and o v the normal to the conchoid, with v z perpendicular to ov for the desired tangent. 196. This interesting curve may be obtained as a plane section of one of the higher mathemat- ical surfaces. If two non - intersecting lines — one vertical, the other horizontal — be taken as guiding lines or directrices of the motion of a third straight line whose inclination to a horizontal plane is to be constant, then horizontal planes will (ait conchoids from the surface thus generated, while every plane parallel to the directrices will cut hyperbolas. From the nature of its plane sections this surface is called the Conchoidal Hyperboloid. (See Fig. 219). THK QUADRATRIX OF niNOSTRATUS. 197. In Fig. Ill let the radius T rotate uniformly about the i;entre; simultaneously with its movement let M N have a uniform motion parallel to itself, reaching A B at the same time with radius T; the locus of the intersection of M N with the radius will be the Quadratrix. Points 68 THEORETICAL AND PRACTICAL GRAPHICS. exterior to the circle may be found by prolonging the radii while moving MN away from A B. As the intersection of M N with OB is at infinity, the former becomes an asymptote to the curve as often as it moves from the centre an additional amount equal to the diameter of the circle; the number of branches of the Quadratrix may therefore be infinite. It may be proved analytically that the curve crosses 0^ at a distance from equal to 2r-4-7r. 198. To trisect an angle, as T a, by means of the Quadratrix, draw the ordinate ap, trisect p T by s and x and draw sc and xni; radii Oc and Om will then divide the angle as desired: for by the conditions of generation of the curve the line MN takes three equi- distant parallel positions while the radius describes three equal angles. THE CISSOID OF DIOCLES. 199. This curve was devised for the purpose of obtaining two mean proportionals between two given quantities, by means of which the duplication of the cube might be effected. The name was suggested by the Greek word for ivy, since "the curve appears to mount along its asymptote in the same manner as that parasite plant climbs on the tall trunk of the pine."' This was one of the first curves invented after the discovery of the conic sections. Let C (Fig. 112) be the centre of a circle, ACE a right angle, NS and MT any pair of ordinates 'parallel to and equidistant from CE; then a secant from A through the extremity of either ordinate will meet the other ordinate in a point of the cissoid. A T and NS give P; AS and MT give Q. The tangent to the circle at B will be an asymptote to the curve. It is a somewhat interesting coincidence that the area between the cissoid and its asymptote is the same as that between a cycloid and its base, viz., three times that of the circle from which it is derived. 200. Sir Isaac Newton devised the following method of obtaining a cissoid by continuous motion: Make AV=AC; then move a right-angled triangle, of base = FC, so that the vertex F travels along *Le8Ue.' Qeometrical Analysis. 1821, THE CISSOID.—THE TRACTRIX. 69 the line DE while the edge JK always passes through V; then the middle point, L, of the base FJ, will trace a eissoid. This construction enables us readily to get the instantaneous centre and a tangent and normal; for Fn is normal to FC — the path of F, while nV is normal, to the motion of / toward / V; the instantaneous centre n is therefore at the intersection of these normals. For any other point as P we apply the same principle thus: With radius AC and centre P obtain x; draw Px, then Vz parallel to it; a vertical from x will meet Vz at the instantaneous centre y, whence the normal and tangent result in the usual way. The point y does not necessarily fall on nV. Since nV and FJ are perpendicular to JV they are parallel." So also must Vz be parallel to Px, regardless of where /' is taken. 201. Two quantities 7/1 and n will be mean proportionals between two other quantities a and b if m'^^na and n''=mb; that is, if m''=rt^6 and if n^=ab-. If 6 = 2 a we will find, from the relation m^ = a'^h, that m will be the edge of a cube whose volume equals 2 a'. To get two mean proportionals between quantities, r and b, make the smaller, r, the radius of a circle from which derive a eissoid. Were APR the derived curve we would then make Ct equal to the second quantity, b, and draw B t, cutting the eissoid at Q. A line A Q would cut off on Ct a distance Cv equal to m, one of the desired proportionals; for m' will then equal r^6, as may be thus shown by means of similar triangles: ' Cv: MQ::CA: MA whence C'^'= "j^r-" (1) Ct:MQ::CB:BM " Ct=-'^~^ (2) M Q : M A::SX:AN:: /AN.BN.A X, whence M Q = ^^ ^ ^^ ^- ^ ^ (3) From (2) we have M Q = ^ ^ (4) " (3) " " MQ-'='J^-^^iM} (5) Replacing M Q^ in equation (1) by the product of the second members of equations (4) and (5) gives Cv^ (i-e., vi^)^r''b. By interchanging 7- and b we obtain /(, the other mean proportional; or it might be obtained by constructing similar triangles having r, b and m for sides. THE TKACTRIX. « 202. The Tractrix is the involute ^)f tlie curve called the Gatenai~y (Art. 214) yet its usual con- struction is based on the fact that if a series of tangents be drawn to the curve, the portions of such tangents between the points of tangency and a given line will be of the same length ; or, in other words, the intercept on the tangent, between the directrix and the curve, will be constant. A practical and very close approximation to the theoretical curve is obtained by taking a radius Q R (Fig. 113) and with a centre a, a short distance from R on Q R, obtaining b, which is then joined with a. On a b a centre c is similarly taken for anothfer arc of the same radius, whence c c^ is obtained. A sufficient repetition of this process will indicate the curve by its enveloping tangents, or a curve may actually be drawn tiingent to all these lines. Could we take a, b, c, etc., as mathematically consecutive points the curve would be theoretically exact. The line Q S is an asymp- tote to the curve. 70 THEORETICAL AND PRACTICAL GRAPHICS. The area between the completed branch RPS and the lines QR and QS would be equal to a quadrant of the circle on radius Q R. M § -=ili.D 203. Tlie surface generated by revolving the trac- trix about its asymptote has been employed for the foot of a vertical spindle or shaft, and is known as Schiele's Anti - Friction Pivot. The step for such a pivot is shown in sectional view in the left half of the figure. Theoretically, the amount of work done in overcoming friction is the same on all equal areas of this surface. In the case of a bearing of the usual kind, for a cylindrical spindle, although the pressure on each square inch of surface would be constant, yet, as unit areas at diflcrent distances from the centre would pass over very different amounts of space in one revolution, the wear upon them would be necessarily unequal. The rationale of the tractrix form will become evident from the following Fi^. lis. consideration : If about to split a log, and having a choice of wedges, any boy would choose a thin one rather than one with a large angle, although he might not be able to prove by graphical statics the exact amount of advantage the one would have over the other. The theory is very simple, how- ^j.g.. n^,. ever, and the student may profitably be introduced to it. Suppose a ball, c, (Fig. 114) struck at the same instant by two others, a and b, moving at rates of six and eight feet a second respectively. On a c and b c prolonged take c e and c h equal, respectively, to dx and eight units of some scale; complete the iiarallogram having these lines as sides; then it is a well-known principle in mechanics* that cd — the diagonal of this parallel- ogram—will not only represent the direction in which the ball c will move, but also the distance — in feet, to the scale chosen — it will travel in one second. Evidently, then, to balance the effect of balls a and b upon c, a fourth would be necessary, moving from d toward c and traversing dc in the same second that a and b travel, so that impact of all would occur simultaneously. These forces would be represented in direction and magnitude (to some scale) by the shaded triangle c'd'e', which illustrates the very important theorem that if the three sides of a triangle — taken like c'e', e'd', d'c', in such order as to bring one back to the initial vertex mentioned — represent in magnitude and direction three forces acting on one point, then these forces are balanced. Fig. lis. Constructing now a triangle of forces for a broad and thin wedge, (Fig. 116) and denoting the force of the supposed equal blows by F in each triangle, we see that the pressures are greater for the thin wedge than for the other; that is, the less the inclination to the vertical the greater the pressure. A pivot so shaped that as the p-essure between it and its step increased the area to be traversed diminished would therefore, theoretically, be the ideal; and the rate of change of curvature of the tractrix, as its generating point approaches the axis, makes it, obviously, the correct form. *Por a demonstration the student may refer to Kankines Applied Mechanics, Art 51. UNIVERSITY THE TRACTRIX.— WITCH OF AGNES I.-CAR TESIAN OVALS. 71 204. Navigator's charts are usually made by Mercator^s projection (so-called, not being a projection in the ordinary sense, but with the extended signification alluded to in the remark in Art. 2). Maps thus constructed have this advantageous feature, that rhumb lines or Joxodromics — the curves on a sphere that cut all meridians at the same angle — are represented as straight lines, which can only be the case if the meridians are indicated by parallel lines. The law of convergence of meridians on a sphere is, that the length of a degree of longitude at any latitude equals that of a degree on the equator multiplied by the cosine (see foot-note, p. 31) of the latitude; when the meridians are made non- convergent it is, therefore, manifestly necessary that the distance apart of originally equi- distant parallels of latitude must increase at the same rate; or, otherwise stated, as on Mercator's chart degrees of longitude are all made equal, regardless of the latitude, the constant length repre- sentative of such degree bears a varying ratio to the actual arc on the sphere, being greater with the increase in latitude; but the greater the latitude the less its cosine or the greater its secant; hence lengths representative of degrees of latitude will increase with the secant of the latitude. Tables have been constructed giving the increments of the secant for each minute of latitude; but it is an interesting fact that they may be derived from the Tractrix thus: Draw a circle with radius QR, centre Q (Fig. 113); estimate latitude on such circle from R upward; the intercept on QS between consecutive tangents to the Tractrix will be the increment for the arc of latitude included between parallels to Q, Fig. 120) is constant. The consecutive intersections of refracted rays give also a caustic, which, for a circle, is the evolute of a Cartesian Oval. The proof of this statement t involves the property upon which is based the most convenient method of drawing a tangent to the Cartesian, viz., that the normal at any j)oint divides the angle between the focal radii into parts whose sines are proportional to the factors of those radii in tlie equation. If, then, we have obtained a point G on the outer oval from the relation lu p' ± np" = k c, we may obtain the tangent at G by laying off on p' and p" distances proportional to m. and n, as Gr and Gh, Fig. 118, then l)isecting rh at j and drawing the normal Gj, to whicli the desired tangent is a perpendicular. At a point on the inner oval the distance would not be laid oft' on a focal radius produced, as in the case illustrated. * American Journal of Mathematics, 1878. t Salmon. Higher Plane Curves. Art. 117. 74 THEORETICAL AND PRACTICAL GRAPHICS. CASSIAN OVALS. 212. In the Cassian OvaU or Ovals of Cassini the points are connected with two foci by the relation p'p" = k^, i.e., the product of the focal radii is equal to some perfect square. These curves have already been alluded to in Art. 114 as plane sections of the annular torus, taken parallel to its axis. Fig-. iSi. Fi.g. 123. In Art. 158 one form — the Lemniscate — receives special treatment. For it the constant k'' must equal m\ the square of half the distance between the foci. When k is less than m the curve becomes two separate ovals. 213. The general construction depends on the fact that in any semicircle the square of an ordinate equals the product of the segments into which it divides the diameter. In Fig. 122 take F^ and F, as the foci, erect a perpendicular F^S to the axis FiF^, and on it lay off F^R equal to the constant, k. Bisect F^ F^ at and draw a semicircle of radius R. This cuts the axis at A and B, the extreme points of the curve; for k.^ = FiAxFiB. Any other point T may be obtained by drawing from F^ a circular arc of radius F^ t greater than F^ A ; draw t R, then R x perpen- dicular to it; xF^ will then be the p", and F^t the p', for four points of the curve, which will be at the intersection of arcs struck from i^j and F^ as centres and with those radii. To get a normal at any point T draw T, then make angle F.^Ts^=^6 = FiT 0; Ts will be the desired line. THE CATENARY. 214. If a flexible chain, cable or string, of uniform weight per unit of length, be freely sus- pended by its extremities, the curve which it takes under the action of gravity is called a Catenary, from catena, a chain. A simjjle and practical method of obtaining a catenary on the drawing-board would be to insert two pins in the board, in the desired relative position of the points of suspension, and then attach to them a string of the desired length. By holding the board vertically the string would assume the catenary, whose points could then be located with the pencil and joined in the usual manner with the irregular curve. Otherwise, if its points are to be located by means of an equation, we take axes in the plane of the curve, the i/-axis (Fig. 123) being a vertical line through the lowest point T of the catenary, while the a; -axis is a horizontal line at a distance m below T. The quan- tity m is called the parameter of the curve, and is equal to the length of string which represents the tension at the lowest point. THE CATENARY.— THE LOGARITHMIC SPIRAL. 75 ml- — "N The equation of the catenary' is then 2/=^\^«'° + e ■"] logarithms' and has the numerical value 2.7182818 +. By taking successive values of x equal to m, 2 m, 3 m, etc., we get the following values for y: x^ m...y = — ( « + ) which for m =^ unity becomes 1 .54308 x^2m...y = ^[e + ^,j . = 4m..., = |(e^+l.) " " " " To construct the curve we therefore draw an arc of radius B = m, giving T on the axis of y as the lowest point of the curve. in which e is the base of Napierian Fig-, 123. 3.76217 10.0676 27.308 m For x=OB=^m we have i/ =5 P = 1.54308 ; for x == a = ^ we have ?/ = a m = 1.03142. The tension at any point P is equal to the weight of a piece of rope of length B P = P C + m. At the lowest point the tangent is horizontal. The length of any arc TP is proportional to the angle 6 between TC and the tangent P V at the upper extremity of the arc. 215. If a circle R L B be drawn, of radius equal to m, it may be shown analytically that tangents PS and Q R, to catenary and circle respectively, from points at the same level, will be parallel: also that PS equals the catenary -arc Pi- T; S therefore traces the involute of the catenary, and a.s S B always equals RO and remains perpendicular to PS (angle ORQ being always 90°) we have the curve TSK fulfilling the conditions of a tractrix. (See Art. 202.) If a parabola, having a focal distance m, roll on a straight line, the focus will trace a catenary having m for its parameter. The catenary was mistaken by Galileo for a parabola. In 1669 Jungius proved it to be neither a parabola nor hyperbola, but it was not till 1691 that its exact mathematical nature was known, being then established by James Bernouilli. THE LOGARITHMIC OR EQUIANGULAR SPIRAL. 216. In Fig. 124 we have the curve called the Logarithmic Spiral. Its usual construction is based on the property that any radius vector, as p, wliich bisects the angle between two other radii, OM and ON, is a mean jDroportional between them; i.e., p^ = S' = M X N. If M and G are points of the spiral we may find an intermediate point K by drawing the ordinate K to a, semicircle of diameter OM+OG; a perpendicular through G to G K will then give D, another point of the curve, and this construction may be repeated indefinitely. Radii making equal angles with each other are evidently in geometrical progression. This spiral is often called Equiangular from the fact that the angle is always the same between iRankine. Applied Mechanics. Art. 17!). 2In the expression 102=100 the quantity "2" is called the logarithm of 100, It being the exponent of the power to which 10 must be raised to give 100. Similarly 2 would be the logarithm of 64, were 8 the bane or number to be raised to the power indicated. 76 THEURETIVAL AND PRACTICAL GRAPHICS. a radius vector and the tangent at its extremity. Upon this property is based its use as tlie out- line for spiral cams and for lobed wheels. The curve never reaches the pole. The name loyarithmic spiral is based on the property that the angle of revolution is proportional to the logarithm of the radius vector. This is expressed by p = a', in wliich is the varying angle, and a is some arbitrary constant. To construct a tangent by calculation, divide the hyperbolic logarithm ^ of the ratio M : K (which are any two radii whose values are known) by the angle Ijctween these radii, expressed in circular measure;^ the quotient will be the tangent of the constant angle of obliquity of the spiral. 217. Among the more interesting ])roperties of this curve are the following; Its involute is an equal logarithmic spiral. ^^'ere a light placed at the pole, the caustic — whether by Tcflection or refraction — would be a logarithmic spiral. The discovery of these i)roperties of recurrence led James Bernouilli to direct that this spiral be engraved on his tomb, with the inscription — Eadem Mutaia, Resurgo, which, freely trans- lated, is — / shall arise the same, though changed. Kepler discovered that the orbits of the planets and comets were conic sections having a focus at the centre of the sun. Newton proved that they would have described logarithmic spirals as they travelled out into space, had the attraction of gravitation been inversely as the cube instead of the square of the distance. THE HYPERBOLIC OR RECIPROCAL SPIRAL. 218. In this spiral the length of a radius vector is in inverse ratio to the angle through which it turns. Like the logarithmic spiral, it has an infinite number of convolutions about the pole, which it never reaches. The invention of this curve is attributed to James Bernouilli, who showed that Newton's conclusions as to the logarithmic spiral (see Art. 217) would also hold for the hyperbolic spiral, the initial velocity of projection determining which trajectory was described: To obtain points of the curve divide a circle ?n5 8 (Fig. 125) into any number of equal parts, and on some initial radius m lay off some unit, as an inch; on the second radius 2 take —^ ; on the third -^, etc. For one - half the angle the radius vector would evidently be 2 On, giving a point .3 outside the circle. The equation to the curve is , = a 6, in which r is the radius vector, a some numerical con- stant, and is the angular rotation of r (in circular measure) estimated from some initial line. >To get the hyperbolic logarithm of a number multiply its common logarithm by 2.3026. 21n circular measure 360° = 2irr, which, for r = 1, becomes 6.28318; J80 ° = 3.14169- 90o = 15708; 60° = 1.0472; 45° =0.7854; 30° = 0..5236: 10 = 0.0174.533. THE HYPEKBOLIO SPIRAL.— TH E LITUUS. 77 The curve has an asymptote parallel to the initial line, and at a distance from it equal to units. X-lg. iSS. To construct the spiral from its equation take as the pole (Fig. 126); OQ as the initial line; a, for convenience, some fraction, as -: and as our unit some quantity, say half an inch, that will make - of convenient size. Then, tukini; QO as the initial line, make P =^ - = 2", and draw I'R parallel to OQ for the asymptote. For ^ = 1, that is, for arc K H ^ radius OH, we have r =: - ~^2", giving // for one ])()int of the spiral. Writing the equation in the form r = --^- and expressing various values of in circular measure we get the following: (9 = 80° = ()..-)236; r = 0M= 3 '.'8 + : fl = 45° = 0.7854; r = 0N= 2 "55; e = 90° = 1.2708; r= S= 1'.'2 + : ^ = 180° = 3.14159; r = 0T= .6306, etc. The tangent to the curve at any point makes with the radius vector an angle <^ which is found hy analysis to sustain to the angle tlie following trigonometrical relation, tan ^=0; the circular measure of ^ may therefore he found in a ta}>le of natural tangents, and the corresponding value of ohtained. THE IJTUrs. — THE IONIC VOLUTE. 219. T'he Lituus is a si)iral in which tlie radius vector is inversely proportional to the square root of the angle through which it has revolved. This relation is shown by the equation r = — :=, also •' * a./0 written a' 6 = --,,. When ^ r:^ we find ?• = cc , which makes the initial line an asymptote to the curve. In Fig. 127 take OQ as the initial line, as the pole, a = 2, and £.j our unit 3"; then -' = U". rt For e=90°=7r (in circular measure 1.5708; we have r = 03f=l". 2+. For 6=1 we have the radius T making an angle of 57 ° 29 + with the initial line, and in length e(|ual to - units, 78 THEORETICAL AND PRACTICAL GRAPHICS. 1. e., li". rotating to For ^=45°=^ (or 0.7854) r will be 0R=\".1+. Then 0H = OR for in 4 ^"' "■■^"^^ ■ "" * ■■ ■■ 2 OH the radius vector passes over four 45° angles, and the radius must therefore be one- half what it was for the first 45 Similarly, K -■ 0M=~, etc. described, this rela- was for _ om: ~ 2 ' tion enabling the student to locate any number of points. To draw a tangent to the curve we employ the relation tan = 2 6, ^ being the angle made by the tangent line with the radius vector, while 6 is the angular rotation of the latter, in circular measure. Architectural Scrolls. — The Ionic Volute. The Lituus and other spirals are occasionally employed as volutes and other architectural ornaments. In the former application it is customary for the spiral to terminate on a circle called the eye, into which it blends tangentially. Usually, in practice, circular-arc approximations to true spiral forms are employed, the simplest AliillllllllllllilllllllillllBI of which, for the scroll on the capital of an Ionic column, is probably the following: Taking AO P, the total height of the volute, at sixteen of the eighteen "parts" into which the module (the unit of proportion ^= the semi - diameter of the column) is divided, draw the circular eye with radius equal to one such part, the centre dividing A P into segments of seven and nine parts respectively. Next inscribe in the eye a square with one diagonal vertical; parallel to its sides draw (see enlarged square mnop) 2 — 4 and 3 — 1, and divide each into six equal parts, which number up to twelve, as indicated. Then (returning to main figure) the arc A B has centre 1 and radius 1 — A. With 2 as a centre draw arc B C; then CD from centre 3, etc. In the complete drawing of an Ionic column the centre of the eye would be at the intersection of a vertical line irom the lower extremity of the cyma reversa with a hori- zontal through the lower line of the echinus. To complete the scroll a second spiral would be required, constructed according to the same law and beginning at Q, where ^ Q is equal to one -half part of the module. BRUSH TINTING AND SHADING. 79 CHAPTBB VI. TINTING — FLAT AND GRADUATED. — MASONRY, TILING, WOOD GRAINING, RIVER-BEDS AND OTHER SECTIONS, WITH BRUSH ALONE OR IN COMBINED BRUSH AND LINE WORK. 220. Brush-work, with ink or colors, is either flat or graduated. The former gives the effect of a flat surface parallel to the paper on which the drawing is made, while graded tints either show curvature, or — if indicating flat surfaces — represent them as inclined to the paper, i.e., to the plane of projection. For either, the paper should be, as previously stated (Arts. 41 and 44) cold-pressed and stretched. The surface to be tinted should not be abraded by sponge, knife or rubber. 221. The liquid employed for tinting must be free from sediment; or at least the latter, if present, must be allowed to settle, and the brush dipped only in the clear portion at the top. Tints may, therefore, best be mixed in an artist's water-glass, rather than in anything shallower. In case of several colors mixed together, however, it would be necessary to thoroughly stir up the tint each time before taking a brushful. A tint prepared from a cake of high-grade India ink is far superior to any that can be made by using the ready-made liquid drawing inks. 222. The size of brush should bear some relation to that of the surface to be tinted; large brushes for large surfaces and vice versa. The customary error of beginners is to use too small and too dry a brush for tinting, and the reverse for shading. 223. Harsh outlines are to be avoided in brush work, especially in handsomely shaded drawings, in which, if sharply defined, they would detract from the general e9"ect. This will become evident on comparing the spheres in Figs. 1 and 4 of Plate II. Since tinting and shading can be successfully done, after a little practice, with only pencilled limits, there is but little excuse for inking the boundaries; but if, for the sake of definiteness, the outlines are inked at all it should be before the tinting, and in the finest of lines, preferably of " water - proof " ink; although any ink will do provided a soft sponge and plenty of clean water be applied to remove any excess that will "run." The sponge is also to be the main reliance of the draughtsman for the correction of errors in brush work; the water, however, and not the friction to be the active agent. An entire tint may be removed in this way in case it seems desirable. 224. When beginning work incline the board at a small angle, so that the tint will flow down after the brush. For a flat, that is, a tiniform tint, start at the upper outline of the surface to be covered, and with a brush full, yet not surcharged — which would prevent its coming to a good point — pass lightly along from left to right, and on the return carry the tint down a little farther, making short, quick strokes, with the brush held almost perpendicular to the paper. Advance the tint as evenly as possible along a horizontal line; work quickly between outlines, but more slowly along outlines, as one should never overrun the latter and then resort to "trimming" to conceal lack of skill. ■ It is possible for any one, with care and practice, to tint to yet not over boundaries. The advancing edge of the tint must not be allowed to dry until the lower boundary is reached. 80 THEORETICAL AND PRACTICAL GRAPHICS. No portion of the paper, however small, should be missed as the tint advances, as the work is likely to be spoiled by retouching. Should any excess of tint be found along the lower edge of the figure it should be absorbed by the brush, after first removing the latter's surjilus by means of blotting paper. To get a dark effect several medium tints laid on in succession, each one drying before the next is, applied, give better results than one dark one. The heightened effect described in Art. 72, viz., a line of light on the upper and left-hand edges, may be obtained either (a) by ruling a broad line of tint with the drawing-pen at the desired distance from the outline, and instantly, before it dries, tinting from it with the brush; or (b) by ruling the line with the pen and thick Chinese White. 226. A tint will spread much more evenly on a large surface if the paper be first slightly dampened with clean water. As the tint will follow the water, the latter should be limited exactly to the intended outlines of the final tint. 226. Of the colors frequently used by engineers and architects those which work best for flat effects are carmine, Prussian blue, burnt sienna and Payne's gray. Sepia and Gamboge, are, fortunately, rarely required for uniform tints; but the former works ideally for shading by the "dry" process described in the next article; and its rich brown gives effects unapproachable with anything else. It has, however, this peculiarity, that . repeated touches upon a spot to make it darker produce the opposite effect, unless enough time elapses between the strokes to allow each addition to dry thoroughly. 227. For elementary practice with the brush the student should lay flat washes, in India tints, on from six to ten rectangles, of sizes between 2" X 6" and 6" X 10". If successful with these his next work may be the reproduction of Fig. 128, in which H, V, P and S denote horizontal, vertical, profile and section planes respectively. The figure should be considerably enlarged. The plane V may have two washes of India ink; H one of Prussian blue; P one of burnt sienna, and S one of carmine. The edges of the planes H, V and P are either vertical or inclined 30° to the horizontal. BRUSH TINTING AND SHADING. SI For the section -plane assume n and m at pleasure, giving direction nm, to which JR and TX are parallel. A horizontal, mz, through m gives z. From n a horizontal, ny, gives y on a 6. Joining y with z gives the "trace" of S on V. 228. Figures 129 and 130 illustrate the use of the brush in the representation of masonry. The former may be altogether in ink tints, or in medium burnt umber for the front rectangle of each stone, and dark tint of the same, directly from the cake, for the bevel. Lightly pencilled limits of bevel and rectangle will be needed; no inked outlines required or desirable. The last remark applies also to Fig. 130, in which " quarry - faced " ashlar masonry is represented. If properly done, in either burnt umber or sepia, this gives a result of great beauty, especially effective on the piers of a large drawing of a bridge. The darker portions are tinted directly from the cake, and are puri)osely made irregular and "jagged" to reproduce as closely as possible the fractured appearance of the stone. x"ig-- ISO. Two brushes are required when an "over-hang" or jutting portion is to be represented, one with a medium tint, the other with the thick color, as before. An irregular line being made with the latter, the tint is then softened out on the lower side with the point of the brush having the lighter tint. A light wash of the intended tone of the whole mass is quickly laid over each stone, either before or after the irregularities are represented, according as an exceedingly angular or a somewhat softened and rounded effect is desired. 82 THEORETICAL AND PRACTICAL GRAPHICS. 229. Designs in tiling are excellent exercises, not only for brush work in flat tints, but also ^ in their preliminary construction — in precision of line work. The superbly illustrated catalogues of the Minton Tile Works are, unfortunately, not accessible by all students, illustrating as they do, the finest and most varied work in this line, both of designer and chromo-lithographer; but it is quite within the bounds of possibility for the careful draughtsman to closely approach if not equal the standard and general appearance of their work, and as suggestions therefor Figs. 131 and 132 are presented. 230. In Fig. 131 the ui)per boundary, a d h k, of a rectangle is divided at a, b, c, etc., into equal spaces, and through each point of division two lines are drawn with the 30° triangle, as bx and br through b. The oblique line.s terminate on the sides and lower line of the rectangle. If the work is accurate — and it is worthless if not— any vertical line as mn, drawn through the inter- section, m, of a pair of oblique lines, will pass through the intersection of a series of such pairs. The figure shows three of the possible designs whose construction is based on the dotted lines of the figure. For that at the top and right, in which hwizontal rows of rhombi are left white, we draw vertical lines as s g and m n from the lower vertex of each intended white rhombus, continuing it over two rhombi, when another white one will be reached. The dark faces of the design are to be finally in solid black, previous to which the lighter faces sliould be tinted Avith some drab or brown tint. The pencilled construction lines would necessarily be erased before the tint was laid on. The most opaque effect in colors is obtained by mixing a large portion of Chinese white with the water color, making what is called by artists a "body color." Such a mixture gives a result in marked contrast with the transparent effect of the usual wash; but the amount of white used should be sufficient to make the tint in reality a paste, and no more should be taken on the brush at one time than is needed to cover one figure. Sepia and Chinese white, mixed in the proper proportions, give a tint which contrasts most agreeably with the black and white of the remainder of the figure. The star design and the hexagons in the lower right-hand corner result from extensions or modifications of the construction just described which wi)l become evident on careful inspection. TINTING. — BRUSH SHADING. 83 231. Fig. 132 is a Minton design with which many are familiar, and which affords opportunity for considerable variety in finish. Its construction is almost self-evident. The equal spaces, ah, cd, Yn,n — which may he any width, x, — alternate with other equal spaces be, which may preferably be about 3 X in width. Lines at 45 °, as indicated, complete the preliminaries to tinting. The octagons may be in Prussian blue, the hexagons in carmine, and the remainder in white and black, as shown; or browns and drabs may be employed for more subdued effects. SHADING. 232. For shading, by graduated tints, provide a glass of clear water in addition to the tint; also an a,mple supply of blotting paper. The water- color or ink tint may be considerably darker than for flat tinting; in fact, the darker it is, provided it is clear, the more rapidly can the desired effect be obtained. The brush must contain much less liquid than for flat work! Lay a narro\\' band of tint quickly along the part that is to be the darkest, then dip the brush into clear water and inmiediately apply it to the blotter, both to bring it to a good point and to remove the surplus tint. With the now once -diluted tint carry the advancing edge of the band slightly farther. Kej)eat the operation until the tint is no longer discernible as such. The process may be repeated from the same starting point as many times as necessary to produce the desired effect; \)ut the work should l)e allowed to dr}' each time before laying on a new tint. Any irregularities or streaks can easily be removed after the work dries, by retouching or "stippling" with the point of a fine brush that contains but little tint — scarcely more than enough to enable the brush to retain its point. For small work, as the shading of rivets, rods, etc., the process just mentioned, which is also called "dry shading," is especially adapted, and, although somewhat tedious, gi^•es the handsomest effects possible to the draughtsman. 233. Where a good, general effect is wanted, to be obtained in less time than would be required for the preceding processes, the method of over-lapping flat tints maj' be adopted. A narrower band of dark tint is first laid over the part to be the darkest. When dry this is overlaid by a broader band of lighter tint. A yet lighter wash follows, beginning on the dark portion and extending still farther than its predecessor. The process is repeated with further diluted tints until the desired effect is obtained. Faintly - pencilled lines may be drawn at the outset as limits for the edges of the tints. 84 THEORETICAL AND PRACTICAL GRAPHICS. This method is better adapted for large work, that is not to be closely scrutinized, than for drawings that deserve a high degree of finish. 234. As to the relative position and gradation of the lights and shades on a figure, the student is referred to Arts. 78 and 79 and the chapter on shadows; also to the figures of Plate II, which may serve as examples to be imitated while the learner is acquiring facility in the use of the brush, and before entering upon constructive work in shades and shadows. Fig. 3 of Plate II may be undertaken first, and the contrast made yet greater between the upper and lower boundaries. Fig. 1 (Plate II) requires no explanation. In Fig. 133 we have a wood -cut of a sphere, with the theo- retical dark or "shade" line more sharply defined than in the spheres on the plate. X'J.g-. 133. JFlg-. la-i. 'A drawing of the end of a highly - i)olished revolving shaft, or even of an ordinary metallic disc, would be shaded as in Fig. 134. Fig. 2 (Plate II) represents the triangular -threaded screw, its oblique surfaces being, in mathe- matical language, warped helicoids, generated by a moving straight line, one end of which travels along the axis of a cylinder while the other end traces or follows a helix on the cylinder. The construction of the helix having already been given (Art. 120) the outlines can readily be drawn. The method of exactly locating the shadow and shade lines will be found in the chapter on shadows. Fig. 4 (Plate II), when compared with Fig. 91, illustrates the possibilities as to the representation of interesting mathematical relations. The fact may again be mentioned, on the principle of "line upon line," as also for the benefit of any who may not have read all that has preceded, that the spheres in the cone are tangent to the oblique plane at the foci of the elliptical section. The peculiar dotted eff'ect in this figure is due to the fact that the original drawing, of which this is a photographic reproduction by the gelatine process, was made with a lithographic crayon upon a special pebbled paper much used by lithographers. The original of Fig. 1, on the other hand, was a brush -shaded sphere on Whatman's i)aper. 235. Fig. 5 (Plate II) shows a "Phoenix column," the strongest form of iron for a given weight, for sustaining compression. The student is familiar with it as an element of outdoor construction in Ijridges, elevated railroads, etc. ; also in indoor work in many of the higher office buildings of our great cities. By drawing first an end view of a Phoenix column, similar to that of Fig. 135, we can readily derive an oblique view like that of the plate, by including it between parallels from all points of the former. The proportions of the columns are obtainable from the tables of the company. Fig. 135 is a cross -section of the 8 -segment column, the shaded portion showing the minimum and the other lines the maximum size for the same inside diameter. ^Lg. a.3S. rfff' OF THB UNIVERSITY MATERIALS OF CONSTRUCTION. 85 In a later chapter the proportions of other forms of structural iron will be found. Short lengths of any of these, if shown in oblique view, are good subjects for the ^ig-- iss. brush, especially for "dry" shading, the effect to be aimed at being that of the rail section of Fig. 136. 236. When some particular material is to be indicated, a flat tint of the proper technical color (see Art. 73) should be laid on with the brush, either before or after shading. When the latter is done with sepia it is probably safer to lay on the flat tint first. A darker tint of the technical color should always be given to a cross- section. For blue -printing, a cross -section may be indicated in solid black. WOOD. — RIVER-BEDS. — MASONRY, ETC. 237. While the engineering draughtsman is ordinarily so pressed for time as not to be able to give his work the highest finish, yet he ought to be able, when occasion demands, to obtain both natural and artistic effects; and to conduce to that end the writer has taken pains to illustra.te a number of ways of representing the materials of construction. Although nearly all of them may be — and in the cuts are — represented in black and white (with the exception of the wood -graining on Plate II), yet colors, in combined brush and line work, are preferable. The student will, however, need considerable practice with pen and ink before it will be worth while to work on a tinted figure. 238. Ordinarily, in representing wood, the mere fact that it is wood is all that is intended to be indicated. This may be done most simply by a series of irregular, approximately -parallel lines, as in Fig. 10 or as on the rule in Fig. 17, page 12. Make no attempt, however, to have the grain very irregular. The natural unsteadiness of tlie hand, in drawing a long line toward one continu- ously, will cause almost all the irregularity desired. If a better effect is wanted, yet without color, the lines may be as in Fig. 107, which represents hard wood. In graining, the draughtsman should make his lines toward himself, standing, so to speak, at the end of the plank upon which he is working. The splintered end of a plank should be sharply toothed, in contradistinction to a metal or stone fracture, which is what might be called smoothly irregular. 239. An examination of any piece of wood on which the grain is at all marked will show that it is darker at the inner vertex of any marking than at the outer point. Although this diff"erence is more readily produced with the brush, yet it may be shown in a satisfactory degree with the pen, by a series of after -touches. 240. If we fill the pen with a rather dark tint of the conventional color, draw the grain as in the figures just referred to, and then overlay all with a medium flat wash of some properly chosen color, we get effects similar to those of Plate II. On large timber- work the preliminary graining, as also the final wash, may be done altogether with the brush; as was the original of Fig. 9, Plate II. End views of timbers and planks are conventionally represented by a series of concentric free- hand rings in which the spacing increases with the distance from the heart; these are overlaid with a few radial strokes of darker tint. In ink alone the appearance is shown in Figs. 39 and 115. 241. The color -mixtures recommended by different writers on wood graining are something short of infinite in number; but with the addition of one or two colors to those listed in the draughts- man's outfit (Art. 56) one should be able to imitate nature's tints very closely. 86 THEORETICAL AND PRACTICAL GRAPHICS. COURSED RUBBLE MASONRY Light India Inli. No hard-and-fast rule as to the proportions of the colors can be given. In this connection we may quote Sir Joshua Reynolds' reply to the one who inquired how he mixed his paints. "With brains," said he. One general rule, however; always employ delicate rather than glaring tints. Merely te indicate wood with a color and no graining use burnt sienna, the tint of Figs 7, 8 and 10 of Plate II. Drawing from the writer's experience and from the suggestions of various experimenters in this line the following hints are presented: — In every case grain first, then overlay with the ground tint, which should always be much lighter than the color used for the grain. If possible have at hand a good specimen of the wood to be imitated. Hard Pine: Grain — burnt umber with either carmine or crimson lake; for overlay add a little gamboge to the grain -tint diluted. Soft Pine: Gamboge or yellow ochre with a small amount of burnt sienna. Black Walnut: Grain — burnt umber and a very little dragon's blood; final overlay of modified tint of the same or with the addition of Payne's gray. ^^- ^^'^■ Oak: Grain — burnt sienna; for overlay, the same, with yellow ochre. Chestnut: Grain — burnt umber and dragon's blood; over- lay of the same, diluted, and with a large proportion of gam- boge or light yellow added. Spruce: Grain — burnt umber, medium; add yellow ochre for the overlay. Mahogany: Grain — burnt sienna or umber with a small amount of dragon's blood; dilute, and add light yellow for the overlay. Rosewood: Grain — replace the dragon's blood of mahogany- grain by carmine, and for overlay dilute and add a little Prussian blue. 242. River-beds in black and white or in colors have been already treated in Art. 26, to which it is only neces- sary to add that such sections are usually made quite narrow, and, preferably — if in color — shaded quite abruptly on the side opposite the water. 243. The sections of masonry, concrete, brick, glass and vul- canite, given on page 25 as pen and ink exercises, are again presented in Fig. 137, for reproduction in combined brush and line work. The appropriate color is indicated under each section. 244. Masonry constructions may be broadly divided into rubble and ashlar. In ashlar masonry the bed -surfaces and the joints (edges) are shaped and dressed with great care, so that the stones may not only be placed in regular layers or courses, but often fill exactly some predetermined place, as in arch construction, in which case the determination of their forms and the derivation of the patterns for the stone-cutter involves the application of the Descriptive Geometry of Monge. (Art. 283). RUBBLE MASONRY Light India Init, VULCANITE India Ink. CONCRETE Yellow Ochre, ^"ler. 13S. REPRESENTATION OF MASONRY. 67 Rubble work, however, consists of constructions involving stones mainly "in the rough," but may be either coursed or uncoursed. Fig. 138 is a neat example of uncoursed though partially dressed or "hammered" rubble. In section, as shown in Fig. 137, it is merely necessary to rule section- lines over the boundaries of the stones — a remark applying equally to ashlar masonry. Fi^. 13©. ng-. 140. The other examples in this chapter are of ashlar, mainly "quarry -faced," that is, with the front nearly as rough as when quarried. A beveled or "chamfered" ashlar is shown in Figs. 129 and 140, the latter shaded in what is probably the most effective way for small work, viz., with dots, the effect depending upon the number, not the size of the latter. Only a careful examination of the kind and position of the lines in the other figures on this page will disclose the secret of the variety in the effects produced. For the handsomest results with any of these figures the pen -work — ng-. 3.-3=1. Fig-. 1.4=2. ■m whether dotting or "cross-hatching" — should be preceded by an undertone of either India ink, umber, Payne's gray, cobalt or Prussian blue, according to the kind of stone to be represented. :E'i.s- i-is- ^i^- a-44. r For slate use a pale blue; for brown free -stone either an umber or sepia; while for stone in general, kind immaterial, use India ink. THEORETICAL AND PRACTICAL GRAPHICS. CHAPTER VII, FREE-HAND AND MECHANICAL LETTERING. — PEOPORTIONING OP TITLES. 245. Practice in lettering forms an essential part of the elementary work of a draughtsman. Every drawing has to have its title, and the general effect of the result as a whole depends largely upon the quality of the lettering. Other things being equal, the expert and rapid draughtsman in this line has a great advantage over one who can do it but slowly. For this reason free-hand lettering is at a high premium, and the beginner should, therefore, aim not only to have his letters correctly formed and properly spaced, but, as far as possible, to do without mechanical aids in their construction. When under great pressure as to time it is, however, perfectly legitimate to employ some of the mechanical expedients used in large establishments as "short cuts" and labor -savers. Among these the principal are "tracing" and the use of rubber types. 246. To trace a title one must have at hand complete printed alphabets of the size of type required. Placing a piece of tracing-paper over the letter wanted, it is traced with a hard pencil, the paper then slipped along to the next letter needed, and the process repeated until the words desired have been outlined. The title is then transferred to the drawing by first running over the lines on the back of the tracing-paper with a soft pencil, after which it is only necessary to re -trace the letters with a hard pencil, on the face of the transfer -paper, to find their outlines faintly yet sufficiently indicated on the paper underneath. Carbon paper may also be used for transferring. 247. The process just described would be of little service to a ready free-hand draughtsman, but with the use of rubber types, for the words most frequently recurring in the titles, a merely average worker may easily get results which — in point of time — cannot be exceeded by any other method. When employing such types either of the following ways may be adopted: (a) a light impression may be made with the aniline ink ordinarily used on the pads, and the outlines then followed and the "filling in" done either with a writing -pen* or fine -pointed sable -hair brush; or (b) the impression may be made after moistening the types on a pad that has been thoroughly wet with a light tint of India ink. The drawing -ink must then be immediately applied, free-hand, with a Falcon pen or sable brush, before the type -impression can dry. The pen need only be passed down the middle of a line, as on the dampened surface the ink will spread instantly to the outlines. 248. The educated draughtsman should, however, be able not only to draw a legible title of the simple character required for shop -work, and in which the foregoing expedients would be mainly serviceable, but be prepared also for work out of the ordinary line, and, if need be, quite elaborate, as on a competitive drawing. Such knowledge can only be gained by careful observation of the forms of letters, and considerable practice in their construction. No rigid rules can be laid down as to choice of alphabets for the various possible cases. Common -sense, custom and a natural regard for the "fitness of things" are the determining factors. Obviously rustic letters would be out of place on a geometrical drawing, and other incongruities • Kefer to Art. 27 with regard to the pens to be used for the vaiious styles of letters DESIGNING OF TITLES. 89 will naturally suggest themselves. In addition to the hints in Art. 27 a few general principles and methods may, however, be stated to the advantage of the beginner, who should also refer to the special instructions given in connection with certain specimen alphabets at the end of this work. 249. In the first place, a title should be symmetrical with respect to a vertical centre-line, a rule which should be violated but rarely, and then, usually, when the title is to be somewhat fancy in design, as for a magazine cover. Blementai^lj Plates drawn by QTorflatitif Bau (Ecirlmr ^+ +hE LEADING TECHNICAL SCHDCL Jan. — June, 3001. 250. If it be a complete as distinguished from a partial or .8w6- title it will answer the following questions which would naturally arise in the mind of the examiner: — What is it? — Where done? — By whom? — When? — On what scale? In answering these questions the relative valuation and importance of the linos are expressed by the sizes and kinds of type chosen. This is a point requiring nn)8t careful consideration, as the final effect depends largely upon a proper balancing of values. -^-i — >-■ OF "t-<-^- PERFECTION SUSPENSION BRIDGE -•>K« dE sign Ed hy *=<•• B-DDdwin^ Mackenzie V Cartwright •> i « -s MINNEAPOLIS. MINN. s»— s-5< — Scale 4 Ft. I In. JXin© 13, 2900. Jose Martinez, Del. 251. The "By whom?" may cover two possibilities. In the case of a set of drawings made in a scientific school it would refer to the draughtsman, and his name might properly have considerably greater prominence than in any other case. The upper title on this i)age is illustrative of this point, as also of a symmetrical and balanced arrangement, although cramped as to space, vertically. Ordinarily the " By whom ? " will refer to the designer, and the draughtsman's name ought to be comparatively inconspicuous, while the name of the designer should be given a fair degree of prominence. This, and other important points to be mentioned, are illustrated in the preceding 90 THEORETICAL AND PRACTICAL GRAPHICS. arrangement, printed, like the upper title, from types of which complete alphabets will be found at the end of this work. 252. The abbreviation Del., often placed after the draughtsman's name, is for Delineavit — He drew it — and does not indicate what the visitor at the exhibition supposed, that all good draughtsmen hail from Delaware. 253. The best designed titles are either in the form of two truncated pyramids having, if , pos- sible, the most important line as their common base, or else elliptical in shape. 254. The use of capitals throughout a line depends ujion the style of type. It gives a most unsatisfactory result if the letters are of irregular outline, as is amply evidenced by the words each letter of which is exquisite in form, but the combination almost illegible. Contrast them with the same style, but in capitals and small letters: — M:ei:franttal Braniing, 255. As to spacing, the visible white spaces between the letters should be as nearly the same as possible. In this feature, as in others, the draughtsman can get much more pleasing results than the printer, since the latter usually has each letter on a separate piece of metal, and can not adjust his space to any ])articular combination of letters, such as PA, L V, WA or A V, where a better effect would be obtained l)y placing the lower part of one letter under r-ig-- i-as. the upper part of the next. This is illustrated in Fig. 146, which may ~ V* Vr* \~ V*"/"*"! — J be contrasted with the printer's best spacing of the separate types for / / \ ' / Hi the A and W in the word " Drawings " of the last title. -' J ' — i ■' ' 256. The amount of space between letters will depend upon the length of line that the word or words must make. If an important word has few letters they should be " spaced out," and the letters themselves of the "extended" kind, i. e., broader than their height. The following word will illustrate. The characteristic feature of this tyi)e, viz., heavy horizontals and light verticals, is com- mon to all the variations of a fundamental fonn frequently called Italian Print. X ID Or :e: . When, on the other hand, many letters must be crowded into a small space, a " condensed " style of letter must be adopted, of which the following is an example: Pennsylvania Railpoad. 257. While the varieties of letters are very numerous yet they are all but changes rung on a few fundamental or basal forms, the most elementary of which is the GOTHIC, ALSO CALLED HALF- BLOCK. Letters like B, O, etc., which have, usually, either few straight parts or none at all, may, for the sake of variety as also for convenience of construction, be made partially or wholly angular; in the latter case the form is called Geometric Gothic by some type manufacturers. It is only appropriate for work exclusively mechanical. The rounded forms are preferable for free-hand lettering. s^CALIFORH\^ LETTERING. 91 The following complete Gothic alphabet is so constructed that whether designed in its "con- densed" or "extended" form the proper proportions may be easily preserved. S'igf. l-iT- \ui =iEv:;sry x/ r V < Taking all the solid parts of the letters at the same width as the I, we will find any letter of average width, as U, to be twice that unit, plus the opening between the uprights, which last, being indetei-minate, we may call x, making it small for a "condensed" letter, and broad as need be for an "extended" form. The word march would foot up 5 U + 3, disregarding — as we would invariably — the amount the foot of the R projects beyond the m^in right-hand outline of the letter. In terms of x this makes 5 a; + 13, as U = x + 2. Allowing spaces of 1^ unit width between letters adds 5 to the above, making 5 a; + 18 for the total length in terms of the I. Assuming x equal to twice the unit we would have the whole word equal to twenty -eight units; and if it were to extend seven inches the width of the solid parts would therefore be one -quarter of an inch. Where the width of a letter is not indicated it is assumed to be that of the U. The W is equal to 2U — 1. This relation, liowever, does not hold good in all alphabets. The angular corners are drawn usually with the 45° triangle. The guide-lines show what points of the various letters are to be found on the same level, and should be but faintly pencilled. As remarked in Art. 27, the extended form of Gothic is one of the best for dimensioning and lettering working draivings, and is rapidly coming into use by the profession. 258. The Full -Block letter next illustrated is easier to work with than the Gothic in the matter of preliminary estimate, as the width of each letter — in terms of uuit squares — is evident at a glance. The same word march would foot up twenty -seven squares without allowing for spaces between letters. Calling the latter each two we would have thirty -five squares for the same length as before (seven inches), making one -fifth of an inch for the width of the solid parts. For convenience the widths of the various letters are summarized: 1 = 3; C,G,0,Q,S,Z = 4; A,B, D,E,F, J,L, P,R,T,& = 5; H,K,N,U, V,X, Y = 6; M = 7; W = 8. 259. In case the preliminary figuring were only approximate and there were but two words in the line, as, for example, Mechanical Drawing, a safe method of working would be to make a fair allowance for the space between the words, begin the first word at the calculated distance to the left of the vertical centre-line, complete it, then work the second word backward, beginning with the 92 THEORETICAL AXD PJiACTICAL GRAPHICS. G aa far to the right of the reference line as the M was to the left. On completing the second word any difference between the actual and the estimated length of the words, due to over- or under- width of such letters as M, W and I, will be merged into the space between the words. ^-i-:H.Hilli ^H--- -■■i--;-f>-r ••+ {^T^sn ■1 f ! ; X. ■ l /T ■ T ' ''\7! . ■ I I ixTT'iTr T . ■ f; TTTi rl i T i T; . Tn t > \' . t I . i iSl-.x \liJ ^igr- ISO. With three words in a line the same method might be adopted, the middle word being easily placed half way between the others, which, by this method of construction would not only begin correctly but also terminate where they should. 260. Note particularly that the top of a B is always slightly smaller than the bottom; xT^.i-as. similarly with the S. This is made necessary by the fact that the eye seems to exaggerate r->j /^ the upper half of a letter. To get an idea of the amount of difference allowable compare ^^ ^ the following equal letters printed from Roman type, condensed. Although not so important in the E, some difference between top and bottom may still to advantage be made. Another refines ment is the location of the horizontal cross-bar of an A slightly below the middle of the letter. 261. While vertical letters are most frequently used, yet no handsomer effect can be obtained than by a well - executed inclined letter. The angle of inclination should be about 70°. Beginners usually fail sadly in their first attempt with the A and V, one of whose sides they give the same slant as the upright of the other letters. In point of fact, however, it is the imaginary (though, in the construction, pencilled) centre-line which should have that inclination. See Fig. 150. In these forms — the Roman and Italic Roman — the union of the light horizontals or "seriffs" with the other parts is in general effected by means of fine arcs, called "fillets," drawn free-hand. On many letters of this alphabet some lines will, however, meet at an angle, and only a careful examination of good models ■will enable one to construct correct forms. Upon the size of the fillets the appearance of the letter mainly depends, as will be seen by a glance at Fig. 151, which repro- x-ig-. iBi. duces, exactly, the N of each of two leading alphabet books. If the fillets ■^ -^ -m -T- round out to the end of the spur of the letter, a coarse and bulky appear- / ^/ / W^. ance is evidently the result; while a fine curve, leaving the straight horizontals projecting beyond them, gives the finish desired. This is further illustrated by No. 23 of the alphabets appended, a type which for clearness and elegance is a triumph of the founder's art. As usually constructed, however, the D and R are finished at the top like the P. lJ5v^ "" n Hi □ 94 THEORETICAL AND PRACTICAL GRAPHICS. 262. The Roman alphabet and its inclined or italic, form are much used in topographical work. A text-book devoted entirely to the Roman alphabet is in the market, and in some works on topographical drawing very elaborate tables of proportions for the letters are presented; these answer admirably for the construction of a standard alphabet, but in practice the proportions of the model would be preserved by the draughtsman no more closely than his eye could secure. Usually the small letters should be about three- fifths the height of the capitals. Except when more than one- third of an inch in height these letters should be entirely free-hand. 263. When a line of a title is cwved no change is made in the forms of the letters; but if of a vertical, as distinguished from a slanting or italic type, the centre-line of each letter should, if produced, pass through the centre of the curve. Italic letters, when arranged on a curve, should have their centre-lines inclined at the same angle to the normal (or radius) of the curve as they ordinarily make with the vertical. 264. jVji alphabet which gives a most satisfactory appearance, yet can be constructed with great rapidity, is what wo may call the "Railroad" type, since the public has become familiar with it mainly from its frequent use in railroad advertisements. The fundamental forms of the small letters, with the essential construction lines, are given in rectangular outline in the complete alphabet on the preceding page, with various modifications thereof in the M'ords below them, showing a large number of possible effects. At least one plain and fancy capital of each letter is also to be found oij the same page, with in some instances a still larger range of choice. No handsomer efiects are obtainable than with this alphabet, when brush tints are employed for the undertone and shadows. 265. For ^ rapid lettering on tracing -cloth, Bristol board or any smooth - surfaced paper a style long used abroad and increasing in favor in this country is that known as Round Writing, illustrated liy Fig. 152, and for which a special text -book and pens have been prepared by F. Soennecken. The pens are stubs of various widths, cut off obliquely, and when in use should not, as ordinarily, be dipped into the ink, but the latter should be inserted, by means of another pen, between the top of the Soennecken pen and the brass "feeder" that is usually slipped over it to regulate the flow. The Soennecken Round Writing Pens are also by far the best for lettering in Old English, German Text and kindred types, addition of a few straight lines to an ordinary title will become ■st^. xss. Sl^imSmititiq E'n Fig. iS3. en a^y IQ -■s M' :h '1 eenanieai D rawmd the The improvement due to evident by comparing Figs. 153 and 154. The judicious use. of "word ornaments," such as those of alphabets 33, 42, 49, and of several of the other forms illustrated, will greatly enhance the appearance of a title with- out materially increasipg the time expended on it. This is illustrated in the lower title on page 89. lemen hary Plate- — n — ee h an lea u rawi n o DESIGNS FOR BORDERS. 95 96 THEORETICAL AND PRACTICAL GRAPHIC IS. 266. Borders. Another effective adjunct to a map or, other drawing is a neat border. It should be strictly in keeping with the drawing, both as to character and simplicity. On page 95 a large number of corner designs and borders is presented, one -third of them orig- inal designs, by the writer, for this work. The principle of their construction is illustrated by Fig. 155, in which the larger design shows the necessary preliminary lines, and the smaller the complete corner. It is evident in this, as in all cases of interlaced designs, that we must first lay off each way from the corner as many equal distances as there are bands and spaces, and lightly make a network of squares — or of rhombi, if the angles are acute — by pencilled construction -lines through the points of division. 267. Shade lines on borders. The usual rule as to shade lines applies equally to these designs, thus: Following any band or pair of lines making the turns as one ^"igr- a-ES. piece, if it runs horizontally the loiver line is the heavier, while in a ^ '~7~y'_ vertical pair the right-hand line is the shaded line. This is on the 'i-- assumption that the light is coming in the direction usually assumed for s — ^- mechanical drawings, i. e., descending diagonally from left to right. n 5|4{3 S .'^ n^ i II M In case a pair of lines runs obliquely, the shaded lines may be determined by a study of their location on the designs of the j)late of borders. It need hardly be said that on any drawing and its title the light should be supposed to come from but one direction throughout, and not be shifted; and the shaded lines should be located accordingly. This rule is always imperative. In drawing for scientific illustration or in art work it is allowable to depart from the usual strictly conventional direction of light, if a better effect can thereby be secured. 268. A striking letter can be made by drawing the shade line onh', as in Fig. 146, page 90, which we may call "Full -Block Shade -Line," being based upon the alphabet of Fig. 148, page 92, as to construction. Owing to its having more projecting parts it gives a nuich handsomer effect than the The student will notice that the light comes from different directions in the two examples. These forms are to the ordinary fully - outlined letters what art work of the "impressionist" school is to the extremely detailed and painstaking work of many; what is actually seen suggests an equal amount not on the paper or canvas. 269. While a teacher of draughting may well have on hand, as reference works for his class, such books on lettering as Prang's, Becker's and others equally elaborate, yet they will be found of only occasional service, their designs being as a rule more highly ornate than any but the specialist would dare undertake, and mainly of a character unsuitable for the usual work of the engineering or architectural draughtsman, whose needs were especially in mind when selecting types for this work. The alphabets aispended afford a large range of choice among the handsomest forms recently designed by the leading type manufacturers, also containing the best among former types; and with the "Railroad," Full -Block and Half- Block alphabets of this chapter, proportioned and drawn by the writer, supply the student with a practical "stock in trade" that it is believed will require but little, if any, supplementing. COPYING PROCESSES. — DRAWING FOR ILLUSTRATION. 97 CHAPTBB nil. BLUE. PRINT AND OTHER COPYING PROCESSES.— METHODS OF ILLUSTRATION. 270. While in a draughting office the process described below is, at present, the only method of copying drawings with which it is absolutely essential that the draughtsman should be thoroughly acquainted, he may, nevertheless, find it to his advantage to know how to prepare drawings for reproduction by some of the other methods in most general use. He ought also to be able to recognize, usually, by a glance at an illustration, the method by which it was obtained. Some brief hints on these points are therefore introduced. Obviously, however, this is not the place to give full particulars as to all these processes, even were the methods of manipulation not, in some cases, still "trade secrets"; but the important details concerning them, that have become common property, may be obtained from the following valuable works: Modem Heliographic Processes* by Ernst Lietze; Photo- Engraving, Etching and Lithography,^ by W. T. Wilkinson; Modern Reproductive Graphic Processes,* by Jas. S. Pettit, and Photo -Engraving, by Carl Schraubstadter, Jr. THE BLUE -PRINT PROCESS. 271. By means of this process, invented by Sir John Herschel, any number of copies of a draw- ing can be made, in white lines on a blue ground. In Arts. 43 and 45 some hints will be found as to the relative merits of tracing- cloth and "Bond" paper, for the original drawing. A sheet of paper may be sensitized to the action of light by coating its surface with a solution of red prussiate of potash (ferrocyanide of potassium) and a ferric salt. The chemical action of light upon this is the production of a ferrous salt from the ferric compound; this combines with the ferrocyanide to produce the final blue undertone of the sheet; while the portions of the paper from which the light was intercepted by the inked lines, become white after immersion in water. The proportions in which the chemicals are to be mixed are, apparently, a matter of indiffer- ence, so great is the disparity between the recipes of different writers; indeed, one successful draughtsman says: "Almost any proportion of chemicals will make blue -prints." Whichever recipe is adopted — and a considerable range of choice will be found in this chapter — the hints immediately following are of general application. 272. Any white paper will do for sensitizing that has a hard finish, like that of ledger paper, so as not to absorb the chemical solution. To sensitize the paper dissolve the ferric salt and the ferrocyanide in water, separately, as they are then not sensitive to the action of light. The solutions should be mixed and applied to the paper only in a dark room. Although there is the highest authority for "floating the paper to be sensitized for two minutes on the surface of the liquid," yet the best American practice is to apply the solution with a soft flat brush about four inches wide. The main object is to obtain an even coat, which may usually • Published by the D. Van Nostrand Company, New York, t American Edition revised and published by Edward L. Wilson, New York. 98 THEORETICAL AND PRACTICAL GRAPHICS. be secured by a primary coat of horizontal strokes followed by an overlay of vertical strokes; the second coat applied before the first dries. If necessary, another coat of diagonal strokes may be given to secure evenness. The thicker the coating given the longer the time required in printing. A bowl or flat dish or plate will be found convenient for holding the small portion of the solution required for use at any one time. The chemicals should not get on the back of the sheet. Each sheet, as coated, should be set in a dark place to dry, either "tacked to a board by two adjacent corners,'' or "hung on a rack or over a rod," or "placed in a drawer — one sheet in a drawer," — varying instructions, illustrating the quite general truth that there are usually several almost equally good ways of doing a thing. 273. To copy a drawing, place the prepared paper, sensitized side up, on a drawing-board or printing -frame on which there has been fastened, smoothly, either a felt pad or canton flannel cloth. The drawing is then immediately placed over the first sheet, inked side up, and contact secured between the two by a large sheet of plate glass, placed over all. Exposure in the direct rays of the sun for four or five minutes is usually sufficient. The progress of the chemical action can be observed by allowing a corner of the paper to project beyond the glass. It has a grayish hue when suflficiently exposed. If the sun's rays are not direct, or if the day is cloudy, a jiroportionately longer time is required, running up in the latter case, from minutes into hours. Only experiment will show whether one's solution is "quick" or "slow;" or the time required by the degree of cloudiness. A solution will print more quickly if the amount of water in it be increased, or if more iron is used; but in the former case the print will not be as dark, while in the latter the results, as to whiteness of lines, are not so apt to be satisfactory. Although fair results can be obtained with paper a month or more after it has been sensitized, yet they are far more satisfactory if the paper is prepared each time (and dried) just before using. On taking the print out of the frame it should be immediately immersed and thoroughly washed in cold water for from three to ten mi antes, after which it may be dried in either of the ways previously suggested. If many prints are being made, the water should be frequently changed so as not to become charged with the solution. 274. The entire process, while exceedingly simple in theory, varies, as to its results, with the experience and judgment of the manipulator. To his choice the decision is left between the follow- ing standard recipes for prejjaring the sensitizing solution. The "parts" given are all by weight. In every case the j)otash should be pulverized, to facilitate its dissolving. No. 1. (From Le Genie Civil.) „, . ,^ f Red Prussiate of Potash 8 parts. Solution Ao. 1. 1 Water 70 parts. f Citrate of Iron and Ammonia 10 parts. SoMwa No. 2. ■{ ^,^ «« ^ i Water 70 parts. Filter the solutions separately, mix equal quantities and then filter again. No. 2. (From U. S. Laboratory at Wlllett's Point). f Double Citrate of Iron and Ammonia 1 ounce. Solution No. i. -{ ,-, . ( Water 4 ounces. ( Red Prussiate of Potassium 1 ounce. Solution No. 2. i ... , . Water 4 ounces. Stock Solution. BLUE-PRINT PROCESS. 99 No. 3. (Lietze's Method). 5 ounces, avoirdupois, Red Prussiate of Potash. 32 fluid ounces Water. "After the red prussiate of potash has been dissolved— which requires from one to two days — the liquid is filtered. This solution remains in good condition for a long time. Whenever it is required to sensitize paper, dissolve, for every two hundred and forty square feet of paper, j 1 ounce, avoirdupois, Citrate of Iron and Ammonia, i 4} fluid ounces Water, and mix this with an equal volume of the stock solution. The reason for making a stock solution of the red prussiate of potash is, that it takes a con- siderable time to dissolve and because it must be filtered. There are many impurities in this chemical which can be removed by filtering. Without filtering, the solution will not look clear. The reason for making no stock solution of the ferric citrate of ammonia is that such solution soon becomes moldy and unfit for use. This ferric salt is brought into the market in a very pure state, and does not need to be filtered after being dissolved. It dissolves very rapidly. In the solid form it may be preserved for. an unlimited time, if kept in a well -stoppered bottle and protected against the moisture of the atmosphere. A solution of this salt, or a mixture of it with the solution of red prussiate of potash, will remain in a serviceable condition for a number of days, but it will spoil, sooner or later, according to atmospheric conditions. . . . Four ounces of sensitizing solution, for blue prints, are amply sufficient for coating one hundred square feet of paper, and cost about six cents." For copying tracings in blue lines or black on a white ground, one may either employ the recipes given in Lietze's and Pettit's work, or obtain paper already sensitized, from the leading dealers in draughtsmen's supplies. The latter course huis become quite as economical, also, for the ordinary blue- print, as the preparing of one's own supply. For copying a drawing in any desired color the following method, known as Tilhefs, is said to give good results: "The paper on which the copy is to appear is first dipped in a bath con- sisting of 30 parts of white soap, 30 parts of alum, 40 parts of English glue, 10 parts of albumen, 2 parts of glacial acetic acid, 10 parts of alcohol of 60°, and 500 parts of water. It is afterward put into a second bath, which contains 50 parts of burnt umber ground in alcohol, 20 parts of lampblack, 10 parts of English glue, and 10 parts of bichromate of potash in 500 parts of water. They are now sensitive to light, and must, therefore, be preserved in the dark. In preparing jjaper to make the positive print another bath is made just like the first one, except that lampblack is substituted for the burnt umber. To obtain colored positives the black is replaced by some red, blue or other pigment. In making the copy the drawing to be copied is put in a photographic printing frame, and the negative paper laid on it, and then exposed in the usual manner. In clear weather an illumination of two minutes will suffice. After the exposure the negative is put in water to develop it, and the drawing will appear in white on a dark ground; in other words, it is a negative or reversed picture- The paper is then dried and a positive made from it by placing it on the glass of a printing- frame, and laying the positive paper upon it and exposing as before. After placing the frame in the sun for two minutes the positive is taken out and put in water. The black dissolves off without the necessity of moving back and forth." 100 THEORETICAL AND PRACTICAL GRAPHICS. PHOTO -AND OTHER PROCESSES. 275. If a drawing is to be reproduced on a different scale from that of the original, some one of the processes which admits of the use of the camera is usually employed. Those of most importance to the draughtsman are (1) wood engraving; (2) the "wax process" or cerography; (3) lithography, and (4) the various methods in which the photographic negative is made on a film of gelatine which is then used directly — to print from, or indirectly — in obtaining a metal plate from which the impressions are taken. In the first three named above the use of the camera is not invariably an element of the process. All under the fourth head are essentially photo -processes and their already large number is constantly increasing. Among them may be mentioned photogravure, collotype, phototype, autotype, photo- glyph, albertype, heliotype, and heliogravure. WOOD ENGRAVING. 276. There is probably no process that surpasses the best work of skilled engravers on wood. This statement will be sustained by a glance at Figs. 14, 15, 20-24, 134, 136, and those illustrating mathematical surfaces, in the next chapter. Its expensiveness, and the time required to make an illustration by this method, are its only disadvantages. Although the camera is often employed to transfer the drawing to the boxwood block in which the lines are to be cut, yet the original drawing is quite as frequently made in reverse, directly on the block, by a professional draughtsman who is supposed to have at his disposal either the object to be drawn or a photograph or drawing thereof. The outlines are pencilled on the block, and the shades and shadows given in brush tints of India ink, re -enforced, in some cases, by the pencil, for the deepest shadows. The "high lights" are brought out by Chinese white. A medium wash of the latter is also usually spread upon the block as a general preliminary to outlining and shading. The task of the engraver is to reproduce faithfully the most delicate as well as the strongest effects obtained on the block with pencil and brush, cutting away all that is not to appear in black in the print. The finished block may then be used to print from directly, or an electrotype block can be obtained from it which will stand a large number of impressions much better than the wood. CEROGRAPHY. 277. For map -making, illustrations of machinery, geometrical diagrams and all work mainly in straight lines or simple curves, and not involving too delicate gradations, the cerographic or "wax process" is much employed. For clearness it is scarcely surpassed by steel engraving. Figures 86, 90 and 107 are good specimens of the effects obtainable by this method. The successive steps in the process are (a) the laying of a thin, even coat of wax over a copper plate; (b) the transfer of the drawing to the surface of the wax, either by tracing or — more generally — by photography; (c) the re -drawing or rather the cutting "of these lines in the wax, the stylus removing the latter to the surface of the copper; (d) the taking of an electrotype from the plate and wax, the deposit of copper filling in the lines from which the wax was removed. Although in the preparation of the original drawing the lines may preferably be inked, yet it is not absolutely necessary, provided a pencil of medium grade be employed. LITHOGRAPHY. — PHO TO-ENGRA VING. 101 Any letters desired on the final plate may be also pencilled in their proper places, as the engraver makes them on the wax with type. A surface on which section -lining or cross-hatching is desired may have that fact indicated upon it in writing, the direction and number of lines to the inch being given. Such work is then done with a ruling machine. Errors may readily be corrected, as the surface of the wax may be made smooth, for recutting, by passing a hot iron over it. , LITHOGRAPHY. — PHOTO - LITHOGRAPHY. — CHROMO - LITHOGRAPHY. 278. For lithographic processes a fine-grained, imported limestone is used. The drawing is made with a greasy ink — known as "lithographic" — upon a specially prepared paper, from which it is transferred under pressure, to the surface of the stone. The un- inked parts of the stone are kept thoroughly moistened with water, which prevents the printer's ink (owing to the grease which the latter contains) from adhering to any portion except that from whicli tlie impressions are desired. Photo -lithography is simply lithography, with the camera as an adjunct. The positive might be made directly upon the surface of the stone by coating the latter with a sensitizing solution; but, in general, for convenience, a sensitized gelatine film is exposed under the negative, and by subsequent treatment gives an image in relief which, after inking, can be transferred to the surface of the stone as in the ordinary process. Chromo- lithography, or lithography in colors, has been a very expensive process, owing to its requiring a separate stone for each color. Recent inventions render it probable that it will be much simplified, and the expense correspondingly reduced. The details of manipulation are closely analogous to those of ink prints. When colored plates are wanted, in which delicate' gradations shall be -indicated, chromo - litho- graphy may preferably be adopted; although "half-tones," with colored inks, give a scarcely less pleasing effect, as illustrated by Figs. 7-10, Plate II. But for simple line -work, in two or more colors, one may preferably employ either cerography or photo - engraving, each of which has not only an advantage, as to expense, over any lithographic process, but also this in addition — that the blocks can be used by any printer; whereas lithographing establishments necessarily not only prepare the stone but also do the printing. , PHOTO - ENGRAVING. — PHOTO - ZINCOGRAPHY. 279. In this popular and rapid process a sensitized solution is spread upon a smooth sheet of zinc, and over this the photographic negative is placed. Where not acted on by the light the coat- ing remains soluble and is washed away, exposing the metal, which is then further acted on by acids to give more relief to the remaining portions. Except as described in Art. 281 this process is only adapted to inked work in lines or dots, which is reproduced faithfully, to the smallest detail. Among the best photo - engravings in this book are Figs. 10 and 11, 50, 79 and 80. 280. The following instructions for the preparation of drawings, for reproduction by this process, are those of the American Society of Mechanical Engineers as to the illustration of papers by its members, and are, in general, such as all the engraving companies furnish on application. "All lines, letters and figures must be perfectly black on a white ground. Blue prints are not available, and red figures and lines will not appear. The smoother the paper, and the blacker the ink, the better are the results. Tracing-cloth or paper answers very well, but rough paper — even 102 THEORETICAL AND PRACTICAL GRAPHICS. Whatman's — gives bad lines. India ink, ground or in solution, should be used; and the best lines are made on Bristol board, or its equivalent with an enameled surface. Brush work, in tint or grading, unfits a drawing for immediate use, since only line work can be photographed. Hatching for sections need not be comj)leted in the originals, as it can be done easily by machine on the block. If draughtsmen will indicate their sections unmistakably, they will be properly lined, and tints and shadows will be similarly treated. The best results may be expected by using an original twice the height and width of the proposed block. The reduction can be greater, provided care has been taken to have the lines far enough apart, so as not to mass them together. Lines in the plate may run from 70 to 100 to the inch, and there should be but half as many in a drawing which is to be reduced one -half; other reductions will be in like proi)ortion. Draughtsmen may use photographic prints from the objects if they will go over with a carbon ink all the lines which they wish reproduced. The photographic color can be bleached away by flowing a solution of hi -chloride of mercury in alcohol over the print, leaving the pen lines only. Use half an ounce of the salt to a pint of alcohol. Finally, lettering and figures are most satisfactorily printed from type. Draughtsmen's best efforts are usually thus excelled. Such letters and figures had therefore best be left in pencil on the drawings, so they will not photograph but may serve to show what type should be inserted." To the above hints should be added a caution as to the use of the rubber. It is likely to diminish the intensity of lines already made and to affect their sharpness; also to make it more difficult to draw clear-cut lines wherever it has been used. It may be remarked with regard to the foregoing instructions that they aim at securing that uniformity, as to general appearance, which is usually quite an object in illustration. But where the preservation of the individuality and general characteristics of one's work is of any importance what- ever, the draughtsman is advised to letter his own drawings and in fact finish them entirely, himself, with, perhaps, the single exception of section -lining, which may be quickly done by means of Day^s Rapid Shading Mediums or by other technical processes. 281. Half Tones. Photo -zincography may be employed for reproducing delicate gradations of light and shade, by breaking up the latter when making the photograpliic negative. The result is called a half tone, and it is one of the favorite processes for high-grade illustration. Figs. 95 and 130 illustrate the effects it gives. On close inspection a series of fine dots in regular order will be noticed, or else a net- work, so that no tone exists unbroken, but all have more or less white in them. The methods of breaking up a tone are very numerous. The first patent dates back to 1852. The principle is practically the same in all, viz., between the object to be photographed and the plate on which the negative is to be made there is interposed a "screen" or sheet of thin glass, on which the desired mesh has been previously photographed. In the making of the "screen" lies the main difference between the variously - named methods. In Meissenbach's method, by which Figs. 95 and 130 were made, a photograph is first taken, on the "screen," of a pane of clear glass in which a system of parallel lines — one hundred and fifty to the inch — has been cut with a diamond. The ruled glass is then turned at right angles to its first position and its lines photographed on the screen over the first set, the times of exposure differing slightly in the two cases, being generally about as 2 to 3. This process is well adapted to the reproduction of "wash" or brush -tinted drawings, photo- graphs, etc. The object to be represented, if small, may preferably be furnished to the engraving company and they will photograph it direct. PHOTOGRAPHIC ILLUSTRATIVE PROCESSES. 103 GELATINE FILM PHOTO - PROCESSES. 282. As stated in Art. 275, in which a few of the above processes are named, a gelatine film may be employed, either as an adjunct in a method resulting in a metal block, or to print from directly; in the latter case the prints must be made, on special paper, by the company preparing the film. In the composition and manipulation of the film lies the main difference between otherwise closely analogous processes. For any of them the company should be supplied with either the original object or a good drawing or photographic negative thereof Not to unduly prolong this chapter — which any sharp distinction between the various methods would involve, yet to give an idea of the general principles of a gelatine process we may conclude with the details of the preparation of a heliotype plate, given in the language of the circular of a leading illustrating company. Figs. 1- — 5 of Plate II illustrate the effect obtained by it. "Ordinary cooking gelatine forms the basis of the positive plate, the other ingredients being bichro- mate of potash and chrome alum. It is a pecularity of gelatine, in its normal condition, that it will absorb cold luater, and swell or expand under its influence, but that it will dissolve in hot water. In the preparation of the plate, therefore, the three ingredients just named, being combined in suitable propor- tions, are dissolved in hot water, and the solution is poured upon a level plate of glass or metal, and left there to dry. When dry it is about as thick as an ordinary sheet of parchment, and is stripped from the drying -plate, and placed in contact with the previously - prepared negative, and the two together are exposed to the light. The presence of the bichromate of potash renders the gelatine sheet sensitive to the action of light; and wherever light reaches it, the plate, which was at first gelatinous or absorbent of water, becomes leathery or waterproof. In other words, wherever light reaches the plate, it produces in it a change similar to that which tanning produces upon hides in converting them into leather. Now it must Ije understood that the negative is made up of trans- parent parts and opaque parts; the transparent parts admitting the passage of light through them, and the opaque parts excluding it. When the gelatine plate and the negative are placed in contact, they are exposed to light with the negative uppermost, so that the light acts through the translucent portions, and waterproofs the gelatine underneath them; while the opaque portions of the negative shield the gelatine underneath them from the light, and consequently those parts of the plate remain unaltered in character. The result is a thin, flexible sheet of gelatine of which a portion is water- proofed, and the other portion is absorbent of water, the waterproofed portion being the image which we wish to reproduce. Now we all know the repulsion which exists between water and any form of grease. Printer's ink is merely grease united with coloring- matter. It follows, that our gelatine sheet, having water applied to it, will absorb the water in its unchanged parts; and, if ink is then rolled over it, the ink will adhere only to the waterproofed or altered parts. This flexible sheet of gelatine, then, prepared as we have seen, and having had the image impressed upon it, becomes the heliotype plate, capable of being attached to the bed of an ordinary printing -jjress, and printed in the ordinary manner. Of course, such a sheet must have a solid base given to it, which will hold it firmly on the bed of the press while printing. This is accomplished by uniting it, under water, with a metallic plate, exhausting the air between the two surfaces, and attaching them by atmospheric pressure. The plate, with the printing surface of gelatine attached, is then placed on an ordinary platen printing-press, and inked up with ordinary ink. A mask of paper is used to secure white margins for the prints; and the impression is then made, and is ready for issue." " The study of Descriptive Geometry possesses an important philosophical peculiarity, quite independent of its high industrial utility. This is the advantage which it so pre-eminently offers in habituating the mind to consider very complicated geometrical combinations in space, and to follow ivith precision their continual correspondence with the figures which are actually traced — of thus exercising to the utmost, in the most certain and precise manner, that important faculty of the human mind which is properly called 'imagination,' and which consists, in its elementary and positive acceptation, in representing to ourselves, clearly and easily, a vast and variable collection of ideal objects, as if they were really before us While it belongs to the geometry of the ancients by the character of its solutions, on the other hand it approaches the geometry of the moderns by the nature of the questions which compose it. These questions are in fact eminently remarkable for that generality which constitutes the true fundamental character of modern geometry ; for the methods used are always conceived as applicable to any figures whatever, the peculiarity of each having only a purely secondary influence.** Auguste Comte: Cours de Philosophic Positive. ' 'A mathematical problem may usually be attacked by what is termed in military parlance the method of 'systematic approach;' that is to say, its solution may be gradually felt for, even though the successive steps leading to that solution cannot be clearly foreseen. But a Descriptive Geometry problem must be seen through and through before it can be attempted. The entire scope of its conditions as well as each step toward its solution must be grasped by the imagination. It must be 'taken by assault.'" George Sydenham Clarke, Captain, Royal Engineers. THE GEOMETRIE DESCRIPTIVE OF GASPARD MONGE. 105 CHAPTER IX. ORTHOGEAPHIC PROJECTION UPON MUTUALLY PERPENDICULAR PLANES. — DEFINITION, CLASSIFICATION AND ILLUSTRATION OP MATHEMATICAL LINES AND SURFACES. 283. In this and nearly all the later chapters of this work the principles of what has been generally known as Descriptive Geometry are either examined or applied. In Art. 19 — which, with Arts. 2, 3 and 14, should be reviewed at this point — reasons are given for calling this science Mong^s Descriptive. Certain German writers call it Monge's Orthogonal Projec- tion. The popular titles Mechanical Drawing, Practical Solid Geometry, Orthographic Projection, etc., are . usually merely indicative of more or less restricted applications of Monge's Descriptive to some special industrial arts; and working drawings of bridge and roof trusses, machinery, masonry and other constructions, are simply accurately scaled and fully dimensioned projections made in accordance with its principles. Monge's service to mathematics and graphical science, which, according to Chasles*, inaugurated the fifth epoch in geometrical history, consisted, not in inventing the method of representing objects by their projections — for with that the ancients were thoroughly familiar, but in perceiving and giv- ing scientific form to the principles and theorems which were fundamental to the special solutions of a great number of graphical problems handed down through many centuries, and many of which had been the monopoly of the Freemasons. Emphasizing in this chapter the abstract principles of the subject, treating it as a pure science, and giving a general outline of the field of its application, its more practical and commercial side is left for the next chapter, including the modifications in vogue in the draughting offices of leading mechanical engineers. Statements without proof are given whenever their truth is reasonably self-evident. FUNDAMENTAL PRINCIPLES. 284. The orthographic projection of a point on a plane is the foot of the perpendicular from the point to the plane. " s'lg-. xss. In Fig. 156 the perpendiculars Pp' and Pp give the pro- jections, p' and p, of the point P. The planes of projection are shown in their space posi- tion; one, H, horizontal, the other, V, vertical. The projection, p, on H, is called the plan or horizontal projection, (h.p.) of P. The point p' is the elevation or ver- tical projection (v. p.) of P. Projections on the vertical plane are denoted by small letters with a single accent or "prime." Projections on H are small letters unaccented. A point may be named by its space -letter, the capital, or by its projections; thus, we may speak of the point P or of the point pp'. •Apercu Hlstorique sur I'Oiigin et le Developpement deg Methodes en G6om6trle. 106 THEORETICAL AND PRACTICAL GRAPHICS. We shall call Pp the H- projector of P, since it gives the projection of P on H. Similarly, Pp' is the Y-projectoT of P. A projector -plane is then the plane containing both projectors of a point, and is evidently perpendicular to both V and H by virtue of containing a line perpendicular to each. 285. V and H intersect in a line called the ground line, hereafter denoted by G. L. They make with each other four diedral angles. The observer is always supposed to be in the first angle, viz., that which is above H and in front of V. We shall call it Q,, or the first quadrant. Q, is then the second quadrant, back of V but above H. Q, is below Q^, while Q, is immediately below the first angle. Points R and S, in the second and third quadrants respectively, have their elevations, r' and a', on opposite sides of G. L., while their plans, r and s, are on the back half of H. The point T, in Q^, has its v. JJ. at t', below G. L., and its h. p. at t, in front of G. L. 286. Transition from pictorial to orthographic view. — Rotation. In making a drawing in the ordi- nary way, and not pictorially, we suppose the planes V and H brought into coincidence by revolution about G. L., the upper part of V uniting with the back part of H, while lower V and front H merge in one. The arrow (Fig. 156) shows such direction of revolution, after which any vertical ■ projection, as p', is found at p\, on a line pz j)erpendicular to G. L. and containing the plan p. This is inevitable, from the following consideration: Any point when revolved about an axis describes a circle whose centre is on the axis and whose plane is perpendicular to the axis ; but as the pro- jector-plane of P contains p' — the point to be revolved, and is perpendicular to G. L. (the axis) because perpendicular to both H and V, it must be the plane of rotation of p', which can therefore only come into H somewhere on pz (produced). In Fig. 157 we find Fig. 156 represented in the ordinary way. Only the pro- jections of the points appear. V and H are, as usual, considered indefinite in ex- tent, and their boundaries have disappeared. A projector- plane is shown only by the line, perpendicular to G. L., in which its intersections with V and H coincide. 287. A point {pp') in the first angle has its h. p. below G. L., and its v. p. above. For the third angle the reverse is the case, the plan, s, above, and the eleva- tion, s', below. The second and fourth angles are also opposites, both projections, rr', being above G. L. for the former, and both, tt', below for the latter. 288. For any angle the actual distance of a point from H, as shown by any H- projector Pp (Fig. 156), is equal to p' z (either figure) — the distance of the v. p. of the point from G. L. Simi- larly, the V-projector of a point, as Rr' (Fig. 156), showing the actual distance of a point from V, is equal to the distance of the h. p. of the point from G. L. If one projection of a point is on G. L. the point is in a plane of projection. If in H, the elevation of the point will be on G. L. ; similarly, the plan is on G. L. if the point lies in V. 289. The projection of a line is the line containing the projections of all its points. The projection of a straight line will be a straight ^^s- ^ss. line; for its extremity -projectors, as ^ a and Bb (Fig. 158) would determine a plane perpendicular to H and containing A B; in such plane all other H- projectors must lie; hence meet H in ab, which is straight because the intersection of two planes. In Fig. 159 we see the line A B of Fig. 158, orthographic- ally represented. 290. Prcjecting planes. — Traces. A plane containing the projectors of a straight line is called a projecting plane of the line. (ABba, Fig. 158). x^ig-. xev. ri x'ig-. isa. FUNDAMENTAL PRINCIPLES OF MONGERS DESCRIPTIVE. 107 ngr- iso. A N -projecting plane of a line contains the line and is perpendicular to the vertical plane. The lA- projecting plane of a line is the vertical plane containing it. Traces. The intersection of a surface by a given line or surface is called a trace. The trace of a line is a point; of a surface is a line. If on H it is called a horizontal trace (h.t.); on V, a vertical trace, abbreviated to v.t. 291. The projection of a cxirve is in general a curve, the trace — upon V or H — of the cylindrical surface* whose elements are the pro- jectors of the points of the curve. (See Figs. 160 and 161). The projection of a plane curve will be equal and parallel to the original curve, when the latter is parallel to x'lgr- isi- the plane on which it is projected. 292. All lines, straight or curved, lying in a plane that is perpendicular to V, will be projected on V in the v. t. of the plane. A similar remark applies q. to the plans of lines lying in a vertical plane. Fig. 162 illustrates these statements. The plane P' Q P, being perpendicular to V — as shown Fig-- ISS. by the h. t. {P Q) being perpen- dicular to G. L — all points in x'igr- 1©3- the plane, as A,B,C,D,E,F, will have their projections on V in the trace P' Q. The plane M'NM is vertical, since MN — its vertical trace .M — is perpendicular to G. L. ; its h.t. (NM) therefore contains the h. p. of every point in the plane. True size of figures in planes not parallel to V or H. Since the triangle ABC and the curve D E F are not parallel to H or V their exact size and shape would not be shown by their projections; these could, however, be readily obtained by rotating their plane into H, about the trace PQ, or into V about P' Q. Such rotation, called rabatment^, is described in detail in Art. 306. 293. Traces of lines. Fig. 163 shows that the h. t. of a line A B is at the intersection of the plan a 6 by a perpendicular to G. L. from s', where the elevation a'b' crosses G. L. Similarly, the v.t. of the line is on its V. p., immedi- ately below r, the intersection of q. the ground line by the plan ab; hence the rule : To find the hor- izontal trace of a line prolong the vertical projection until it meets the ground line; thence draw a perpendicular to G. L. to meet the plan of the line. An analogous construction gives the vertical trace of the line. Fig. 164 illustrates by the ordinary method the same projections and traces as in Fig. 163. ^■ig-- ie-i. •See Kemark, Art. 8. tFroni the French rabaitement. 108 THEORETICAL AND PRACTICAL GRAPHICS. x-igr- iss. ^•ig:. iss. 294. The angle between a line and H or V. The inclination, 6, of a line to H, is that of ths line to its plan ah. The angle (^) between a line and V, is that between it and its v. p., a' h'. 295. Any horizontal line has a plan, a b (Fig 165), equal and parallel to itself Its elevation, a'U, is parallel to G. L., and at the same distance from it as the line from H. The line makes the same angle with V that its plan makes with G. L. Such line can evidently have no h. t. Its v. t. would be found by the rule in the preceding article. In the extreme case of perpendicularity to V the v. p. bi A B would reduce to a point. 296. A line parallel to V but oblique to H has its h. p. parallel to the ground line; makes with the same angle that its v. p. does with G. L. ; equals its v. p., and has Fig-. IST-. H its E'igr. iss. h. t. found by the usual rule. (Fig. 167). 297. A vertical line has no v. t.; is projected on H in a point; has its v. p. parallel and equal to itself (See CD, Fig. 168). A line parallel to both V and H is parallel to their intersection, has no traces, and each pro- jection equals the line. (See M N, Fig. 168). 169 shows, orthographically, the lines A B, CD and MN of the two preceding figures. 298. Representation of planes. A plane is represented by its traces. Like H and V, any plane is considered indefinite in extent when drawn in the usual way; though our pictorial diagrams show them bounded, to add to the appearance. A horizontal plane has but one trace, that on V. A plane parallel to V, as MNKO, Fig. 170, has no vertical E'lgr- ±S3- ^^^ c a » ^„^^ w! s> ■^ d' cid h f % 6 i!^l 1 trace, and its horizontal trace MN is parallel to G. L. A plane will have parallel traces when it is parallel to G. L. but oblique to both H and V. {RS, R'S', Fig. 170). Fig. 171 shows the planes of Fig. 170, as usually rei»resented. The traces of a plane not parallel to G.L. must meet at the same point on G. L. 299. Planes perpendicular to a plane of projection. If a ^-ig-. 171. FlC- IVO. p' 9' P' i \ ^ ^ G ^^ 'V =^ \. ^^ ^\^ A ^ r'' Fis- ±vz. plane is perpendicular to a plane of projection its trace on the other plane is perpendicular r , " ^ to G. L. This is the only case in which the angles between the ground line and the traces of a plane equal the dihedral P angles made with H and V by the ^ ** plane. Such equality may be thus shown : The dihedral angle between two planes equals the plane angle between two lines, cut from the planes by a third plane perpendicular to both ; plane P' QR (Fig. 172) is by hypothesis perpendicular to V; hence the ground line and P' Q are the lines cut by plane V from two other planes to which it is perpendicular; and their angle 6 is, therefore, the measure of the dihedral angle between H and P'QR. Similar reasoning applies to N'SN. A"^-^ UNivERoiry FUNDAMENTAL PRINCIPLES OF MONGERS DESCRIPTIVE. 109 300. Plane determined by lines. — Lines drawn in a plane. Fig. 173 illustrates all the possibilities. (a) Any line parallel to H is necessarily a horizontal line; but when also contained by a plane it is called a horizontal of the plane. It is obviously parallel to the h. t. {Q R) of the plane. (b) Any line parallel to V can have that fact discovered by the parallelism of its plan to G. L. ; jf^y" v.p. of horizontal IFig-. S-T-S. but when contained by a plane we shall call it a V-parallel of the plane. Such line will evidently be parallel to the v. t. of the plane in which it lies. (c) The oblique lines CD and E F, with those just described, illustrate the additional fact that the traces of lines that lie in a plane will be found on the traces of the plane. This furnishes the following — and usual — method of determining a plane, i.e., by means of two lines known to lie in it: Prolong the lines until they meet the plane of projection; their H -traces joined give the h. t. of the plane. Similarly for the vertical trace of the plane. (d) Conversely, to assume a line in a plane assume its traces on those of the plane. (e) Two intersecting lines or two parallel lines determine a plane. Three points not in a straight line, or a point and a straight line may be reduced to either of the foregoing cases. 301. Lines of declivity. A line lying in a plane' and making with H or V the same angle as the plane, is called a line of declivity. Figs. 175 and 176 show a line of declivity with respect to H. Both the line and its plan must be perpendicular to the h. t. of the plane; for the inclin- ation of a line to its plan is that of the line to H ; and if, at the same time, Fi.s.±vs. the measure of a dihedral angle, such lines must, by ele- mentary geometry, be perpendicular to the intersection of the planes. Fig. 177 gives a line of declivity with respect to V. 302. Limiting angles. Were the plane P'QR (Fig. 175) rotated on its line of declivity (ab, a'h') it would make an increasing angle ^ig-- 177. with H until perpendicular to it. If, then, a jjlane con- tain a line, its inclination -limits ^re 90° and that of the line. On the other hand, the line of declivity might be turned in the plane, until Q — g--; horizontal. The limits of the inclinations of lines in a plane are therefore 0° and that of the plane. A plane may make 90° with H and be parallel to V, or vice versa; or it may be perpendicu- lar to both H and V; the limits of the sum of its inclinations to H and V are thus 90° and 180°. If parallel to G. L., but inclined to H and V, the sum of the inclinations of the plane is the lower limit, 90°. 110 THEORETICAL AND PRACTICAL GRAPHICS. 303. Lines perpendicular to planes. ^ig-. ITS. If a plane is equally inclined to both H and V, but cuts G. L., its traces will make equal angles with the latter. If a right line, as A B (Fig. 178), is perpendicular to a plane P' Q R, its projections will be perpendicular to the traces of the plane. For the plan of the line lies in the h. t. of its H- projecting plane; the latter plane is — from its definition — perpendicular to H; is al^ perpendicular to the given plane by virtue of containing the given line; hence is perpendicular to the h. t. of the given plane, since such h. t. is the intersec- tion of the latter with H. The same principles apply to the relation of the elevation of the line to the v. t. of the plane. 304. Profile planes. Any plane perpendicular to both H and V is called a profile plane. Such plane (P, Fig. 179) is used when side or end views of an object are to be projected. To bring V, H and P into one plane we suppose the latter first rotated into V about their line of intersection, L, then both V and P about G. L. into H. Projections on the profile plane are usually lettered with a double accent, the same as for any point revolved into or parallel to V. When, as in Fig. 179, all the projectors of a line A F lie in the same projector -plane, both projections of the line will be perpendicular to G. L. The most convenient method of dealing with such line is to project it upon a profile plane and revolve- the latter into V; or the profile projector -plane through the line might be directly revolved into V, carrying the line with it. The former method is illustrated in Figs. 179 and 180. In this, as in many other constructions, we make use of the following principles : (a) All projections of one point on two or more vertical planes will be at the same height above H. (b) If rotation occur about an axis that is perpendicular to H, each arc, V__ described about that axis by a point revolved, will be projected on H as an equal circular arc ; similarly as to V, if the axis is perpendicular to it. Since in Figs. 179 and 180 we rotate upon a vertical axis, a projector, as A a", will be seen in a s, drawn through the plan of A and perpendicular 1^^ to the h. t. of P. From s a circular arc, ss^ (Fig. 180), from centre L, will be the plan of the arc described by a" of Fig. 179. From «, a vertical to the level of a' gives a". Similarly the projection /" is obtained, which, joined with a", gives a"f", the profile* view of the line AF. 305. As far as our view of what is in the first angle is concerned, the rotation just described amounts, practically, to the turning of H and V through an angle of 90°, so that instead of facing V we see it "edgewise," / y^'f ','""']' as a line 31 o; while H appears also as a line, GLs^. We thus get an '^ ^' ' "^' "end view" into the angle. All figures lying in profile planes are then seen in their true form. In Fig. 181 let us start with the entire system of angles thus turned. The ground line appears as a point, T; H and V as lines; and two lines, AB and CD — each of which lies in a profile plane— are shown in their true length and inclination. FUNDAMENTAL PRINCIPLES OF MONGERS DESCRIPTIVE. Ill Peroendiculars to H from A,B,C and D, give their revolved plans a i, &,,<;, and di- V- projectors give d', c', etc., the heights of the elevations of the points. Revolving the whole system into its usual position, remembering, meanwhile, that the profile plane, P, turns on a vertical axis (as in Fig. 182) which divides P into parts which are on opposite sides of the axis both before and after revolution, we find Cj at c; d^ at d; a, at x. Assuming S S' as the plane of the line CD, and that the plane oi A B is RP', at a given distance TQ from SS', we find a' and b' on R P' at the same level as A and B; while a is derived from x, and h from 6, as shown. The elevations c' and d' on SS' are on the level of C and D respectively. 306. Rabatment, and analogous rotations. The term rabatment, already employed (Art. 292) to indicate the rotation of a plane about one of its traces until it comes into a plane of projection, is also used to denote the rotation of a point or line into H or V about an axis in such plane. Restoration to an original space -position will be called counter -rabatment or counter-revolution. ng-. les. In Fig. 183 we have a, as the rabatment of A into H, about an axis mn. Bai is equal to BA, i.e., to the actual distance of A from the axis, and which is evidently the hypothenuse of the triangle A B a, whose altitude is the H- projector {Ad) of the point, and whose base is the h. p. (a B) of the real distance, A B. Were the axis parallel to and not in H we would state the prin- ciple thus: In revolving a point about, and to the same level with, an horizontal axis it will be found on a perpendicular drawn through the h. p. of the point to the h. p. of the axis, and at a distance from the latter equal to the hypothenuse of a right-angled triangle whose altitude equals the difference of level of point and axis, and whose base is the h. p. of the real distance. Were the axis in or parallel to V, the base of the triangle constructed would be the v. p. of the desired distance, and the altitude would be — in the first case — the V- projector of the point, and — in the second case — the difference of distances of point and axis from V. In any case, the vertex of the right angle, in the triangle constructed, is the projection of the original point on that plane of pro- jection in which or parallel to which the axis is taken. In usual position the foregoing construction would appear thus: With the point given by its pro- jections, (a a', Fig. 184), let fall a perpendicular through a to mn; prolong this ^^'^^' indefinitely; make a the vertex of the right angle in a right triangle of base Ba, altitude a's; then the hypothenuse of such triangle, used as a radius of arc Q a^tti (centre B) gives a, as the revolved position desired. 307. In applying the foregoing principles in the following problems we shall frequently find it convenient to employ the right cone as an auxiliary surface. All the elements of such a cone are equally inclined to its base, and a tangent plane to the cone makes with the base the same angle as the elements. The element of tangency is a line of declivity of the . plane with resj)ect to the base of the cone. If the base of the cone is on H, the h. t. of a tangent plane will be tangent to the base of the cone; similarly for its v. t. were the base in V. 308. Prob. 1. Prom the projections of a line to determine (a) its traces; (b) its actual length; (c) its inclinations, 6 and 4>, to H and V respectively. (a) The traces of the given line, when it is oblique to both H and V, as in Fig. 186, are found by the rule given in Art. 293. 112 THEORETICAL AND PRACTICAL GRAPHICS. It will show the true length on 61 :::iv!h.f. 4.... ...4W i i i ! VOL \ \ (b) The actual length of a Une may be found either (1) by rabatment into H or V, or (2) by rotation until parallel to H or V. By the first method rabat the Hne on its plan aft as an axis. H in «!&,, the distance aa^ equalling a' — the original height of the point A above H; similarly, 6 6, equals b' n. Notice that a^a and 6,6 must be perpendicular to the axis, and that each is the projection of a circular arc, described by the point revolved. The point where a revolved line meets the axis of rotation is common to both the original and the revolved positions of the line. In illustration see h. t. and v. t.. Fig. 186. If we make a' a" and 6' 6" perpendicular to a' b' and equal to ao and bn respectively, we have in a" b" the real length, shown on V. By rotation till parallel to a plane of projection, as H, either extremity of the line may be brought to the level of the other, when the new plan will show the actual length. Thus, (Fig. 187), tmng Fis. iar. a horizontal azis, (the V- projector of 66') a' may be brought to a", at the level of 6', by an arc, centre 6', radius a'b'. The circular arc a' a" thus described has its h. p. in o,«, the distance from V having been constant during the rotation, since the axis was perpendicular to V. In 0,6 we then have the real length sought. If we rotate the line on a vertical axis through a until 6 reaches 6,, we will find the v. p. of the revolved point at 6", on its former level. The new pro- jection, a'b", is again the real length, now projected on V. (c) The inclinations, and ^, to H and V respectively. Either of the foregoing constructions for showing the real length of a line solves also the problem as to inclination. Thus, in Fig. 186, the rabatted lines make with their axes of rabatment the angles sought. In Fig. 187 we have a'b" inclined 6° to H, while 6a, makes with aa^ the angle ^. When the^ line lies in a profile plane the traces, length and inclination are found by means of the operation described in Art. 305 and illustrated by Fig. 181. In that figure, were cd and c' d' given, we would carry c and d about T as a centre to c, and d^, whence perpendiculars to their former levels would give C and D; joining the latter we would have CD, whose v. t. is at z; h. t. (not shown) at a distance Ty above T; while 6 and ^ are seen in actual size at y and 2. 309. Prob. 2. To determine the projections of a line of given length, having given its angles, 19 and , vnth H and V respectively. If with the line we generate a E'lg'- isa. vertical right cone of base angle 6, four elements could be found on the cone, each of which would make with V an angle equal to <^ and therefore fulfill all the conditions. The sum of 6 and <^ can obviously not exceed 90°, and when equalling that limit there could be but two solutions on a given cone, and both would lie in a profile plane. For data take length of line, 2"; (9 = 44°; made ivith H and V respectively, by a given plane; (b) the angle between the traces of the plane. From the properties of the cone and its tangent plane mentioned in Art. 307, we may solve the problem by generating a cone with a line of declivity of the plane, and ascertaining the inclination of such line. (a) In Fig. 194 let P'QR be the plane. The projections a'b' and ab — the latter perpendicular to RQ — represent a line of declivity of the plane with respect to H. With it, and about a' a as an axis, gene- rate a semi-cone. When the generatrix reaches V, either at b"a' or a'n, its inclination to G. L. shows the base angle of the cone, and therefore the inclination of the given plane. With respect to V the construction is analo- gous. A line of declivity with respect to V Tias its v.p. ic'd') at 90° to P'Q, (Fig. 195). Using d, on the h.t. of the plane, as the vertex of a semi -cone of horizontal axis dd', we find the base of the cone tan- gent to P'Q at c'. Carrying c' to the ground line, about d' as a, centre, ■and joining it with d gives the angle <^ sought. This problem might also be readily solved by rabatting the line of declivity into a plane of projection. Thus making d'd" perpendicular to c' d' and equal to d'd, we find the angle between c' d' and c'd". For a plane parallel to G. L. use the auxiliary profile plane, rotating its line of intersection with the given plane as in Fig. 196. (b) The angle between the traces is obtained by rabatment of the given plane about either of its traces. In Fig. 195, using trace RQ as an axis and rotating QP' about it, any point, c', thus turned wiU describe an arc projected in a perpendicular through c io QR. Q being on the axis is constant during this rotation, and the distance from it to c' will be the same after as before revolution; therefore cut cci by^an arc, centre Q, radius Qc'; the desired angle is (3 ox c^QR. ' rigr. iss. ng-- LSS. m' N ^ r M N / 116 THEORETICAL AND PRACTICAL GRAPHICS. FlS- IS'7. 319. Prob. 12. To obtain the traces of a plane, having given its inclinations, and <^, to H and V respectively. This is, obviously, the converse of Prob. 11 and is, practically, the same construction in reverse. The required plane will be tangent, simultaneously, to two semi -cones of base angles and <^, and having axes (a) in V and H respectively, and (b) in the same profile plane. Assume in V any vertical line a' a" as the axis of the 6 -cone, and draw from any point of it, as a', a line a'b", at 6° to G. L.; use a"b", the plan of this line, as radius of the base of the vertical semi -cone, to which the desired h. t. of the required plane will be tangent. The line a"s shows the perpendicular distance from a to the point of intersection of the two elements of tangency of the required plane with the cones; hence the generatrix of the ^-cone must, when in H, be at an angle c^ with G. L., and tangent to arc ski of radius a"s, centre a", and is there- fore x^c". Draw x,yx for the half- base of the ^-cone. The h. t. of the plane sought is then RQ, drawn through c" and tangent to the base of the ^-co.ne; while the v. t. is Q P', tangent to base x,y x. For the limits of + refer to Art. 302. 320. Prob. 13. To draw two parallel planes at a given distance apart. Parallel planes have parallel traces. To draw (Fig. 198) a plane at a distance of i" ^^s- ^^s- from plane P'QR, rotate a line of declivity of P'QR into V at a'b". Draw c'd", parallel to and }" from a'b", to represent (in V) the line of declivity of the plane sought. It meets a a' (prolonged) at c', through which draw c' N parallel to P'Q. From N draw the trace MN, parallel to RQ. M'NM then fulfills the conditions. 321. Prob. Uf. To obtain the line of intersection of two planes. As two points determine a line, we have merely to find two points, each of which lies in both planes, and join them. In Fig. 199 the line in space which would join a', the intersection of the vertical traces of the planes, with b, the corresponding point on the ^-igr- iS9. horizontal traces, would be the required line. Its projections are ab, a'b'. Were the H- traces of the planes parallel, the line of intersection would be horizontal. Were the V- traces parallel, ^^sr- 200. their line of intersection would be a V- parallel of each plane. If both planes were parallek-to G. L., a profile plane might advantageously be employed in the solution. The line sought would be parallel to G. L. 322. Prob. 15. To find the point of intersection of a line and plane find the line of intersection of the given plane by any aux- iliary plane containing the line; the given line will meet such line of intersection in the daeired point. In Fig. 200 mnv' is an auxiliary vertical plane through the given line ab, a'b'. The given MONGE'S DESCRIPTIVE.— ELEMENTARY PROBLEMS. 117 plane, P'QR, intersects the plane mnv' in the line mn, m'n'. The elevations a'b' and m'n' meet at o'^the v.p. of the point sought, whose h. p. must then be on a vertical through o' and on both the other projections of the lines, hence at o. 323. Prob. 16. To show the actual size of the angle between two intersecting planes. A plane perpendicular to the line of intersection of the planes will cut from them lines whose plane angle equals the dihedral angle sought. Let a'b', ab be the line of intersection of the planes P'QR and M' N M. Any line mn, drawn perpendicular to ab, may be taken as the h.t. of an auxiliary plane perpendicular to the line of intersection. The part sn, included be- tween the H -traces of the given planes, will — with the lines cut from the lat- ter — form a triangle whose vertical angle is that desired. The altitude of this triangle is a line projected on cb, and — in space — is a perpendicular from c io AB. To ascertain its length rabatthe H- projecting plane oi AB into V, about a' a as an axis. The point c appears at c", and the line AB at a'b". The altitude sought ap- pears in actual size at c"d", perpen- dicular to a'b'. From c lay off on cb the length cd, equal to c"d"; sdiU, or /3, is the required angle. A vertical from d" to G. L., thence an arc to cb (from centre a), gives d, the plan of the vertex of the angle in space. When both planes are parallel to G. L. use an auxiliary profile plane in solving. 324. Prob. 17. To determine the angle between a line and a plane let fall a perpendicular from any point of the line to the plane; -the angle between the perpendicular and the given line will be the complement of the desired angle. The principles involved are those of Arts. 303 and 311. 325. Prob. IS. To show in its true length the distance from a point to a plane. If the plane is perpendicular to H or Y the distance is evident without any construction. Thus, in Fig. 202, a perpendicular through the point to the plane is seen in its true length, o's', because in space it is parallel to V. Similarly, tr is the actual length of the perpendicular from tt' to the vertical plane v'fh. When the plane is oblique to both H lanes, wliile the straight directrix ^'ig-. ssi. is perpendicular to the planes of the circles and passes through the middle point of the line joining their centres. In oblique or skew arch construction one of the best known methods' ie that in which the sofiit of the arch is a corne de vachc, for which, obviously, only one-half of the surface would be employed. (Fig. 221). CYLINDROID OF FRKZIKR. COKNE DE VACUE. DOUBLE CUKVKD SURFACES. THE TORUS. 362. DoMe Curved Surfaces are surfaces that cannot be generated by a right line. 363. Double Curved Surfaces of Revolution. The sphere is the most familiar example under this head, the generatrix being a semi -circle and the axis its diameter. After it come the ellipsoids — the prolate spheroid and the oblate spheroid — generated by rotating an ellipse about its major or minor axis E-isr- 322- respectively; the paraboloid of revolution, generatrix — a parabola, axis — that of the curve; the hyperboloid of revolutum, of separate nappes, formed by rotating the two branches of an hyperbola about their transverse axis; and the tm-us — annular or not — generated by revolving a circle about an axis in its plane but not a diameter. (See Fig. 222; and also Arts. 112-114). 364. The revolution of other plane curves — as the involute, tractrix, conchoid, gives double curv.ed surfaces of frequent use in architectural constructions and the arts.' 365. Dovhle Curved Surfaces of Transposition. Of the innumerable surfaces possil)le under this head we need only mention here the serpentine, generated by a sphere whose centre travels along a helix; the ellipsoid of three unequal axes, which would result from turning an ellipse about one of its axes in such manner that, while remaining an ellipse, its other axis should so vary in length that its 'Congruent figures, if superposed, wiU coincide throughout. «For a. comparison of the relative merits of these methods refer to Skew Archea, by E. W. Hyde (Van Nostrand's Science Series, No. 15). For full treatment of the plane sections see Wiener's DarsteUende Oeometrie. 'See Note, p. 64; also Art. 203. 128 THEORETICAL AND PRACTICAL GRAPHICS. extremities would trace a secontl ellipse; the elliptical paraboloid, whose plane sections perpendicular to the axis are ellipses, while sections containing the axis are parabolas; the elliptical hyperboloid of one nappe, generated by turning a variable hyperbola about its real axis so that its arc shall follow an elliptical directrix; the elliptical hyperboloid of two nappes, analogous to the two -napped hyperboloid S-lgr- 223. Fis- 22'S:. Fig-- 22S. SERPENTINE. ELLIPSOID. ELLIPTICAL HYPERBOLOID of revolution, but having elliptical instead of circular sections perpendicular to its axis; and the cyclide,* whose lines of curvature (see Art. 381) are all circles, and each of whose normals intersects two conies — an ellipse and hyperbola — whose planes are mutually perpendicular, and having the foci of each at the extremities of the transverse axis of the other. ■Fi-S- 22©. X-lg-. 22'r. ELLIPTICAL HYPERBOLOID OP TWO NAPPES. THE CYCLIDE OF DUPIN. The cyclide is the envelope of all spheres (a) having their centres on one of the conies and (b) tangent to any sphere whose centre is on the other. The torus is a special form of cyclide. TUBULAR SURFACES. — QUADRIC SURFACES OR CONICOIDS. 366. Among the surfaces we have described, the torm and serpentine belong to the family called tubular, since each is the envelope of a sphere of constant radius. 367. Surfaces whose plane sections are invariably conies are called conicoids or quadrics. These are the cone, cylinder and sphere; ellipsoids; hyperboloids of one and two nappes; elliptic and hyperbolic paraboloids. In theory, all conicoids are ruled surfaces; but on some the right lines are imaginary, while they are real on the cone, cylinder, hyperbolic paraboloid and warped hyperboloid. TANGENTS AND NORMALS TO CURVES. — TANGENT CURVES. 368. A tangent to a curve is a right line joining two consecutive points of the curve. If tangent at infinity it is called an a.symptote. A normal to a curve is a right line perpendicular to the tangent at the point of tangency. Both tangent and normal to a plane curve lie in its plane. Two curves are tangent to each other when they have the same tangent line at any common point. *ror mathematical, optical and other properties of the cyclide see Salmon's Qeometry of Three Dimensiom, and the writings of J. Clerk Maxwell. GENERAL DEFINITIONS. 129 369. If a curve is the intersection of a surface by a plane, the tangent to it at any point will be the intersection of the plane of the curve by the tangent plane to the surface at the given point. The tangent at any point of a non -plane curve, when the latter is the intersection of two sur- faces, is the line of intersection of two planes, each of which is tangent — at the given point — to one of the surfaces. LINES AND PLANES, TANGENT AND NORMAL TO SURFACES. 370. A straight line joining two consecutive points of a surface is a tangent to it. 371. A tangent plane to a surface at any point is the locus of all the right lines tangent to the surface at that point. 372. A right line perpendicular to a tangent plane at the point of tangency is a normal to the surface. 373. Any plane containing the normal cuts from the surface a normal section. TANGENT PLANES TO RULED SURFACES. 374. Since any element of a ruled surface fulfills the condition (Art. 370) necessary to make it a tangent to the surface, it may be taken as one of the two right lines necessary to determine a plane, tangent to the surface at any point of the element. For a doubly ruled surface the two elements through the point would determine the tangent plane. 375. In general, the line which — with an element — would determine a tangent plane to a ruled surface at a given point, would be a tangent, at that point, to any curve passing through the latter and lying on that surface. 376. A plane, tangent to a developable surface at any point, would, therefore, be determined by the element containing the point, and, preferably, by a tangent to the base at the extremity of the element, since to such a surface a tangent plane has line and not merely point contact. The element of tangency belongs to both the nappes which constitute a complete cone, develop- able helicoid or analogous surface; but the plane that is tangent to the surface along so much of the element as lies on one nappe is a secant plane to the other nappe. 377. The tangent plane at any point of a warped surface is found by Arts. 374 and 375, and is also, usually, a secant plane, tangency being along an entire element only in special cases. (See Art. 469). TANGENT PLANES TO DOUBLE CURVED SURFACES. 378. A tangent plane to a double curved surface has, usually, but one i)oint of contact with it. If a normal to the surface can readily be drawn then the tangent plane may be determined most simply on the principle that it will be perj)endicular to the nonnal, at the given point. This method is especially applicable in problems of tangency to a sphere, since the radius to the point of tangency is the normal to the surface; also to any tubidar surface, since such may be regarded as generated by the motion of a sphere, and at any point of the circle of contact of sphere and tubular surface they would have a common tangent plane. In general, by taking the two simplest curves that could be drawn through the point of desired tangency and upon the surface, the tangent plane would be determined by the tangents to these curves at that point. Methods are given in Chapter V for drawing tangents to the more important mathematical curves, among which we would find nearly all of the plane sections of the surfaces defined in this chapter. 130 THEORETICAL AND PRACTICAL GRAPHICS. TANGENT SURFACES. — INTERSECTING SURFACES. 379. (a) Two surfaces are tangent to each other at a given point if at that point they have a common tangent plane. Any secant plane passing through a point of tangency of two tangent surfaces will cut them in sections that have a common tangent line. (b) Developable surfaces will be tangent along a common element if they have a common tangent plane at one point of such element, since the element is one of the determining lines of the tangent plane to each surface. (c) Raccordment, or the mutual tangency of warped surfaces along a common element, exists when at three different points of their common element they have common tangent planes. (d) Double curved surfaces usually have point contact only, with single or double curved sur- faces to which they are tangent. When, however, the tangent surfaces are con -axial and also surfaces of revolution they will have a circle of tangency; and with surfaces of transposition a curve of tangency is also possible. In any case, however, condition (a) above must be fulfilled. (e) The line of interpenetration of two intersecting surfaces may be found by cutting them by a series of auxiliary surfaces; the two sections of the former, cut by any one of the latter, will meet (if at all) in points of the desired line of intersection. The surfaces should be so located with respect to H and V as to facilitate the drawing of the auxiliary sections; and the latter should, if possible, be straight lines or circles, in preference to other forms less easy to represent. RADII AND LINES OF CURVATURE. — OSCULATORY CIRCLE AND PLANE. — GEODESICS. 380. Osculatory circle. — Radius of curvature. — Osculating plane. A circle is osadatory to a non- circular curve when it has three consecutive points in common with it. Its radius is called the radius of curvature of the curve, for the middle one of the three points. When a circle is osculatory to a non- plane curve its plane is called an osculating plane. 381. Line of curvature. It is ascertained by analysis that among all the possible normal sections at any point of a surface two may be found, mutually perpendicular, whose radii of curvature are resjiectively the maximum and minimum for that point; such sections are called principal sections, and their radii the pnncipal radii for that point. If upon any surface a line be so drawn that the tangent to it at any point lies in the direc- tion of one of the .principal sections at that point, the line is called a line of curvature. Since at every point of a surface two principal sections are possible, there may also be drawn through each point two lines of curvature, intersecting each other at right angles. Such curves are shown in white lines on several of the preceding figures illustrating mathematical surfaces. 382. Geodesies. At every point of a geodesic line on a surface the osculating plane is normal to the surface. Either the greatest or shortest distance between two points of a surface would be measured on the geodesic passing through them. Since the maximum or minimum distance between two points on a sphere would be measured on the great circle containing them, such circle would be the geodesic on that surface. The geodesic between two points on a cylinder would be a helix. On any other developable surface it would obviously be the space -form of the straight line which joined the given points on the development of the surface. MECHANICAL DRAWING S.— WORKING DRAWINGS. 131 CHAPTER X. PROJECTIONS AND INTERSECTIONS BY THE THIRD -ANGLE METHOD.— THE DEVELOPMENT OF SURFACES FOR SHEET METAL PATTERN MAKING. — PROJECTIONS, INTERSECTIONS AND TANGENCIES OF DEVELOPABLE, WARPED AND DOUBLE CURVED SURFACES, BY THE FIRST -ANGLE METHOD. 383. The mechanical drawings preliminary to the construction of machinery, blast furnaces, stone arches, buildings, and, in fact, all architectural and engineering projects, are made in accordance with the principles of Descriptive Geometry. When fully dimensioned they are called working drawings. The object to be represented is supposed to be placed in either the first or the third of the four angles formed by the intersection of a horizontal plane, H, with a vertical plane, V. (Fig. 228). The representations of the object upon the planes are, in mathematical language, projections,* and are obtained by drawing perpendiculars to the planes H and V from the various points of the object, the point of intersection of each such projecting line with a plane giving a projection of the original point. Such drawings are, obviously, not "views" in the ordinary sense, as they lack the perspective effect which is involved in having the point of sight at a finite distance; yet in ordinary parlance the terms top view, horizontal projection and plan are used synonymously; as are front view and front elevation with vertical projection, and side elevation with profile view, the latter on a plane perpendicular to both H and V and called tlie profile plane. Until the last decade of the first century of Descriptive Geometry (1795-1895) problems were solved as far as possible in the first angle. As the location of the object in the third angle — that is, below the horizontal plane and behind the vertical — results in a grouping of the views which is in a measure self- interpreting, the Third Angle Method is, however, to a considerable degree supplant- ing the other for machine-shop work. The advantageous grouping of the projections which constitutes the only — though a quite suf- ficient — ^justification for giving it special treatment, is this: The front view being always the central one of the group, the top view is found at the top; the view of the right side of the object appears on the right; of the left-hand side on the left, etc. Thus, in Fig. 228 (a), with the hollow block BDFS as the object to be represented, we have ades for its horizontal projection, c'd'e'f for its vertical projection, f"e"s"x" for the side elevation; then on rotating the plane H clockwise on G. L. into coincidence with V, and the profile plane P about QR until the projection f"e"s"x" reaches f"'e"'s"'x"', we would have that location of the views which has just been described. The lettering shows that each projection represents that side of the object which is toward the plane of projection. 384. The same grouping can be arrived at by a different conception, which will, to some, have advantages over the other. It is illustrated by Fig. 228 (b), in which the same object as before is • For the convenience of those who hare to take tip this subject without previous study of Descriptive Geometry the Third-Angle section of this chapter is made complete In itself, by the re-statement of the principles involved and which have been treated at somewhat greater length in the previous chapter; although a review of such matter may be by no means disadvantageous to those who have already been over the fundamentals. 132 THEORETICAL AND PRACTICAL GRAPHICS. supposed to be surrounded by a system of mutually -perpendicular transparent planes, or, in other words, to be in a box having glass sides, and on each side a drawing made of what is seen through that side, excluding the idea, as before, of perspective view, and representing each point by a per- E-lgr. 22S. " (a.) p (15) pendicular from it to the plane. The whole system of box and planes, in the wood -cut, is rotated 90° from the position shown in Fig. 228(a), bringing them into the usual position, in which the observer is looking perpendicularly toward the vertical plane. Fig:- 22S. 385. In Fig. 229 we may illustrate either the First or the Third Angle method, as to the top view of the object; ades in the upper plane being the flan by the latter method, and a^d^e^s^ by the former. ORTHOGRAPHIC PROJECTION OF SOLIDS. 133 Disregarding Q TXN we have the object and planes illustrating the first-angle method through- out, the lettering of each projection showing that it represents the side of the object farthest from the plane, making it the exact reverse of the third -angle system. In the ordinary representation the same object would be represented simply by its three views as in Fig. 280. In the elevations the short- dash lines indicate the invisible edges of the hole. The arcs show the rotation which carries the profile view into its proper place. FJ-S- 231. 386. For the sake of more readily contrasting the two methods a group of views is shown in Fig. 231, all above G. L., illustrating an object by the First Angle system, while all below HK represents the same object by the Third Angle method. When looking at Figures 1, 2, 3 and 4 the observer queries: What is the object, in space, whose front is like Fig. 1, top is like Fig. 2, left side is like Fig. 3 and right side like Fig. 4? For the view of the left side he might imagine himself as having been at first between G and H, looking in the direction of arrow N, after which both himself and the object were turned, together 134 THEORETICAL AND PRACTICAL GRAPHICS. to the right, through a ninety -degree arc, when the same side would be presented to his view in Fig. 3. Similarly, looking in the direction of the arrow M, an equal rotation to the left, as indicated by the arcs 1-2, 3-4, 5-6, etc., would give in Fig. 4 the view obtained from direction M. His mental queries would then be answered about as follows: Evidently a cubical block with a rectangular recess — r'v'd'c' — in front; on the rear a prismatic projection, of thickness ph and whose height equals that of the cube; a short cylindrical ring projecting from the right face of the cube; an angular projecting piece on the left face. In Fig. 2 the line rv is in short dashes, as in that view the back plane of the recess r'v'd'c' would be invisible. In Fig. 4 the back plane of the same recess is given the letters, v"d", of the edge nearest the observer from direction M. To illustrate the third angle method by Fig. 231 we ignore all above the line H K. In Fig. 5 we have the' same front elevation as before, but above it the view of the top; below it the view of the bottom exactly as it would appear were the object held before one as in Fig. 5, then given a ninety- degree turn, around a'b', until the under side became the front elevation. Fig. 7 may as readily be imagined to be obtained by a shifting of the object as by the rotation of a plane of projection; for by translating the object to the right, from its position in Fig. 5, then rotating it to the left 90° about b'n', its right side would appear as sliown. 387. For convenient reference a general resume of terms, abbreviations and instructions is next presented, once for all, for use in both the Third Angle and First Angle methods. (1) H, V, P the horizontal, vertical and profile planes of projection respectively. (2) H - projector the projecting line which gives the horizontal projection of a point. (3) V- projector the projecting line giving the projection of a point on V. (4) Projector-plane the profile plane containing the projectors of a point. (5) h. p the horizontal projection or plan of a point or figure. (6) v. p the vertical projection or elevation of a point or figure. (7) h. t horizontal trace, the intersection of a line or surface with H. (8) V. t vertical trace, the intersection of a line or surface with V. (9) H- traces, V- traces plural of horizontal and vertical traces respectively. • (10) G. L .... ground line, the line of intersection of V and H. (11) V -parallel a line parallel to V and lying in a given plane. (12) A horizontal any horizontal line lying in a given plane. (13) Line of declivity the steepest line, with respect to one plane, that can lie in another plane. (14) Rabatment revolution into H or V about an axis in such plane. (15) Counter - rabatment or revolution . restoration to original position. 388. For Problems relating solely to ike Point, Line and Plane. Given lines should be fine, continuous, black ; required lines heavy, continuous, black or red ; construction lines in fine continuous red, or short- dash black; traces of an auxiliary plane, or invisible traces of any plane, in dash -aud- itor Problems relating to Solid Objects. (V) Pencilling. Exact ; generally completed for the whole drawing before any inking is done ; the work usually from centre lines, and from the larger — and nearer — parts of the object to the smaller or more remote. (2) Inking of the Object. Curves to be drawn before their tangents ; fine lines uniform and drawn before the shade lines* shade lines next and with one setting of the pen, to ensure uniformity. On tapering shade lines see Art. 111. (3) Shade Lines. In architectural work these would be drawn in accordance with a given direction of light. In Am,erican machine- yhop practice the right-hand and lower edges of a plane surface are made shade lines if they separate it from invisible surfaces. Indicate curvature by line-shading if not otherwise sufficiently evident. (See Fig. 288'. THE CONSTRUCTION AND FINISH OF WORKING DRAWINGS. 135 (4) Invisible lines of the object, black, invariably, in dashes nearly one-tenth of an inch in length. (5) Inking of lines other than of the object. "When no colors are to be employed the following directions as to kind of line are those most frequently made. The lines may preferably be drawn in the order mentioned. Centre lines, an alternation of dash and two dots. : Dime?ision lines, a dash and dot alternately, with opening left for the dimension. Extension lines, for dimensions placed outside the views, in dash-and-dot as for a dimension line. Ground line, (when it cannot be advantageously omitted) a continuous heavy line. Construction and other explanatory lines in short dashes. (6) When using colors the centre, dimension and extension lines may be fine, continuous, red; or the former may be blue, if preferred. Const-i-uction lines may also be red, in short dashes or in fine continuous lines. Instead of using bottled inks the carmine and blue may preferably be taken directly from Winsor and Newton cakes, "moist colors." Ink ground from the cake is also preferable to bottled ink. Drawings of developable and warped surfaces are much more eflTective if their elements are drawn in some color. (7) Dimensions and Arrow. Tips. The dimensions should invariably be in black, printed free-hand with a writing- pen, and should read in line with the dimension line they are on. On the drawing as a whole the dimensions should read either from the bottom or right-hand side. Fractions should have a horizontal dividing line; although there is high sanction for the omission of the dividing line, particularly in a mixed number. Extended Gothic, Roman, Italic Koman and Reinhardt's form of Condensed Italic Gothic are the best and most generally used types for dimensioning. The arrow- tips are to be always drawn free-hand, in black; to touch the lines between which they give a distance; and to make an acute instead of a right angle at their point. 389. Working drawing of a right pyramid; base, an equilateral triangle 0.9" on a side; altitude, x. Draw first the equilateral triangle ah c for the plan of the base, making its sides of the pre- scribed length. If we make the edge a b perpendicular to ^^- ssa. the profile plane, 01, the face vab will then appear in profile view as the straight line v"h". Being a right pyra- mid, with a regular base, we shall find ,v, the plan of the vertex, equally distant from a, b and c; and v a, vb, vc for the plans of the edges. Parallel to G. L. and at a distance apart equal to the assigned height, x, draw mv" and nc" as upper and lower limits of the front and side elevations; then, as the h. p. and v. p. of a point are always in the same perpendicu- lar to G. L., we project v, a, b and c to their resjjective levels by the construction lines shown, obtaining v'.a'b'c' for the front elevation. Projectors to the profile plane from the points of the plan give 1, 2, 3, which are then carried, in arcs about 0, to L, 5, 4, and projected to their proper levels, giving the side elevation, v"b"c". As the actual length of an edge is not shown in either of the three views, we employ the fol- ^'^•^^3. lowing construction to ascertain it; Draw vVi perpendicular to vb, and make it equal to x; v^b is then the real length of the edge, shown by rabatment about vb. The development of the pyramid (Fig. 238) may be obtained by drawing an arc ABCA^ of radius =•», 6 (the true length of edge, from Fig. 232) and on it laying off the chords AB, BC, CAi equal to ab, be, ca of the plan; then V-ABCA^ is the j)lane area which, folded on VC and VB, would give a model of the pyramid represented. 136 THEORETICAL AND PRACTICAL GRAPHICS. 390. Working drawing of a semi -cylindrical pipe: outer diameter, x; inner diameter, y; height, z. For the j-)lan draw concentric semi -circles aed and bsc, of diameters x and y respectively, join- E-lgr- 23-i- ing their extremities by straight lines ah, cd. At a distance apart of z inches draw the upper and lower limits of the elevations, and project to these levels from the points of the plan. In the side view the thickness of the shell of the cylinder is shown by the distance between e"f' and s"t" — the latter so drawn as to indicate an invisible limit or line of the object. The line shading would usually be omitted, the shade lines generally sufficing to convey a clear idea of the form. 391. Hfilf of a hollow, hexagonal prism. In a semi -circle of diameter a d step off the radius three times as a chord, giving the vertices of the plan abed of the outer surface. Parallel to b c, and at a distance from it equal to the assigned thickness of the prism, draw ef, terminating it on lines (not shown) E'lg-. S35. drawn through b and c at 60° to ad. From e and / draw eh and fg, parallel respectively to a 6 and c d. Drawing a'c" and m't" as upper and lower limits, project to them as in preceding problems for the front and side elevations. 392. Working drawing of a -hollow, prismatic block, standing obliquely to the vertical and profile planes. Let the block be 2"x3"xl" outside, with a square open- ing 1 " X 1 " X 1 " through it in the direction of its thickness. Assuming that it has been required that the two -inch edges should be vertical, we first draw, in Fig. 236, the plan asxb, 3"xl", on a scale of 1:2. The inch -wide opening through the centre is indicated by the short- dash lines. n' n" t For the elevations the upper and lower limits are drawn 2" apart, and a, b, s, x, etc., projected to them. The ^^- 23©- elevations of the opening are between levels m'm'' and k'k", one inch apart and equi-distant from the upper and lower outlines of the views. The dotted construction lines and the lettering will enable the student to recognize the three views of any point without difficulty. 393. In Fig. 287 we have the same object as that illustrated by Fig. 236, but now represented as cut by a vertical plane whose horizontal trace is vy. The parts of the block that are actually cut by the plane are shown in section -lines in the elevations. This is done here and in some later examples merely to aid the beginner in understanding the views; but, in engineering prac- tice, section- lining is rarely done on views not perpendicular to the section plane. ^^ ^ x^ \ „]^Av . A^ ._^ i^i_. ^/^' _^i^'\\ ! ^ J^y^\ /^ . t^'-V^ \ 1 \ ^V" \ \ \ ,y^ ^ 1 i /NvrA \ 1 1 ! ! ' L . 1 1 M 1 ! , i i ! 1 ■ . h 'li \ 1 : 1 !x s _. a \ i\ 1 i s" 1 ch m' 1 Ij' I ni' \ !.)■" ! ■L-.J.- d i i i 1/ ?._ k \ I d" ;c" e" PROJECTION OF SO LIDS.— WORKING DRAWINGS. 137 is customary X-lg-. EST. SECTIONAL VIEW 394. Suppression of the ground line. In machine drawing it is customary to omit the ground line, since the forms of the various views — which alone concern us — are independent of the distance of the object from an imaginary horizon- tal or vertical plane. We have only to remember that all elevations of a point are at the same level; and that if a ground line or trace of any vertical plane is wanted, it will be perpendicular to the line joining the plan of a point with its projection on such vertical plane. (Art. 286.) 395. Sections. Sectional vieivs. Although earlier defined (Art. 70), a re-statement of the distinction between these terms may well precede problems in which they will be so frequently employed. When a plane cuts a solid, that portion of the latter which comes in actual contact with the cutting plane is called the section. A sectional view is a view perpendicular to the cutting plane, and showing not only the section but also the object itself as if seen through the plane. When the cutting plane is vertical such a view is called a sectional elevation; when horizontal, a sectional plan. 396. Working drawing of a regular, pentagonal pyramid, hollow, truncated by an oblique plane; also the development, or "pattern," of the outer surface below the cutting plane. For data take the altitude, at 2"; inclination of faces, 6° (meaning any arbitrary angle) ; inclination of section plane, 30°; distance between inner and outer faces of pyramid, ^". (1) Locate « and »' (Fig. 238) for the plan and elevation of the vertex, taking them sufficiently apart to avoid the overlapping of one view upon the other. Through V draw the horizontal line ST, regarding it not only as a centre line for the plan but also as the h. t. of a central, vertical, refer- ence plane, parallel to the ordi- nary vertical plane of projection. PLAN 138 THEORETICAL AND PEACTICAL GRAPHICS. SECTIONAL VIEW (The student should note that for convenient reference Fig. 238 is repeated on this page.) On the vertical line vv' (at first indefinite in length) lay off v' s' equal to 2", for the altitude (and axis) of the pyramid, and through s' draw an indefinite horizontal line, which will contain the V. p. of the hase, in both front and side views. Draw v' b' at 0° to the horizontal. It will represent the v. p. of an outer face of the pyramid, and b' will be the v. p. of the edge a 6 of the base. The base abode is then a regular jientagon circumscribed about a circle of centre v and radius ■yt = s'&'. Since the angle avb is 72° (Art. 92) we get a starting corner, a or b, by drawing v a or vb at 36 ° to ST, to intercept the vertical through b'. The plans of the edges of the pyramid are then v a, vb, v c, vd and v c. Project d to d' and draw v' d' for the elevation of V d; similarly for v e and v c, which happen in this case to co- incide in vertical projection. For the inner surface of the pyramid, whose faces are at a perpendicular distance of ^" from the outer, begin by drawing g' I' parallel to and \" from the face projected in b'v'; this will cut the axis at a point t' which will be the vertex of the inner sur- face, and g' t' will represent the elevation of the inner face that is parallel to the face avb — v'b' ; while gh, vertically above g' and included between va and vb, will be the plan of the lower edge of this face. Complete the pentagon gh — k for the plan of the inner base; project the comers to b' d' and join with t' to get the ele- vations of the interior edges. The section. In our figure let G' H' be the section plane, sit- uated perpendicular to the ver- tical plane and inclined 30° to the horizontal. It intersects v' d' in p', which projects upon vd at p. Similarly, since G' H' cuts the edges v' c' and v' e' at points projected in o', we project from the latter to v c and ve, obtaining o and q. A like construction gives m and n. The polygon mnopq is then the j^lan of the outer boundary of the section. The inner edge g' t' is cut by the section plane at I', which projects to both vh and v g, giving the parallel to mn through I. The inner boundary of the section may then be completed either by determining all its vertices in the same way or on the principle that its sides will be parallel to those of the outer polygon, since any two planes are cut by a third in parallel lines. The line m'p' is the vertical projection of the entire section. PROJECTION OF SOLIDS. — WORKING DRAWINGS. 139 (2) The side elevation. This, might be obtained exactly as in the five preceding figures, that is, by actually locating the side vertical, or profile, jDlane, projecting upon it and rotating through an arc of 90°. In engineering practice, however, the method now to be described is in far more general me. It does not do away with the profile plane, on the contrary presupposes its existence, but instead of actually locating it and drawing the arcs which so far have kept the relation of the views constantly before the eye, it reaches the same result in the following manner : A vertical line .S" T' is drawn at some convenient distance to the right of the front elevation ; the distance, from S T, of any point of the plan, is then laid off horizontally from S' T', at the same height as the front elevation of the point. For, as earlier stated, S T was to be regarded as the horizontal trace of a vertical plane. Such plane would evidently cut a profile plane in a vertical line, which we may call S'T', and let the S' T' of our figure represent it after a ninety -degree rotation has occurred. The distances of all points of the object, to either the front or rear of the vertical plane on S T would, obviously, be now seen as distances to the left or right, respectively, of the trace S' T' , and would be directly transferred with the dividers to the lines indicating their level. Thus, e" is on the level of e', but is to the right of S'T' the same distance that e is above (or, in reality, behind) the plane ST; that is, e" d" equals exi. Similarly d" h" equals ib; n"x" equals nx. It is usual, where the object is at all symmetrical, to locate these reference planes centrally, so that their traces, used as indicated, may bisect as many lines as possible, to make one setting of the dividers do double work. (3) True size of the section. Sectional view. If the section plane G' H' were rotated directly about its trace on the central, vertical plane S T, until parallel to the paper, it would show the section m' p' — mnopq in its true size; but such a construction would cause a confusion of lines, the new figure overlapping the front elevation. If, however, we transfer the plane G' H' — keeping it parallel to its first position during the motion — to some new position S" T", and then turn it 90° on that line, we get nViniO^p^q^, the desired view of the section. The distances of the vertices of the section from S" T" are derived from reference to ST exactly as were those in the side elevation; that is, m,a;i = ma;= m"x-". We thus see that one central, vertical, reference plane, S 2] is auxiliary to the construction of two important views ; -S" T" represents its intersection with the profile or side vertical plane, while S" T" is its (transferred) trace upon the section plane G'H'. For the remainder of the sectional view the points are obtained exactly as above described for the section; thus c'Ci^i is perijendicular to S" T" ; e^Ui equals ew, and c^Ui equals cu. (4) To determine the actual length of the various edges. The only edge of the original, uncut pyramid, that would require no construction in order to show its true length, is the extreme right- hand one, which — being parallel to the vertical plane, as shown by its plan vd being horizontal — is seen in elevation in its true size, v'd'. Since, however, all the edges of the pyramid are equal, we may find on v'd' the true length of any portion of some other edge, as, for example o'c', by taking that part of v'd' which is intercepted between the same horizontals, viz.: o"'d'. Were we compelled to find the true length of o'c', oc, independently of any such convenient relation as that just indicated, we would apply one of the methods fully illustrated by Figs. 183, 184 and 187, or the following " shop " modification of one of them : Parallel to the plan o c draw a line yz, their distance apart to be equal to the difference of level of o' and c', which diff'erence may be obtained from either of the elevations ; from tht. plan o of the higher end of the line draw the common perpendicular of, and join / with c, obtaining the desired length fc. (5) To show the exact form of any face of the pyramid. Taking, for example, the face ocdp, revolve op about the horizontal edge cd until it reaches the level of the latter. The actual distance 140 THEORETICAL AND PRACTICAL GRAPHICS. of from c, and of j) from d will be the same after as before this revolution, while the paths of o and p during rotation will be projected in lines or and pw, each perpendicular to cd; therefore, with c as a centre, cut the perpendicular or by an arc of radius fc — just ascertained to be the real length of oc, and, similarly, cut puo by an arc of radius dw = p'd'; join r with c, w with d. draw w r and we have in cdwr the form desired. (6) The development of the outer surface of the truncated pyramid. With any point F as a centre (Fig. 239) and with radius equal to the actual length of an edge of the pyramid (that is, equal to v'd', Fig. 238) draw an indefinite arc, on which lay off the chords DC, C B, B A, A E, ED, equal respectively to the like -lettered edges of the base abcde; join the extremities of these chords with V: then on DFlay off DP=d'p'; make C 0= EQ= d'o'" = ih& real length of c' o' ; also BN= j4 ilf = d'm'" = the actual length of a'm' and h'n'; join the points P, 0, etc., thus obtaining the development of the outer boundary of the section. The pattern A,BiC D E^ of the base is obtained from the plan in Fig. 238, while N Mq^p^o^ is a duplicate of the shaded part of the sectional view in the same figure. (7) In making a model of the pyramid the student should use heavy Bristol board, and make allowance, wherever needed, of an extra width for overlap, slit as at x, y and z (Fig. 239). On this Fig-. 233. overlap put the mucilage which is to hold the model in shape. The faces will fold better if the Bristol board is cut half way through on the folding edge. 397. For convenient reference the characteristic features of the Third Angle Method, all of which have now been fuUy illustrated, may thus be briefly summarized : (a) The various views of the object are so grouped that the plan or top rieio comes above the front elevation; that of the bottom below it; and analogously for the projections of the right and left sides. (b) Central, reference planes are taken through the various views, and, in each view, the distance of any point from the trace of the central plane of that view is obtained by direct transfer, with the dividers, of the distance between the same point and reference plane, as seen in some other view, usually the plan. 398. To draw a truncated, pyramidal block, having a rectangular recess in its top; angle of sides, 60°; lower base a rectangle 3" X 2", having its longer sides at 30° to the horizontal; total height tSj"; recess 1^" x ^", and \" deep. (Fig. 240.) The small oblique projection on the right of the plan shows, pictorially, the figure to be drawn. PROJECTION OF SO LIDS. — WORK ING DRAWINGS. 141 The plan of the lower base will be the rectangle abde, 3" x 2", whose longer edges are inclined 30° to the horizontal. Take AB and mn as the H- traces of auxiliary, vertical planes, perpendicular to the side and end faces of the block. Then the sloping face whose lower edge is de, and which is inclined 60° to H, will have d^y for its trace on plane mn. A parallel to mn and ^" from it will give s,, the auxiliary projection of the upper edge of the face sved, whence sv — at first indefinite in length — is derived, parallel to de. Similarly the end face btsd is obtained by projecting db upon AB at 6i, drawing b^z at 60° to AB and terminating it at s^ by CD, drawn at the same height (yV) as before. A parallel to bd through -s.^ intersects vSi at s, giving one corner of the plan of the upper base, from which the rectangle stuv is completed, with sides parallel to those of the lower base. As the recess has vertical sides we may draw its plan, o pqr, directly from the given dimen- sions, and show the depth by short -dash lines in each of the elevations. The ordinary elevations are derived from the plan as in preceding problems; that is, for the front elevation, a'u's'd', by verticals through the plans, terminating according to their height, either on a' d' or on u's', -^" above it. For the side elevation, e"v"t"b", with the heights as in the front elevation, the distances to the right or left of s" equal those of the plans of the same points from si, regarding the latter as the h. t. of a central, vertical plane, parallel to V. The plane S T of right section, perpendiadar to the axis KL, cuts the block in a section whose true size is shown in the line -tinted figure gih^ktl,, and whose construction hardly needs detailed treatment after what has preceded. The shaded, longitudinal section, on central, vertical plane KL, also interprets itself by means of the lettering. • 142 THEORETICAL AND PRACTICAL GRAPHICS. The true size of any face, as auve, may be shown by rabatment about a horizontal edge, as ae. As V is actually yV above the level of e, we see that ve (in space) is the hypothenuse of a triangle of base ve and altitude ^". Construct such a triangle, v v.^e, and with its hypothenuse v^e as a radius, and e as a centre, obtain Vi on a perpendicular to ae through v and representing the path of rotation. Finding ttj similarly we have au,v^e as the actual size of the face in question. If more views were needed than are shown the student ought to have no difficulty in their construction, as no new principles would be involved. 399. To draw a hollow, pentagonal prism, 2" long; edges to be horizontal and inclined 35° to V; base, a regular pentagon of 1" sides; one face of the prism to be inclined 60° to H; distance between inner and outer faces, \". In Fig. 241 let HK be parallel to the plans of the axis and edges; it will make 35° with a horizontal line. Perpendicular to H K draw mn as the h. t. of an auxiliary, vertical plane, upon which we may suppose the base of the prism projected. In end view all the faces of the prism would be seen as lines, and all the edges as points. Draw a,6i, one inch long and at 60° to mn, to represent the face whose inclination is assigned. Completing the inner and outer pentagons, allowing ^" for the distance between faces, we have the end view complete. The plan is then PROJECTION OF SOLIDS. — WO R KING DRAWINGS. 143 obtained by drawing parallels to H K through all the vertices of the end view, and terminating all by vertical planes, a d and g h , parallel to m n and 2 " apart. The elevations will be included between horizontal lines whose distance apart is the extreme height z of the end view; and all points of the front elevation are on verticals through their j)lans, and at heights derived from the end view. The most expeditious method of working is to draw a horizontal reference line, like that of Fig. 243, which shall contain the lowest edge of each elevation ; measuring upward from this line lay off, on some random, vertical line, the distance of each point of the end view from a line (as the parallel to mn through 6, in Fig. 241, or xy in Fig. 243) which repre- sents the intersection of the plane of the end view by a horizontal plane containing the lowest point or edge of the object; horizontal lines, through the j^oints of division thus obtained, will contain the projections of the comers of the front elevation, which may then be definitely located by vertical lines let fall from the plans of the same points. For example, e' and /', Fig. 241, are at a height, z, above the lowest line of the elevation, equal to the distance of e^ from the dotted line through 6,; or, referring to Fig. 243, which, owing to its greater complexity, has its construction given more in detail, the distance upward from M to line G is equal to (j^g.^ on the end view; from M to Q equals q^q.^, and similarly for the rest. Since the profile plane is omitted in Fig. 241 we take M' N' to represent the trace upon it of the auxiliary, central, vertical plane whose h. t. is MN; as already explained, all points of the side elevation are then at the same level as in the front elevation, and at distances to the right or left of M' N' equal to the perpendicular distances of their plans from MN. For example, e"«" equals e s. The shade lines are located on the end view on the assumption that the observer is looking toward it in the direction HK. 400. Prajections of a hollow, pentagonal prism, cut by a vertical plane oblique to V. Letting the data for the prism be the same as in the last problem, we are to find what modification in the appear- ance of the elevations would result from cutting through the object by a vertical plane PQ (Fig. 242) and removing the part hxdi which lies in front of the plane of section. Each vertex of the section is on an edge of the elevation and is vertically below the point where P Q cuts the plan of the same edge ; the student can, therefore, readily convert the elevations of Fig. 241 into reproductions of those of Fig. 242 by drawing across the plan of Fig. 241 a trace P Q, similarly situated to the P Q of Fig. 242. Supposing that done, refer in what follows to both Figures 241 and 242. Since P Q contains h we find h ' as one corner of the section. Both ends of the prism being vertical, they will be cut by the vertical plane P Q in vertical lines; therefore h'l' is vertical until the top of the prism is reached, at I'. Join I' with x', the latter on the vertical througii x — the intersection of P Q with the right-hand top edge ed, e'd'. From x "the cut is vertical until the interior of the prism is reached, at o', on the line 5-4. We next reach lo' on edge No. 4. The line o' w' has to be parallel to x'V (two parallel planes are cut by a third in parallel lines); but from w' the interior edge of the section is not parallel to I'h', since PQ is not cutting a vertical Old, but the inclined, interior surface. The other points hardly need detailed description, being similarly found. The side elevation is obtained in accordance with the principle fully described in Art. 896 and summarized in Art. 397 (b). M' N' represents the same plane as MN; e" s" equals es, and anal- ogously for other points. 401. In his elementary work in projections and sections of solids the student is recommended to lay an even tint of burnt sienna, medium tone, over the projections of the object, after which 144 THEORETICAL AND PRACTICAL GRAPHICS. any section may be line -tinted; and, if he desires to further improve the appearance of the views, distinctions may be made between the tones of the various surfaces by overlaying the burnt sienna with flat or graded washes of India ink. 402. Projections of an L- shaped block, after being ait by a plane oblique to both V and H; the block also to be inclined to V and H, and to have running through it two, non-communicating, rectangular openings, whose directions are mutually perpendicular. If the dotted lines are taken into account the front elevation in Fig. 243 gives a clear idea of the shape of the original solid. The end view and plan give the dimensions. Requiring the horizontal edges of the block to be inclined 30° to V, draw the first line xy dX 60° to the horizontal; the plans of all the horizontal edges will be perpendicular to xy. Let the inclination of the bottom of the block to H be 20°. This is shown in the end view by drawing m^p^ at 20° to xy. All the edges of the end view of the object will then be parallel or perpendicular to m,p, and should be next drawn to the given dimensions. WORKING DRAWINGS. — PLANE SECTIONS OF SOLIDS. 145 from of 4" The central opening, 6,rf,n,o„ through the larger part of the block, has its faces all \ the outer faces. In the plan this is shown by drawing the lines lettered of at a distance from the boundary lines, which last are indicated as 1^" apart. The opening qir,8,t, has three of its faces i" from the outer surfaces of the block, while the fourth, q,rj, is in the same plane as the outer face A , e 1 . The cutting plane X Y gives a section which is seen in end view in the lines c,gr„ ^,jl and kj,; while in plan the section is pro- jected in the shaded portion, obtained, like all other parts of the plan, by perpendiculars to xy from all the points of the end view. For the front eleva- tion draw first the "reference line." To provide against overlapping of projections the reference line should be at a greater distance below the lowest point, I, of the plan, than the greatest height (a,a^) of the end view above xy. Then on MW lay off from M the heights of the vari- ous horizontal edges of the block, deriving them from the end view. Thus a,a^ is the height of A a' from M; from M to level B equals ^i^.^, etc. Next project to the level A from points a a of the plan, getting edge a' a' of the elevation, and similarly for all the other comers of the block. Notice that all lines that are parallel on the object will be parallel in each projec- tion (except when their projections coin- cide) ; also that in the case of sections, those outlines will he parallel which are the inter- section of parallel planes by a third plane. These principles may be advantageously employed as checks on the accuracy of the construction by points. The construction of the side elevation is left to the student. 146 rilEORETICAL AND PRACTICAL GRAPHICS. FRONT ELEVATION. With section made by vei-liLal plane P Q Reference line SIDE ELEVATION. With section by plane S T. Shade lines on this view are located for pictorial effect and not in accord- ance with shop rule. WORKING DRAWINGS. — PLANE SECTIONS OF SOLIDS. 147 403. Projections and sections of a block of irregular form, with two mutually perpendicular openings through it, and with equal, square frames projecting from each side. In Fig. 244 the side elevation shows clearly the object dealt with, while we look to the end view for most of the dimensions. The large central opening extends from w,«;j to x^y^. The width of the main portion of the block is shown in plan as 2\", between the lines lettered a e. The square frames project ^" from the sides, while the width of the central opening between the lines wx is f". Two section planes are indicated, S T across the end view, and P Q^a vertical plane — across the plan ; the section made by plane '''' analogy to that of Fig. 252. The plan V b being parallel to the base line shows that v' b' is the actual length of that edge. By carrying vc,, where it becomes parallel to V, and then projecting c, to c" we get v' c" for the true of edge v'c'. x^. sss. To illustrate another method make vVi = v'o; then v^d by rabatment into H. is the real length of v'd', shown 154 THEORETICAL AND PRACTICAL GRAPHICS. •E^g. 2S-i, For the development take some point v^ and from it as a centre draw arcs having for radii the >ascertained lengths of the edges. Thus, letting v^A represent the initial edge of the development, take A aa a, centre, ad as a radius, and cut the arc of radius v^d at D; then Av^D is the develop- ment of the face avd, a'v'd'. With centre D and radius do obtain C on the arc of radius v'c"; similarly for the remaining faces, completing the development v^ — AD... A,. The shaded area v^ — TP...T is the development of that part of the pyramid above the oblique plane s'p', found by laying off, on the various edges as seen in the development, the distances along those edges from the vertex to the cutting plane; thus v.^N = v'n', the real length of v'm'; v.jP = v^p^■, the length of v' p' ; v^^S = v's', the only elevation showing actual length. 418. The development of an oblique cone. The usual method of solving this problem gives a result which, although not mathematically exact, is a sufficiently close approximation for all practical purposes. In it the cone is treated as if it Were a pyramid of many sides. The length of any element is then found as in the last problem. Thus in Fig. 254 an element t; c is carried to v c" about the vertical axis v o. In Fig. 255 we have v'.a g for the elevation of the cone, and — ahc.g for the half plan. Make ob" = ob; then v' b" is the real length of the element whose plan is o b. Similarly, c, d, e and / are carried by arcs io ag and there joined with v'. For the development make v^A equal and parallel to v' a, and at 1^ any distance from it. With i;, as a centre draw arcs with radii equal to the true lengths of the elements; then, as in the pyramid, make A B = arc ab ; B C = arc b c , et-c. The greater the number of divisions on the semi -circle ab...g the more closely will the develop- ment approximate to theoretical exactness. 419. The five regular convex solids, with the forms of their developments, are illustrated in Figs'. 256-265. They have already been defined in Art. 345, and that five is their limit as to number is thus shown: The faces are to be equal, regular polygons, and the sum of the plane angles forming a solid angle must be less than four right angles; now as the angles of equilateral triangles are 60° we may evidently have groups of three, four or five and not exceed the limit; with squares there can be groups of three only, each 90°; with regular pentagons, their interior angles being 108°, groups of three ; while hexagons are evidently impracticable, since three of their interior angles would exactly equal four right angles, adapt- ing them perfectly — and only — to plane surfaces. (See Fig. 131.) r-igr- ESS. orig-. ssy. ■F1.S- 2Ee. TETRAHEDRON. Flgr- 2S9. DODECAMEORON. E'igr- 2SO. OCTAHEDRON. ICOSAHEORON. THE FIVE REGULAR CONVEX SOLIDS. 155 The dihedral angles between the adjacent faces of regular solids are as follows: 70° 31' 44" for the tetrahedron; 90° for the cube; 109° 28' 16" for the octahedron; 116° 35' Fig-- SSI- 54" for the dodecahedron; and 138° 11' 23" for the icosahedron. A sphere can be inscribed in each regular solid and can also as readily be circumscribed about it. The relation between d, the diameter of a sphere, and e, the edge of an inscribed regular solid, is illustrated graphically by Fig. 266, but may be otherwise expressed as follows: d : e :: v^3" : n/^; for the aibe d : e :: ^~S : 1 d : e :: "^ 2 : 1 ; " " dodecahedron e = the greater segment of the edge r-ig-- 2es. ^^^' % i i a TETRAHEDRON, For the tetrahedron " " octahedron of an -inscribed cube when the latter has been medially divided, that is, in extreme and mean ratio, Figr- sea. F igr- 26-^. OCTAHEDRON. DODECAHEDRON. For the icosahedron e = the chord of the arc whose tangent is d; i. e., the chord of 63° 26' 6". Reference to Figs. 256-260 and the use of a set of cardboard models which can readily be made by means of Figs. 261-265 will enable the student to verify the following statements as to those ordinary views whose construction would naturally precede the solution of problems relating to these surfaces. ngr.sss. In all but the tetrahedron each face has an equal, opposite, parallel face, and except in the cube such faces have their angular points alter- nating. (See Figs. 260, 267, 268.) The tetrahedron projects as in Fig. 256, U])on a plane that is par- allel to either face. The cube projects in a square upon a plane parallel to a face, while on a plane perpendicular to a body diagonal it projects as a regular hexagon, with lines joining three alternate vertices with the centre. The octahedron, which is practically two equal square pyramids with a common base, projects in a square and its diagonals, upon a plane perpendicular to either body diagonal; in a rhombus and shorter diagonal when the plane is parallel to one body diagonal and at 45° with the other Figr- ssT. i^er- sea. • ^i^- ass. rCOSAHEDRCN. two; and (as in Fig. 267) in a regular hexagon with inscribed triangles (one dotted) when it is projected upon a plane parallel to a face. The dadMohedron projects as in Fig. 268 whenever the plane of projection is parallel to a face. 156 THEORETICAL AND PRACTICAL GRAPHICS. Fig. 260 represents the icosahedron projected on a plane parallel to a face, and Pig. 269 when the projection -plane is perpendicular to an axis. 420. The Developable Helicoid. When the word hdicmd is used without qualification it is under- stood to indicate one of the warped helicoids, such as is met with, for example, in screws, spiral sigr- a-ro. staircases and screw propellers. There is, however, a developable helicoid, and to avoid confusing it with the others its characteristic property is always found in its name. As stated in Art. 346, it is generated by moving a straight line tangentially on the ordi- nary helix, which curve (Art. 120) cuts all the elements of a right cylinder at the same angle. Fig. 209 illustrates the completed surface pictorially; Fig. 270 shows one orthographic projection, and in Fig. 271 it is seen in process of generation by the hypothenuse of a right-angled triangle that rolls "tangentially on a cylinder. The construction just mentioned is based on the property of non- plane curves that at any point the curve and its tangent make the same angle with a given plane; if, therefore, the helix, beginning at a, crosses each element of the cylinder at an angle equal to obp in the rolling triangle, the hypothenuse of the latter will evidently move not only tangent to the cylinder, but also to the helix. The following important properties are also illustrated by Fig. 271: (a) The involute* of a helix and of its hori- zontal projection are identical, since the point b is the extremity of both the rolling lines, o b and p b. (b) The length of any tangent, as mb, is that of the helical arc m a on which it has rolled. (c) The horizontal projection b q of any tan- gent b m equals the rectification of an arc a q which is the projection of the helical arc from the initial point a to the point of tangency m. The development of one nappe of a helicoid is shown in Fig. 273. It is merely the area between a circle and its involute; but the radius p, of the base circle, equals r sec^ 0,^ in which r is ' • For full treatment of the involute of a circle refer to Arts. 186 and 187. t This relation is due to considerations of curvature. At any point of any curve its curvature is its rate of departure from its tangent at that point. Its radius of curvature is that of the oscutatory circle at that point. (Art. 380.) Now from the nature of the two uniform motions imposed upon a point that generates a helix (Art. 120) the curvature of the latter must be uniform; and if developed upon a plane by means of its curvature It must become a circle — the only plane curve of uniform curvature. The radius of the developed helix will, obviously, be the radius of curvature of the space helix. Following Warren's method of proof in establishing its value let o, 6 and c (Fig. 272) be three equi-distant points on a helix, with h on the foremost element; then a' c' is the elevation of the circle containing these points. One diameter of the circle a'b'e' is projected at b'. It is the hypothenuse of a right-angled triangle having the chord 6c, b'c', for its base. Let 2p be the diameter of the circle a'b'c'; 2r = 6d, that of the cylinder. Using capitals for points in space we have BC^ =2pX6«; also be' = 2rx6fi; whence, dividing like members and substituting tiigonometric functions (see note p. 31), we have p = r sec^ft in which 3 is the angle between the lineBC and its projection. Let e be the inclination of the tangent to the helix at b'. If, now, both A and C approach B, the angle p will approach 9 as its limit; and when A, B and C become consecutive points we will have p = rsec2 9 = the radius of the oseulatory circle =• the radius of curvature. For another proof, involving the radius of curvature of an ellipse, see Olivier, Cours de Oeomitrie Descriptive, Third Ed., p. 197. THE INTERSECTION OF SURFACES. 157 Fig. 27-^:. the radius of the cyUnder on which the helix originally lay, and 6 is the angle at which the helix crosses the elements. To de- termine p draw on an elevation of the cylinder, as in Fig. 274, a line rt b, tangent to the helix at its fore- most point, as in that position its inclination 6 is seen in actual size; then from 0, where a b crosses the extreme element, draw an in- definite line, OS, par- allel to cd, and cut it at m by a line am that is perpendicular to a 6 at its intersection with the front element e/ of the cylinder; then '^6. For we have K DEVELOPED HELICOID _ "'"^^J^— ^^^'*' om =■ p ■= r see oa = on sec 6 = r sec ; and on (= r) : oa :: o a : om ; whence om = r sec^ 6 = p. The circumference of circle p equals 2 n- ?• sec 6, the actual length of the helix, as may be seen by developing the cylinder on which the latter lies. The elements which were tangent to the helix maintain the same relation to the develojied helix, and appear in their true length on the development. The student can make a model of one nappe of this surface by wrapping a sheet of Bristol board, shaped like Fig. 273, upon a cylinder of radius r in the equation r sec' 6 = p; or a two- napped helicoid by superposing two equal circular rings of paper, binding them on their inner edges with gummed paper, making one radial cut through both rings, and then twisting the inner edge into a helix. THE INTERSECTION OF SURFACES. 421. When plane-dded surfaces intersect, their outline of interpenetration is necessarily composed of straight lines; but these not being, in general, in one plane, form what is called a twisted or warped polygon; also called a gauche polygon. 422. If either of two intereecting surfaces is curved their common line will also be curved, except under special conditions. 423. When one of the surfaces is of uniform cross section — as a cylinder or a prism — its end view will show whether the surfaces intersect in a continuous line or in two separate ones. In Cases a, b, c, d and g of Fig. 275, where the end view of one surface either cuts but one limiting line of the other surface or is tangent to one or both of the outlines, the intersection will be a continvmta line. Two separate curves of intersection will occur in the other possible cases, illustrated by e and /, in which the end view of one surface either crosses both the outlines of the other or else lies wholly between them. j^^ bed « / ^ A cylinder will intersect a cone or another cylinder in a plane curve if its end view is tangent to the outlines of the other surface, as in d and e'Is'. s-tb. g, Fig. 275. Two cones may also intersect in a plane curve, but as the conditions to be met are not as readily illustrated they will be treated in a special problem. (See Art. 439). 158 THEORETICAL AND PRACTICAL GRAPHICS. Refer ence Line 424. In general, the line of intersection of two surfaces is obtained, as stated in Art. 379, by passing one or more auxiliary surfaces, usually planes, in such manner as to cut some easily con- structed sections — as straight lines or circles — from each of the given surfaces; the meeting -points of the sections lying in any auxiliary surface will lie on the line sought. The application of the principle just stated is much simplified whenever any face of either of the surfaces is so situated that it is projected in a liiie. This case is amply illustrated in the problems most immediately following. The beginner will save much time if he will letter each projection of a point as soon as it is determined. 425. The intersection of a vertical triangular prism by a horizontal square prism; also the developments. The vertical prism to be 1^" high and to have one face parallel to V; bases equi- lateral triangles of 1" side. The horizontal prism to be 2" long, its basal edges f", and its faces inclined 45° to H; its rear edges to be parallel to and y from the rear face of the horizontal prism. The elevations of the axes to bisect each other. Draw e i horizontal and 1" long for the plan of the rear face of the vertical prism. Complete the equilateral triangle egi and project to levels 1^" apart, obtaining e' f g'h', i'j', on the elevation. Construct an end view g"i"j"h", using t" j" to represent the reference line et, transferred. The end view of the horizontal prism is the square a"h"c"d", having its diagonal horizontal and upper and lower bases of the other prism, and with its comer from i" j". The plan and front elevation of the horizontal rived from the end view as in preceding constructions. Since the lines eg and gi are the plans of vertical faces P> their intersection by the edges a, b, c and d of the hori- n, m, I, p, q, r — and project to the elevations of the same edge a a meets the other prism at o and k, which project o' and k'. Similarly for the remaining points. The development of the vertical prism is shown in the shaded rectangle EJ', of length Sgi and altitude e' f. (See Art. 411). The openings 0Jp^q^r^ and i,Z,m, n, are thus found: For p,, which represents p', make G P = g p, the true distance ofp' from g'h'; then Pp^ = xp'. Similarly, 0G = og, and Oq, = yq'. midway between the b" one -eighth inch prism are next de- of one prism we note zontal prism — as at edges. Thus the to the level of a" at THE INTERSECTION OF PLANE-SIDED SURFACES. 159 The right half of the horizontal prism, o' a' c' q', is developed at r^b^b^r, after the method of Art. 412. 426. The intersection of two prisms, one vertical, the other horizontal, each having an edge exterior to the other. The condition made will, as already stated (Art. 423), make the result a single warped polygon. x-ig-. 377-. Let abed, 1" x ^", be the plan of the vertical prism, which j^-Jin? stands with its broader faces at some convenient angle cwg to V. From it construct the front and side elevations, taking a reference plane through d for the latter. Let the horizontal prism be triangular (isosceles section) one face inclined 45° to H; another 30° to H; the rear edge to be' \" from that of the vertical prism. Begin by locating g" one -fifth inch from the right edge, draw J" g" at 45°, making it of sufficient length to have /" exterior to the other prism; then f'e" at 80° to H, terminated at e" by an arc of centre g" and radius g"f"; finally e" g". The edges e', f and g' of the front elevation are then projected from e", /" and g". The rear edge g in the plan meets the face ad at s, which projects to s' on the elevation of the edge through g'. Moving forward from s, the next edge reached, of either solid, is a, of the vertical prism. To ascertain the height at which it meets the other prism we look to the end view, finding q" for the entrance and t" for the exit. Being on the way up from g" to e" we use q", reserving t" until we deal with the face g" f". Projecting q" over to q' on edge a a' draw q' s', dotted, since it is on a rear face. Returning to a and moving toward b we next reach the edge e, whose intersection p with a b is then projected to edge e' at p' and joined with q'. For the next edge, b, we x'lgr- zv@. obtain o' from the side eleva- tion, projecting from the inter- section of /"e" by the edge b". Moving from b toward w, projecting to the front eleva- tion from either the plan or the side elevation according as we are dealing with a hori- zontal edge or a vertical one, we complete the intersection. The development of the vertical prism is shown in Fig. 278. As already fully described, rZrf, = perimeter abed in Fig. 277; aQ=a'q'; b = b' o' ; ax=ax (of Fig. 277); a; ,S' = vertical distance of s' from a'c', etc. 160 THEORETICAL AND PRACTICAL GRAPHICS. Although not required in shop work the draughtsman will find it an interesting and valuable r'lr- seo. exercise to draw and shade either solid after the removal of the other ; also to draw the common solid. The former is illustrated by Fig. 279; the latter by Fig. 280. 427. The intersection of two prisms, one vertical, the other oblique hut with edges parallel to V. Let ahcd a' r' (Fig. 281) be the plan and elevation of the vertical prism. Let the oblique prism be (a) inclined 30° to H; (b) have its rear edge //' back of the axis of the vertical prism; (c) have its faces inclined 60° and 30° respectively to V; (d) have a rectangular base 1|" X f". These conditions are fulfilled as follows: Through some point o' of the edge e' o' draw an indefinite line, o'/', at 30° to H, for the elevation of the rear edge, and //, also indefinite in length at first but yV' back of s, for the plan. ,'-'' Take a reference plane MN through s, and, as in Art. 397 (b), construct an auxiliary elevation on M N, transferring it so that it is seen as a perpendicular to o'/', thus obtaining the same view of the prisms as would be had if looking in the direction of the arrow. To construct this make o"/" equal to yV'; draw /"t" at 60° to MN, and on it com- plete a rectangle of the given dimensions, after which lay off the points of the pentagonal prism at the same distances from MN in both figures. Project back, in the direction of the arrow, from /", g"-, h" and i" to the front elevation, and draw g' i' and the opposite base each perpendicular to o'/' and at equal distances each side of o'. For the intersection we get any point n' on an oblique edge, as g', by noting and projecting from THE INTERSECTION OF PLANE-SIDED SURFACES. 161 x-ig-. aes. S"ig-- 2S3. n where the plan grgr meets the face c d. For a vertical edge as c' m' look to the auxiliary elevation of the same edge, as c", getting I" and m" which then project back to I' and m'. The development need not again be described in detail but is left for the student to construct) with the reminder that for the actual distance of any corner of the intersection from an edge of either prism he must look to that projection which shows the base of that prism in its true size: thus the distance of I' from the edge h' is h"l". 428. The intersection of pyramidal surfaces by lines and planes. The principle on which the inter- section of pyramidal surfaces by plane -sided or single curved surfaces would be obtained is illustrated by Figs. 282 and 283. (a) In Fig. 282 the line ah, a'b', is supposed to intersect the given pyra- mid. To ascertain its entrance and exit points we regard the elevation a' h' as representing a plane perpendicular to V and cutting the edges of the pyramid. Project m', where one edge is cut, to m, on the plan of the same edge. Ob- taining n and o similarly we have mno as the plan of the section made by plane a'h'. The plan ah meets mno at s and t, the plans of the points sought, which then project back to a'h' at s' and t' for the elevations. As ah, a'h', might be an edge of a pyramid or prism, or an element of a conical, cylindrical or warped surface, the method illustrated is of general appli- cability. (b) In Fig. 283 the auxiliary planes are taken vertical, instead of perpendicular to V as in the last case. The plane MN cuts a pyramid. To find where any edge v' o' pierces the plane MN pass an auxiliary vertical plane xz through the edge, and note a: and z, where it cuts the limits of MN ; project these to x' and z on the elevations of the same limits; draw x'z', which is the elevation of the line of intersection of the original and auxiliary planes, and note s', where it crosses v' o'. Project s' back to s on the plan of v'o'. If a side elevation has been drawn, in which the plane in question is seen as a line M" N", the height of the points of intersection can be obtained therefrom directly. 429. The intersection of two quadrangular pyramids. In Fig. 284 the pyramid v.efgh is vertical; altitude v' z' ; base efgh, having its longer edges inclined 30° to V. The ohlique pyramid. Let s'y', the axis of the oblique pyramid, be parallel to V but inclined 0° to H, and be at some small distance (approximately v k) in front of the axis of the vertical pyramid; then sc will contain the plan of the axis, and also of the diagonally opposite edges sa and sc, if we make — as we may — the additional requirement that a'c', the diagonal of the base, shall lie in the same vertical plane with the axis. Instead of taking a separate end view of the oblique pyramid we may rotate its base on the diagonal a'c' so that its foremost corner appears at h" and the rear corner at d", whence b' and d' are derived by perpendiculars b" b' and d" d', and then the edges s' h' and s' d'. For the plans h and d use sc as the trace of the usual reference plane, and offsets equal to h' b" and d'd", as previously. The angle a' c' d', or , is the inclination of the shorter edges of the base to V. The intersection. Without going into a detailed construction for each point of the outline of interpenetration it may be stated that each method of the preceding article is illustrated in this 162 THEORETICAL AND PRACTICAL GRAPHICS. problem, and that there is no special reason why either should have a preference in any case except where by properly choosing between them we may avoid the acute angle — a kind of intersection which is always undesirable. intersection of two lines at a very In the interest of clearness only the visible lines of the inter- section are indicated on the plan. (a) Auxiliary plane perpendicular to V. To find m, the intersection of edge s d with the face v h e, take s' d' as the trace of the auxiliary plane containing the edge in question; this cuts the limiting edges of the face at i' and n' whic^ then project back to the plans of the edges at i and 1). Drawing ni we note m, where it crosses s d, and project m to m' on s' d'. Had ni failed to meet sd within the limit of the face vhe we would conclude that our assumption that s d met that face was incorrect, and would then proceed to test it as to some other face, unless it was evident on inspection ^^^.^ that the edge cleared ^^\ the other solid entirely, _^^„ as is the case with sb, ■-"' / s'b', in the present in- /'' stance. By using s',b' as an auxiliary plane the student will obtain a graphic proof of fail- ure to intersect. (b) Auxiliary plane vertical. This case is illustrated by using vg as the trace of an auxiliary vertical plane containing the edge vg,v'g'. Thinking this edge may possibly meet the face sba we proceed to test it on that assumption. The plane vg crosses sa at I, and 86 at p; these project to I' on s' a' and to p' on s'b'; then p'l' meets v' g' at q', which is a real instead of an imaginary intersection since it lies between the actual limits of the face considered. From q' a vertical to vg gives q. The order of obtaining and connecting the points. The start may be with any edge, but once under way the progress should be uniform, and each point joined with the preceding as soon as obtained. Two points are connected only when both lie on a single face of eac'h pyramid. THE INTERSECTION OF SINGLE CURVED SURFACES. 163 ■Fis- 2SS- Supposing that q' was the point first found, a look at the plan would show that the edge sa of the oblique pyramid would be reached before vh ow the other, and the next auxiliary plane would therefore be passed through « a to find uu' ; then would come vh and ad. Running down from m on the face scic we find the positions such that inspection will not avail, and the only thing to do is to try, at random, either a plane through vh or one through sc; and so on for the remaining points. The developments. No figure is furnished for these, as nearly all that the student requires for obtaining them has been set forth in Art. 396, Case 6. The only additional points to which attention need be called are the cases where the intersection falls on a face instead of an edge. For example, in developing the vertical pyramid we would find the development of j' by drawing v' j', prolonging it to o', and projecting the latter to o, when fxo would be the real distance to lay off from / on the development of the base; then laying off the real length of v' j' on v' o' as seen in the development we would have the point sought. Similarly, for tt', draw vx; make V2x, = vx, and ■y^i;, = altitude v' z' ; then v^x, is the true length of t7a; (in space); also, making v^t^^vt and drawing t^ti, we find v^t^ to lay off in its proper place on the development of the same face vfg. 430. An elbow or T- joint, the intersection of two equal cylinders whose axes meet. Taking up curved surfaces the simplest case of intersection that can occur is the one under consideration, and which is illustrated by Fig. 285. The conditions are those stated in Art. 423 for a plane intersection, which is seen in a' b' and is actually an ellipse. The vertical piece appears in plan as the circle mq. To lay off the equidistant elements on each cylinder it is only necessary to divide the half plan of one into equal arcs and project the points of division to the elevation in order to get the full elements, and where the latter meet a' b' to draw the dotted elements on the other. The development of the horizontal cylinder is shown in the line -tinted figure. The curved boundary, which represents the developed ellipse, is in reality a sinusoid. (Refer to Art. 171). The relation of the developed ele- ments to their originals, fully de- scribed in Art. 120, is so evident as to require no further remark, except to call attention again to the fact that their distances apart, e,/i, f^g,, etc., equal the rectification of the small arcs of the plan. 431. To turn a right angle with a pipe by -J pj 'T a four-piece elbow. This problem would arise in carrying the blast pipe of a furnace around a bend. Except as to the number of pieces it differs but slightly from the last problem. Instead of one joint or curve of intersection there would be three, one less than the number of pieces in the pipe. (Fig. 286). 164 THEORETICAL AND PRACTICAL GRAPHICS. Fig-. 237. Let oqs show the size of the cylinders employed, and be at the same time the plan of the ■vertical piece o's'n'a'. Until we know where a' n' will lie we have to draw o' a' and s' n' until they meet the elements from S' and T', and get the joint mM' as for a two-piece elbow. On mM' produced take some point v', use it as a centre for an arc t'xyt" tangent to the extreme elements; divide this arc, between the tangent points, into as many equal parts as there are to be joints in the turn; then tangents at x and y — the intermediate points of division — will determine the outer limits of the joints at a', h' and /. Draw a'v', finding n' by its intersection with ss'; then n' V parallel to a'b', and similarly for the next piece. The developments of the smaller pieces would be equal, as also of the larger. One only is shown, laid out on the developed right section on v' x. The lettering makes the figure self- interpreting. 432. The intersection of two cylinders, when each is partially exterior to the othei: The given con- dition makes it evident, by Art. 423, that a continuous non- plane curve will result. Let one cylinder be ver- tical, 2" in diameter and 2" high. This is shown in half plan in hkl, and in front and f side elevations between hori- zontals 2" apart. Let the second cylinder be horizontal ; located midway be- tween the upper and lower levels of the other cylinder; its diameter ^". On the side elevation draw a circle a" b" c" d" of f" diameter, locating its centre midway between k" I" //and kj^N', and in such posi- tion that a" shall be exterior to k" k^. The elevation of the horizontal cylinder is then pro- jected from its end view, and is shown in part without con- struction lines. The curve of intersection is obtained by selecting particular elements of either cylinder and noting where they meet the other surface. The foremost element of the vertical cylinder is k . . .k' n' m'. Its side elevation, k"k^, meets the circle at n" and m", which give the levels of n' and m' respectively. On the horizontal cylinder the highest and lowest elements are central on the plan and meet the vertical cylinder at e, which projects down to the elements d' and b'. The front and rear elements, c and a, would be central on the elevation. The vertical line drawn from the intersection of element c with the arc hkl gives the right-hand point of the curve of intersection, at the level of a'. Any element us g x may be taken at random, and its elevation found in either of the following ways: (a) Transfer gz, the distance of the element from MN, to s" x on the side elevation, and draw xg" and g" y', to which last (prolonged) project g at g'; or (b) prolong gx to meet a THE INTERSECTION OF SINGLE CURVED SURFACES. 166 semi-circlo on ac at g"' ; make a'y'^xg'" and draw y' g'. The same ordinate xg"\ if laid otF below a, would obviously give the other element which has the same plan gx, and to which g projects to give another point of the desired curve. 433. The intersection of a vertical cone by a horizontal prism. Let the cone have an altitude, ww', of 4"; diameter of base, 3". (As the cylinder is entirely in front of the axis of the cone only one- half of the latter is represented.) s-i^. sea. For the cylinder take a diameter of I"; length 3^^"; axis parallel to V, |" above the base of the cone, and |" from the foremost element. Draw n s parallel to p' r' and f" from it; also g' m' horizontal and 4" from the base; their intersection s is the centre of the circle a" d' c"m', of f" diameter, which bears to the element p' r' the relation assigned for the cylinder to the foremost element ; said circle and jj'ww' are thus, practically, a side elevation of cylinder and cone, superposed upon the ordinary view. The dimensions chosen were purposely such as to make one element of the cone tangent to the cylinder, that the curve of intersection might cross itself and give a mathematical "double point." The width d h, of the plan of the cylinder, equals m' d'. The plan of the axis (as also of the highest and lowest elements, a' and c') will be at a distance \\\\\\\m ,/ sg' from iv. Any element as x' y' h' is shown in plan parallel to pq, and at a distance from it equal either to h' y' if on the rear or to h' x' if on the front. The element through v, on which /' falls, is not drawn separately from bf in plan, since vf and m' g' are so nearly equal to each other; but / must not be considered as on the foremost element of the cylinder, although it is apparently so in the plan. For the intersection pass auxiliary horizontal planes through both surfaces; each will cut from the cone a circle whose intersection with cylinder-elements in the same plane will give points sought. A horizontal plane through the element a' would be represented by a'o', and would cut a circle of radius o' z' from the cone. In plan such circle would cut the element a at point 1, and also at a point (not numbered) symmetrical to it with respect to w Q. Similarly, the horizontal plane through the element x' h' cuts a circle of radius I' h' from the' cone; in plan such circle would meet the elements x and y in two more points (5 and 8) of the curve. As the curve is symmetrical with respect to wQw' the construction lines are given for one -half only, leaving the other to illustrate shaded effects. The small shaded portion of the elevation of the cylinder is not limited by the curve along which it would meet the cone, but by a random curve which just clears it of the right-hand element of the cone. 434. To find the diameter and inclination of a cylindrical pipe that will make an elbow with a conical 166 THEORETICAL AND PRACTICAL GRAPHICS. •pipe on a given plane section of the latter. Let vab be a vertical cone, and cd the elliptical plane section on which the cylindrical piece is to fit. The diameter of the desired cylinder will equal the shorter diameter of the ellipse c d. To find this bisect cd at e ; draw fh horizontally x^e'- seo. through e, and on it as a diameter draw the semi- ,^ circumference fgh; the ordinate eg is the half width of / I \ -the cone, measured on a perpendicular to the paper at e, • \ \ and is therefore the radius of the desired cylinder. / I \ In Fig. 290, the base N G equals twice s^er- sso. ge oi Fig. 289. At first indefinite perpen- diculars are erected at N and G, on one of which a point C is taken as a centre for an arc of radius equal to cd in Fig 289. The angle <^ being thus determined is next laid off in Fig. 289 at c, and cdN"G" made the exact duplicate of CDNG, com- pleting the solution. The developments are obtained as in Arts. 120 and 191. 435. To determine the conical piece which will properly connect two unequal cylinders of circular section, whose axes are parallel, meeting them either (a) in circles or (b) in ellipses; the planes of the joints being parallel. (a) When the joints are circles. To determine the conical frustum be he prolong the elements e b and he to v ; develop the cone v ...eh as in Art. 418, and on each element as seen in the develop- ment lay off the real distance from v to the upper base b c. Thus the element whose plan is v^h is of actual length vk^ and cuts the upper base at a distance v n from the vertex, which distance is therefore laid on vlci wherever the latter ap- pears on the development. (b) When the joints are ellipses. Let the elliptical joints no and qr be the bases of the conical piece qnor. To get the development complete the cone by prolonging qn and or to w; prolong qr and drop a perpendicular to it from w; find the minor axis of the ellipse qr as in tlie first part of Art. 434 and having con- structed the ellipse proceed as in Art. 418, since in Fig. 255 the arc abc.g is merely a special case of an ellipse. 436. The projections and patterns of a bath-tub. Before taking up more difiicult problems in the intersection of curved surfaces one of the most ordinary applications of Graphics is introduced, partly by way of illustrating the fact that the engineer and architect enjoy no monopoly of practical projections. In Fig. 292 the height of the main portion of the tub is shown at a'd'. Let it be required that the head end of the tub be a portion of a vertical right cone whose base angle c'b'a' equals the flare of the sides, such cone to terminate on a curve whose vertical projection is o'n'z'a'. Draw x'ig-. ssa.. THE INTERSECTION OF SINGLE CURVED SURFACES. 167 ng. 2S2. two lines, b' I' and c' i', at first indefinite in length and at a distance a' d' apart. Take a' d' vertical, and regard it not only as the projection of the elements of tangency of the flat sides with the conical end, but also as the elevation of part of the axis, . prolonging it to represent the latter. Use v, the plan of the axis, as the centre for a semicircle of radius vc, whose diameter e cZ is the width of the bottom of the tub. Project c to c'; make angle v'c'd> equal to the predetermined flare of the sides; prolong v' c' to b' and o'; project b' to b on vc prolonged and draw arc abm with radius bv, obtaining avi for the width of the plan of the top. The plan of one-half the curve o' n' z' a' is shown at onzm and is thus found: Assume any element v'x'y'; prolong it to z'; obtain the plan vxy and project z' upon it at z. Similarly for n and as many inter- mediate points as it might seem desirable to obtain. Assuming that the foot of the tub is composed of an oblique cone whose section, his, with the bottom is equal to ecd, and whose base angle is h'i'k', we project i to i', draw i' k' at the given angle to the base, project k' to k, and through the latter draw the semicircle rkq with radius bv, obtaining the plan of the upper base. Joining the tajigent points r and s, h and q, we have rs and hq as the elements of tangency of sides with end. Their elevations coincide in h'l', which meets k' i' at v", whose plan is v^ on hq. S^ig. SS3. The development. Fig. 293 is the development of one -half of the tub. EM equals b' c'; VO equals v' o'; VZ equals v'z", the true length of v'z', obtained, as in previous constructions, by car- rying z to Zi, thence to level of z'. Similarly at the other end. (Reference Articles 191, 408, 418.)- 437. The intersection of a vertical cylinder and an oblique cone, their axes intersecting. Let MBd and M'R'P'N' be the projections of the cylinder; v'.a'b' and v.anbm those of the cone. The axes meet o' at an angle 6 which is arbitrary. 168 THEORETICAL AND PRACTICAL GRAPHICS. X'ig'. 23-i. The ellipse anbm is supposed to be constructed by one of the various methods employed when the axes are known; nnd in this case we get the length of mn from a' h' and its position from n', while ab is vertically above a'h'. (a) Solution by auxiliary vertical planes. Any vertical plane vis will cut elements from the cy Under at e and I; also, ' from the cone, elements which meet the base at s and t. Project s and t to s' and'f', join the latter with the vertex v' and note I' and e' (just below d') where they cross the vertical projection of the elements from I and e; these will be points in the desired curve of intersection. By assuming a sufficient number of vertical planes through V the entire curve can be determined. (b) Solution by auxiliary spheres. If two surfaces of revo- lution have a common axis they will intersect each other in a circle whose plane is perpendicular to that axis.* This property can be advantageously applied in problems of inter- section. With o' — the intersection of the axes — as a centre, we may draw circles with random radii o'f, o'i, and let these represent spheres. The sphere f'g'w intersects the cone in the circle f g'; the cylinder in the circle h'k'. These circles inter- sect each other at x in a common chord whose extremities are points of the curves sought. They are both j:)rojected in the point X. A second pair of circular sections, lying on the same auxiliary sphere, are seen at pq and rw, their intersection z being another point in the solution. The point y results from taking the smaller sphere. 438. Intersection of a cylinder and cone, their axes not lying in the sartxe plane. In Fig. 295 let the cylinder be vertical and the cone oblique, the axis of the latter being parallel to V and inclined 6° to H, and also lying at a distance x back of the axis of the cylinder. The auxiliary surfaces employed may preferably be vertical planes through the vertex of the cone, since each will then cut elements from both cylinder and cone. Thus, vfe is the h. t. of a vertical jjlane which cuts ev,e'v' from the cone, and the vertical element through / from the cylinder; these meet in vertical projection at /', one point of the desired curve. The plan of the intersection obviously coincides with tha,t of the cylinder. ^'igr- ESS. * By the definition of a surface of revolution (Art. 340) any point on it can generate a circle about its axis. If, then, two surfaces have the same axii, any point common to both surfaces would generate one and the same circle, -which must also lie oji both surfaces and therefore be their line of intersection. THE INTERSECTION OF SINGLE CURVED SURFACES. 169 ft/ i x'ier- sss. >v i:::;^*' 439. Conical elbow; right cones imeting at a given angle and having an elliptical joint. This is one of the cases mentioned in Art. 423 as not admitting of illustration t^ in the same way as when dealing with surfaces of uniform cross section, but a plane intersection is nevertheless secured as with cylinders by making the extreme dements of the cones intersect. Let vx in Fig. 296 be the axis of one of the cones. If xyz is the required angle between the axes bisect it by the line ym, and draw the joint cd parallel to such bisector. The right cone which is to meet abed on cd must be capable of being cut in a section equal to cd by a plane making an angle 6 with its axis, and must obviously have the same base angle as the original cone; since, however, the upper portion vdc of the given cone fulfills these conditions we may emj^loy it instead of a new cone, rotating it about an axis p t which is per- pendicular to the plane of the ellipse dc and passes through its centre. The point 0, in which the axis v x meets the plane d c, will then appear at s, by making op=rzps; 'sv', drawn parallel to yz, will be the new direction of vo; and an arc from centre d with radius cv will give v', which is then joined with d and c to complete the construction. If the length of the major axis of the elliptical joint had been assigned, as c/ for example, that length would have first been laid off from some point e on the extreme element and parallel to ym, then from / a parallel to V e, giving g on vc ; then g h parallel and equal to ef, gives the joint in its proper place. 440. Right canes intersecting in a nan -plane curve; axes meeting at an oblique angle. Let one cone, v'.a'b', (Fig. 297) be vertical; the other, oblique, its axis meeting v' 0' at an angle 0. The plane a' b' of the base of the vertical cone cuts the other cone in an ellipse whose longer axis is e'f. As in Art. 434 determine g' h', the semi-minor axis of this ellipse. Project e', g' and /' up to e, g and /; make 170 THEORETICAL AND PRACTICAL GRAPHICS. gh, and gh, each equal to g'h'; then on ef and h^h, as axes construct the ellipse eh^fk, as in Art. 131. Tangents from v^ to the ellipse complete the plan of the oblique cone. (a) The curve of intersection, found by auxiliary planes. In order that each auxiliary plane shall contain an element (or elements) of each cone, it must contain both vertices and therefore the line v'v", which joins them; hence its trace on the plane e' a' b' must pass through the trace, t' t, of such line on that plane. Take tx &s the horizontal trace of one of these auxiliary planes. It cuts elements starting at i and I on the base of the oblique cone. One of the elements cut from the other cone is v p, which in vertical projection (v' p') crosses the elevations of the other elements at q' and r', two points of the curves sought. Since the extreme elements of the cones are parallel to V we will have c' and d' — the intersections of their elevations — for two more points of the curve. Having found other points by repeating the same process the curve c'q'rd' is drawn through them, and the cones may then be developed as in Art. 191. (b) Method by auxiliary spheres. Since the axes intersect we may use auxiliary spheres as in Case (b) of Art. 437. Thus, with o' — the intersec- tion of the axes — as a centre, take any radius o' k and regard arc kyz as rep- resenting a portion of a sphere which cuts the cones in k s and y z. These meet at w, one point of the curve of intersection c' q' d'. 441. Intersecting cones, bases in the same plane but axes not. Let v.kbfg and e.sQhj be the plans of the cones; v.'p'd' and e.'Q'c' their elevations. As argued in Case (a) of the last problem, the auxiliary planes must con- tain the line joining the vertices; their H- traces would therefore, in the gen- eral case, pass through the trace of that line upon the plane of the bases; but, in the figure, both vertices hav- ing been taken at the same height above the bases, the line which joins them must be horizontal, hence parallel to the H -traces of the auxiliaries: that is, X Y, ST, QR, etc., are parallel to V e. c'p/ It happens that the trace MN of the foremost auxiliary plane is tangent to both bases, hence contains but one element of each cone and determines but one point of the desired curve. These elements, a e and b v, meet at n, while their elevations intersect at n'. THE INTERSECTION OF SINGLE CURVED SURFACES. 171 Each of the other planes, except X Y, being secant to both bases, will cut two elements from each cone, their mutual intersections giving four points of the curve of interpenetration. Thus, in plane P, the element e meets v k in q and v d in x, while element h e gives I and m on the same elements. The plane X Y being tangent to one base while secant to the other gives but two points on the curve sought. Order of connecting the -points. Starting with any jjlane, as M N, we may trace around the bases either to the right or left. Choosing the former we find, in the next plane, the point h to the right of a on one base, and d similarly situated with respect to h on the other ; therefore m, on he and dv, is the next point to connect with 7k Elements oe and fv give the next point, then ue and gv locate s, after which those from j and w give the last before a return movement on the base of the v-cone. As nothing new would result from retracing the arc gfd we continue to the left from w, although compelled to retrace on the other base, since planes beyond j would not cut the -w-cone. The element ue is therefore taken again, and its intersection noted with an element whose projection happens to be so nearly coincident with v x that the latter is used. Continuing along arcs och and ikb we reach the plane MN again, the curves ilx and qnm crossing each other then at n — the point lying in that plane. Such point is called a double point, and occurs on non- plane curves of intersection at whatever point of two intersecting surfaces they are found to have a common tangent plane. Tracing to the left from a and to the right from h the elements e and d v are reached, in the plane OP. Their intersection x is joined with n on one side and with the intersection oi Se and gv on the other. Soon the tangent plane X F is again reached and a return movement necessitated, during which the arc XSQOa is retraced, while on the other base the counter-clockwise motion is continued to the initial point b, completing the curve. Visibility. The visible part of the intersection in either view must obviously be the intersection of those portions of the surfaces which would be visible were they separate, but similarly situated with respect to H and V. In plan the point n lies on visible elements, and either arc passing through it is then visible till it passes (becomes tangent to, in projection) an element of extreme contour as at m or t, when it runs from the upper to the under side of the surface and is concealed from view. The point w would be visible on the »-cone but for the fact that it is on the under side of the e - cone. A similar method of inspection will determine the visible portions of the vertical projection of the curve, which will not be identical with those of the plan. In fact, a curve wholly visible in one view might be entirely concealed in the other. 442. The intersection of a vertical cylinder and an oblique cone, their axes in the same plane. If in Art. 440 the vertex v' were removed to infinity the i;-cone Avould become a vertical cylinder; the line v' v" would become a vertical line through v" ; t would be vertically above v" ; but the method of solving would be unchanged. 443. In general, any method of solving a problem relating to a cone will apply with equal facility to a cylinder, since one is but a special case of the other. The line, so frequently used, that passes through the vertex of a cone in the one problem is, in the other, a parallel to the axis of the cylinder. Planes containing both vertices of cones become jjlanes parallel to both axes of cylinders. In view of the interchangeability of these surfaces it is unnecessary to illustrate by a separate figure all the possible variations of problems relating to them. 172 THEORETICAL AND PRACTICAL GRAPHICS. 444. Intersection of two cones, two 'pyramids, or of a cone and a ■pyramid, wtien neither the hoses nor axes lie in one plane. One method of solving this problem has been illustrated in Art. 429, where the intersection was found by using auxiliary planes that were either vertical or perpendicular to V; we may as easily, however, employ the method of the last problem, viz., by taking auxiliary planes so as to contain both vertices. This will be illustrated for. the problems announced, by taking a cone and pyramid ; and, for convenience, we will locate the sur- faces so that one of them will be ver- tical, and the base of the other will be perpendicular to V, since the problem can always be reduced to this fonn. Let the cone v'.a'b', v.cdB, (Fig. 299) be vertical, and the pyramid o'. r' q' p', o.rqp, inclined. We will assume that the projec- tions of the pyramid have been found as in j^receding problems, from assigned data, using oo.^, o' p', (taken perpen- dicular to the base r' q') as the refer- ence line. Join the vertices by the line v' o', V 0, and prolong it to get its traces, ss' and tt', upon the planes of the bases. All auxiliary planes containing the line vo, v'o', must intersect the planes of the two bases in lines pass- ing through such traces. Prolong r' q' to meet the plane a' b' at X. Project up from X, get- ting yz for the plan of the intersection of the two bases. AV'e may assume any number of auxiliary planes, some at random, but others more definitely, as those through edges of the pyramid or tangent to the cone. Taking first one through an edge, as or, we have trz for its trace on the pyramid's base, then zs for its trace on H. The elements cv and dv which He in this plane meet the edge or at e and /, giving two points of the curve. These project to o' r' at e' and /'. FIRST ANGLE METHOD. 173 The plane sy, tangent to the cone along the element uv, has the trace yt on the base of the pyramid, and cuts lines jo and ko from its faces. These meet vu at two more points of the curve, their elevations being found by projecting _; to j' and k to k', drawing o' j' and o' k', and noting their intersections with v'u'. To check the accuracy of this construction for either point, as I, draw i;!;, perpendicular to vu and equal to v'u', join t), with u, and we have in v v^u the rabatment of a half section of the cone, taken through the element vu and the axis; then 11^, parallel to vv^, wiU be the height of I' above the base a'b'. With one exception, any auxiliary plane between sy and sz will give four points of the inter- section. The exception is the plane s Y, containing the edge o q, and which, on account of hap- pening to be vertical, requires the following special construction if the solution is made wholly on the plan: Rabat the plane into H; the elements it contains will then appear at Av, and Bv^, while the edge oq will be seen in o^q^ (by making o Oi = o' 0, and qqi^ q' Q)', elements and edge then meet at /; and iVj which counter -revolve to / and N. We might, however, get elevations first, as /', by the intersection of element A' v' with edge o' q' ; then / from J'. In the interest of clearness several lines are omitted, as of certain auxiliary planes, hidden por- tions of the ellipses, and the curves in which srq (the rear face) cuts the cone. The student should supply these when drawing to a larger scale. * See the latter part of this chapter for further problems on the intersection of surfaces. MONGE'S DESCRIPTIVE GEOMETRY.- FIRST ANGLE METHOD. 445. In this method — the first, and so long the only one employed, and whose use would probably be still universal but for the reason given in Art. 383 — the object is located in front of the vertical plane and above the horizontal, as illustrated in Art. 385. While acquaintance with what has preceded in this chapter would be an advantageous prelim- inary to the study of the First Angle treatment of figures, yet it is not absolutely essential; but if for any reason, as, for example, with reference to its applications to perspective or stone cutting, the First Angle Method is taken up in advance of the other, it is assumed that the student will first thoroughly fimiiliarize himself with Arts. 284-330, and 335-379. 446. To determine one projection of a point on a given surface, having given the other. This problem is of frequent recurrence and is illustrated in Figs. 300 and 301, in which the more familiar sur- faces are shown in their most elementary positions, i. e.^ with axes either perpendicular to or parallel to a plane of projection. The required projection is in each case enveloped in a small circle. Where two solutions are possible both are given. The general solution of this problem, for all surfaces, is as follows : Through the given projection draw a line on the surface, preferably a straight line, but otherwise the simplest curved section possible; obtain the other projection of this auxiliary line and project upon it from the given projection. (a) Right cone, axis vertical. In No. 1 of Fig. 300 the element v' x' is drawn through the given projection a'. Projecting x' down to x and y we draw the plans vx and vy and project a' upon each. (b) Right cone, axis vertical. Solution by auxiliary circle. In No. 2 draw through the given pro- jection a' the line m' n' parallel to the v. p. of the base. It represents a circle of diameter m' n', which is seen in full size in plan, and upon which a' projects in the two possible solutions. (c) Right cone, axis parallel to the ground line. As before, a' represents the given projection. The 174 THEORETICAL AND PRACTICAL GRAPHICS. element v' a' meets the base at ^', whose real distance in front of or to the rear of the vertical diameter of the base is seen at i' k', found by rotating the semi-base on b' c' as an axis until it is seen in full size at b'f'c'. The counter-revolution is shown in plan, and the two solutions indicated. t— i/ W P--- (d) Right cylinder, axis paralkl to the ground line. In No. 4 the two rectangles represent the projections of the cylinder on H and V. To find the plans of the elements whose common eleva- tion passes through a' rotate the end of the cylinder into V, using as an axis the vertical line t c' . (The vertical -diameter method of the last case would answer equally well.) The arcs show the paths of the various points. Then m' n' projects to both p" and q", which are transferred to p and q by arcs from r and s. Fi-s- 301. (^e) Sphere. In Fig. 301 a horizontal section through the given projection, a', cuts a circle seen in full size in plan, upon which a' projects in the two solutions. (f) Annular torus. This surface, also known as the anchor ring, is generated by revolving a circle about an axis in its plane but not a diameter. It has the same mathe- matical properties whether the axis is ex- terior to the circle or is a tangent or a chord; but obviously there would be no hole in the surface except in the former case. In Fig. 301 one -half of the ring is shown in plan and elevation, either shaded section showing the size of the generating circle. The axis is a vertical line through o. The axis being vertical, a horizontal plane through a' will cut two circles from the torus, of radii equal respectively to o' p' and o' q'. These are seen full size in plan, and upon them a' projects. 447. As the representation of any surface of revolution, when its axis is oblique to one or both planes of projection, necessitates the drawing of the oblique projection of a circle, the solution of the latter problem is a natural preliminary to constructions involving the former. 448. To obtain the prcjection of a circle when its plane is oblique to the plane of projection. Proof that such projection is an ellipse. In Fig. 302 let abcd,...a'c', be the projections of a circle lying in a horizontal plane. Using as an axis of rotation the diameter b d, which is perpendicular to V, let CIRCLES AND CYLINDERS OBLIQUE TO PROJECTION-PLANE. 176 us suppose the plane of the circle to rotate through an angle 6; QRC will then represent its new position. Since the axis is perpendicular to V any points, as a and e, of the original circle, will describe arcs parallel to V and therefore seen in their true size in elevation, as at a' A' and e' E', while their plans, a A and e E, will be parallel to G.L; their new positions. A, E, are then, evidently, the intersections of verticals from A' and E' with horizontals through a and e. In Analytical Geometry the " greater auxiliary circle " of an ellipse has for its diameter the major axis of the latter curve; and, by analysis, the relation is established that, when measured on the same perpendicular to such major axis, an ordinate of the circle will be to the corresponding ordinate of the ellipse as the major axis of the ellipse to its minor axis. If, there- fore, we can establish this relation between the circle abed and the curve Ab C d, the latter must be an ellipse. In the elevation we have, from similar triangles, the pro- portion o' E' : o' m :: o' A' : o' 8. But o' £" = o'e' = ex; o' m =^ Ex; o' A' = o' a' =^ a ; and o' a ^ o A : the proportion may Flgr- 303. / therefore be written ex: Ex:: oa: oA::2oa(^bd):2oA(=AC). 449. Working drawing of a hor- izontal cylinder 4" long, 2" in diame- ter, axis 1" above H and inclined 30° to V. Draw first the plan cgsx (Fig. 303) which is simply a rect- angle 2" X 4", with longer sides at 30° to the ground line. The ends eg and sx are circles 2" in diame- ter and vertical. Rotate the base eg about the vertical tangent e.a'b' until it takes the position c^/i, when its elevation g" n" c' d" will equal the circle of which it is the pro- jection. The centre of such circle will be at the height (1") assigned for the axis. Note various points, q, as (/,(/", /,/", d,d", and then, by a construction in strict analogy to that of the last problem, counter- revolve them into the original plane. Their paths of rotation will be hor- izontal arcs, seen in full size in plan, but as horizontal straight lines in elevation, as g" g', /"/', d" d'. 176 THEORETICAL AND PRACTICAL GRAPHICS. The new elevations are then the intersections of verticals from g , f, d to the levels at which they rotated, giving points of the ellipse g' f d' c'. The other end of the cylinder might have been obtained similarly, but the figure illustrates the use of the horizontal diameter sx, s'x', as an axis, when qq^ shows the distance to lay off above and below v' to get the levels of the elements whose common plan is pq; that is, for p' cf and for p"'q"'. 450. To project a circle when its plane is oblique to both planes of projection; also to draw a tangent at a given point. To avoid multii^licity of lines we will assume that in Fig. 304 P Q P' — the jjlane of the circle — has been already determined from assigned inclinations, by means of Art. 319. Let it be required that the circle lying in that plane shall have a given radius (o d), and that its centre shall be at assigned distances, b' b and on, from H and V respectively. To fulfill the condition as to height of centre draw a' b' at assigned height b' b above G. L., to represent the v. p. of a hor- izontal of the plane. On the plan a 6 of such horizontal note the jjoint o which fulfills the con- dition as to assigned distance (on) from V; this will be the h. p. of the centre and projects to a'. By Art. 306 rabat o into H, about PQ as an axis. It takes the position Oj, about which draw a circle with the prescribed radius o d. The diameter d,/,, which is parallel to the axis P Q, remains so during counter-revolution, and at df (passing through the original o) becomes the major axis of the ellipse, since it is the only diameter which is horizontal and therefore projected on H in its actual size. Project d and / to d' and /' on a' b', since its elevation must evidently be parallel to G. L. The minor axis of an ellipse being always perpendicular to the major must in this case be the space - position of c , e i . It will be part of the line of declivity (Art. 301) cut from plane PQP' by an auxiliary vertical plane RSP', and which appears at mRj when the latter plane is carried into H about RS us an axis. (R,S=SP'). On such line we find o^ representing o, and make 0^6, = df for the auxiliary view of the minor axis sought, whence c and e are derived by counter-revolution. We find c' at height c'y^^c^c; similarly e'z = e^e. But one diameter can be parallel to V, and in this case it must be a V- par- allel (Art. 300) of PQP; therefore through o' (and bisected by it) draw g' h' parallel to P' Q and of length dj. Its plan gh is parallel to G. L. To draw a tangent / at any point, as t, find ^ ^ 192 THEORETICAL AND PRACTICAL GRAPHICS. Make J 'P, P Q, QR, each equal to j'x". Also make h" J and JK equal to h" G ; then, with Z at the same level as /, we see that PZ and QK are positions of j' h" differing from each other by a semi -revolution, and that if we rotate PZ to PJ and then make j' A, P X, Q N, R L, each equal to the original length a" x" {=x" G), the curve LN..DG will be a conchoid, whose revolution about Wx" will give the new locus of the point a". The determination of the cmve AEHN, traced upon this new surface by the point A, is stated in the next article in the form of a general jiroblem, and it need only be further remarked as to the surface in question, that as Wx" is an asymptote to the conchoid, the helicoid will, at its limit, become a plane, tangent to a meridian plane of the surface. 485. To determine the line of intersection of a helicoid tvith a con- axial surface of revolution. Illustrating from the upper elevation in Fig. 326, let a" j' h" B be the directrix, and YB any element of the helicoid. Let LNDG be the meridian section of a surface of revolution ha-\ang the same axis, Wx", as the helicoid. Rotate the element VB to YC, when it and LNDG are seen in true relation to one another, being parallel to V. They intersect at D, which counter -revolves to E on the original position of the element. s-ig-. 3ST. The same process may be repeated with other elements until a suflRcient number of points have been determined for the drawing of a fair curve. 486. Helicoids of axially - expanding pitch. If the generating line of a helicoid have a uniform turn- ing motion about the axis, combined with a varying rate of motion in the direction of the axis, it is said to have an axially expanding pitch. With its generatrix intersecting the axis such a helicoid has been constructed for the acting surface of a screw propeller, with the idea that the water upon which it acted would be followed up by the ele- ments which had set it in motion, at a rate in some degree approximating the accelerated move- ment of the receding mass. 487. To draw a helicoid of axially expanding pitch. In Fig. 326 the axis nv is divided into parts n m, m r, etc., that are in some ratio to each other.; in arithmetical progression in this case. Horizontal lines are then drawn through the points of division, upon which — as for the ordinary helix — the points a, b, c, d, etc., are projected from the plan to obtain the helix shown, the elements in plan making equal angles with each other. Since the ordinary helix develops into a straight line, it is obvious that a helix of the kind under consideration will develop into a curve. 488. The conchoidal hyperholoid of Catalan. In accordance with the definition of Art. 359 draw any vertical line a" e', ab (Fig. 327) for one directrix; let the horizontal directrix be the line WARPED HELICOIDS. 193 m'n', mn, and assume 45° for the inclination of the elements to the vertical directing line. In ar we have the plan of that element that meets a" e' nearest to its middle point. Carry r to r, whence project to m'n' at o', when o'a', at 45° to the vertical, will bo the element in revolved position. After counter-revolution its vertical projection becomes a'r'. On the vertical directrix lay off, for convenience, eqtnd spaces from a', as a' b', h'c', etc. Since the elements are all equally inclined they will be parallel to each other if rotated till parallel to V- hence the dotted parallels through a', b'....h' will represent a few in such position, their common plan being then apj. In counter-revolution s, reaches o, projects to o' and joins with b', for the space -position of b' s'. Similarly c' t' becomes c' v', cu; and from y' we have y, then z and z'. By making r' d' equal to r' a', and working downward from d' as we have upward from a', a second surface of the same nature is formed, on the same directrices and with elements having projections coincident with those of the first surface. The two surfaces can best be distinguished from each other by the use of colors. The elements that are visible in front of m n on the plan are on that portion of the surface which is visible above m' 7i' in the elevation, and would be in the same color; while the part behind mn and below m'n' would represent the other visible portions, and would have the other color. The student should note, however, that the same part of one element is not visible in both views, in the latter case, but is in the former. By prolonging an element, as f o', no, to meet any limiting horizontal plane, P' Q', we obtain a point k' k of the conchoidal arc Ikx' (Art. 193) in which such plane would cut the surface. The other element o'j' passing through o' will meet the same plane in a point j', which gives on the plan ao a, point j of aji, a part of the other branch of the conchoid. A tangent jjJane to the conchoidal h3'perboloid at any point would most simply be determined by the element containing the point, and the tangent to a conchoidal section through the point, a method for drawing which has been given in Art. 195. 489. The cylindroid of Prezier. This surface may be readily constructed by the student without other illustration or definition than that already given in Art. 360. In Fig. 220 abed is the plan of a cylindroid, and the oblique figure is an enlarged elevation of the same. A tangent plane at any given point would most conveniently be determined }jy means of an auxiliary hyperbolic paraboloid raccording with the cylindroid along the element through the point, V , being the plane director; while the directrices would be the tangents to the elliptical bases at the extremities of the element. 490. Either (a) through an exterior point, or (b) parallel to a given line, it is, in general, possible to pass an infinite number of tangent planes to a ivaiped surface. In the former case they would all be tangent to a cone having the given point for its vertex, and for elements the tangents drawn through the point to the curves cut from the surface by planes containing the point, since any such tangent would — with the element through the point of tangency — determine a tangent plane. In the latter case they would be the possible tangent planes to a cylinder whose elements are tangent to the curves in which the surface is intersected by a system of planes parallel to the line. 491. If a given line be prolonged to meet a warped surface, the element through their inter- section will — with the given line — determine a tangent plane to the surface and containing the line. (Art. 469). If the line can meet more than one element, whether because the surface is doubly -ruled or by reason of intersecting the surface more than once, or when both of these conditions exist simultane- ousl}', each element that it meets would — with the given line — determine a tangent plane. 194 THEORETICAL AND PRACTICAL GRAPHICS. TANGENT PLANES TO DOUBLE CURVED SURFACES. 492. Art. 378 presents the principles on which problems under this head are solved. Refer to cases (e) and (/) of Art. 446, if necessary, as to the projections of points on double curved surfaces. 493. A plane, tangent to a sphere at a given point, is perpendicular to the radius drawn to that XT^- 3SS. point ; hence in Fig. 328 we make a plane P' Q P tangent to the P sphere at oo' by drawing the contact - radius o c, o'c', and then, by Art. 317, passing a plane perpendicular to it at its extremity, oo'; od, o' d' — a "horizontal" (Art. 300) of the plane sought — giving d', through which P' Q is drawn perpendicular to o'c'. 494. A plane, tangent to an annular torus at a given point, would be determined by the tangents to the circular sections con- taining the point; or as in the last problem, by being made perpendicular to the radius of the circular section in the meridian plane through the point. Illustrating only the first method, we take through the point pp', Fig. 329, a meridian section cd (eleva- tion unnecessary), and a parallel, p,px. Carrying the former about the axis c until parallel to V it appears at ^^- ®^®- a'p"b', ah, and p' reaches p", at which the tangent p"s" is drawn. The trace «, of the tangent counter -revolves to s, through which the h.t. of the desired plane (Ps Q) is drawn, parallel to np, which is not only a tangent at p to the parallel Pipx, but also a horizontal of the plane sought. The V. t. of the plane joins Q with n', the v. t. of the horizontal np. 495. A plane, tangent to a sphere and containing a given line, may be found on this principle: A plane through the centre of the sphere and perpen- dicular to the line would cut the sphere in a great circle, the line in a point; either tangent that could be drawn from the point to the circle would — with the given line — determine a plane fulfilling the conditions. If the given line is tangent to the sphere there is but one solution, and if it intersects it the problem is impossible. In Fig. 330 let ab, a'b' be the given line, and oo' the centre of the given sphere. The plane ed'c' is then drawn, containing oo' and perpendicular to the given line, determined by means of the horizontal o c, o'c'. (Art. 300). The line and plane intersect at ss'. (Art. 322). After rabatment into H about ed' as an axis we find s at «,, and o at o,, the latter the centre of the revolved great circle cut from the sphere by plane ed'n'. (Art. 306). Draw the tangent s, <,. It meets ed' at i, which is not only the h.t. of the tangent, and as such may be joined with the like trace of ab, a'b' to give PQ, but being also a constant point during rotation may be joined with s to give the plan of the tangent when in its true position. Then t^ projects upon is at (, whence o t aa the contact radius. Projecting- i to i' and joining with s' we have the v. p. of the tangent, upon which t projects at TANGENT PLANES TO SPHERES. 195 Fier- 330. «', whence o't' follows for the v. p. of ot. P' Q passes through m', the v. t. of the given line, and is perpendicular to o't', either of which condition is, with Q, sufficient to determine it. 496. A plane, containing a given line and tangent to a given sphere, may also be found hy means of an auxiliary cone, thus: Make any point of the given line the vertex of a right cone which is tangent to the sphere; then either tangent to their circle of contact, drawn from the trace of the given line upon the plane of that circle, will — with the given line — determine a plane meeting the requirements. Should the given line happen to be parallel to the plane of the circle of contact of cone and sphere, the required plane would be determined by (a) the given line, and (b) a line parallel to the given line and tangent to the circle of contact. The axis of the auxiliary cone may pref- erably be parallel to either H or V, so that the base may be projected as a straight line on that plane. In Fig. 331, deciding that the axis of ■ the auxiliary cone shall be horizontal, we take for its vertex vv' that point of the given line ah, a'b', that is at the same level as the centre oo' of the sphere; then ved is the plane of the tangent cone, and ed that of its circle of contact. The plane of e d meets the given line at ss'. Using for an axis the line ed (regarded now aa the diameter of the circle of contact) we find s and said circle appearing at s, and dt^e, when brought into a horizontal plane. Draw the tangent s^c and project c upon q' o' at c'; then in counter-revolution the tangent line becomes sc, s'c', and the tangent point t, t', whence ot, o't' follows for the contact radius. Through the traces (x' and n) of the given line the traces of the required plane are last drawn, perpendicular to o't' and I respectively. 497. A third metfiod of determining a plane through a line and tangent to a sphere is to envelope the sphere by two tangent cones whose vertices are on the given line. The common chord of their circles of contact will pierce the surface of the sphere at points, either of which will — with the given line — determine a plane fulfilling the conditions. 498. A tangent plane to any surface of revolution, at a given point P, is perpendicular to the meridian (axial) plane containing the point. For a tangent at P to a parallel of the surface would be perpendicular to its radius, and also to a line drawn through P parallel to the axis; hence any plane through such tangent would be jjerpendicular to the plane that would be determined X"ier- 331- ''"^^-v / i---^\A " ^j4^ -:^^V_4V>^ ,pi(i25— ^M 1 ' i \''-^^\j\ i i ( . of the same edge. Similarly, got m from m', n from n', and complete the shaded plan of the section. Another way of getting all points of the section bid one, is to use the intersections of the trace R L with the H -traces of the various faces. Thus, by the first method, get from o', with which to start; then, as e rf is the h. t. of the face ved, we shall have j as one point of the intersection of that face with the given plane, and jo for the line itself, op being that portion of it which lies within the limits of the face considered. In like B< / T "^^^^-^^^^ ^ \ \ / 1 I \ '■ \ ^.-■—' 6 ? \\ 7 ! /%^^. -''' -'" ' "^-^ No ^ y E 1 iy /\X^ ^3 e z' : 4' -s- sea. ing problem. '"^ (2) Using ttvo planes of projection we would lay out the given faces a and b in either (in H, in Fig. 352). For convenience let the ground line be perpendicular to one of the edges, as OP. Lay out the angle K' P equal to the given angle C. Next rotate the face a about P as an axis until R falls on PK , when its then position jS" will also be a point of the third face c; while S' P, SO, will be the projections of the space -edge. S' Q is the plane of the third side c. To find its inclination (A) to the face 6 draw Sy perpendicular to Q, for the plan of a line of declivit}' of the plane. Carry y into V at o and there connect with S', when in S'oS we find in its true size the angle sought. Make yE^S'o and draw OE for the edge of c, after development of the latter into H. OS, being the intersection of a and c, may be employed as in Art. 323, to find the angle B. 529. Given, two sides and the angle opposite one of them. From the fact that there are, usually, two solutions possible, this is called the ambiguous case. (1) Illustrating first by a solution on two planes of projection (Fig. 353) take the given sides, a and 6, in H, with the ground line, R Q, at 90° to their common edge OP. Let the angle A be the other given element of the problem. Draw q y, at 90 ° to Q, as the h. p. of a line of declivity of the face c; carry y io o from centre q, and make qor^=A; ro meet- ing the vertical 5 r at a point of the v. t. of the face c. Draw Q r. Rotate face a upon P. The arc described by R cuts Q r at S" and S', either of which, connected with 0, would give the space- position of the edge common to faces a and c. If the former be used, draw s e perpendicular to OQ and make eE,^S"e", the true length of the line of declivity se. Then Q E^ is the face c for one solution, while QOE^, analogously found, is its value for the S'- solution. The angle C is either S"P Q or S'P Q, while B is found as in Art. 323. (2) Using only the horizontal plane in the solution, (Fig. 354) lay out a and h upon H, and anywhere on the face b draw a line og at 90 ° to Q, for the h. t. of a vertical plane containing the angle A. Make ogi equal to said angle, terminating gi hy & perpendicular to og at 0. Given a, ft, C ^*' OF THE '»'^ UNIVERSITY SOLUTION OF SPHERICAL TRIANGLES. 209 E'lg-. SS3. Qtven a, b,A Draw jQ at 90" to OP, as the h.t. of a plane in which some point j of face a would rotate. Rotate such plane into H about j Q as an axis. As i was a point vertically above o and TO the face c, its new position A: is one point of the trace of c upon the plane j Q; hence QkF represents c upon the rotation -plane of j. The arc j S T, described by j, cuts Q F at S and T, either of which, joined to 0, fulfills the conditions. Completing only the S- solution, draw So, and in So Q we have the angle C. Draw the line syE perpendicular to the edge Q; revolve Ss to sSj about sy aa an axis; then by joining s, with y and carrying it by an arc, centre y, to E on sy produced, we shall have in £■ the second edge of face c. Os being the plan of the united OR and R^, make tr, at 90° to it, for the h.t. of the plane containing angle B. As before, drop perpendiculars rp and xz on the faces c and a, and use them as radii with which to get I, the rabatted vertex of angle B. Then Ix and Ir complete the solution. 630. As earlier stated, the remaining cases may by means of the polar triangle be made to take the form of those already solved, although they are solved in the following articles without •"'v, y' recourse to that expedient. ~~"-,^ _,--'' 531. Given, one side mid the adjacent angles. ' (1) Solved upon but one plane, H, let b, the given side (Fig. 355) be taken in the plane of projection used. At any point of P, as s, draw s r at 90 ° to P, and s m making the given angle C with s r. Simi- ^^s- sss. larly, at a random i)oint e of Q, lay out the other given angle, A. In am and ei we see lines of declivity of the desired faces a and c, after rotation into the plane of b. Take some point m on sm. When in its space -position the height of m above the face 6 is sn. Make ef^sn and find i, a point of equal height with m but on face c. Then i k and m k are the projections of horizontal lines in the faces O'' c and a, and being at the same level must intersect on the line of intersection of those faces; hence join with k to obtain the plan of the space-edge. Solve the remainder as in earlier problems. f*, 210 THEORETICAL AND PRACTICAL GRAPHICS. E^er. 35S. M v' \K n pX^'' \ \ \ ■ ^ , shows that ii a E is a chord, a t must be a tangent. To project a meridian of longitude. Let the meridian to be projected make an angle with the plane of the primitive meridian, WNES, upon which it is to be projected. Draw SP at 6° to SN; then P is the centre and PS the radius of the arc NTS in which the meridian projects. This is established as follows: Let Fig. 363 represent a top view of a sphere; the point of sight on the equator; MB the plan of a x-ig-- sss. meridian making 6° with the primitive, m WE; then mb \b the plan of the circle in which MB is stereographically projected; and P, bisecting m b, is the centre of the projection. Now as MOB is a right angle we have P = Pb; hence angle P B equals Pb 0, or (f>. But <^ = 6 + /3; and as we have NOB = p, it follows that PON equals 6. Draw the arc o 5 in Fig. 362. Then P, the centre of the meridian NTS, is distant from the centre of the sphere an amount P which is the tangent of the inclination of the meridian to the primitive; while S P, the radius of the projection, is the secant of such inclination. 548. To project a small circle stereographically when its plane makes an angle of 90° with the primitive. x'ig-. ss-i. In Fig. 364 let XM he a small circle whose plane is per- pendicular to the primitive KNm; then the projectors QX and Q M give x m for the stereographic projection as seen in plan. Since R X equals Q M the lines R M and Q X will meet at x. Bisect mx at 0. Draw Mo and M N. The angle R Mm = R M Q = ^° ; hence M== ox ^= om. The angles are equal; also angles j8. We then have NQM + NmQ = NMQ + oMm^dO°; hence iVMo = 90°, and Mo is a tangent to arc Mt. We see, then, that M, the radius of the projection, equals the tangent of the polar distance Mt; while the centre of a small circle's projection is distant from the centre N of the primitive an amount No equal to the secant of said polar distance. SPHERICAL PROJECTIONS. 215 Fig:. ssT-. N 549. If 6 is the angle between the primitive and any circle, great or small, the poles of such a a circle will project upon the line of measures at distances which are, respectively, tan ~ and cot ^ the former for the pole farthest from the centre of projection. For in Fig. 365, which is a plan of the sphere, Q is the centre of projection; pNE the primitive; P and R the poles of circles WL, IJ, MX, all inclined 6° to the primitive; p and r, the projections of the poles, and Np and Nr to have values as just stated. The angle NQr is 90°. Arc QR equals EX. Hence QPR=h6, and pN=tanie. Nr obviously then equals cot i 6. 550. Having given the pole of a circle, to project the circle. Let P (Fig. 366) be the pole of a small circle whose polar distance is 60 °. Prolong QP io q and lay off" arcs q X and q Y each equal to 60°. Project X at a; and Y at y and draw a circle on zy a& a. diameter. This will be the projection sought. E"ig:. 3SS. For a great circle the arcs q X and q Y would be made 90 °, but other- wise the solution would be unchanged. 551. Stereographic equatorial projection. In this pro- jection the meridians project as straight lines, and the parallels as circles, concentric with the primitive. The radius of any projection will be the tangent of one -half the polar distance. Thus, in Fig. 367, let the circle WNES represent the equator and let M R he a parallel of latitude. We shall then have nr equal to the tangent of one -half the arc N R. 552. Although distortion of j'orm is obviated by using stereographic projection, that of areas is quite considerable near the centre of the map as compared with the outside. GNOMONIC. — NICOLISI'S GLOBULAR. — DE LA HIRe's.— SIR HENRY JAMES'. 553. Gnomonic projection is believed to have been the first method employed in projecting the sphere, it dating back to Thales, one of the seven wise men of Greece. Although used chiefly for celestial charts, it derived its present name from its serviceability in the graduation of gnomons. It has been employed to some extent for representing the polar regions. This projection is made upon a tangent plane to the sphere, the eye being taken at the centre of the latter. Every great circle will project in a straight line, while small circles parallel to the primitive will project in concentric circles whose radii are the tangents of their polar distances. As the great circle parallel to the primitive will project at infinity, this method will evidently not answer for an entire hemisphere. 554. Nicolysi''s Globular Projection.* This method of representation, for it is not a true projection, is largely employed in making terrestrial maps. As meridians and parallels appear as circular arcs, it has in that respect the same advantage as stereographic projection over others less conveniently constructed. It lacks, however, the ortho- morphic property of the stereographic. It was invented in 1660 by J. B. Nicolisi, of Paterno, Sicily. To represent the meridians and parallels by this method draw a circle for the primitive meridian; M / ;" I 71 Jr • Following Germain, the term glotndar Is here applied to the method which first received the name. Having been Intro- duced into England in 1794 by Arrowsmlth it has been erroneonsly accredited to him. 216 THEORETICAL AND PRACTICAL GRAPHICS. a horizontal diameter for the projection of the equator, and a vertical diameter for that of the central meridian. Then, for ten -degree intervals, divide the quadrants into nine equal parts, number- ing each way from the equator. Also divide the horizontal and vertical radii into nine equal parts, numbering each way from the centre. The parallels are then drawn as circular arcs through like- numbered divisions each side of the equator, while the meridians are circular arcs containing the poles and the divisions on the equator. 555. De La Hire's Perspective Prelection. This projection, often erroneously termed globular, was devised by De La Hire in 1701. In it the eye is taken on the prolonged axis of the primitive, and at a distance from the surface of the sphere equal to the sine of 45°. Its classification is obviously under zenithal projection. With a radius of unity the sine of 45° = v/i. Hence, in Fig. 368, if Cr=nX=x/J, then Ox and X E, which are the projections of the 45°- arcs NX and X E, will be equal. Other equal arcs will have projections very nearly equal. This is its only prac- tical advantage, as it is neither orthomorphic nor equivalent, and involves elliptical projections for all circles not parallel to the primitive. 556. Sir Henry James' Perspective Projection is an interesting case of zenithal, devised for the purpose of reducing the misrepresentation to a minimum. Like De La Hire's, the eye is taken exterior to the sphere, but in this case at a distance equal to one -half the radius. For a hemisphere this is regarded as the best possible system of projection. By taking a plane of projection parallel to the ecliptic and touching one of the tropics, or, in other words, by adding a 23 J °- zone to the hemisphere. Colonel James obtained America, Europe, Asia and Africa in one projection, claiming it to include "two -thirds of the sphere." This has been shown by Captain A. K. Clarke to be an underestimate, the exact figures being seven -tenths; while the same writer shows that for minimum distortion with the new primitive the eye should be at a distance of \\r outside the surface, instead of ^. PROJECTION BY DEVELOPMENT. — CYLINDEIC. — CONIC. — POLYOONIC. 557. With the eye at the centre of the sphere we may project the various circles of the latter upon either a cylinder that is tangent or secant to the sphere, or upon a tangent or secant cone. By then developing the auxiliary surface we will have in the one case a cylindric and in the other a cmiic projection. 658. In square cylindric prelection the auxiliary cjdinder is tangent along the equator. The meridians then appear as straight lines perpendicular to the rectified equator, while the parallels — which projected as circles — develop into straight lines at 90° to the meridians, the distance of each from the equator being the tangent of its latitude. This projection is only occasionally used, the exaggerations involved being too great to make it serviceable except for a short distance each side of the equator. 659. Mercatar^s projection (also called a reduced chart) differs from the last described only as to the spacing of parallels. This spacing is, however, so effected that on the resulting map the angles are preserved between any two curvilinear elements of the sphere; in other words, Mercator's is an orthomorphic projection. Since meridians actually converge on a sphere at such rate that the length of a degree of longi- tude at any latitude equals that of a degree on the equator multiplied by the cosine of the latitude, it is obvious that when they are represented as non- convergent the distance apart of originally SPHERICAL PROJECTIONS. 217 equidistant parallels of latitude should increase at the same rate; or, otherwise stated, as on Merca- tor's chart degrees of longitude are all made equal, regardless of the latitude, the constant length representative of such degree bears a varying ratio to the actual arc on the sphere, being greater with the increase in latitude; but the greater the latitude the less its cosine or the greater its secant; hence lengths representative of degrees of latitude will increase with the secant of the latitude. The increments of the secant for each minute of latitude can be ascertained from tables. Navigators' charts are usually made by Mercator's projection, since upon them (as upon the square cylindric) rhumh lines or loxodromics — the curves on a sphere that cross all the meridians at the same angle — are represented as straight lines. A loxodromic not being also a geodesic, the mariner takes for his practical shortest course between two points the portions of those difTerent loxodromics which most nearly coincide with the great -circle arc through the points. 560. In conic projection, if the auxiliary cone be tangent along a parallel of latitude, the meridians will project as elements of the cone; the parallels into circles. On the development the parallels become concentric arcs on the sector into which the cone develops, the radius of each being the slant -height distance from the parallel to the vertex. The meridians obviously develop into radii of the sector. 561. If tangent to the sphere near the equator the vertex of a cone is inconveniently remote. Even when tangent along a parallel of latitude more medially situated this method gives undue distortion, except for a narrow zone on which the parallel of contact* is central. Many methods have been devised for the purpose of obviating these difficulties, a few of which are next briefly mentioned. 562. Mercator suggested the substitution of a secant for a tangent cone, choosing its position with reference to the balancing of certain errors. By this method a large' map of Europe was made in 1554. Euler carried out the same idea with greater exactness, fulfilling his self-imposed conditions that the errors at the northern and southern limits should not only equal each other, but also the maximum error near the mean parallel. 563. Bonne, in 1752, applied the following method (its invention is variously accredited) which was later adopted (1803) by the French War Department, and has been extensively used in European topographical work: Assuming a central meridian and a central parallel, a cone is made tangent to the sphere on the parallel. The central meridian is then rectified on the element tangent to it, and using the cone's vertex as a centre circular arcs are drawn through (theoretically) consecu- tive points of the developed meridian. The zones between the consecutive parallels on the sphere then develop in their true areas upon a plane. Each meridian is drawn upon the map so as to cut each develo])ed parallel at the same point as on the sphere. The parallel of tangency cuts each meridian at a right angle. Bonne's method evidently comes under the head of equivalent projections, as it preserves the area though not the form of all elementary quadrilaterals. When extended to include the whole earth in one view the map has a peculiar shape, some- what like a crescent with full, rounded ends, and quite broad at the centre. By taking the equator for the "central parallel" a projection results, due to Sanson, called sinusoidal by d'Avezac, and often credited to Flamstead. When applied to the entire sphere it resembles two equal and opposite parabolas with their extremities joined. 564. Polyconic Projection. A method largely used in England and employed by the United States Government on its Coast and Geodetic Survey, is based upon the use of a separate tangent cone for each jjarallel to be developed. 218 THEORETICAL AND PRACTICAL GRAPHICS. 565. In Rectangular Polyconic Projection the rectangularity of the quadrilaterals between meridians and parallels is preserved. It is thus constructed: In the elevation, Fig. 369, let b"b', d"d', f'f be parallels of latitude; their plans will be the concentric circles shown in the upper figure. Nf, V'd', v'b', are the elements of cones, tangent to the sphere on the indicated parallels. Taking the meridiian Edh N as the central meridian, rectify it at E" D B in the lower figure, getting the lengths from E'd'b'. Make E"F^E'f', etc. Through F an arc of radius Nf is the development of the parallel /"/'• The arc through B has radius v'b'. On the plan draw meridians Ni, No, etc. Then lay off on each developed parallel the distances included on it between the meridians just drawn. Thus, DK and KM equal the rectified arcs dk and km. B TZ equals the true length of htz. When the parallel of tangency is so near the equator as to make the vertex of the auxiliary cone inconveniently remote, tables are employed giving the rectilinear coordinates of points on the developed parallels. 566. Equidistant Polyconic Projection is a modification of the method just described, resulting in a representation in which two parallels will include equal arcs on all meridians. This method is used in Government work, for small areas. To draw it a central meridian E" F D B (Fig. 369) and a central parallel DKMW are drawn as in the rectangular polyconic system, and the meridians also found in the same manner, or by the use of tables. From the points D, K, M, W, where the central parallel intersects the meridians, the equal lengths FD, D B, are laid off on the meridians, giving points through which the other parallels may be drawn. 567. Ordinary Polyconic Prcjection. This method sacrifices the rec- tangular intersection of meridians with parallels (except on the central meridian) in order to preserve the lengths of the degrees on the parallels. RECTANGULAR POLYCONIC Drawing the usual central meridian in its true length, the parallels are developed as for the rectangular polyconic; but on each parallel the degrees of longitude are laid off in their actual lengths, and points thus obtained through which to draw the meridians. This method is in general use by the U. S. Government for the maps of its Coast Survey. 568. The foregoing is as extended an excursion into this attractive field as the limits of this treatise will permit, but it should be understood to be but a glance, and that a large number of interesting methods must go unnoticed, the student being referred to the authorities earlier mentioned, in case he wishes to pursue the subject further. SHADES AND SHADOWS. 219 CHAPTER XIII. SHADES AND SHADOWS OF MISCELLANEOUS SUKFACES. 569. The shadows cast by an object which is illumined by either the sun or some other source of light are, in the mathematical sense, projectiojis, and the rays of light become the projectors. 670. The shade of an object is that part of its own surface which receives no direct rays from the source of light, while the shadow is the darkened portion of some other surface from which the original object excludes the light. The rays through all points of a given line will determine either a plane of rays or a cylinder of rays, according as the line is straight or curved. 571. The line of shade on an object is the boundary between the illumined and the unillumined portions, and its shadow forms the boundary of the shadow cast by the object. If the object is curved, the line of shade is the line of contact of a tangent cylinder of rays, each element of which would be tangent to the object at a point at which the cylinder and the FLg. sTo. object would have a common tangent plane of rays. For convex plane -sided surfaces the line of shade is the warped polygon formed by the edges contained by non- secant planes of rays. 572. It is the province of Descriptive Geometry, in its application to this topic, merely to determine the rigid outlines of shadows and shades. The delicate effects of cross and reflected lights, which always exist in nature in greater or less degree, can only be theorized about in a general way and can be most suc- cessfully imitated in draughting by working from a model or by the aid of photographs. Figs. 370 and 371, which are half-tone reproductions of photo- graphs of plaster models, while illustrative mainly of shades as distin- guished from shadows, also show the absence on double curved surfaces of those rigid lines of demarcation to which theoretical constructions lead. Yet the ability to correctly locate the geometrical lines of shade and boundaries of shadows is as essential an element of the draughts- man's education as a knowledge of the laws of perspective; since, lack- ing either, he could neither make an intelligent visit to an art or architectural exhibition nor so work up an original design that it could bear critical examination. 573. A conventional rule, much employed for throwing machine drawings into sharp relief, is to Fig-. 37X. 220 THEORETICAL AND PRACTICAL GRAPHICS. make the right-hand and lower boundaries of a flat surface shade (i. e., heavy) hnes, provided that they separate visible from invisible surfaces. When, however, they are located with reference to some source of light, as in architectural and other drawings, the shade lines are those which could cast shadows, and their determination usually requires more thought, as some of the succeeding problems will show. 574. Direction of Light. A direction quite often (though not necessarily) assumed for the light is that of the body - diagonal of a cube whose faces are parallel to the planes of projection; that diagonal, in particular, which descends from left to right in approaching the vertical plane. It is illustrated by the arrow in Fig. 372, the source of light being assumed to be the sun, whose rays may for all practical purposes be regarded as parallel. 575. The ray FR (Fig. 372), being the body -diagonal of the cube, will project xa. ER on the base, or in .4i2 on the back. ER and AR, being diagonals of squares, make 45° with the edges of the cube, which has led to the expression "light at forty-five degrees," for the conventional direction. But the ray itself makes an angle (^) of 35° 16' with either E R or A R, as may be established thus: Taking the edge of the cube as unity we have ER^V2; also tan ° to H is found by duplicating the last procedure in every detail, s^t^t^ being then the triangle whose altitude t^t^ is laid off vertically from T', and toward which 3'2' and 5'6' converge. The remaining construction is as follows: Vertical visual planes are drawn through Sj and all points of the object. To avoid complicating the lines only a few of these are shown, s^G^, s,P,, SiF^, s^J. Taking K^P^Si as illustrative of all, we draw its vertical trace kkp. The line of heights for the point P being 9-10, project P upon the latter at 10; then 10-T' is the perapective of 9-P,, and its intersection p with the vertical kk is the perspective sought. K^, in the same visual plane, has its height projected from K to z, upon the line of heights through Z; then z 7" gives k. The perspectives of all the other points might be similarly found; but with two or three points thus obtained we may find the various edges by means of the vanishing points, thus: Starting with e, for example, prolong i'e to meet at / the trace of visual plane s,F,; then fT', stopping at g on trace SiG,. As gk and fe have the same vanishing point, we find k as the intersection of gi' and z T'. Then T'k prolonged gives I and c on traces of visual planes (not drawn) through L, and C, . The point d being found independently, we join it with c for the edge c d, for which we might also find a vanishing point thus: Obtain the base angle of a right triangle of base C^D^, and altitude equal to height of C above R; then use this angle (which we may call /3) and the direc- tion DiCi exactly as 6 and F^J were used in the construction giving vanishing point i'. 614. Perspective of Shadows. These might be obtained from their orthographic projections in every case, but usually a shorter method is employed. Both ways are illustrated in Fig. 389. The object whose perspective and shadows are to be constructed is a hollow rectangular block, PERSPECTIVE OF SHADOWS. 233 FJ^. seo. Perspective Plane W^^ ==»*R Transfen-ed whose plan is fbcd, and whose height is seen at A B. The corner b being in the perspective plane, we have in ^ -B the perspective of the front edge. The vanishing points R and L having been found from s, as in the last problem, draw A R and B R, and terminate them on the trace n g of the visual plane s m. Similarly, terminate A L and B L upon the trace dk of the vertical visual plane sf. Then FR and CL give the rear corner E, etc. The shadmo. Let c'm' be the orthographic elevation of the edge whose plan is c. Then if a ray of light through c (c') has the projections cx^, c' X, we shall have x^ for the shadow of c c'. In the same way the shadow might be completed in orthographic pro- jection, a portion only, being, however, actually indicated. Then, treating x^ like any other point whose perspective is desired, we would find r — the vanishing point of hori- zontal lines parallel to mx-j, and draw Dr for the persjjec- tive of the plan of a ray; then x, the intersection of Dr with the trace oa; of the vertical visual plane sx,, is the perspective of the shadow of C. CE being horizontal, its shadow on H is in reality parallel to it, and, perspectively, has the same A-anishing point; hence draw from x toward L to complete the visible portion of the shadow. CD being a vertical line, has its shadow Dx in the direction of the projection of rays on H. 615. The perspectives of shadows, without preliminary construction of their orthographic projections, are thus obtained: In Fig. 389, with c' X and ex, as the orthographic projections of a ray, draw s'r', St, for the parallel visual ray, when r' is seen to be the vanishing point of rays. Then r is obviously the vanishing point of horizontal projections of rays; and for shadows on horizontal planes the two points thus found are sufficient. For the shadow of C we have merely to take the direct ray Cr', and the plan Dr oi the same ray, and note their intersection, x. For shadows on a set of parallel planes that are not horizontal, we would replace r by the vanishing point of projections of rays on the planes in question. 616. Perspective by the method of scales, (a) In Fig. 390 let s be the point of sight, mn the horizon, and m and n vanishing points of diagonals. Attention is called again, by way of review, to the fact that the real position of the eye in front of the i)erspective plane is shown by either ms or ns, and that all horizontal lines inclined 45° to V will converge to m or n. Plmie 234 THEORETICAL AND PRACTICAL GRAPHICS. We now apply the properties of the 45 "-triangle thus: To cut off any distance, perspectively, upon a perpendicular to V, lay off the same distance parallel to V and draw a diagonal. If the room is to be twenty -two feet deep, make Ch equal to that number of units and draw the diagonal hn, cutting off the perpendicular from C at i, making Ci the perspective of the given depth. The rectangle A B G D having been laid off in the perspective plane, from given dimensions and to the same scale, draw from A, D and B toward s, terminating these perpendiculars on a rectangle obtained by drawing id and ij, then jj and df parallel to the corresponding sides of the larger rectangle. (b) Reduced vanishing points. In case the point of sight has been taken at such a distance from the perspective plane as to throw m and n beyond convenient working limits, we may get the same Figr- 390. h.— c e g result by bisecting or trisecting sn and taking the same proportion of the distance to be laid off. Thus the point i might be obtained by bisecting Ch and drawing a line to the middle of sn. We might equally well lay off from C toward h any other fraction of Ch, and draw thence to a point on the horizon whose distance from s was the same fraction of sn. (c) The ■perspective of the steps. Let it be required to draw a flight of three steps leading to a doorway in the left wall. If the lowest step is to be six feet from the front of the room, make Da equal six feet and draw am, cutting i)d at a corner of the step in question. If the steps are to be three feet wide, make Dh equal to three units, and draw bs for the trace of the vertical plane of the sides of the steps. Making ac three feet, draw cm, getting point 9, which should be even with the first corner found. PERSPECTIVE BY THE METHOD OF SCALES. 235 The widths of the steps being laid off from c at e and g, and their heights at 1, 2, 3 on D^, their perspective is completed by a process which should need no further description. (d) The doorway in the left wall. Assuming this to be the same width as the landing, which — as seen at gr — is evidently four feet; and also that the walls of the hallway are in the planes of the front and back of the landing, draw vertical lines from the left-hand corners of the latter, terminating them by a perpendicular Ps drawn from a point P whose height (ten units) is that of the top of the doorway. 3-4 shows the height of step from landing to hallway. The perspective of the door is obtained in this case on the supposition that it is open at an angle of 54°, for which a vanishing point (not shown) lies on sm prolonged, and from which a line J'v gives the direction v J, and similarly H' H for the top of the door. To find / we may draw D K at 54° to a vertical line (the 45°- angle indicated is an error, should be 36°) so as to represent a four- foot door swung through the proper arc, when by project- ing up DK to 4-i, which is the level of the bottom of the door, a perpendicular Ls will cut J'v at /. Then a vertical line from / will cut the line H' at H Were the door actually open 45°, the edge Jv would pass through m. The hallway on the right has its corner 7 at a distance of thirteen feet from C, and is seven feet high. The width of the passage may be ascertained by the student. The method of getting the perspective of a door by means of an auxiliary circle is shown in Fig. 391. (e) The location of the light, I. To locate the light five feet from the right wall, move five units from B, to E, when Es will be the trace, on the ceiling, of the vertical plane containing the light. If the light is to be five feet below the ceiling, mark off five units down from E, when Gs will be a horizontal line giving the level of I. Finally, to have the light a definite distance back, say eighteen feet, make Bo eighteen units on the front edge of the ceiling; draw on and get t, when txl^y will be a plane at the required depth, and its intersection I with Gs will be the position of the light. (f) The shadoics. As in any other shadow construction, we have to note, in any case, where a direct ray through a point meets the projection of the same ray. All horizontal prelections of rays will pass through l^, which is the projection of the light on the floor. For the triangular block FM, we take a direct ray IS, through any point of the edge casting the shadow, and ^,8 for the projection of the same ray; then 8 is the shadow of the point selected. At F, where the edge meets the floor, the shadow begins, hence F-8 is the direction and FN the extent of the shadow cast on the floor, and, obviously, NM that received by the side wall. (g) The shadow of the door. JHll^ is a vertical plane of rays containing JH. It cuts the left wall in a line Wz, found by continuing the trace from ?, to meet Dd, erecting therefrom a vertical line and cutting it at TF by a ray from I through H. The shadow of Jv on the landing has the same vanishing point as Jv. When it meets the side wall it joins with v, since there the line casting the shadow meets the surface receiving it. The shadow of J is at the intersection of ray I J with the trace (not drawn) of the plane of rays JHl upon the top of the block. This would be found thus: Where the h. t. of said plane cuts the edge 9-10 draw a vertical line, and from the intersection of the latter with the top edge of the landing draw a line to the point below I on the line 14-s. This will give the direction of the shadow of HJ on the landing, since the line 14 -s is at the level of the top of the landing. (h) The shadows of the steps. The plane yEGl has the trace l^p on the floor; and if on the vertical pg we lay off" distances equal to the heights of the steps and draw vanishing lines to s, these will cut II ^ at points which may be regarded as the projections of the light upon the planes 236 THEORETICAL AND PRACTICAL GRAPHICS. ■S'Lg. 3Si. of the tops of the steps, and should be used in getting the directions of the shadows of vertical lines upon said tops. The direction of the shadow of the vertical edge at 9 is given by Z,9, which is made definite as to length by a ray from I through its upper extremity. The shadow on the floor is then parallel to the front edge till it meets the side wall, where it joins with the end of the line casting the shadow. The vertical edges of the second and third steps would cast shadows whose directions would be found by means of the points above /; on 12-s and 13-«, and which would run obliquely across the tops, instead of covering them entirely, as shown, incorrectly, by the engraver. 617. The ■p^spective of a right lunette, the intersection of two semi -cylindrical arches of unequal heights. Let A MB be the front of one of the arches and DNC the opposite end, at a distance back which may be found by drawing from the vanishing point of diagonals, T, a line TD to meet A B pro- duced, giving a point whose distance from A is that sought. Let the smaller passage be at a distance A X back of A, and equal to X F in width. Continue the vertical plane on AD to the level eS of the highest element of the smaller arch, ■ and in that plane construct Pmno — the perspective of half of a square whose sides equal X Y. In this draw the perspective of a semicircle Pkgo. At ee', del', etc., we see the amounts by which the elements of the side cylinder extend past the plane Aen F io their intersection with the main arch, and these in perspective are ordinates of the curve o'e"g'. For any one, as hh', draw hS cutting the semicircle Pho at o and g. Horizontals through these points will be those elements of the smaller cylinder that lie in the hori- zontal plane bb'S; and the perpendicular b' S cuts them at the points o' and g' of the intersection. 618. The perspective of a door, found by means of an auxiliary circle in perspective. Let QP, Fig. 391, be, perspectively, the width of the given door. Construct PKO, the perspective of the circle that P would describe as the door opened. If Q P were to swing to Q3, the prolongation of the latter would give 1 on the horizon for its vanishing point, which joins with J for the direction of the top edge, the latter being then limited at 2 by a vertical through 3. ^ Similarly, KQ prolonged to the horizon, gives a vanishing point from which a line through / gives the top edge //. 619. The perspective of a groined arch. If the axes of two equal cylinders intersect, the cylinders themselves will intersect in plane curves, ellipses. In the case of arches intersecting under these con- ditions the curves are called groins, and the arches groined arches, when that part of each cylinder is constructed which is exterior to the other, as in Figs. 392 and 393. Were G joined with M in Fig. 392 and the line then moved up on the groin curves Go and Mg to h, it would generate between those curves one -quarter of the surface of a cloistered arch, but the curves would still be called groins. • Fig. 392 represents in oblique projection that portion of. the structure which is seen in perspec- tive above the pillars in Fig. 393. THE PERSPECTIVE OF A GROINED ARCH. 237 The pillars are supposed to be square, and to stand at the comers of a square floor set with square tiles. Each pillar rests on a square pedestal and is capped by an abacus of the same size except as to thickness. Taking the perspective plane coincident with the faces of the pedestals and abaci, make a bit, Fig. 393, of any assumed size; prolong bl until Iv n^- sss- equals the height assigned to the pillar; then complete the front of the abacus on qv as an edge, to given data. Locate on < Z a point whose distance from I equals that of the front face of pillar from the perspective plane, and draw therefrom the perpendicular hGs; also draw the diagonal I h d, giving h for a starting corner on the base of pillar. A parallel to 1 1 through h is cut by the diagonal t Ed at E. The same diagonal gives G and two points on the diagonally -opposite pillar, coiTesponding to E and G. The vertical line through h is cut by a diagonal from the f- comer of the abacus at the point where that edge meets the abacus, and the completion of the perspective of the top is identical with that just described for its base. The prolongation of hv meets a diagonal from q at the point where the front semicircle begins on the top of the abacus. Joining it with the corresponding point on the other abacus and bisecting' such line gives the centre of the front curve, which may be drawn with the compasses, as the circle is parallel to the perspective plane. Similarly for the back semicircle. The perspectives of the groiiis and side semicircles. As the cylinders on which these curves lie are either parallel to or perpendicular to the paper we may refer to them as the parallel and perpen- dicular cylinders, respectively. The elements of the latter will converge in perspective to the point of sight, as 7)1 c and y L e, Fig. 393. On the other cylinder they will be parallel in perspective. A horizontal plane of section, as that through a b (Fig. 392) or B T (Fig. 393) will cut a square abed from the outside of the structure, and two elements from each cylinder. Either diagonal of this square, as ac (Fig. 392) will cut the elements in points of the groin. In Fig. 393 the diago- nal B d cuts the elements m s and y s a.i c and e, two points of the groin. These points also belong to elements of the parallel cylinder, and the latter, if drawn, will meet the side walls in points of the side semicircles. This is shown in Fig. 392 by drawing the element fi to meet the trace be at i. In perspective this is seen in the horizontal element through e (Fig. 393) which meets the perspective perpendicular B s at point / of the side curve. A number of planes should be treated like B T to give enough points for the accurate drawing of the curves. The tiled floor is made of squares whose diagonal is seen in true size at gj. By laying off the latter on a and drawing diagonals to the vanishing points d, the floor is rapidly laid out. The shadow of the left-hand front pillar and of its abacus. Assume r^ and r for the vanishing point of rays and of their horizontal projections, respectively, remembering that these points must be on the same vertical line, since a ray and its plan determine a vertical plane, which can intersect another vertical plane only in a vertical line. The ray from I to r , meets its projection & r at the shadow of I on the floor, whence a perpen- dicular to s would give the direction of the shadow of Is. Join h with r; it runs off the pedestal at s, whence a ray to r, will give the shadow s" on the floor, from which s"r is one boundary of ,*% 238 THEORETICAL AND PRACTICAL GRAPHICS. the shadow of the pillar. This meets the further pedestal, upon whose front the shadow runs up vertically, being the shadow of a vertical line. On the toji of the pedestal the shadow continues E-lgr- 333. :*«!. toward r till it meets the face of the pillar, where it again runs vertically until merged in the shadow of the abacus q r. The shadmv of the ahacm qv on the diagonally -opposite pillar consists of a horizontal shadow cast by a small portion of the lower front edge; a vertical portion cast by qv; and a jwrtion running SOME PRINCIPLES OF DESIGN AND CRITICISM. 239 from u parallel to S7\ and cast by a part of the edge running from q toward s. The plane of rays through the edge last mentioned would be perpendicular to the paper, and its trace necessarily parallel to that of the projecting plane of the ray of light through the eye. The front semicircle casts a shadow on the rear pillar, whose centre is found by extending the plane of the front of the pillar sufficiently to catch a ray drawn through the centre of the original curve. The radius of the shadow would be the (imaginary) shadow of the radius of the front semicircle. The shadow of the arch- curve on the interior of the perpendicular cylinder is found thus: Pass planes of rays parallel to the axis of the cylinder. R N is the trace of one such plane. It is parallel to sr^. It cuts the element xS from the cylinder, and the point n from the curve casting the shadow. The ray nr^ meets the element xS in a point of the curve of shadow. The shadow begins at the point where a plane of rays, parallel to R N^, is tangent to the face curve of the arch. The shadow, P Q, of the side semicircle, is found by taking planes of rays parallel to the axis of the parallel cylinder. The traces of such planes upon the side face of the structure, (which is per- pendicular to V), involve the location of the vanishing point of projections of rays on profile planes. The ray of light through the eye (which meets the perspective plane at rj, projected on the profile jjlane through the eye, appears at So; hence o, at the level of ?•[) and on the vertical line through s, is the vanishing point sought. Lines radiating from o, as o iv, are perspective traces of planes of rays. Each cuts the arch curve in a point as w, and the cylinder in an element as ik. The ray wri through the point then meets the element in a point of the shadow. As k is not on the real part of the cylinder, it is useful only in connection with other points similarly found, to determine the shape of the curve PQ. 620. Some hints as to planning a drawing, and on intelligent criticism of works of art. At this point, though we grant acquaintance on the part of the student with the principles of this and the preceding chapter, upon whose correct application the success of the architect or artist so largely depends, there is no certainty that he could make a drawing which should not only be mathematically correct but also pleasing to the eye, or that he could pass just criticism on the work of others. The artistic sense, to be cultivated, must be innate; and originality or inventive- ness can only in small degree be inculcated by either precept or example. Yet one who is neither the "born artist" or "natural architect" can, by the mastery of a few cardinal principles, be not only guarded against the making of glaring errors, but also have his interest in works of art materially enhanced. In bringing this chapter to a close it seems advisable, therefore, to give a few hints with regard to the more important points upon which successful work depends. In the first place, the location of the point of view is by no means immaterial. It is not well to attempt to include too much in the angle subtended by the visual rays to the extreme outlines of the object drawn, and the frequently -recommended angle of 60° may safely be taken, as, in most cases, the maximum for pleasing effect. Nor should the eye be taken too near the perspective plane, since this involves a degree of convergence amounting to positive distortion, as all must have noticed in photographs of architectural subjects taken at too short range. In case of an error in first loca- tion of view -point, one can diminish the convergence of the lines without reduction in the size of the perspective result, by a simultaneous increase of the distance of the eye from the perspective plane and of the latter from the object. Usually, and, in particular, in an architectural perspective, the eye should not be opposite the centre of the structure, a more agreeable efifect resulting from a lack of rigid geometrical symmetry and balance. Nor should the lines of the structure make equal angles with the perspective plane. 240 THEORETICAL AND PRACTICAL GRAPHICS. When viewing a picture, the endeavor should be made to put one's self at the point of sight selected by the artist. In fact, one will instinctively make the attempt so to do; but, to succeed, it is well to bear in mind the principles previously set forth as to the location of vanishing points. Among other essential preliminaries to which careful thought must be given is the quality called " Balance " by artists. It is the adjustment of the various elements of a picture, so that while leaving no doubt as to its purpose there shall not be over -emphasis of its main feature, but a general interest maintained in the various accessories, the office of the latter to be, evidently, how- ever, contributory to the central idea or object. Probably no better example of balance can be found, to say nothing of its illustration of the other requisites of a good picture, than Hofman's well-known painting of Christ preaching on the shore of Galilee. (National Gallery, Berlin). When a drawing has been well planned as to its geometrical character and arrangement, and its main lines pencilled in, the next point to be considered is the style of finish. For architectural work there may be all degrees, from the barest outline drawing in black and white, to the most highly finished water color work, with sky effects, and foliage, water and figure "incident." To secure the sketchy effect which is so desirable, and avoid the harsh exactness of geometrical diagrams, all inked lines should be drawn free-hand except in work upon which considerable free- hand shading is intended. And even when the inked lines are ruled, they may preferably be in a succession of dashes of varying length, rather than in continuous lines. Whatever guide lines are required for the boundaries of surfaces that are to be "rendered" (i. e., brush -tinted) in water colors, should be ruled in pencil only. Probably the most practical as well as pleasing style of work is that in which each stroke of pen or brush suggests, by its location or its weight, quite as much as it actually represents, — impres- sionist work, in technical language. Scarcely second to correct planning and outlining is the chiaroscuro, or light and shade effect due to the values or intensities of the tones given to the various surfaces. Not exactly synonymous with it, yet dependent upon it, is the quality termed atnosphere, upon which the effect of distance largely depends. When well rendered, the foreground, background and middle distance are harmo- niously treated, and the idea conveyed is the same as by a view in nature, changing from the clearness and sharp definition of that which is nearest, to the hazy air and general indistinctness of detail of the remote. The architect ordinarily has considerably less to do with these qualities than the artist, the element of time usually being, for him, of too great importance to permit of the highest finish of which he may be capable; but the fundamental principles upon which they depend, so far as they apply to plane surfaces, with which he is mainly concerned, are, the following: Illuminated surfaces, parallel to the perspective plane, and at different distances, are lightest at the front, and get darker in tone as they recede. When unilluminated, the exact opposite is the rule. On an illuminated surface seen obliquely, the lightest part is nearest the observer. This rule is also reversed, like the preceding one, for a surface in the shade when viewed obliquely. When the surface receiving a shadow is of the same nature (material) as that casting it, the shade should be darker than the shadow. The intensity of a shadow diminishes as the shadow lengthens. Should the student wish to go thoroughly into the technique of free-hand sketching in black and white, he should obtain Linfoot's Picture Making with Pen and Ink; while for the beginner in color work as applied to architectural subjects, F. F. Frederick's Rendering in Sepia is admirably adapted. With these, and Delamotte's Art of Sketching from Nature he will find himself fully advised on every point that can arise in connection with the free-hand part of his professional work. ORTHOGRAPHIC PROJECTION UPON A SINGLE PLANE. 241 CMAJPTEB XV. AXONOMETRIC (INCLUDING ISOMETEIC) PROJECTION.— ONE -PLANE DESCRIPTIVE GEOMETRY. 621. When but one ])lane of projection is employed there are but two applications of ortho-. graphic projection having special names. These are Axonometric (known also as Axometric) Projection, and One- Plane Descriptive Geometry or Horizontal Projection. AXONOMETRIC PROJECTION. — ISOMETRIC PROJECTION. 622. Axonometric Projection, including its much - employed special form of Isometric Projection, is applicable to the representation of the parts or s'lgr- ss-i. "details" of machinery, bridges or other con- structions in which the main lines are in direc- tions that are mutually perpendicular to each other. An axonometric drawing has a pictorial effect that is obtained with much less work than is involved in the construction of a true perspective, yet which answers almost as well for the conveying of a clear idea of what the object is; while it may also be made to serve the additional purpose of a working drawing, when occasion requires. 623. Fundamental Problem. — To obtain the orthogra-phic projection of three mutually perjtendicu- lar lines or axes, and the scale of real to pirojected lengths. Let ab, be and b d (Fig. 394) be the projections of three lines forming a solid right angle at b. . Let the line ab be inclined at some given angle 6 to the plane of projection. Locate a vertical plane parallel io ab and pro- ject the latter upon it at a'b', at 6° to the horizontal. Since the plane of the other two axes is perpendicular to ab, a'b', its traces will be P'd' R. (Art. 303). In order to find either c or d we need to know the inclination of the axis having such point for its extremity. Supposing -ft given for cb, draw b' C at p° to GL; project C to c, and draw arc c,c, Join a with c; then ac is the trace of the plane of the axes ba and be, and being perpen- dicular to the third axis we may draw the latter as the line ebd, making 90° with ac. centre h, obtaining c. 242 THEORETICAL AND PRACTICAL GRAPHICS. CsiTTy d to rf, about b; project tZ, to D and join the latter with b'. Then Db' is the triie length, and b'DL (or ) the inclination, of the third axis, b d. Lay off a'n', D s' and C<', each one inch. Their projected lengths on the horizontal are respec- tively a'n, Ds and Ct. The latter are then the lengths, representative of inches, for all lines parallel to ab, be and b d respectively. 624. To make an axonometric projection of a one-inch cube, to the scale just obtained. Although not absolutely necessary it is cmtomary to take one axis vertical. Taking the a 6- axis vertical, the cube in Fig. 394 fulfills the conditions. For BA equals a'n; B D" equals Ds, and B C" equals Ct, while the angles at B equal those at b. The light being taken in the usual direction, i.e., parallel to the body - diagonal of the cube {C" R), the shade lines indicated are those which separate illumined from unillumined surfaces, and are those which could, therefore, cast shadows. 625. , The axonometric projection of a vertical pyramid, of three -fourths -inch altitude and inch -square base, to the same scale as the cube. The pyramid in Fig. 394 meets the requirements, xwyz having been made equal to C"BD"X; while the altitude m M, rising from the intersection of the diago- nals of the base, equals three -fourths a'n, the inch - representative for the vertical axis. 626. To draw curves in axonometric projection obtain first the projections of their inscribed or cir- cumscribed polygons, or of a sufficient number qf secant lines; then sketch the curve through the points on these new lines which correspond to the points common to the curves and lines in the original figure. This will be illustrated fully in treating isometric projection. 627. Isometric Projection. — Isometric Drawing. When three mutually perpendicular axes are equally inclined to the plane of projection they will obviously make equal angles (120°) with each other in projection. This relation led to the name "isometric," implying equal measure, and also obviates the necessity for making a separate scale for each axis. The advantages of this method seem to have been first brought out by Prof. Farish of England, who presented a paper upon it in 1820 before the Cambridge Philosophical Society of England. 628. In practice the isometric scale is never used, but, as all lines parallel to the axes are equally foreshortened, it is customary to lay off' their given lengths directly upon the axes or their parallels, the result showing relative .position and proportion of parts just as correctly as a true projection, but being then called an isometric draicing, to distinguish it from the other. It would, obviously, be the projection of a considerably larger object than that from which the dimensions were taken. Lines parallel to the axes are called isometric lines. Any plane parallel to, or containing two isometric axes, is called an isometric plane. Figr- 3S5- 629. To make an isometric drawing of a cube of three -quarter -inch edges. Starting with the usual isometric centre, 0, (Fig. 395) draw one axis vertical, and on it lay oS A equal to three -fourths of an inch. OC and OB are then drawn with the 30 "-triangle as shown, made equal in length to A, and the figure completed by parallels to the lines already drawn. One body -diagonal of the cube is perpendicular to the paper at 0. 630. To draw circles and other curves isometrically, employ auxiliary tan- gents and secants, obtain their isometric representations, and sketch the curves through the proper points. In Fig. 396 we have an isometric cube, and at MO'P'N the square, which — by rotation on MN and by an elongation of If P'— becomes transformed into M P N. The circle of centre S' then ISOMETRIC DRAWING. 243 x"igr. 3SS. becomes the ellipse of centre S, whose points are obtained by means of the four tangencies d', F, E and G, and by making gn equal to gn', hm equal to h'm', etc. 631. Tfie isometric circle may be divided into parts corresponding to certain arcs on the original, either (1) by drawing radii from S' to 31 N, as those through b', c', d', (which may be equidistant or not, at pleasure) and getting their isometric representatives, which will intercept arcs, as hd', d'e, which are the isometric views of b'd', d'e'; or (2) by drawing a o! semicircle x i y on the major axis as a diameter, letting fall perpendiculars to xy from various points, and noting the arcs as 1-2, 2-3, that are included between them and which correspond to the arcs ij, j k, origi- nally assumed. 632. Shade lines on isometric drawings. While not universally adhered to, the conventional direction for the rays, in isometric shadow construction, is that of the body -diagonal C R of the cube (Fig. 39-5). This makes in projection an angle of 30° with the horizon- tal. Its projection on an isometrically- horizontal plane — as that of the top — is a horizontal line CB; while its projection CA, on the isometric representation of a vertical plane, is inclined 60 ° to the horizontal. 633. To illustrate the principles just stated Fig. 397 is given, in which all the lines are isometric, with the exception of Z)z and its parallels, and ST. The drawing of non - isometric lines will be treated in the next article, but assuming the objects as given whose shadows we are about to construct, we may start with any line, as Dz. The ray Dd is at 30°. Its pro- jection d^d is a horizontal through the plan of D. The ra}- and its projection meet at d. As the shadow begins where the line meets the plane, we ha\e 2 d for the shadow of D z. This gives the direction for the shadow of any line parallel to D z, hence for y v, which, however, soon runs into the shadow of B C. As b is the intersec- tion of the ray B b with its projection 6,6, it is the shadow of B, and 6,6 that of 6 1 -B. Then bv is parallel to B C, the line casting the shadow being parallel to the plane receiving it. In accordance with the principle last stated, de is equal and parallel to D E, and ef to E F. At / the shadow turns to g, as the ray J F, run back, cuts MG at /', and /'(? casts the /^r-shadow. 244 THEORETICAL AND PRACTICAL GRAPHICS. ^igr, 3©Q. S-lg-. 3SS. Then gh equals G H, and hh^ is the shadow of Hh. The projection jm catches the ray Mm at m. Then mf, equal to 3//', completes the construction. The timber, projecting from the vertical plane PQR, illustrates the 60° -angle earlier mentioned. Kk' being perpendicular to the vertical plane, its shadow Kl is at 60° to the horizontal, and Klk is the plane of rays containing said edge. Its horizon- tal trace catches the ray from k' at k. Then nk, the shadow of n'k', is horizontal, being the trace of a ver- tical plane of rays on an isometrically- horizontal plane. The construction of the remainder is self-evident. Letting S T represent a small rod, oblique to isometric planes, assume any point on it, as u; find its plan, M,; take the ray through u and find its trace w. Then (Sw is the direction of the shadow on the vertical plane, and at r it runs ofif the vertical and joins with T. 634. Timber framings, drawn isometrically, are illustrated by Figures 398 and 399. In Fig. 398 the pieces marked A and B show one form of mortise and tenon joint, and are drawn with the lines in the custom- ary directions of isometric axes. The same pieces are represented again at C and D, all the lines having been turned through an angle of 30°, so that while maintaining the same relative direction to each other and being still truly isometric, they lie dif- ferently in relation to the edges of the paper— a matter of little importance when dealing with comparatively small figures, but affecting the appearance of a large drawing very materially. 635. Non-isometric lines. — Angles in isometric planes. In Fig. 399 a portion of a cathedral roof truss is drawn isometrically. Three pieces are shown that are not parallel to isometric lines. To represent them correctly we need to know the real angles made by them with horizontal or vertical pieces, and use isometric coordinates or "offsets" in laying them out on the drawing. In the lower figure we see at the actual angle of the inclined piece Mf to the horizontal. Oifsets, fl and I C, to any point C of the inclined piece, are laid ofif in isometric directions at f'V ISOMETRIC DRAWING.— NON-ISOMETRIC LINES. 245 and I'C, when C'f'l' (or 6') is the isometric view of 6. A similar construction, not shown, gave the directions of pieces D and D'. . Much depends on the choice of the isometric centre. Had N been selected instead of B, the top surfaces of the inclined pieces would have been nearly or quite projected in straight lines, render- ing the drawing far less intelligible. The student will notice that the shade lines on Fig. 399 are located for effect, and in violation of the usual rule, it having been found that the best appearance results from assuming the light in such direction as to make the most shade lines fall centrally on the timbers. 636. Non- isometric lines. — Angles not in isometric planes. To draw lines not lying in isometric planes requires the use of three isometric offsets. As one of the most frequent aj^plications of isometric drawing is in problems in stone cutting, we may take one such to advantage in illustrating constructions of this kind. Fig. 400 shows an arched passage-way, in plan and elevation. The surface no, r'l'n'o' is verti- tical as far as n'o', and conical (with vertex J, C") from there to n"o". The vertical surface on nn is tangent at n' to the cylinder n'f'e'o". Similarly, mm is vertical torn', and there changes into the cylinder m'g'h'. The radial bed h' g' is indicated on the plan (though not in full size) by parallel lines at hcjigzb. The bed a'h' is of the same form as b'g', being symmetrical with it. In Fig. 401 we have an enlarged drawing of the key- stone with the plan inverted, so that all the faces of the stone may be correctly represented as seen. The isometric drawing is made to correspond, that is, it represents the stone after a 180 "-rotation about an axis periiendicular to the paper. The isometric block in which h' the keystone can be inscribed is shown in dotted lines, its di- mensions, derived from the projections, being length, A Aj=aa ; breadth, AB=a'h'; height, AO^a'p. The top surface a'b' becoming the lower in the isometric, reverses the direction of the lines. Thus, a' is seen at A, and b' at B. To get D make A U=a'u, then UD^^ud'. Make C symmetrical with D and join with B, and also D with A. WQ equals w'q', for the ordinate of the middle point of the arc. D E is not an isometric plane, hence to reach E from A we make AT=a't; Te" — te', and e"E=ay (the dis- tance of e from the plane a b). The remainder of the construction is but a duplication of one or other of the above processes. The principle that lines that are parallel on the object will also be parallel on the drawing may be frequently availed of in the interest of rapid construction or for a check as to accuracy. x-ic:' -iOO- Y k' V i' S^d^^fc ( /X/V-^A \ ^ n" t c' J n' o' 1 1 1 1 % / / / / \ \ \ \ / / \ k' l' t' *" /" " ~^^ / a X z b \m / h n - \ n / / ' e f "'X / " \^ / /d 4 b \" 246 THEORETICAL AND PRACTICAL GRAPHICS. HORIZONTAL PROJECTION OR ONE -PLANE DESCRIPTIVE GEOMETRY. 637. One -Plane Descriptive Geometry or Horizontal Projection 'is a method of using orthographic projections with but one plane, the fundamental principle being that the space - position of a point is known if we have its projection on a plane and also know its distance from the plane. Thus, in Fig. 402, a with the subscript 7 shows that there is a point A, vertically above a and x-ier- -ios- at seven units distance from it. The significance of 6, is then evident, and to show the line in its trUe length and inclination we have merely to erect perpen- diculars ft A and B h, of seven and three units respectively, join their extremities, and see the line A B in true length and inclination. In this system the horizontal plane alone is used; One- plane Descriptive is o, '*» therefore applied only to constructions in which the lines are mainly or entirely •horizontal, as in the mapping of small topographical or hydrographical surveys, in which the curva- ture of the earth is neglected; also in drawing fortifications, canals, etc. The plane of projection, usually called the datum or reference plane, is taken, ordinarily, below all the points that are to be projected, although when mapping the bed of a stream or other body of water it is generally taken at the water line, in which case the numbers, called indices or refer- ences, show depths. 638. A horizontal line evidently needs but one index. This is illustrated in mapping contour lines, which represent sections of the earth's surface by a series of equidistant horizontal planes. Figr- -ios. +. HILL CONTOUR In Fig. 403 the curves indicate such a series of sections made by planes one yard, metre or other unit apart, the larger curve being assumed to lie in the reference or datum plane, and there- fore having the index zero. The profile of a section made by any vertical plane MN would be found by laying off — to any assumed scale for vertical distances — ordinates from the points where the plane cuts ^^^- "*®*- the contours, giving each ordinate the same number of units as are in the index of the curve from which it starts. Such a section is shown in the shaded portion on the left, on a ground line PQ, which represents MN transferred. 639. The steepness of a plane or surface is called its slope or declivity. A line of slope is the steepest that can be drawn on the surface. A scale of slope is obtained by graduating the plan of a line of slope so that each unit on the scale is the projection of the unit's length on the original line. Thus, in Fig. 404, if m n and o B are horizontal lines in a plane, one hav- ing the index 4 and the other 9, the point B is evidently five units above A, and the five equal divisions between it and A are the projections of thoSe units. HORIZONTAL PROJECTION OR ONE-PLANE DESCRIPTIVE. 247 The scale of slope is often used as a ground line upon which to get an edge view of the plane. Thus, if B B' is at 90° to B A, and its length five units, then B' A is the plane, and <^ is its inclination. The scale of slope is always made with a double line, the heavier of the two being on the left, ascending the plane. As no exhaustive treatment of this topic is proposed here, or, in fact, necessary, in view of the simplicity of most of the practical applications and the self-evident character of the solutions, only two or three typical problems are presented. 640. To find the intersection of a line and plane. Let *15 "30 be the line, and X Y the plane. X-xg-. -iOE. Figr- -ios. Draw horizontal lines in the plane at the levels of the indexed points. These, through 15 and 30 on X F, meet horizontal lines through a and & at e and d; ed is then the line of intersection of X F and a plane containing ab; hence c is the intersection of the latter with X Y. The same point c would have resulted if the lines a e and b d had been drawn in any other direction while still remaining parallel. 641. To obtain the line of inter- section of tico planes, draw two hori- zontals in each, at the same level, and join their points of intersection. In Fig. 406 we have mn and qn as horizontals at level 15, one in each plane. Similarly, xy and ys are horizontals at level 30. intersect in y n. Were the scales of slope parallel, the planes would intersect in a horizontal line, one point of which could be found by introducing a third plane, oblique to the given planes, and getting its intersection with each, then noting where these lines of intersection met. 642. To find the section of a hill by a plane of given slope. Draw, as in the problem of Art. 640, horizontal lines in the plane, and find their intersections with contours at the same level. Thus, in Fig. 403, the plane XY cuts the hill in the shaded section nearest it, whose outlines pass through the points of intersection of horizontals 10, 20, 30 of the plane, with the like -numbered contour lines. The planes 248 THEORETICAL AND PRACTICAL GRAPHICS. CHAPTER XVI. OBLIQUE OB CLINOGBAPHIC PBOJECTION. — CAVALIEE PEESPECTIVE. — CABINET PEOJECTION. MILITABY PEESPECTIVE. 643. If a figure be projected upon a plane by a system of parallel lines that are oblique to the plane, the resulting figure is called an oblique or clinog7-aphic projection, the latter term being more frequently employed in the applications of this method to crystallography. Shadows of objects in the sunlight are, practically, oblique projections. In Fig. 407, ABnm is a rectangle and mxyn its oblique projection, the parallel projectors Ax and By being inclined to the plane of projection. 644. When the projectors make 45° with the plane this system is known either as Cavalier Perspective, r'lgr. -aoT. Cabinet Projection or Military Perspective, the plane of projection being vertical in the case of the first two, and horizontal in the last. 645. Cavalier Perspective. — Cabinet Projection. — Military Perspective. As just stated, the projectors being inclined at 45° for the system known by the three names above, we note that in this case a line perpendicular to the plane of pro- jection, as Am or B n (Fig. 407), will have a projection equal to itself. It is, therefore, unnecessary to draw the rays for lines so situated, as the known original lengths can be directly laid out on lines drawn in the assumed direction of projections. Let abed .n be a cube with one face coinciding with the vertical plane. If the arrow m indi- cates one direction of rays making 45° with V, then the ray hn, parallel to m, will give h as the projection of n, and from what has preceded we should have ch equal to en, and analogously for the remaining edges, giving abcd.i for the cavalier perspective of the cube. Similarly, EKH is a correct projection of the same cube for another direction of projectors, and we may evidently draw the oblique edges in any other direction and still have a cavalier perspec- tive, by making the projected line equal to the original, when the latter is perpendicular to the plane of projection. 646. Oblique projection of circles. Were a circle inscribed in the back face of the cube D K G (Fig. 407) the projectors through the points of the circle would give an oblique cylinder of rays, whose intersection with the vertical plane DX would be a circle, since parallel planes cut a cylinder in similar sections. We see, therefore, that the ol^lique projection of a circle is itself circular when the plane of projection is parallel to that of the circle. In any other case the olilique projection of a circle may be found like an isometric projection (see Art. 631), viz., by drawing chords of the circle, and tangents, then representing such auxiliary lines in oblique view and sketching the curve (now an ellipse) through the proper points. Fig. 408 illustrates this in full. OBLIQUE PROJECTION. — CAVALIER PERSPECTIVE. 249 647. Oblique -prelection is even better adapted than isometric to the representation of timber fram- ings, machine and bridge details, and other objects in which ^^s- -ioe. straight lines — usually in mutually perpendicular directions — predominate, since all angles, curves, etc., lying in planes jjar- allel to the paper, appear of the same form in projection, while the relations of lines perpendicular to the paper are preserved by a simple ratio, ordinarily one of equality. 648. When the rays make with the plane of projection an angle greater than 45°, oblique projections give effects more closely analogous to a true perspective, since the fore- shortening is a closer apjjroximation to that ordinarily exist- ing from a finite point of view. This is illustrated by Fig. 409, in which an object A B D E, known to be 1" thick, has its depth represented as only \" in the second view, instead of full size, as in a cavalier perspective, the front faces being the same size in each. Provided that the scale of reduction were known, abcdkf would answer as well for a working drawing as a 45 "-projection. 649. By way of contrast with an isometric view the timber framing represented by Fig. 398 is Fig', ^li. 3Figr. -iiO. B W A drawn in cavalier perspective in Fig. 410. Eeference may advantageously be made, at this point, to Figs. 44, 45 and 46, which are oblique views of one form. The keystone of the arch in Fig. 400, whose isometric view is shown in Fig. 401, appears in oblique projection in Fig. 411; the direction of lines not j^arallel to the axes of the circumscribing prism being found by "offsets" that must be taken in Figr- -iia. axial directions. 650. Shadmvs, in oblique projection. As in other pro- jections, the conventional direction for the light is that Ai< of the body -diagonal of the oblique cube. The edges to draw in shade lines are obvious on inspection. (Fig. 412). 651. An interesting application of oblique projection, earlier mentioned, is in the drawing of crystals. Fig. 414 illustrates this, in the representation of a form common in fluorite and called the tetrahexahe- dron, bounded by twenty -four planes, each of which fulfills the condition expressed in the formula 250 THEORETICAL AND PRACTICAL GRAPHICS. 00 : n : 1 ; that is, each face is parallel to one axis, cuts another at a unit's distance, and the third x-igr- -ai3. at some multiple of the unit. * The three axes in this system are equal, and mutually per- pendicular; but their projected lengths are a a', bb', cc'. The direction of projectors which was assumed to give the lengths shown, was that of EN in Fig. 413, derived by turning the jDerj^endicular CN through a horizontal angle C NM^ 18° 2Q', and then elevating it through a vertical angle MNS=9° 28'; values that are given by Dana as well adapted to the exhibition of the forms occurring in this system. The axes once established, if we wish to construct on them the form oo : 2 : 1, we lay off on XAHP]D110]^ each (extended) one-half its own (projected) length; thus cc" and c' c'" each equal oc'; bb" equals ob, etc. Then draw in light lines the traces of the various faces on the planes of the axes. Thus, a'b" and a"b each represent the trace of a plane cutting the c-axis at infinity, and the other axes at either one or two units distance; the former intercepting the two units on the 6 -axis and the one on the a -axis, while for a"b it is exactly the reverse. Through the intersection of a'b" and a"b' a line is drawn parallel to the c-axis, indefinite in length at first, but determinate later by the intersection with it of other edges similarly found. The student may develop in the same manner the forms oo:3:l; oo:2:3; co:3:4; oo : 4 : 5. PRACTICAL DRAUGHTING'OFFICE PROBLEMS. 261 CHAPTER Xril. END-POST BRIDGE DETAIL, UPPER CHORD.— SPUR GEAR, APPROXIMATE INVOLUTE OUTLINES. HELICAL SPRINGS.— STANDARD BOLTS, SCREWS AND NUTS. — TABLE OF PROPORTIONS. Supplementing the working drawing.? found in the earlier pages, a few others of frequent occur- rence are given in this chapter, in concluding the main portion of the book. In the Appendix will be found one or two more, not needing descriptive text,— an 100- pound rail section and an Allen -Richardson valve, the latter of the proportions employed on one of the locomotives drawing the celebrated "Exposition Flyer" in 1892; also a table of the proportions of washers. It need hardly be said that the illustrations are not to be copied by transfer with dividers, but that to get all the benefit intended from their presentation they should have their proportions reduced to scale by the student, in which case the work becomes in greater degree constructive, and in closer analogy to that of the shop draughting -room, which is so frequently from free-hand, dimensioned sketches. 652. Detail of a Bridge. — Upper-Chord Pod- Connection. — A. bridge or roof truss is an assemblage of pieces of iron or wood, so connected that the entire combination acts like a single beam. Figs. 415 and 416 are what are called "skeleton diagrams" of bridge trusses, each piece or "member" of the truss being represented by a single line. A BCD and A' B' C D' are the trusses proper, the Fig:, -iis. M ^ig-. -il©- A e g fc 1 n p r D 1 \ \ \ X \l/ X /I / S E s / / I J N i c ( 9 C B c y X X X \. 1 ' 1 1 former being for an overhead track and the latter for a roadway running through the bridge. In each case the upper part — called tbe upper chord — (A D, B'C) sustains compression, and is made of "built beams," formed by riveting together various plates and lengths of structural iron in such manner as to form one practically continuous column. The lower chords (B C, A' D') sustain tension, and are made of bars of high tensile strength. The members that connect the chords are called either ties or struts according as the strain in them is tensile or compressive. Collectively they form the weh of the truss. In the form of truss illustrated — which is only one of many which have commended themselves to, the profession — the vertical pieces or "posts," Be,Jg, etc., sustain compression, and are therefore "built" columns. They divide the trapezoid into parts called jmnels, which has given the name panel system to this largely - employed arrangement of bridge members. All the diagonal members in both figures, excepting A' B' and C D\ are tension bars or rods. B b and C s are struts whose sole office is to keep the posts A b and D s vertical ; said posts then conveying to the masonry whatever weights are transmitted through the truss to A and D resjjectively. 252 THEORETICAL AND PRACTICAL GRAPHICS. Fig. 417 is a perspective view of the connection at A between the post A h, the upper chord AD, and the four diagonal- bars that are projected in A B. The working drawings required for such connection are shown in the three views on the opposite page. Three analogous views would be required for the connection at the foot of the same post. ^^^hen — as in the case from which our example is taken — there are two railroad tracks overhead, iFigr. -iiT-. ^}jg members of the middle truss will usually have different propor- tions from those in the outside trusses, and a separate set of three views has therefore to be made for each of its post connections, so that the smallest number of shop drawings for one such bridge — after making all allowance for the symmetry of the structure with reference to the central plane MX — would consist of twenty such groups of three as are illustrated by Fig. 419. The uj)per projections (Fig. 419) are obviously a front and a side elevation. The lower figure may preferably be regarded as a plan of the object inverted, since that conception is somewhat more natural than that of the post in its normal position, while the draughtsman lies on his back and gazes up at it from beneath. Fig. 418 shows the inverted plan on a somewhat smaller scale, and, although presented mainly to illustrate the contrast between views with shade lines and without, contains one or two serviceable dimensions that are omitted on the other plate. 658i General description. Referring to the wood cut as well as the orthographic projections, we find the upper chord to be composed of a long cover-jDlate, 18" X ^", riveted to the top angles of two vertical channel bars set back to back; each channel being 15" high and weighing 200 pounds per yard. The cross section of the upper chord is shown in solid black, with just enough space intended between plate and channels to show that they are not all in one piece. Perpendicular to the vertical faces of the channels and through holes cut therein runs a cylinder called a "pin," A" in diameter and 21 J" "between shoulders" (as marked on the plan), that is, between the planes where the diameter is reduced and a thread turned, on which connection can be made with the corresponding post in the next truss. Four diagonal bars are sustained by the pin, the latter passing through holes, ' called " eyes," in the bars. Two of the bar heads are between the channels. Two plates are inserted between each of the outside bars and the nearest channel not only to prevent the bar from touching the angle, as at h, but also to relieve the metal nearest the pin from some of the strain. The longer plate miiFE is next the channel. The other, nmop, has a kind of hub cast on it which rounds up to the bar head, as shown in the side elevation. The vertical post is made up of an I-beam and two channels, as shown by the black sections on the plan. :Fig-. -iie. UPPER-CHORD POST-CONNECTION. 253 -2^ -14- ,-2—. is'x Vk cover plate -l^«- Railrnad Bridge Past CDnnEctian UppEP CliDrd, When drawing the above the student should make horizontal dividing lines In all fractions. A brush tint of Prussian blue for all the metal parts win enhance the appearance of the drawing very materially, but the "previous lining; should be, obviously. In best waterproof Ink. Centre, dimension, and extension lines should be in continuous red lines, unless for blue printing. In which case all lines will be black. 254 THEORETICAL AND PRACTICAL GRAPHICS. Between the upper chord and the top of the post is a three-quarter-inch plate, seen hest on the plans at iflt. It is nicked out 4", near the nuts K, so as to clear the two middle bars S which come between the channels. A 6"x3" angle -iron runs from outside to outside of channels, and is held by rivets and by the bolts marked H. A shorter piece of the same kind is fastened by bolts K to the plate, and, by rivets to the web of the I-beam. 664. Hints as to drawing the bridge post connection. Draw the main centre lines first; then the plan and side elevations simultaneously, as the horizontal centre line of the plan represents the same vertical reference -plane as the vertical centre line of the side elevation, and one spacing of the dividers may be made to do double work. The solid sections should be drawn first of all; then the pins, bars, and cap plates of the post in the order named. The parts already drawn should next be represented on the front elevation by • t-^ ' C:.. . ^'| ^^Z3 DECK BEAM 15 CHANNEL 200 LBS. PER YARD. 5 X 21 ^''' Cs i. An arc from centre i, with a radius -' / ^-''i^ of one-fourth Ci, will give the centre s of the outline hij. Draw the "circle of centres" through s, from centre C. Then with si in the dividers, and from centre / find q, which use in turn for arc g J Cf and so continue. The width of rim, vw, is often made, by a "shop" rule, equal to three - fourths the pitch. Reuleaux gives for it .the fol- lowing formula : vw=^ 0.4 P -|- .12. Diametral pitch is very frequently used instead of circular pitch, and is simply the number of teeth per inch of pitch -circle diameter. 658. Helical Springs. Draw first (Fig. 423) a central helix acfm..T, as follows: Divide aa,— ,# >?^ ."JVc? 256 THEORETICAL AND PRACTICAL GRAPHICS. E'lg. -3=23. Flgr. -434. which is the fitch, or rise in one turn — into any number of equal parts, and the semi -circumference A EM into half as many equal divisions ; then each point marked with a capital on the half i)lan gives two elevations (denoted by the same letter small) by a process which is self-evident. If the spring is circular in cross -section draw a series of circles having centres on the helix, and whose diameters equal that of the spring; then the outlines of the spring will be curves that are tangent to the circles. If the spring be small the curvature of the helix may be ig- nored, and a series of parallel straight lines employed instead, drawn tangent to circular arcs as in Fig. 424. The upper half of the figure gives the method of construction, while the lower shows the spring in section, and surrounding a solid cylindrical core. 659. Springs of rectangular cross- section. Fig. 426 shows a spring of this descrip- tion, formed by moA'ing the rectangle ah cd helically, each point describ- ing a helix which can be constructed as described in the last article. When any considerable number of turns of the same helix has to be drawn, it will save time if the draughtsman will shape a strip of pear-wood into a templet, i.e., a piece whose outline conforms to a line to be drawn or an edge to be cut, using it then as a curved ruler to guide his pen. This is the preferable method for all large work. 660. Square-threaded screws. If in- stead of spirally twisting a rectangu- lar bar the same kind of surface be cut upon a cylinder of wood or metal, we shall have a square - threaded screw. This is illustrated by the upper part of Fig. 425, and its construction is self- evident after what has preceded. On a larger scale the curva- ture of the helices would have to be indicated. The upper view is an elevation of a small double -threaded square screw, generated by winding two equal rectangles simul- taneously around the axis. The central figure is an elevation of a single -threaded screw. The lower figure is a sectional view of the nut for the single- threaded screw, and evidently presents a surface identical with which fits it. jflg-. -SiSS. that of the back half of the screw SCREW-THREADS.— BOLTS.— NUTS. 257 ■F^S- -^SB. 661. Triangular -threaded screics. — United States Standard. — The proportions devised by Mr. William Sellers of Philadelphia have been so generally adopted as to be known as the United States Standard. They are given in the table on the next page. Fig. 427 shows a section of the Sellers screw. It is blunt on the thread, and also at the root. The part op B which is removed from the point may be regarded as filled in at Nst. AB being the pitch (P), the widths op, 8t, are each one -eighth of P. With N equal to the number of threads per inch, and D the outside diameter of the screw or bolt, the value of d — the diameter at the root — may be obtained from the formula d = Z)- (1.299 H-JV). Other proportions are as follows: The pitch is equal to The depth of thread equals 0.65 P. For bolts BCJLT -i — -D--r ■> 0.24 v'/)+0.625 — 0.175. and nuts, whether hexagonal or square, the "width across flats," or shortest distance between parallel faces, equals 1.5 D, plus one -eighth of an inch for rough or unfinished surfaces, or plus one - sixteenth of an inch for " finished," i. e., machined or filed to smoothness. The depth of nut equals the diameter of the bolt, for "rough" work. Tables should be consulted for the proportions of finished pieces. Fig. 428 is a drawing, to reduced scale, of a finished ^" bolt.' The elevations show a bevel or chamfer, such as is usually given to a finished bolt or nut. On the plans this is indicated by the circles of diameter p q, the latter usually a little more than three- fourths of the diameter a d. To draw the lines resulting from chamfering proceed thus: On a view showing "width across flats," as that of the nut, draw the chamfer lines z u, o i, at 30° ^^- ^^s. r-igr- -iso. x-ig-- -isi- to the top, and cutting ofl" rri the desired amount. Draw circles on the plans, with diameter equal to uo. Pro- B ject p and q to P and Q p and draw Px and Qj/ at 30° to the top. Make Nk on the nut equal to n y on the head. On the latter draw a parallel to P Q, and as far from it as ou is from vi. The arcs limit- ing the plane faces have their centres found by "trial and error," three points of each curve being known. When drawn to a small scale screws may be PLAN OF NUT represented by either of the conventional methods illustrated by Figs. 429, 430 and 431. NUT. 258 THEORETICAL AND PRACTICAL GRAPHICS. DIMENSIONS OF BOLTS AND NUTS, UNITED STATES STANDARD ( SELLERS SYSTEM) Proportions of Bolt Dimensions of Nuts Rough and Finished Dimensions of Bolt Heads Rough and Finished Outside Diam. D At Root of Thread N= Number of Threads per inch. Width ( f ) of Flat. Across w ■ Flats Across / w \ Corners Across Corners Depth Across w Flats Across Corners Across Corners Depth |[1l^ If 11^ Diam. Area of Nut of Hea^d R F R R R F R F R R R F 1 4 .185 .026 20 .0062 1 2 7 16 37 " 64 7 10 1 4 3 IB 1 2 7 16 37 64 10 JL 4 Id 5 .240 .045 18 .0074 19 32 17 32 11 18 5 6 5 16 1 4 19 32 17 32 11 16 5 6 19 64 1 4 3 8 .294 .067 16 .0078 H 16 5 8 51 64 63 64 3 8 .5 16 11 16 5 8 51- 64 63 64 11 32 5 16 7 16 .344 .092 14 .0089 25 32 23 32 9 10 1^4 7 16 3 8 25 32 23 32 9 10 4i 25 64 3 8 1 2 .400 .125 13 .0096 7 .8 13 16 1 iH 1 2 7 16 i 13 16 1 iM 7 16 7 16 9 16 .454 .161 12 .0104 31 32 29 32 l| li 9 16 1 2 31 32 29 32 li i§ 31 64 1 2 5 8 ' .507 .201 11 .0113 1^ 1 li li 5 8 9 16 ik 1 ih li 17 32 9 16 3 4 .620 .301 10 .0125 li li ir; ■164 3 4 11 16 li ll^ ife li9 l64 5 8 11 16 1 8 .731 .419 9 .0138 ii^ if li 2^ 7 8 13 16 lf6 if li 2^ 23 32 13 16 1 .837 .55 8 .0156 i| ll^ ll 9I9 264 1 15 16 1| ll^ ll 2i 13 16 15 16 li .940 .693 7 .0178 lis ^16 If 232 2^ -^16 li ll^ lis If n3 ^^32 2f« 29 32 ll^ li 1.065 .89 7 .0178 2 ll5 ■^16 ^16 053 ^64 li 111 2 lis ■^16 -^16 2i 1 111 if 1.160 1.056 6 .0208 2il 2i ^ 8 2i q3 ^32 If 14 2^ ^16 2^ ^8 2i 3| 1| Ife 1| 1.284 1.294 6 .0208 2| 2il 2f 3i li 14 23^ 2f6 2! o23 ^64 Ife Ife If 1.389 1.515 4 .0227 2^ 2| 9 31 3f If ^16 2ll 2| 931 ''32 o5 ^8 4 1^ •■^16 i! 1.491 1.746 5 .0250 2| -^16 q3 q57 3m If lii ^16 2f 2M q 3 3 16 q57 i3 Is lil ^16 if 1.616 2.051 5 .0250 2i 2| ql3 A 5 1| ilj ^16 2ii 2| 3I3 A^ li ll_3 •■■16 2 1.712 3.301 4| .0277 3i 3ii 3| 42? *64 2 lis •^16 3§ 3f6 3f *64 4 ll5 ■^16 2i 1.962 3.023 4 .0277 3i 3^ 4 4SI *64 2i 2^ ^16 3| 3 re ^1^ ^61 ^64 If ■^16 2-^ 2.176 3.718 4 .0312 3| ol3 3 16 4| k31 064 2i 2i-6 3| „13 ''re 4i °64 ^16 ^16 2f 2.426 4.622 4 .0312 ^ 4fa 42? *32 6 2f 2U ^16 *i ^fe ^i 6 2i ■^8 2^ ^16 3 2.629 5.428 3| .0357 4^ *8 4^6 °8 ft I'' 3 <>15 ''le 4| 4ii 5| 6i 2* ^16 9 15 ''le 3i 2.879 6.509 3i .0357 5 *16 k13 °16 71 '16 3i 3il 5 4I5 *ie 4^ 7ii 2i ^ 2 Sfe 3| 3.100 7.547 3i .0384 5f «;5 5i« 6m 7i 3| 3i^ 5| 5 16 66-1 7i 2il 3i^ 3| 3.317 8.641 3 .0413 5f Ol6 21 6^ 8i 3! ol3 3i6 5! rH »16 fig] °32 8i 2| 311 4 3.567 9.993 3 .0413 6i 6 fa 73 ^32 8*1 ^64 4 3ii 6| 6ii 7 3 *'64 3f6 q]5 3 16 4} 3.798 11.329 2| .0435 6| 6ii ^16 9ll 4i 4f6 6l 61^ 7-8 '16 9li 3i ill 4 4.028 12.742 2f .0454 6| „13 6i6 731 '32 9f 4| ^1^ 6| «i 7i 9| ^16 4ll 4 4.256 14.226 2f .0476 7i 7 3 7 16 ol3 loi 4 4^ *16 n 7f6 «i loi o5 ^8 4li *16 5 4.480 15.763 2i .0500 75 ' 8 'I6 q27 10*2 ^"64 5 4L5 75 ' 8 7i! »32 A" 64 ql3 4 15 4 4.730 17.570 2i .0500 8 7i q9 "32 111 5i K 3 5 re 8 7i q9. ^32 Hi 4 <;3 OI6 4 4.953 19.267 4 .0526 8| 81^ q23 111 5i 5ii 8| 81I q23 ^38 111 ^3 *I6 Sfe k3 St 5.203 21.261 2| .0526 8| 8M io| 12| 5f °16 8| < io| 12 1 4^ *8 5ii 6 5.243 23.097 2i .0555 H 9i^ ■^"32 i2li 6 rIS 5 16 9i ^h loi' i2ii . 9 4i6 k15 5 16 APPENDIX Section of Staadarcl Rail, one hundred pounds to the yard, Page 261 Sectional View of Allen - Richardson Slide Valve, Page 262 Figures serviceable for variation of problems in Projection, Sections, Conversion of views from one system of projection to another. Shadows, Perspective, etc., Page 263 Note to Art. 113, on the Sections cut from the Annular Torus by a Bi- tangent Plane. Note to Art. 131, on the Projection of a Circle in an Ellipse, Page 264 The Nomenclature and Double Generation of Trochoids, Pages 265-272 Alphabets and Ornamental Devices for Titles, Pages 273-286 Index, Pages 287-292 List of Reference Works. Page 293 APPENDIX. 261 \ P. R. R. STANDARD RAIL SECTION. 100 LBS. PER YARD. Draw the above either full size or enlarged 50 % . In either case draw section lines in Prussian blue, spacing not less than one - twentieth of an inch. Dimension lines, red. Dimensions and arrow heads, black. Ivettering and numerals either in Extended Gothic or Reinhardt Gothic. 262 APPPJNDIX. ^ei". ALLEN-RICHARDSON SLIDE VALVE. Draw either full size or larger. Section lines in Prussian blue, one -twentieth of an. inch apart. Dimension lines, red. Dimensions and arrow heads, black. Lettering and numerals either in Extended Gothic or Keinhardt Gothic. ,Xf 264 APPENDIX. PROOF THAT A BI-TANGENT PLANE TO AN ANNULAR TORUS CUTS IT IN TWO EQUAL CIRCLES. (sEE ART. 113). Let d m z (j and nlhb be the plans of the curves of section, d'z' their common elevation. The plane MN cuts the equa- tors of the surface at the points g, h, m, n; and if the sections are circles their diame- ters must obviously equal gm or h7i. Take gk equal to one -half g m. Let R denote the radius of the generating circle of the torus. Then gk = R + oh = a'o'. We have also ok = R. Assume any horizontal plane PQ, cut- ting the torus in the circles e cv and t x y, and the plane MN in the line ev, e'. This line gives e and / on one of the curves, y and V on the other. Theit elevation is e'. If ed m is a circle we IriUSt have ek = gk = a'o'. Drawee; make kd parallel to a, and e r perpendicular to It. Then, as the difference of level of the points k and e is seen at s'e', we will have \/ke'' ■+■ s'l;'^ for the true length of the line Whose plan is ke; and k e' + s'e'' is to equal g k'. In the right triangle spk we have ke^ = 2)k' + 8 2>\ Then ke' + s'e" =pk' + S2)^^- s'e'''. The second member becomes a'o" by substitution and reduction. For pk'( = o's") employ [{s' e". a'o" — s'e'.' R') -i- R'"], derived from triangles o's'e' and o'a'b'; and as ps = rs — rp = rs — R, we have sp' = (r s — Ry' = i^/os' —or' — i?)'= (\/o"VM- a'u"—~V7"' — R)'. R and a'u' disappear by using values derived from the triangle a'u'c'. NOTE TO ART. 131, ON THE ELLIPSE AND ITS AUXILIARY CIRCLES. The relation of T to t and Tj is thus shown analytically: Representing lines by letters, let OA(=OB=FC) = a; OC=b; OF=OF, = c, a con.stant quan- tity; OS^x; ST^y; ST^^y'; FT^p; F,T=p'. Then p + p' =^2a= \/y' + {x+ cy + i/y' + (x — c)', which, after squaring, and substituting f for a'' — c'', gives b'' x' + a'' y' = a''b^, the well- b' known equation to the ellipse; written also y'^^ -^^(a'' — x') . . . (I). Li the circle AEBK we have OT,=r = a= V S 2\' + OS' ■=V{y'y' + x'; whence {y' Y = a^ — x' . . . {'I). Dividing (1) by (2), remembering that x is the same for both r, and T, we have Tpyi'^-^z' whence y.y' -.-.b-.a; that is, the ordinate of the ellipse is to the ordinate of the circle as the semi- conjugate axis is to the semi - transverse. But in the similar tri- angles T,SO, T,Tt, we have ST: S T,:: of. T,; that is, y.y' : -.b-.a, the relation just estabhshed otherwise for a point of an ellipse. ^^'?L£''Ufo:r(\^ THE NOMENCLATURE AND DOUBLE GENERATION OF TROCHOIDS. THE NOMENCLATURE AND DOUBLE GENERATION OF TROCHOIDS. [The anomalies and inadequateness of the pre-existing nomenclature of trochoidal curves led to an attempt on the part of the writer to simplify the matter, and the following paper is, in substance, that presented upon the subject before the American Association for the Adrancenient of Science, In 1887. Two brief quotations from some of the communications to which it led will indicate the result. From Prof. Francis Keuleaux, Director of the Royal Polytechnic Institution, Berlin : •'/ agree with pleasure to your discrimination of major, minor and medial hijpoirochoids and will in future apply these novel designations.^^ From Prof Kichard A. Proctor, B.A., author of Oeometry of O/cloids, etc.: "Your system seems complete and satisfactory. I was conscious that my otcn suggestions were but partially corrective of the manifest anomalies in former nomenclatures.^* The final outcome of the investigation, as far as technical terms are concerned, appears on page 59, in a tabular arrange- ment suggested by that of Kennedy, and which is both a modification and an extension of his ingenious scheme. The property of double generation of trochoids, when the tracing-point is not on the circumference of the rolling circle, is even at present writing not treated by some authors of advanced text-books who nevertheless emphasize it for the epi-, hypo- and peri-cycloid. This fact, and the importance of the property both in itself and as leading to the solution of a vexed question, are my main reasons for introducing the paper here in nearly its original length; although to the student of mathematical tastes the original demonstration presented may prove to be not the least interesting feature of the investigation. The demonstrations alone might have appeared in Chapter V — their rightful setting had this been merely a treatise on plane curves, but they wonld there have unduly lengthened an already large division of the worlc, while at that point their especial significance could not, for the same reason, have been sufficiently shown.] That would be an ideal nomenclature in which, from the etymology of the terras chosen, so clear an idea could he obtained of that which is named as to largely anticipate definition, if not, indeed, actually to render it superfluous. This ideal, it need hardly be said, is seldom realized. As a rule we meet with but few self-explanatory terms, and the greater their lack of suggestiveness the greater the need of clear definition. Instances are not wanting of ill-chosen terms and even actual misnomers having become so generally adopted, in spite of an occasional protest, that we can scarcely expect to see them replaced by others more appropriate. Whether this be the case or not, we have a right to expect, especially in the exact sciences, and preeminently in Mathematics, such ' clearness and comprehensiveness of definition as to make ambiguity impossible. But in this we are frequently disappointed, and notably so in the class of curves we are to con- sider. Toward the close of the seventeenth century the mechanician De la Hire gave the name of Roulette — or roll -traced curve — to the path of a point in the plane of a curve rolling upon any other curve as a base. This suggestive terra haa been generally adopted, and we may expect its complementary, and equally self - interpreting term, Glissette, to keep it company for all time. By far the most interesting and important roulettes are those traced by points in the plane of a circle rolling upon another circle in the same plane, such curves having valuable practical applications in mechanism, while their geometrical properties have for centuries furnished an attractive field for investigation to mathematicians. The terms Cycloids and Trochoids have been somewhat indiscriminately used as general names for this class of curves. As far as derivation is concerned they are equally appropriate, the former being from /cfeXos, circle, and efSos, form; and the latter from Tp6xos, wheel, and eiSos. .Preference has, however, been given to the term Trochoids by several recent writers on mathematics or mechanism, among them Prof R. H. Thurston and Prof. De Volson Wood ; also Prof. A. B. W. Kennedy of England, the translator of Reuleaux' Theoretische Kinematik, in which these curves figure so largely as cen- troids. Adopting it for the sake of aiding in establishing uniformity in nomenclature I give the following definition: If two circles are tangent, either externally or internally, and while one of them remains fixed the other rolls upon it without sliding, the curve described by any point on a radius of the rolling circle, or on a radius produced, will be a Trochoid. Of these curves the most interesting, both historically and for its mathematical properties, is the cycloid, with which all are familiar as the path of a point on the circumference of a circle which rolls upon a straight line, i. e., the circle of infinite radius. The term "cycloid" alone, lor the locus described, is almost universally employed, although it is occasionally qualified by the adjectives right or common. Of almost equally general acceptation, although frequently inappropriate, are the adjectives curtate and prolate, to indicate troehoidal curves traced by points respectively loithout and within the circumference of the rolling circle (or generator as it will hereafter be termed) whether it roll upon a circle of finite or infinite radius. As distinguished from curtate and prolate forms all the other trochoids are frequently called common. Should the fixed circle (called either the base or director) have an infinite radius, or, in other words, be a straight line, the curtate curve is called by some the curtate cycloid; by others the curtate trochoid; and similarly for the prolate forms. Since uniformity is desirable I have adopted the terms which seem to have in their favor the greater number of the authorities consulted, viz., curtate and prolate trochoid. It should also be further stated here, with reference to this ■word " trochoid," that it is usually the termination of the name of every curtate and prolate form of troehoidal curve, the termination cycloid indicating that the tracing point is on the circumference of the generator. With the base a straight line the curtate form consists of a series of loops, while the prolate forms are sinuous, like a wave line; and the same is frequently true when the base is a circle of finite radius; hence the suggestion of Prof. Clifibrd that the terms looped and wavy be employed instead of curtate and prolate. But we shall see, as we proceed, that they would not be of universal applicability, and that, except with a straight line director, both curtate and prolate curves may be, in form, looped, wavy, or neither. And we would all agree with Prof Kennedy that as substitutes for these terms "Prof. Cayley's kru-nodal and ac-7iodal hardly seem adapted for popular use." It is therefore futile to attempt to secure a nomenclature which shall, throughout, suggest both the form of the locus and the mode of its construction, and we must rest content if we completely attain the latter desideratum. We have next to consider the trochoids traced during the rolling of a circle upon another circle of finite radius. At this point we find inadequacy in nomenclature, and definitions involving singular anomalies. The earlier definitions have been summarized as follows by Prof. R. A. Proctor, in his valuable Geometry of Cycloids: — f epicycloid ) " The ■) f- is the curve traced out by a point in the circumference of a circle which rolls without sliding ( hypocycloid j ( external ) a fixed circle in the same plane, the two circles being in i \ contact." ( internal ) As a specific example of this class of definition I quote the following from a more recent writer : — "If the gen- erating circle rolls on the circumference of a fixed circle, Instead of on a fixed line, the curve generated is called an epicycloid if the rolling circle and the fixed circle are tangent externally, a hypocycloid if they are tangent internally." (Byerly, Differential Calculus, 1880.) • In accordance with the foregoing definitions every epicycloid is also a hypocycloid, while only some hypocycloids are epicycloids. Salmon {Higher Plane Curves, 1879) makes the following explicit statement on this point: — "The hypo- cycloid, when the radius of the moving circle is greater than that of the fixed circle, may also be generated as an epicycloid." To avoid any anomaly Prof. Proctor has presented the following unambiguous definition: — (■ epicycloid ) . " An i ^18 the curve traced out by a point on the circumference of a circle which rolls without sliding ( hypocycloid j ° on a fixed circle in the same plane, the rolling circle touching the \ ._ 'j [■ of the fixed circle.' ■ outside 1 inside j This certainly does away with all confusion between the epi- and Ay^o-curves, but we shall find it inadequate to enable us, clearly, to make certain desirable distinctions. By some writers the term external epicycloid is used when the generator and director are tangent externally, and, similarly, internal epicycloid when the contact is internal and the larger circle is rolling. Instead of internal epicycloid we often find external hypocycloid used. It will be sufficient, with regard to it, to quote the following from Prof. Proctor : — " It has hitherto been usual to define it (the hypocycloid) as the curve obtained when either the convexity of the rolling circle touches the concavity of the fixed circle, or the concavity of the rolling circle touches the convexity of the fixed circle. There is a manifest want of symmetry in the resulting classification, seeing that while every epicycloid is thus regarded as an external hypocycloid, no hypocycloid can be regarded as an internal epicycloid. Moreover, an external hypocycloid is in reality an anomaly, for the prefix 'hypo,' used in relation to a closed figure like the fixed circle, implies interiorness." To avoid the confusion which it is evident from the foregoing has existed, and at the same time to conform to that principle which is always a safe one and never more important than in nomenclature, viz., not to use two words where one will suffice, I prefer reserving the term " epicycloid " for the case of external tangency, and substituting the more recently suggested name pericycloid for both "internal epicycloid" and "external hypocycloid." The curtate and prolate forms would then be called peritrochoids. By the use of these names and those to be later presented we can easily make distinctions which, without them, would involve undue verbiage in some cases, and, in others, the use of the ambiguous or inappropriate terms to which exception is taken. And the necessity for such distinctions frequently arises, especially in the study of kinematics and machine design. Take, for example, problems like many in the work of Keuleaux already mentioned, relating to the relative motion of higher kinematic pairs of elements, the centroids being circular arcs and the point - paths trochoids. In such cases we are quite as much concerned with the relative position of the rolling and fixed circles as with the form of a point-path. In solving problems in gearing the same need has been felt of simple terms for the trochoidal profiles of the teeth, which should imply the method of their generation. Although they have not, as yet, come into general use, the names pericycloid and peritrochoid appear in the more recent editions of Weisbach and Reuleaux, and will undoubtedly eventually meet with universal acceptance. Yet strong objection has been made to the term " pericycloid " by no less an authority than the late eminent mathematician. Prof. W. K. Cliflbrd, who nevertheless adopted the "peritrochoid." I quote the following from his Elements of Dynamic: — "Two circles may touch each other so that each is outside the other, or so that one includes the other. In the former case, if one circle rolls upon the other, the curves traced are called epicycloids and cpitrochoids. In the latter case, if the inner circle roll on the outer, the curves are hypocycloids and hypotrochoids, but if the outer circle roll on the inner, the curves are epicycloids and peritrochoids. We do not want the name pericycloids, because, as will be seen, every pericycloid is also an epicycloid; but there are three distinct kinds of trochoidal curves." As it will later be shown that everj' pen -trochoid can also be generated as an epi- trochoid we can scarcely escape the conclusion that the name ^:>e)-»»Q. The centre R will then be found at R,, and the tracing point P at Pj. The point of contact, Q, will then be the instantaneous centre of rotation for Pj, and Pj Q will, therefore, be a normal to the trochoid for that particular position of the tracing point. The motion of P is evidently circular about R, while that of R is in a circle about F. The curve P P, P, P^ is that portion of the hypotrochoid which is described while P describes an arc of 180° about R, the latter meanwhile moving through an arc of 108 ° about F, the ratio of the radii being 3 : 5. Now while tracing the curve indicated the point P can be considered as rigidly connected with a second point, p, about which it also describes a circle, p meanwhile (like R) describing a circle about F. Such a point may be found as follows: — Take any position of P, as P,, and join it with the corresponding position of R, as R,; also join RV to F. Let us then suppose P^R and RjF to be adjacent links of a four-link mechanism. Let the remaining links, Kp, and pjPj, be parallel and equal to PjR, and RjF respectively. Taking F as the fixed point of the mechanism let i^s suppose Pj moved toward it over the path P.;P3....P5. Both Rj and pj will evidently describe circular arcs about F; while the motion of Pj with respect to pj will be in a circular arc of radius PjPj. We may, therefore, with equal correctness, consider p, as the centre of a generator carrying the point P, , and p, F a new line of centres, intersected by the normal P, Q in a second instantaneous centre, q, which, in strictest analogy with Q, divides the line of centres on which it lies into segments, p,y and Fj, which are the radii of the second generator and director respectively; q being, like Q, the point of contact of the rolling and fixed circles for the instant that the tracing point is at P,. The second generator and director, having p^g and jF respectively for their radii, are represented in their initial positions, p being the centre of the former, and p. the initial point of contact. The second generator rolls in the opposite direction to the first. It is important to notice that whereas the tracing point is in the first case within the generator and therefore traces the curve as a prolate hypotrochoid, it is without the second generator and describes the same curve as a curtate hypo- trochoid. If we now let R and F denote no longer the centres, but the radii, of the rolling and fixed circles, respec- tively, we have for the first generator and director 2 R > F, and for the second 2 R < F. It occurred to mo that a distinction could very easily be made between trochoids generated under these two opposite relations of radii, by using the simple and suggestive term major hypotrochoid when 2 R is greater than F, and minor hypotrochoid when the opposite relation prevails. We would then say that the preceding demonstration had estab- lished the identity of a major prolate with a minor curtate hypotrochoid. Similarly the identity of major curtate and minor prolate forms could be shown. If the tracing point were on the circumference of the generator tlie trochoids traced would be, by the new nomen- clature, major and minor hypo - cycloids. It is worth noticing that for both hypo -cycloids and hypo -trochoids the centre F is the same for both generations, and that the radius F is also constant for both generations of a hypo - cycloid, but variable for those of a hypo - trochoid. DOUBLE GENERATION OF HYl'OTROCHOIDS. Having given the radii of generator and director for the construction of a hypo - trochoid, the method just illustrated will always give the lengths of the radii of the second rolling and fixed circles. The accuracy of the values thus obtained may be checked by simple formulae derived from the same figure, as follows : — Radii being given for generation as a major hypotrochoid, to find corresponding values for the identical minor hypo- trochoid. Let Fj denote the radius F Q [ := F m ] of the first director. " r2 '■ " " F(? [=F,n ] " " second " " r " " " RjQ [=E??i] " " first generator. " p " " " p2? [= />M ] " " second " " tr " " tracing radius of the first generation, i. e., the distance R2P2 (°'' B P) of tracing point from centre of first generator. Let ip equal the second tracing radius ^ pjP, = p P. From the similar triangles Q F ? and Q R j Pj we have Fj : F, : : fo- : r whence i>\ = c . . (1) Fi [ir] also p = Fj — ir = tr r "{7-^} • '^' and tp = P2P2 = FR2 = d, the distance between the centers of first generator and director (3) If the radii be given for a minor hypotrochoid then FQ : pj^a ■• ^1 ■ Pilt from which we have, as before, radius of given jixed circle X given tracing radius fixed radius desired = '- ; — ; '- (4) radius of given generator and, similarly, formulae (2) and (3) give the radius of desired generator and the corresponding tracing radius. With the tracing point on the circumference of the generator, if we let R = radius of the latter for a major hypo- cycloid and r correspondingly for the minor curve, then for a major hypocycloid R = F — r (5) " a minor " ?• = F — R (6) For the curves intermediate between the major and minor hypotrochoids, viz., those traced when the diameter of the rolling circle is exactly half that of the fixed circle, a separate division seems essential to completeness, and for such I suggest the general name of medial hypotrochoids. For these the formulae for double generation are the same as for the "major" and "minor" curves, and similarly derived. With the tracing point on the circumference of the generator these curves reduce to straight lines, diameters of the director. In all other cases the medial hypotrochoids are an interesting exception to what we might naturally expect, being neither looped nor wavy, but ellipses. The failure of the terms "looped" and "wavy" to apply to these medial curves is paralleled by that of the adjectives "curtate" and "prolate," since, contrary to the signification of the latter terms, any ellipse generated as a curtate curve is larger than the largest prolate elliptical hypotrochoid having the same director. And as we have seen that, with scarcely an exception, "curtate" and "prolate" apply equally to the same curve, our only reason for retaining them is the fact of their general acceptation as indicative of the location of the tracing point with respect to the circumference of the rolling circle. Since the medial hypotrochoids are either straight lines or ellipses, we can readily find for them that which we have found it useless to attempt to construct for the other troohoidal curves, viz., simple terms suggestive of their form; in fact the names "straight hypocycloid" and "elliptical hypotrochoid" have long been familiar to us all, and we have but to incorporate them into the nomenclature we are constructing. It only remains to show that a prolate ejo^-trochoid can be generated as a curtate ^eH- trochoid, and vice versa, for which the demonstration is analogous to that given for the hypo-curves and leads to the following formulae, derived from the similar triangles QFy and QRiPj (the values being supposed to be given for the epi-trochoid and desired for the peri - trochoid) : F, (tr) P =^^{5+1} (8) t p = d = distance between centres of given generator and director :^ F", -f- ?• (9) If given as a ^jeri- trochoid and desired as an epi- trochoid the tracing radius will again equal the distance between the given centres (in this case, however ^ E — F) ; the formula of the radius of desired director will be of the same form in (10) as equations (1) and (7) ; but ( F, ) radius of second generator := tr < 1 J. With the tracing point on the circumference of the generator, and letting K ^ radius of the same for a peritrochoid and r for an epitrochoid, we have for the epicycloid r = R — F (11) " " pericycloid R=F-fr , (12) ALPHABETS AND ORNAMENTAL DEVICES FOR TITLES. 11 TTKn^rv-ac-r'vv No. 1. ABCD E FGH I J K LM N OPQRSTUVWXYZ&1234567890 No. 2. 7 21^3^53 7 39Z No. 3. ABCDEFGHIJKLMNOPQRSTUVWXYZ 12345 & 67890 No. 4. 3fpecimef!s ef ffye mpMed Italic fyrm called"f?e//ihardt Gcth/c^ its ^arms fynns and ap/?//cal'iof7s ham^ i>ee/7 ha/id^/pme/y illusti-ated hy Cl^ Relnhardf^ in a speadi -kxf-i;tPo/f £/ev(?kd alm(^sf- exclijs/vely ft; fli/s fcrm. /r/$ /nuch used in en^in- eer//?^, c/iie/'/y on accounf of/fs compacf-ness, and//s leqihilify af/et redi/d/on /py pfyo/v-pn>ce5ses. An in- clined e///pse /sifie basis of many of ^e ieffers. TFie and 5 are peci///dr^ a/so f//e ^. 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ABGBEFGRI JKLANOP QRsravvxYzs. =^ 1 2 3 4 5 * 6 7 5 9 ► No. 48. im TIT 2- MK 1 J No. 49. ;«B6DEF(J5ig^Ln]I]0P ^ 1 2 5 4 5 •:• 7 3 9 ^ No. .50 /A i © § i LP i Of! # U^ (L 0!^ LIS © 0^ 11 g i <$ i i :;7 i i i No. 58. lillabcdefghijklmnopqF stuvwxp., 1234367890 No. 59. No. 60. -tv "ijJ^ -1 •^y>/ -^f^. ' M' 5) N ^ IS. I > I! m X n n -r-i ;. a 4 4i B 7 B 01 ^^ jf ^ & (k ^ i=: '» ©>!«;• — -£x ••>;. 2. -» -^^ -G-^5v.S-. ^ J^J ■^A •wj>H .11^' .# ,# ^ .3f ll -W^ ^ t) S«v J^ INDEX AND LIST OF REFERENCE WORKS. INDEX, Agnesi, Witch of, 205. Algebraic curves and surfaces, 331. Alphabets, Chapter VII and Appendix. Angles, laying out, 61 ; equal to given angle, 85. Anti-parallelogram, linkage, 163. Annular torus, 112-114; 333, 363. Curve of shade on, 590. Arch, semi-circular, voussoir in isometric, 636. Archimedean spiral, 188 ; tangent line, 189 ; cam outline, 190; relation to conical helix, 191. Architect's scale, graduation of, 52. Architectural perspective, exteriors, 612. Architectural perspective, interiors, 616. Architectural scrolls, 219. Asymptote, 134, 152, 197, 199, 202, 205, 218, 368. Auxiliary elevations, 396 (2) ; 404 (a) (c). Axis of homology, 145, 510. Axonometric (including Isometric) projection, 18 ; 621-636. Of three mutually perpendicular lines, 625. Of a one-inch cube, 624, Of a vertical pyramid, 625. Of curves, 626. Band-wheels, guide pulleys for, 452. Bisection of line, 82 ; of angle, 83 ; of arc^ 84. Bi-tangent plane, to torus, 113. Blue-printing, 270-274. Bolts, note-taking on, 25. Drawing of, 661. Proportions, table of, 661. Bonne's conic projection, 563, Borders and corners, designs for, 266, 267. Boscovich definition, and construction of conies based thereon, 121, 126, 138-144. Bow-pen and pencil, 38, iii. Bridge-post, upper-chord connection, 652-654. Bridge sketching, 25. Brilliant points, 587. On surface of revolution, 588. On sphere, with curve of shade, 589. Brushes, choice of, 62. Brush tinting and shading, 220-236. Burmester, relief-perspective model, 154. Catalan, conchoidal hyperboloid of, 196, 359, 488. Cabinet projection, 17, 645. Cardanic circles, 180. Cardioid, as trochoid, 181-183 ; degree, 332. Cartesian ovals, on two foci, 206-208 ; third focus, 209; by continuous motion, 210; relation to caustics, 211 ; degree, 332. Cartography (map making), 534-568. Cassian ovals, 114, 212 ; degree, 332. Catenary, 214 ; degree, 332. Caustics, 211, 217, Cavalier perspective, 17, 645. Cayley, on non-Euclidean geometry, 19 (note). Cylindroid of, 333, 356, 477. Central projection, 7. Centre lines, 25, 65, 388. Centre of the picture, 599. Cenlroids, 159-163. Cerographic process, for illustration, 277. Changed planes of projeclior , 404. Chromo-lithography, 278. Circle, through three points, 86. Perspective of, 61 r. Various problems, 81-106. Cissoid of Diodes, 199-201 ; 332. Class, of line or surface, 334. Clinographic projection, 14, 643-^51. Collinear points, 4. Colored lines, 65 ; 388 (6). Colors, 56 ; conventional, 27, 73. Companion to the cycloid, 170, 171. Common solid of intersecting surfaces, 426. Cone, scalene, 135. Sub-contrary section, 135, 136, Plane sections, 137-144 ; 507. Flat, and homologous conies, 145-153. Right, development, 191. As auxiliary surface, 307, 309, 318, 319, 327, 329^ 519* Properties, 342-347- Point on, given one projection to find other, 446 (a) (b) (c). Oblique, with development, 418. Intersecting cylinder, 433-435. 437* 43*^1 442. As part of bath-tub, 436. Intersecting cone, 439-441, 444. To project, having 6 and for axis, 451. Tangent plane to, 456. Tangent plane, containing exterior point, 457. Tangent plane parallel to line, 458. Intersected by plane, 507, 511. Conchoid of Nicomedes, 193 ; as trisectrix, 194 ; tangent and normal, 195 ; as section of algebraic surface, 196. Conchoidal hyperboloid of Catalan, sections, 196 ; order, 333 ; projections, 359, 488. Conchoidal screw, Holm's, 484. Conical surface, 8. Projection, 8, 9. Helix, 191, 508. Conic sections, 121-144. As homologous figures, 145-153. Conicoids (quadrics), 333, 367. Conoidal surfaces, 354-356. Conoid of Pliicker, 333, 356, 477. Conoidal surfaces, 854-356. Cono-cuneus of Wallis, 333, 355, 468 (b), 473. Conventional representations, 26. Companion to the cycloid, 121 ; 170-172. Compasses, selection, care and use, 36, 37. Comte, Auguste, on Descriptive Geometry, page 104. Come de vache, 333, 361, 475, 476. Corresponding points, 4. Cremona, on nomenclature, 19 (note). Crystal projection, 651. Cube, 345, 419. Curtate Trochoid, 173, 174. Curvature, radius of, 380 ; line of, 381 ; of helix, 420 (note). Curve of shade, 571. On a sphere, 589. On a torus, 590. On a warped surface, 592. Curve of shade on warped helicoid, 596. Curves, non-circular, how drawn, 58. Reverse, 77, Cuspidal edge, 346. Cyclide of Dupin, 333, 365. Cyclo-orthoids, 176. Cylinder, point on, to find projections, 446 (d). Tangent planes, 459-461. (Half) and rectangular abacus (shadows), 584, Cylindrical surface, defined, 8. Projection, 8, 14, 558. Column and abacus, shadows, 584. Cylindroid, of Cayley, 333, 356, 477. Of Frezier, 333, 360, 489. Cycloidal curves, general construction, 179. Cycloid, common, 166-168. Tangent line at given point, 169. Companion to (Roberval's curve of sines), 170. 171. Area between cycloid and base, 172. Curtate form, 173, 174. Prolate form, 175. Degree, of curve or surface, 332. De la Hire's perspective projection, 555. Descriptive properties, 5. Descriptive geometry, defined, 6, 19. Monge's system : First Angle Method, 283-330 ; 446-533. Third Angle Method, 383-444. (See also headings MongCy Intersections^ Developntents) . Design, method of, 23. Detail drawings, 20. Developable surfaces, 120; 191; 344-347. Tangent planes to, 374-6 ; 454-464, Developable helicoid, 187 ; 346 ; 420. Tangent planes to, 462-464. Development of surfaces, 405-420. Right cylinder, 120. Right cone, 191 ; 507. Right pyramid, 389 ; 396 (6). Right prisms, 411-412. Oblique prisms, 414, 415. Oblique cylinder, 416. Oblique pyramid, 417. Oblique cone, 418 ; 521. Regular solids, 419. Developable helicoid, 420, Intersecting surfaces, 425, 426. Bath-tub, 436. Diagonal scales, 53. Diagonals and their vanishing points, 604. Dimensioning, 25, 388. Dinostratus, Quadratrix of, 197, igS. Diodes, Cissoid of, 199-201. Direction of light, 574. Dividers, choice, care and use, 34. Dodecahedron, 345, 419. Doors and doorways in perspective, 616, 618. Double-curved lines, 338 ; tangents to, 369. Double-curved surfaces, 362-365. Tangent planes to, 378, 492-502, Doubly-ruled surfaces, 350. INDEX. Draughtsman's equipment, 28-62. Drawing-board, 46. Drawing-paper, 41, 43, 44. Drawing-pins (thumb-tacks)^ 57. Dupin's CycUde, 333, 365. Duplication of cube, by Cissoid, 201. Edge of regression, 346. Elbow-joint, 430. Elbow, four-piece, 431. Elevations, relation of , 383-386; 404. Ellipse, Boscovich definition, 124. Gardener's, 124, 125, By concentric circles on axes, 131. By rotation of circle, 448. Tangent line to, 130, 132. On conjugate diameters, 133. As plane section of cone, 142, 507. As a centroid, 163. As an hypotrochoid, 180. As projection of circle in plane of given inclina- tion, 450. Ellipsoids'; 363, 365. Elliptical hyperboloids of one and two nappes, 365. Elliptical paraboloid, 365, Engineer's scale, graduation of, 52. Engraving, hand or chemical, 275-281. Envelopes, 335. Epicycloid, ordinary, 179 ; spherical, 338. Epitrochoid, nomenclature, 176 and Appendix ; construction, 179 ; as trisectrix, 184, 185. Equal division of lines, 87. Equiangular spiral. 216. Equidistant polyconic projection, 566. Equilibrium polygon, 203. Equivalent projections, 534, 563. Erasures, on paper and tracing-cloth, 45, 59, 60. Evolute, 187, 211. Expansion curve, 134. Figures, geometrical, defined, i ; properties of, 5. Free-hand drawing, 20-26. Equipment for, 22. Free-hand lettering, 27, Chap. VII and Appendix. Frezier's cylindroid, 333, 360. Projections and tangent plane, 489. Gearing, note-taking on, 25, Proportions, 656-657. Geodesic, 382 ; on cone, 508. Geometry, Descriptive, defined, 6. Monge's Descriptive, 19. Euclidean, Cartesian, Projective, non-Euclid- ean, etc., remarks on nomenclature, page 4. Globular projection, Nicolisi's, 554. Gnomonic projection, 553. Graphical statics, defined, 9 ; illustrated, 203. Greek fret, 67. Groined arch, perspective, 619. Hair-spring dividers, 34. Half-tone illustrations, how made, 281. Harmonic curve. (See Sinusoid). Helical springs, 658, 659. Helicoid, developable, involute sections, 187. Generation of, 346, 420. Development, 420. Helicoids, warped, 357, 358, 47S-487. Right helicoid, 358, 478. Oblique helicoid, 357, 479. General cases, uniform pitch, 480-482. Radially-expanding pitch, 483, 484, Axially-expanding pitch, 486, 487. Intersection by con-axial surface, 485. Tangent planes, 478, 479, Helix, construction of, 120. As sinusoid, 121. On cone, 191, 508. Hexagons, how to draw, 51. Holm's conchoidal screw, 484. Homologous figures, 145-153 ; 510. Homologous space figures (relief-perspective), 154- 156. Horizon, defined, 601 ; as locus, 602. Horizontal projection (One-plane Descriptive), 18, 637-642. Hyperbola, by Boscovich definition, 123, 127, 138- 140. On two foci, 129. Tangent line to, 130. As expansion curve, 134. Plane section of cone, 138-140. Homologous with circle, 150. As a centroid, 163. Hyperbolic paraboloid, 349-352, Projections and tangent plane, 471. As raccording surface, 474. Hyperbolic spiral, 218. Hypocycloids, construction of, 178. Hypotrochoids, nomenclature, 176 and Appendix. Hyperboloids, double-curved. Of revolution, 363. Of transposition, 365, Hyperboloids, warped, Of revolution, 116, 468,470, 513, 514. Of transposition, 349-351. Hyperboloid, conchoidal, 196, 333, 359, 488. Icosahedron, 345, 419. Illustrative processes, 270-282. India ink, 55. India rubber, 59. Inscribed figures, 93-98. Interiors, perspective of, by method of scales. 616. Intersection of plane, with plane, 321 ; with line, 322 ; with right pyramid, 396, 417, 509; with various irregular solids, 393, 398, 400, 402, 403 ; right prism, 412 ; oblique prism, 415 ; oblique cylinder, 416, 512 ; oblique pyramid, 417 ; warped hyperboloid, 513, 514. Intersecting surfaces: Prisms, vertical with horizontal, 425, 426. Vertical and oblique prisms, 427. Pyramidal surfaces, general principles, 428. Vertical with oblique pyramid, 429. Cylinder with cylinder, 430-432. Vertical cone with horizontal cylinder, 433. Cylindrical pipe to make elbow with conical pipe on given joint, 434. To find cone to join unequal circular cylinders, joints either circles or ellipses, 435. Bath-tub, projections and developments, 436. Vertical cylinder with oblique cone, axes inter- secting, 437. Same problem as last, axes non- plane, 438. Conical elbow, angle given, also size of joint, 439- Right cones, axes meeting at oblique angle, non - plane intersection, 440. Oblique cones, bases in same plane, 441. Vertical cylinder and oblique cone, bases in same plane, 442. Two cones, two pyramids, or cone and pyra- mid, neither axes nor bases in same plane. 444- Helicoid intersected by con -axial surface, 485. Pyramids, bases in same plane, only one aux- iliary plane, 515, 516. Intersecting prisms, cylinders, etc., bases in same plane, 517. Surface of revolution, with cylinder, using cylinders as auxiliaries, 518. Surface of revolution and conical surface, using cones as auxiliaries, 519. Sphere by cone whose vertex is at the centre of the sphere, 520. Instantaneous centre, 159. Inverted plan, how used in perspective, 610. Involute, of circle, 186, 187, 420; degree, 332; of logarithmic spiral, 217 ; of helix, 420, (a) and (c). Ionic volute, 219. Irregular curves, 58 ; exercises for, 119-219. Isometric projection and drawing, 18 ; 627-636. Of a cube, 629. Of curves, 630, 631. Shadows and shade lines, 632, 633. Non -isometric lines, angles in isometric planes, 635- Non - isometric lines, angles not in isometric planes, 636. James' perspective projection, 556. Kinematics, 157. Kinematic method of obtaining tangents, 159. Kochansky's method of rectifying semi -circum- ference, 104. Lemniscate of Bemouilli, 114; as motion curve. 158, 164 ; parallel curve to, 192 ; as Cassian oval, 212. Lettering, free - hand, 27, 245-247 ; in general, 245- 269 ; alphabets m Appendix. Light, direction of, 575. Lima^on, the, 181-185 ; degree, 332. Line of shade, 571. Lines of height, use in perspective, 612. Line of curvature, 381. Lines, parallel, 49; perpendicular, 50; kinds and significance, 65 ; examples for ruling pen, 67-118. Line shading, 71, 76, 78, 79 Line tinting, 69, 70. Link-motion curves, 157, 158. Lithography, 278. Logarithmic spiral, 216. Loxodromics (rhumb lines), 204. Machine drawing and design, 23. Map projection (see Spherical Projection). Masonry, 73, 74, 228, 243, 244. Materials, how indicated, 26, 73, 74. Mercator's projection, 204, 559. Metrical properties, 5. Military perspective. 17,645. Monge's Descriptive Geometry, First Angle Method. Definitions and remarks, 19, 283. Fundamental principles, 284-330. Projections of point, 284-288. Projections of line, 289, 291, 292. Projecting planes, 290. Traces of lines, 293. Lines parallel to H or V, 294-297. Representation of planes, 298. Planes, determination of, 300. Lines of declivity, 301. Limiting angles, 302. Line perpendicular to plane, 303. Profile planes, 304 ; lines in, 304, 305. Rabatment and other rotations, 306. Cone as auxiliary surface, 307, 519. Traces, length, inclination of line, 308. Projections of line, having and 4> given, 309. Plane through two lines, 310. Angle between lines, 311. Locating lines in planes, 312. Point in plane, one projection given, 313. Plane through three points, 314. Plane through line, parallel to line, 315. Plane through point, parallel to plane, 316. Plane through point, perpendicular to line, 317. Inclination of plane to H and V, 318. Angle between traces of plane, 318. Plane desired, B and ^ given, 319. Plane parallel to plane, at given distance, 32a Line of intersection of planes, 321. INDEX. Monge's Descriptive, First Angle Method: Intersection of line and plane, 322. Angle between two planes, 323. Angle between line and plane, 324. Distance from point to plane, 325. Line in plane, inclination given, 326. Plane, to contain line and make given angle, 327- Lines of given inclination, and intersecting at given angle, 328. Planes, to be mutually perpendicular, their in- clination given, 329. Common perpendicular to non - plane lines, 330. Given one projection of point on surface, to find the other, 446. To project a circle when inclination of its plane is given, 448. To prove projection of circle an ellipse, 448 ; also see Appendix. To project horizontal cylinder, oblique to V, 449. To project circle whose plane is oblique to both H and V, 450. To project right cone, axis making given angles with H and V, 451. To determine guide pulley to connect band- wheels, 452. To project any solid, having the inclination of its base given, 453. (See also headings Tangency, Intersection^ De- velopable Surfaces^ Warped Surfaces). Monge's Descriptive Geometry, Third Angle Method, 383-444. Motion curves, 157, 158. Mouldings, in oblique projection, 75, 76. Navigator's charts, 204, ssq. Newton {Sir Isaac), method for generating cissoid, 200. Niche, shadow on interior, 591. Nicolisi's globular projection, 554. Nicomedes, conchoid of, 193-196. Non - circular curves, drawing of, 58. Non -plane curves, 334, 338, Normal, to random curve, 88 ; see also 368. Normal, found by instantaneous centre, 159. Normal sections, 373. Normal hyperbolic paraboloid, 475. Note -taking, on riveted work, pins, bolts, screws, nuts, gearing, bridge trusses, etc., 25. Oblique helicoid, 357. Oblique projection, 14, 15, 643-651. Mouldings, 75, 76. Circles, 646. ' Timber framing, 23, 649. Arch voussoirs, 649. Shadows, 650. Crystals, 651. Octahedron, 345,419. One- plane Descriptive Geometry, 18, 637-642. Intersection of line and plane, 640. Intersection of two planes, 641. Section of hill by plane of given slope, 642. Order of a line or surface, 332, 333. Order of laying out and inking in work, 66. Ordinary polyconic projection, 567. Ortho- cycloids, 176. Orthogonal projection. (See Orthographic.) Orthographic projection, 14 ; fundamental prob- lems, 283-330. Orthographic projection of sphere, 541-543. Orthoids, 176. Orthomorphic projection, 534, 546, 559. Osculating plane, 334, 380. Osculating circle, 380, ■ Oval, to construct on given line, log. Ovals, of Cassini, 114, 212 ; Cartesian, 206-21T. Paper, for drawings, 41 ; to stretch, 44 ; division of, 64. Parabola, by enveloping tangents, 68, 128 ; by Boscovich definition, 121, 122, 126; tangent line, 130; as plane section of cone, 141; homologous with circle, 147 ; as envelope, 335- Paraboloid, hyperbolic, 349-352, 471 ; of revolution, 363 ; elliptical, 365. Parallel curves, 192. Parallel lines, how to draw, 49. Parallelogram of forces, 203. Parallel projection, 7, 14-ig, Pen, for right lines, 28-33 '. ^o^ curves, 36-38. Pencils, 54 ; use, 66. Pennsylvania R. R. standard section lines, 74, Peritrochoids, 176, 179 and Appendix. Perpendiculars, how drawn, 50, 81. Perpendiculars, vanishing point of, 603. Perspective, linear, 10, 598-620. Relief, ii, 154-156. Military, 645-647. Cavalier, 645-647. Aerial, 598. Preliminary definitions, 599-604. Of vertical lines, 600. Of parallels to picture plane, 605. By trace and vanishing point, 606. By diagonals and perpendiculars, 607. Of a cube, by above methods, 609. Of a cube, by inverted plan, 610. Of a circle, 611. As applied jn architecture, 612, 616, Of shadows, 614-616; 619. By the method of scales, 616. Of a right lunette, 617. Of a groined arch, 619. General remarks, 620. Phoenix columns, section and shading, 235, Photo -engraving, drawings for, 279. Photogrammetry (Photometrography), 13. Photogravure, 275, 282. Photo -lithography, 278. Photo • processes, for illustration, 275-282. Photo- zincography, 279, 381. Plane curves, 337 : class, 334 ; examples, 121-219. Plane figures, i. Plane problems of straight line and circle, 8i-iog. Perpendicular to line at given point, 81. Bisection of line, 82. Bisection of angle, 83. Bisection of arc, 84. Angle, made equal to given angle, 85. Circle, to pass through three points, 86. Division of line into any number of equal parts, 87. Tangent and normal to random curve, 88. Tangent to circle at given point, 8g. Tangent to circle from exterior point, 90. Tangent to circle whose centre is inaccessible, 91. Construction of regular polygons, 92-100. Circle inscribed in equilateral triangle, 96. To inscribe a circle in any triangle, 97. Inscribed triangle, square, hexagon and octa- gon, 93-95. Inscribed pentagon, 98. Arc- equivalent of straight line, Rankine's method, 102. Rectification of arc, Rankine's method, 103. Rectification of semi-circumference, Kochans- ky's method, 104. Circle, tangent to two lines and a circle, 105. Tangent line to two circles, 106. Arc (radius given) tangent to two lines, 107. * Line through given point which would meet two lines at their inaccessible intersection, 108. Oval, constructed on a given line, 109. Plane sections, of annular torus, 112-114; of de- velopable helicoid, 187 ; of right cone, 135- 144, 507 ; of various solids, 393, 396, 398, 400-403 ; of right prism, 412 ; right pyra- mid, 509; oblique prism, 415; oblique cylinder, 416, 512; oblique pyramid, 417; oblique cone, 511 ; of warped surfaces, 348, 352. 473i479i 513. 5M- Plucker, conoid of, 333, 356, 477, Poinsot,star polyhedra of, 345. Point of sight, 599. Points, collinear, 4 ; corresponding, 4 ; of concourse, 47X (d). Polyconic projection, 564-568. Polyhedra, regular convex, 345, 419 ; star, 345. Principal plane, 135 ; sections, 381 ; radii, 381. Principal diametric planes, 471 (c). Projection, centre of, 2 ; plane of, 3; divisions of, 7-19. Projective geometry, 9 ; see also note, page 4. Projective conies, 145-153. Homologous space figures, 154-156. Projection drawing, First Angle Method, 283-330 ; 405-420 ; 445-522. Third Angle Method, 383-404,421-444. Prolate trochoid, 175. Protractors, 61. Quadratrix of Dinostratus, 197 ; as trisectrix, 198, Quadrics (conicoids) 333, 367. Raccordment, 379 (c), Radial projection, 8, Radius of curvature, 380. Rail section, 118, 255. (See also Appendix). Rankine's methods of approximation. Obtaining arc -equivalent of straight line, 102. Rectifying a given circular arc, 103. Reciprocal spiral, 218. Rectangular polyconic projection, 565. Rectangular projection, 14. Rectification of curves, 103, 104, 413. Regular polygons, 92-100, Regular solids, 345, 419, Relief- perspective, 11,154-156. Rendering, 620. (See also Shading). Reverse curves, 77. Revolution, surfaces of, 34a, 347, 348, 363, 364. Warped hyperboloid, 116, 348, 470, 513. Double -curved hyperboloid, 363. Rhumb lines (loxodromics) 204, 559. River-bed sections, 26, 24a. Riveted work, note-taking on, 25. Roberval, companion to cycloid, 170-172. Rotation, 286, 306, 404. Roulettes. (See Trochoids), Rubber, India, 59. Scales, 52 ; diagonal, 53. Scenographic projection, 10; also Chapter XIV. Sciography, 12. Screws, note -taking on, 25. Square -threaded, 660. Triangular -threaded (general), 357, 480. Triangular -threaded (U. S. standard). 661, Scroll. (See Warped surfaces). Secant, 88. Section lining, 69, 70 ; conventional, 72-74. Sectional view, 70, 395. Sections of wood, masonry, etc., 26, 73, 74, Serpentine, 365. Set - squares, 48. Shade, line of, 571, 589, 590, 592, 596. Shade lines, 67, 115, 576. Shade versus shadow, 570. Shading, with lines, 71, 112 ; with brush, 232-23^ Shadow, of point, 577. Equal to original line, 578. Of line perpendicular to plane, 579. INDEX. Shadow, of parallel lines, 580. Of cube, 575, 581. Of vertical pyramid, 583. Pier and steps, 583. Cylindrical abacus on similarly - shaped col- umn, 584. Rectangular block on vertical semi -cylinder, 585. Vertical, inverted, hollow cone, 586. On the interior of a niche, 591. . Of line on a warped surface, 593, Of helix on surface of screw, 595. Shadows, 12 ; 569-597 ; isometric, 633, 633 ; in oblique projection, 650. Shop drawings, method of making, 383-444. Sinusoid, projection of helix, 121 ; companion tu cycloid, 171 ; degree, 332 ; transformed into directrix of Pliicker conoid, 356. Sinusoidal projection of sphere, 563. Sketching from measurement, 23-25. Skew arch, corne de vache form, 361. Space figure, defined, i. Sphere, brilliant point and curve of shade, 589. Sphere, shading of, 234 ; given one projection of point on, to find tUe other, 446 (e) ; projec- tions of, 534-568. Spherical epicycloid, 338. Spherical projections (cartography), 534-568. Orthomorphic, Equivalent, Zenithal, 534. Orthographic projection, 541. Stereographic projection, 544-552. Gnomonic, 553. Nicolisi's globular, 554. De la Hire's perspective projection, 555. Sir Henry James' perspective projection, 556. Projection by development, 534, 557-568. Square cyhndric projection, 558. Mercator's projection, 559. Conic projection, 560, Bonne's method, 563. Sanson's projection, 563. Sinusoidal, 563, Rectangular polyconic, 564. Equidistant polyconic, 566. Ordinary polyconic, 567. Spherical triangles, solved by projection, 523-533. General definitions and properties, 523-6. Three sides given, to find the angles, 527, Two sides and the included angle given, 528. Given, two sides, and the angle opposite one of them, 529, One side and the adjacent angles given, 531. Two angles given, and the side opposite one, 532- Given, the three angles, 533. Spirals, degree, 332; of Archimedes, 188-191 ; loga- rithmic (equiangular), 216; hyperbolic (re- ciprocal), 218 ; lituus, 219 ; Ionic volute, 219. Springs, circular cross- section, 658; rectangular cross -section, 659. Square- threaded screws, 358, 660. Standard section - lining, 74. Star polyhedra, 345. Statics, graphical, 9, 156 ; equilibrium polygon, 203. Steps and pier, shadows, 583. In perspective, 616 (c). Stereographic projection, 544-552. Circle always projected as circle, 545. Orthomorphic character of^ 546. Meridional projection of a parallel of latitude and meridian of longitude, 547. Projection of small circle whose plane is per- pendicular to the primitive, 548. To project any circle making any angle with primitive, 549. Projection of circle, given its pole, 550. Equatorial projection, 551. Straight - line work, 38-33. Striction, line of, 353. Structural iron, note -taking on, 25 ; sections of, 235i 655. Sub -contrary section of scalene cone, 135, 136, 545. Suppression of ground line, 394. Surface of revolution, brilliant point on, 588. Surfaces, algebraic, transcendental, 331 ; order of, 333 ; class of, 334 ; as envelopes, 335. Of revolution, 339, 340, 363, 364. Chapter X. Of transposition, 339, 341, 349, 365. Ruled, 342-361. Developable, 344-347, 405-464. Warped, 116 ; 348-360, 465-491 ; 513. Doubly -ruled, 350. Double -curved, 112; 362-365, 446; 492-502; 518-521. Intersecting, 379 (e) ; 421-444; 503-521. T -rule, 47, 63. Tangent arc (radius given) to intersecting lines, 107. Tangent circle, to two lines and another circle, 105. Tangent curves, how to draw when heavy, no. Tangent lines, properties of , 88, 368-370. Tangent line to non- mathematical curve, 88. Tangent line by means of instantaneous centre, 159. 'I'angent surfaces, 379 (a) (b) (c) (d). Tangent lines to plane curves, various problems ; To circle, at given point, 89 ; to circle from point without, 90; to circle, centre un- known, 91 ; to two circles, 106 ; to hyper- bola, 130 ; to ellipse, 130, 132, 450, 507, 511 ; to parabola, 130 ; to cycloid, 169 ; to con- choid, 195; to logarithmic spiral, 216; to hyperbolic spiral, 218. Tangent lines, in development, 507. (Fig. 334). Tangent planes, 371, 374-378» 49o- To ruled surfaces, 374-377, 455-464^ 4691 49°. 491. To double-curved surfaces, 378, 492-502. Tangent planes to various surfaces : Cone, at point on surface, 456; containing given exterior point, 457 ; parallel to given line, 458. Cylinder, 459 ; containing exterior point, 460 ; parallel to given line, 461. Developable helicoid, 462 ; containing exterior point, 463 ; parallel to given line, 464. Warped hyperboloid, 470. Hyperbolic paraboloid, 471 (e). Cono - cuneus of Wallis, 473 (b). Plucker conoid, 477 (e). Warped helicoids, 478, 479. Conchoidal hyperboloid, 488, Cylindroid of Frezier, 489. Sphere, at given point on surface, 493 ; contain- ing given line (three methods), 495-497. Annular torus, 494. Surface of revolution, at given point on surface, 498 ; to contain given line and determine by means of auxiliary con -axial surface, 500. Double -curved surface of revolution, tangency on either a parallel or meridian, and to con- tain exterior point, 501 ; perpendicular to given line, 502. Tapering lines ; arcs, in ; other lines, 117. Tetrahedron, 345, 419. Tetrahexahedron, clinographic projection of, 651. Third Angle Method of making working drawings, 383-444. Thumb tacks, 57. Tiling, 229-231. Tinting, in lines, 69, 70; with brush, 22&-231. Titles, planning of, 245-269. Toothed gearing (spur), 25, 656, 657. Torse (see Developable Surfaces). Torus, annular, 112-114, 333, 363* 446 (f). Also Ap- pendix. Torus, curve of shade on, 590. Tracing-cloth, 43, 45. Tractrix, construction, 202 ; outline of anti- friction pivot, 203 ; relation to graduation of Mercator chart, 204; as involute of catenary, 215; de- gree, 332. Transcendental lines and surfaces, 331. Transposition, surfaces of, 341 ; developable, 346; warped. 349-361 ; double -curved, 365. Triangles (set squares), 48-51, Triangular -threaded screws, 357; U. S. standard, with table of proportions, 661. Tri- focal curves, 209. Trigonometric functions, note, page 31. Trihedrals (see Spherical triangles). Trisection of angle ; by epitrochoid, 184, 185 ; by conchoid, 194 ; by quadratrix, 198. Trochoids, 166-191 ; nomenclature, 176 ; cycloid, 166-172 curtate form, 173-174 ; prolate, 175 ; general solution for all trochoids, 179 ; hypotro- choids, 178 ; epitrochoids and peritrochoids, 179; special forms, 180-191 ; for double generation of, see Appendix, Tubular surfaces, 366. Vanishing points, 602 ; reduced, 616 (b). Vanishing point of perpendiculars, 603. Of diagonals, 604. Of lines inclined at various angles to H, 613. Of rays, 614. Versiera (Witch of Agnesi), 205. Visual ray and plane, 599. Volute, Ionic, 219. Wallis, cono-cuneus of, 333., 355, 468 (b), 473, 474. Warped arch (corne de vache), 333, 361, 476. Warped helicoids, 357, 358, 478-487. Warped hyperboloids, 116, 349-351, 468 (a), 470. 513, 514. Warped surfaces, curve of shade on, 592. Shadows upon, three methods, 593. Shade and shadow on screw, 594-596. Warped surfaces (scrolls), 116, 348-361, 465-491. (Refer to the following headings: Warped hyperboloid^ hyperbolic paraboloid^ cono- cuneus^ conoid of Pliicker., corne de vache^ cylindroid of Frezier ^ -warped helicoids., tvarped arck.^ conchoidal hyPerboloid.') White's parallel motion (hypocycloidal), 180, Witch of Agnesi (Versiera), 205, 332. Wood engraving, 276. Wood graining, 26, 237-241. Working drawings, engineering designs : P. R. R. Rail Section, 85 lbs. per yd., 118. P. R. R. Rail, 100 lbs. per yd., see Appendix. Bridge post connection, upper chord, 652-654. Spur gear, 656, 657. Structural iron, sections, 655. Springs, round and rectangular, 658, 659. Screws, square-threaded, 660. Screws, U. S. Standard, 661. Valve, Allen-Richardson, Appendix. Working drawings, by Third Angle Method, 383- 404, 421-444 ; by First Angle Method, 445- 521. Development of surfaces, 405-420, and addi- tional as in Index under special heading. Intersections, 421-444, and additional as under special heading. Working drawings, systems compared, 383-386. General instructions as to order of work, 388. Drawing of right pyramid, to given data, 389. Vertical, semi -cylindrical pipe, 390. Hollow, hexagonal prism, 391. Prismatic block, hollow, oblique to V, 392. Same object as above, cut by plane. 393. Vertical right pyramid, hollow, with section, and development, 396. Truncated pyramidal block, with sections, 3g8. INDEX. Working Drawings: Hollow pentagonal prism, horizontal, inclined to V, 399- Same object as above, cut by vertical plane, 400. Inclined, irregular block, with section, 402. Problem similar to last, two section planes, 403. Projections of object after various supposed ro- tations about vertical or horizontal axes, 404. Developments, 405-420. (See special heading,) Working Drawings: Intersections, 421-444. (See special heading.) Projections of horizontal cylinder, oblique to V, 449- Circle of given diameter, inclination of plane given, 450. Cone of given axis and base, inclinations of axis given, 451. Working Drawings : Guide pulley, to locate between two band wheels, 452. Pyramid of given proportions, inclination of base given, 453. Cone, vertical, cut by plane, with development, 507- Pyramid, vertical, with section, 509. Zenithal projections, 534, 553-556. VALUABLE REFERENCE LITERATURE ON GRAPHICAL SCIENCE. LINEAR AND MACHINE DRAWING. The Engineer and Machinist's Drawing Book ; Delaistre, Cours de Dessin Lineaire : Appleton's Cyclopoedia 0/ Draiving ; Ripper's Machine Drawing and Design ; Clarke's Practical Geom- etry and Engineering Draiving; Low and Bevis, Manual of Machine Drawing and De- sign. PLANE CURVES. Leslie, Geometrical Analysis; Salmon, Higher Plane Curves; Eagles, Constructive Geometry 0/ Plane Curves; Proctor, Geometry 0/ Cycloids ; Burmester, Lehrbuch der Kinematik ; also chap- ters in the Descriptive Geometries of Peschka and Wiener. KINEMATICS AND MECHANISM, Reuleaux, Kinematics of Machinery ; Willis, Principles of Mechanism ; Clifford, Kinematic ; Weisbach, Machinery of Transmission ; Kempe, Ho7u to Draw a Straight Line : Burmester, Lehrbuch der Kinematik; Grant, Odontics ; Goodeve, Principles of Mechanistn. PROJECTIVE GEOMETRY. Poncelet, Traite des Proprietes Proj'ectives des Figures ; Chasles, Traite de Geometrie Supe- rieure; Fiedler, Darstellende Geometrie; Peschka, Darstellende und Projective Geometrie ; Wiener, Darstellenden Geometrie ; Burmester, GrundzUge der reliefperspektive ; Cremona, Elements of Pro- ''eciive Geometrv ; Graham, Geometry of Position. LETTERING. Authors : Prang, Becker, Copley, Ames, Ja- coby, Reinhardt. REPRODUCTION OF DRAWINGS. Lietze, Modern Heliographic Processes; Wil- kinson, Photo • Engravingy Etching and Lith- ography; Petl'it, Modern Reproductive Graphic Processes; Schraubstadter, Photo- Engraving. MONGE'S DESCRIPTIVE GEOMETRY- Authors: Monge, Lacroix, Hachette, Vallee, Lefebure de Fourcy, Leroy, Olivier, De la Gour- nerie, Adhemar, Mannheim, Songaylo, Javary, Caron, Fiedler, Peschka, Wiener, Marx, WooUey, Eagles, Pierce, Church, Warren. SHADES AND SHADOWS. Authors : Adhemar, Olivier, de la Gournerie Tilscher, Koutny, Weishaupt, Warren, Watson. PERSPECTIVE. Authors. Cousinery, Adhemar, de la Gournerie, Bossuet, Wiener, Cassagne, Church, Warren, Ware, Linfoot. AXONOMETRIC AND OBLIQUE PROJECTION. Farish, Isometric Projection (Transactions, Cam- bridge Phil. Spc. J820) ; Sopwith, Treatise on Isometrical Drawing : Burmester, GrundzUge der sckiefen Parallelprojection ; Staudigl, Ax' onometrischen und schiefen Projection. ONE- PLANE DESCRIPTIVE GEOMETRY. Noizet, Memoire sur la Geometrie appliguee au dessin de la fortification ; Angel, Practical Plane Geometry and Projection; Eagles, De- scriptive Geometry, MAP PROJECTION. Germain, Traittf des Projections des Cartes Geographigues ; Craig, Treatise on Projections. I UNIVERSITY JTlllspntFt TheoreticalL graphics. mMQ^izjm i^-'.**.*'^ .*.'< UNIVERSITY OF CAUFORNIAUBRARY 1\^ k/\1 k«^ ^-vx^: ■?Nv^ Xi y« SSl^ >»'' ■ ! . , , >i