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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 â€” but too often does

not â€” 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 â€” illustrations, typog-

raphy and even the quality of the paper- â€” 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 â€” and, in particular, of projective â€”

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 â€” and usually called "practical" â€”

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 â€” an extension of Kennedy's scheme â€” 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 â€” 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. â€” Di-

visions of Projections. â€” Definitions and Appli-

cations of the Sciences Based on Central Pro-

jection, as Projective Geometry, Perspective,

Relief - Perspective, Sciography, Photogram-

metry. â€” 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.

â€” Remarks on the Nomenclature and Differ-

entiation of Geometries.

Pages 1-41

CHAPTER U.

Technical Free - Hand Sketching and Lettering. â€”

Note -Taking from Measurement. â€” Dimension-

ing. â€” Conventional Representations.

Pages 5-10.

CHAPTER til.

The Choice and Use of Drawing Instruments and

the Various Elements of the Draughtsman's

Equipment. â€” General remarks preliminary to

instrumental work.

Pages 11-20.

CHAPTER IV.

Kinds and Signification of Lines. â€” Designs for

Elementary Practice with the Right Line Pen. â€”

Standard Methods of Representing Materials. â€”

Line Shading. â€” Plane Problems of the Right

Line and Circle, including Rankine's and

Kochansky's approximations. â€” Exercises for the

Compass and Bow -pen, including uniform and

tapered curves. â€” The Anchor Ring. ^ The Hy-

perboloid. â€” A Standard Rail Section.

Pages 21 -38.

CHAPTER V. ,,^^

Regarding the Irregular Curve. â€” The Helix. â€”

The Ellipse, Hyperbola and Parabola, by various

methods of construction. â€” Homological Plane

Curves. â€” Relief- Perspective. â€” Link- Motion

Curves, â€” Centroids. â€” The Cycloid. â€” The

Companion to the Cycloid. â€” The Curtate and

Prolate Trochoids. â€” Hypo-, Epi-, and Peri-

Trochoids. â€” Special Trochoids, as the Ellipse,

Straight Line, Limagon, Cardioid, Trisectrix,

Involute and Spiral of Archimedes. â€” Parallel

Curves. â€” Conchoid. â€” Quadratrix. â€” Cissoid.

â€” Tractrix. â€” Witch of Agnesi. â€” Cartesian

Ovals â€” Cassian Ovals. â€” Catenary. â€” Logarith-

mic Spiral. â€” Hyperbolic Spiral. â€” Lituus. â€”

Ionic Volute.

Pages 39-78.

CHAPTER VI.

Brush Tinting, Flat and Graduated. â€” 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. â€” Mechanical Expedients.

â€” Proportioning of Titles. â€” Discussion of

Fonns. â€” Half- Block, Full Block and Railroad

Types. â€” Borders and how to draw them,

(Alphabets in Appendix).

Pages 88-96.

CHAPTER VIII.

ll il 1

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1 III i

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C 2 775 flflS

TM'

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i%:

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Â§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 â€” but too often does

not â€” 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 â€” illustrations, typog-

raphy and even the quality of the paper- â€” 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 â€” and, in particular, of projective â€”

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 â€” and usually called "practical" â€”

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 â€” an extension of Kennedy's scheme â€” 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 â€” 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. â€” Di-

visions of Projections. â€” Definitions and Appli-

cations of the Sciences Based on Central Pro-

jection, as Projective Geometry, Perspective,

Relief - Perspective, Sciography, Photogram-

metry. â€” 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.

â€” Remarks on the Nomenclature and Differ-

entiation of Geometries.

Pages 1-41

CHAPTER U.

Technical Free - Hand Sketching and Lettering. â€”

Note -Taking from Measurement. â€” Dimension-

ing. â€” Conventional Representations.

Pages 5-10.

CHAPTER til.

The Choice and Use of Drawing Instruments and

the Various Elements of the Draughtsman's

Equipment. â€” General remarks preliminary to

instrumental work.

Pages 11-20.

CHAPTER IV.

Kinds and Signification of Lines. â€” Designs for

Elementary Practice with the Right Line Pen. â€”

Standard Methods of Representing Materials. â€”

Line Shading. â€” Plane Problems of the Right

Line and Circle, including Rankine's and

Kochansky's approximations. â€” Exercises for the

Compass and Bow -pen, including uniform and

tapered curves. â€” The Anchor Ring. ^ The Hy-

perboloid. â€” A Standard Rail Section.

Pages 21 -38.

CHAPTER V. ,,^^

Regarding the Irregular Curve. â€” The Helix. â€”

The Ellipse, Hyperbola and Parabola, by various

methods of construction. â€” Homological Plane

Curves. â€” Relief- Perspective. â€” Link- Motion

Curves, â€” Centroids. â€” The Cycloid. â€” The

Companion to the Cycloid. â€” The Curtate and

Prolate Trochoids. â€” Hypo-, Epi-, and Peri-

Trochoids. â€” Special Trochoids, as the Ellipse,

Straight Line, Limagon, Cardioid, Trisectrix,

Involute and Spiral of Archimedes. â€” Parallel

Curves. â€” Conchoid. â€” Quadratrix. â€” Cissoid.

â€” Tractrix. â€” Witch of Agnesi. â€” Cartesian

Ovals â€” Cassian Ovals. â€” Catenary. â€” Logarith-

mic Spiral. â€” Hyperbolic Spiral. â€” Lituus. â€”

Ionic Volute.

Pages 39-78.

CHAPTER VI.

Brush Tinting, Flat and Graduated. â€” 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. â€” Mechanical Expedients.

â€” Proportioning of Titles. â€” Discussion of

Fonns. â€” Half- Block, Full Block and Railroad

Types. â€” Borders and how to draw them,

(Alphabets in Appendix).

Pages 88-96.

CHAPTER VIII.