LI BR AR Y 
 
 OF THE 
 
 UNIVERSITY OF CALIFORNIA. 
 
 GIRT OR 
 
 Received 
 
 Accessions NO._^__^J^^_ . SJulf fto'.._^__ 
 
THE 
 
 DRAWING GUIDE; 
 
 A MANUAL OF INSTRUCTION IN 
 
 INDUSTRIAL DRAWING, 
 
 DESIGNED TO ACCOMPANY THE 
 
 INDUSTRIAL DRAWING SERIES. 
 
 WITH AN INTRODUCTORY ARTICLE ON THE 
 
 PRINCIPLES AND PRACTICE OF ORNAMENTAL ART. 
 
 BY MARCIUS WILLSOST, 
 
 ATJTIIOE or "BCUOOL AND FAMILY BEBIES or READEJIS," "MANUAL or OBJECT LESSONS, 
 
 ETC., ETC. 
 
 UNIVERSITY 
 
 HARPER & BROTHERS, PUBLISHERS, 
 
 FRANKLIN SQUARE. 
 
 1881. 
 

 Entered according to Act of Congress, in the year 1873, by 
 
 HARPER & BROTHERS, 
 In the Office of the Librarian of Congress, at Washington. 
 
CONTENTS. 
 
 Preface Page v 
 
 PART I. ORNAMENTAL ART. 
 
 I. Introductory 13 
 
 II. General Principles of Ornamental Art 18 
 
 Prop. I. The Cardinal Principle in Decoration, 18: Prop. II. Of 
 Angular and Winding Forms, 19: Prop. III. Of Firm and Unbroken, 
 and Fine and Faint Lines, 20 : Prop. IV. Of Construction and Deco- 
 ration, 21 : Prop.V. Of General Forms, 22 : Prop. VI. Of Geometrical 
 Construction, 22: Prop. VII. Of Methods of Surface Decoration, 23: 
 Prop. VIII. Of Proportion in Ornamentation, 24 : Prop. IX. Of Har- 
 mony and Contrast, 25: Prop. X. Of Distribution, Radiation, and 
 Continuity, 26 : Prop. XI. Of Conventional Representations of Natu- 
 ral Objects, 27. 
 III. Ornamental Art among different Nations, and in different Periods of 
 
 Civilization 28 
 
 I. Ornament of Savage Tribes, 28 : II. Egyptian Ornament, 29 : 
 III. Assyrian and Persian Ornament, 30 : IV. Greek Ornament, 31 : 
 V. Pompeian Ornament, 32 : VI. Roman Ornament, 32 : VII. 
 Byzantine Ornament, 33 : VIII. Arabian Ornament, 34: IX. Turk- 
 ish Ornament, 35: X. Moresque or Moorish Ornament, 36: XI. 
 Persian Ornament, 37: XII. East Indian Ornament, 37: XIII. 
 Hindoo Ornament, 38 : XIV. Chinese Ornament, 38 : XV. Celtic 
 Ornament, 39 : XVI. Mediaeval or Gothic Ornament, 39 : XVII. 
 Renaissance Ornament, 41: XVIII. Elizabethan Ornament, 42: 
 Modern Ornamental Art, 42. 
 
 PART II. PRINCIPLES AND PRACTICE OF INDUSTRIAL 
 DRAWING. 
 
 DRAWING-BOOK No. I. 
 
 I. Materials and Directions 47 
 
 II. Straight Lines and Plane Surfaces 51 
 
 Horizontal and Vertical Lines, 51 : Angles and Plane Figures, 52: 
 Principles of Surface Measurement, 53 : Rules, 53-61 : Problems, 
 53 : Diagonals, 55 : Problems, 55 : Two-space Diagonals, 57 : 
 Problems, 59, 60 : Three -space Diagonals, 61 : Problems, 62: 
 Egyptian Patterns, 66, 67: Arabian, 67, 71, 72 : Byzantine, 67, 69 : 
 Pompeian, 67: Grecian, 68-72: East Indian, 69: Braided 
 Work, 68 -.Problems, 64, 66, 68. 
 
 III. Curved Lines and Plane Surfaces 73 
 
 Regular Curves, 73: Irregular Curves, 74: Symmetrical Figures, 
 75 : Problems, 76 : Conventional Leafage, 77 : Problems, 78 : 
 Renaissance Ornament, 78: Bulb Pattern, 79: Problems, 79: 
 Assyrian and Byzantine Patterns, 79-81: Problems, 81: Quarter- 
 foil, 81: Original Designs, 81. 
 
 DRAWING-BOOK No. II. 
 
 Cabinet Perspective Plane Solids 84 
 
 Diagonal Cabinet Perspective, 84 : Elementary Rule, 85 : Solid 
 Contents of a Cube Rule, 86 : Parallelepipeds Rule, 87 : Hatch- 
 ing, 88 : Problems, 89 : Stairs, 90 : Cabinet Frame-work, 91 : 
 
IV CONTENTS. 
 
 Problems, 92 : Scarfing, 92 : Problems, 91 : Brick-work, English 
 Bond and Flemish Bond, 94 : Problems, 90 : Divisions of the Cube, 
 96 : Solid Frets, 97 : Timber Framing, 97 -.Problems, 97 : Mould- 
 ings and Cabinet-work, 98 : Table, 99 : Problems, 100 : Irregular 
 Block Forms, 100: Problems, 102: Pyramidal Structures, 102: 
 Problems,l05 : Fence Frame-work, 1 05 : Problems, 10G : Post-and- 
 Kail Fence, 106 : Arabian Fret, Solid, 107: Problems, 108: Picket 
 Fence, Grecian Frets, Chest with Tray, 108, 109 : Problems, 109 : 
 Solid and Hollow Geometrical Block Forms, 109-111: Bridge-work, 
 112: Cubical Block Patterns, 112 -.Problems, 112. 
 
 DPvAWING-BOOK No. III. 
 
 Cabinet Perspective Curvilinear Solids 114 
 
 Cylinders, Solid, Hollow, and Divided, 114-116: Problems, 116 : 
 Semicircular Arches, 117: Problems, 118: Braces, Straight and 
 Curvilinear, 119 : Quarterfoil, 120 : Problems, 121 : Brackets, 121 : 
 Trefoil and Quarterfoil, 122 -.Problems, 123 -.Conventional Leaf 
 Pattern, 123: Solid Triangle, 124: Curvilinear and Quadrangular 
 Solids, 125: Architectural Band, 126: Problems, 126: Rims of 
 Wheels, 126: Beveled Tub, 127: Hollow Cylinders, 128: Irregular 
 Curved Solids, 128 : Problems, 129 : Curvilinear Frame-works, 129 : 
 Problems, 131 : Large Wheel, with Spokes, 131 -.Problem, 133 : 
 Large Wheel, with Spokes and Double Kim, 133 : Crown-wheel, 135 : 
 Ratchet Wheel, 136 : Windlass, with Spokes, 138. 
 
 DRAWING-BOOK No. IV. 
 
 Cabinet Perspective Miscellaneous Applications 1 43 
 
 I. Different Diagonal Views of Objects 143 
 
 II. Ground-Plans and Cabinet-Plans of Buildings 144 
 
 Problem, 146 : Series of Platform Structures, 147: Problem, 149. 
 
 III. Cylindrical Objects in Vertical Positions 149 
 
 I. Ellipses on Diagonal Bases 150 
 
 II. Ellipses on Rectangular Bases 152 
 
 Ellipses in Vertical Positions, 153 : Rule, 155 : Problems, 15 5: 
 Hollow Cylinders, 156 -.Problems, 159 : Horizontal Wheel with 
 Spokes, 161 : Vertical Tub with Twenty-four Uniform Staves, 162 : 
 Problems, 163: Crown-wheel, with Axis vertical, 164: Tub 
 beveling upward, 166: Beveled Octagonal Tub, 168. 
 
 III. Arches in Diagonal Perspective 1 69 
 
 IV. Semi-diagonal Cabinet Perspective 1 71-1 78 
 
 V. Shadows in Cabinet Perspective 179-187 
 
 APPENDIX. 
 ISOMETRICAL DRAWING. 
 
 I. Elementary Principles 189 
 
 II. Figures having Plane Angles 191 
 
 III. The Drawing of Isometrical Angles 194 
 
 IV. The Isometric Ellipse, and its Applications 197 
 
 V. Miscellaneous Applications 202 
 
 VI. Table for drawing Circles in Isometrical Perspective 205 
 
 Isometric Plates, I. to VIII. inclusive 207-221 
 
PREFACE. 
 
 Ix presenting to the public the first four numbers of TOE 
 INDUSTRIAL DRAWING SERIES, a few words of explanation 
 are needed. More than thirty years ago the undersigned 
 prepared a work on Perspective, Architectural, and Landscape 
 Drawing, for the use of an Institution with which he was 
 then connected ; but, as the work was designed for a local 
 purpose only, it has long been out of print. It is not, there- 
 fore, to the writer, a new subject which he has now taken 
 in hand, but the elaboration of an art which, from boyhood, 
 he has indulged in as a pastime, with constantly enlarging 
 views of its importance in the business of both a practical- 
 ly useful and disciplinary education. 
 
 A few years ago our attention was specially called to the 
 subject of Isometrical drawing, which had been brought for- 
 ward in England, and there highly recommended for the use 
 of mechanics, architects, etc., and for all purposes in which 
 working drawings are desirable. But the strict mathemat- 
 ical accuracy required in the guiding slope lines, which must 
 be drawn to a particular angle, and for the drawing of 
 which no means were suggested beyond ordinary pencil rul- 
 ing, placed this valuable method of representing objects be- 
 yond the reach of all except the most accurate draughtsmen, 
 and thus rendered it almost wholly useless for all practical 
 purposes, and especially for school uses. 
 
 This difficulty in the ruling, however, we were enabled to 
 overcome by the preparation of " Isometrical Drawing-Pa- 
 per," printed from stone in fine tinted lines accurately drawn 
 
VI PREFACE. 
 
 to the required angle. We then proceeded to prepare a 
 somewhat elaborate work on Isometrical Drawing, in which, 
 we have the assurance to believe, we were able to extend 
 and simplify the principles of the art ; but when the draw- 
 ings were all made, and the book was ready for the press, it 
 occurred to us that a still more easy system of industrial 
 drawing, more nearly approaching linear perspective in ap- 
 pearance, and equally practical with isometrical drawing, 
 might be invented ; and the result has been the system of 
 Cabinet Perspective, which is now offered to the public in 
 the Second, Third, and Fourth Numbers of the " Industrial 
 Drawing Series." If we are not greatly mistaken, this sys- 
 tem of Cabinet Perspective, which is so very simple in plan, 
 and so easy of execution as to render its more valuable 
 portions capable of being understood and practiced by the 
 children in our primary schools, will give to the subject of 
 industrial drawing, in its application to the representation of 
 solids of every variety of form, a value hitherto unknown. 
 
 While we regard it, however, as better for most industrial 
 drawings, especially for school purposes, than Isometrical 
 Perspective, yet the latter has some very valuable adapta- 
 tions ; and, as it can be easily applied by those who under- 
 stand Cabinet Perspective, we have given an exposition of 
 its principles in the Appendix to the present volume. 
 
 A peculiarity in the plan of the system now offered to the 
 public consists in placing the drawings which are to be im- 
 itated, or which are to serve as models for suggesting orig- 
 inal designs, on paper printed with fine lines one eighth of 
 an inch apart, and in furnishing the pupil with similarly 
 printed red or pink-lined paper on which to make his draw- 
 ings. These lines cross each other at right angles, vertically 
 and horizontally. Any draughtsman will see at a glance 
 with what facility and accuracy a figure may be copied from 
 the Drawing-Book on paper thus prepared ; how readily it 
 may be enlarged to any extent, or diminished, in true pro- 
 
PREFACE. Vll 
 
 portion ; and how easy it is, with the aid of such paper, to de- 
 sign new patterns and models, and draw them in perfect sym- 
 metry in all their parts. Draughtsmen are often obliged to 
 rule paper in a similar manner, for their own use, in making 
 intricate patterns ; and it is perfectly evident that the vast 
 variety of decorative designs which we find among the re- 
 mains of Egyptian, Grecian, Roman, Byzantine, and Arabian 
 art, was formed upon paper, or papyrus, ruled by pencil in 
 this identical manner, although not on the scale which we 
 have used. Indeed, these ancient patterns could not pos- 
 sibly have been executed with the accuracy which they ex- 
 hibit without such aid. They show the accurate direction 
 of the diagonal and other oblique lines, which are so easily 
 formed upon such ruling. For all purposes of illustrating 
 industrial art, the two kinds of ruled drawing-paper both 
 Cabinet and Isometrical will be found invaluable. Their 
 varied applications will be seen throughout the Industrial 
 Drawing Series. 
 
 In Drawing-Book No. I. we have taken up, in an element- 
 ary manner, the subject of Decorative Design both on ac- 
 count of its being well adapted to elementary practice in 
 drawing, and because of its importance in nearly all depart- 
 ments of industrial art. 
 
 In our drawing-lessons under this head, we have aimed, 
 in the first place, to furnish a variety of such copies as are 
 most suitable for elementary exercises in training the hand 
 and the eye, while at the same time they shall be adapted to 
 cultivate a correct taste for that which combines harmony of 
 design with grace and beauty of form. Hence, instead of 
 thinking it desirable that we should originate all of our 
 figures for the drawing exercises, we have selected them, in 
 great part, from the best examples of the decorative art of 
 all ages, being parts, or wholes, of patterns which have stood 
 the test of time, the only true standard of taste. By this 
 course we are not only able to give a very great variety of 
 
Vlll PREFACE. 
 
 excellent patterns for imitation, and for suggestion in de- 
 signing, but we are also enabled to impart to the pupil some 
 general knowledge of those principles of form and propor- 
 tion which govern all true art decoration ; and in the intro- 
 ductory articles we have given brief sketches of the growth 
 and development of these principles in different nations and 
 in different periods of civilization. Should the Series be car- 
 ried so far as we now anticipate, we hope, in higher numbers, 
 to greatly enlarge upon the designs here given ; to show the 
 application of industrial drawing to various specific forms of 
 industry ; and also to illustrate the Harmonies of Color, as 
 applied to decorative art. 
 
 But we would, furthermore, call special attention to the 
 new method of representing objects, called CABINET PER- 
 SPECTIVE, as illustrated in Drawing-Books Numbers II., III., 
 and IV., and embracing both plane -and curvilinear solids in 
 almost every variety of form and position. This kind of per- 
 spective, when carried out by the use of the ruled drawing- 
 paper, enables us to construct all kinds of working drawings 
 for artisans drawings which, instead of giving a geometrical 
 representation of but one side of a rectilinear object, present 
 in one view three sides, at the same time avoiding the appear- 
 ance of distortion, and giving, with perfect accuracy, the same 
 as Isometrical Perspective, the dimensions of the objects rep- 
 resented, according to whatever scale the draughtsman may 
 adopt. Moreover, the principles of the system are so simple 
 that a child can understand them ; while any one who can 
 draw straight lines by the aid of a ruler, and curved lines 
 by the aid of a pair of compasses, can apply them. 
 
 As indicating something of the scope of the system, as ap- 
 plied to solids, we have represented, under this head, within 
 the narrow limits which we have assigned to ourselves, such 
 objects as cabinet frame-works of various forms; tables; cu- 
 bical, hexagonal, octagonal, and other blocks, either entire, 
 or variously cut and divided; crosses; star figures; boxes; 
 
PREFACE, x 
 
 English bond and Flemish bond forms of brick-laying ; pil- 
 lars and their mouldings ; pyramids, obelisks, etc. ; post and 
 board, post and rail, and picket fences ; various forms of the 
 solid Grecian fret, and other architectural ornaments ; frame- 
 work of bridges ; cylinders, solid or hollow, entire, or various- 
 ly cut and divided, and in both horizontal and vertical po- 
 sitions ; arches, both pointed and semicircular ; braces and 
 brackets, both plain and curvilinear; solid quarterfoils and 
 trefoils ; wheels, in sections, and entire with crown-wheel, 
 ratchet-wheel, chain-pulley wheel, etc. ; windlass ; vertical 
 and beveled tubs, both circular and octagonal ; ground-plans 
 and elevations of buildings ; tenon and mortise work ; scarf- 
 jointing of timbers; stairways; platforms; ellipses; rings, etc., 
 etc., and all drawn to definite dimensions, while the measure- 
 ments are indicated by the drawing-paper itself. By this 
 system the study of drawing, in its application to the indus- 
 trial arts, is rendered one of the exact sciences, wholly me- 
 chanical in execution, and as accurate in its delineations as 
 geometry itself. We have here presented only an elementary 
 exposition of the system, designed for school purposes ; but 
 the system itself is so simple, that, with the helps here given, 
 the intelligent teacher and pupil will find little difficulty in 
 carrying out the application of its principles to any extent 
 which may be desired. 
 
 For several of the rules and principles of Ornamental Art, 
 and also for many of the designs in Drawing-Book Xo. I., 
 we are indebted to the " Grammar of Ornament" by Owen 
 Jones. It may, perhaps, be thought that it was not especial- 
 ly desirable to preface an Elementary Drawing Series with 
 a statement of the general principles of Art Decoration, and 
 an account of the Leading Schools or Periods of Art, for the 
 reason that such information will seldom be appreciated by 
 beginners in drawing. But to teachers at least and not 
 merely teachers of drawing we may hope that these intro- 
 ductory pages will be of some value; and if they shall serve 
 
 A2 
 
X PREFACE. 
 
 merely to enlarge the ideas of both teachers and pupils as to 
 the magnitude and importance of the subject of art repre- 
 sentation, they will thereby have done a good service to the 
 cause of education. 
 
 We would take this occasion to impress upon educators, 
 and those who have the management of our Public Schools, 
 the extreme desirability that all the school instruction in ele- 
 mentary industrial drawing shall be given by the ordinary 
 teachers ; and that professional drawing-masters shall be em- 
 ployed, if at all, only in the training of the teachers them- 
 selvesin a general superintendence of the whole subject of 
 art instruction in all the schools of a city, or county, or even 
 larger district or in giving instruction to advanced stu- 
 dents in the higher Schools of Design. The teachers in our 
 Public Schools are competent to give all the instruction re- 
 quired by their classes in industrial drawing ; and care should 
 be taken that pupils do not get the idea that they are re- 
 quired to do something which their teachers themselves can 
 not do. 
 
 MARCIUS WILLSON. 
 
 VINELAND, N. J., June 5, 1873. 
 
PART I. 
 
 PRINCIPLES AND PRACTICE 
 
 OP 
 
 ORNAMENTAL ART. 
 
L ^StomifiK 
 
 INTRODUCTORY. 
 
 WE desire to offer to the public a few introductory re* 
 marks on Ornamental Art, a subject which we have en- 
 deavored to illustrate, in a very elementary manner, in the 
 first book of our Industrial Drawing Series. 
 
 We are aware that those who have given the subject but 
 little attention entertain very erroneous ideas of the im- 
 portance and value of a knowledge of the principles and 
 practice of decoration, as applied to the products of human 
 industry. A very little reflection, however, must convince 
 the most utilitarian, that, in an advanced stage of society, 
 decoration enters so fully into all works of art as to consti- 
 tute, in perhaps a majority of cases, the greater part of 
 their market value. We see the principle illustrated in the 
 importance that is attached to surface ornamentation in the 
 manufacture of carpets, and oil-cloths, and matting, and 
 wall-paper, and curtains; in printed cloths, and other arti- 
 cles designed for dress ; in crochet and tapestry work ; in 
 the elegant forms required for vases, and all crockery and 
 earthenware ; alike in the fine sculpture of the most delicate 
 ornaments and the chiseling of stone for public and private 
 dwellings; in all mouldings of wood, and iron, and other 
 ornamental work in architecture ; and it is found to enter 
 into all plans and patterns of utensils and tools, and into 
 all objects of art which may be deemed capable of improve- 
 ment by giving to them increased beauty of form and pro- 
 portion. Indeed, all the vast variety of form and color 
 which we observe in the works of man, beyond the require- 
 ments of the most barren utility, is, simply, ornamentation. 
 Beginning with the savage, with whom ornament precedes 
 dress, it has been the study of man in all ages not only to 
 make art beautiful, but to improve upon nature also. The 
 
14 INDUSTRIAL DRAWING. 
 
 subject is thus seen to embrace all that, in industrial art, 
 marks the advance of civilization ; and decoration may be 
 taken as a true exponent, in every stage of its development, 
 of the progress of society ; for the comforts and the elegan- 
 cies of life are ever found to grow together. 
 
 Inasmuch, therefore, as ornamentation enters so largely 
 into the daily life of civilized society as to be every where 
 recognized, studied, admired, and practiced, it would seem 
 not only appropriate, but very desirable, that its elementary 
 principles, at least, should find a place at the beginning of 
 every system of public instruction and, where they prop- 
 erly belong, in the study and practice of Industrial Drawing. 
 
 England is so decidedly a manufacturing country, that 
 art education has there long been deemed a national neces- 
 sity : and it is not only thought important that the manufac- 
 turer should understand the laws of beauty, and the princi- 
 ples of design, in order that his products may command a 
 ready market, but that the artisan also the mere workman 
 in art shall possess something of the skill which comes 
 from educated taste. More than thirty years ago a British 
 Association for the Advancement of Art, composed of the 
 chief nobility, capitalists, bankers, merchants, and manufac- 
 turers of the kingdom, sent out the declaration and appeal, 
 that, without a pre-eminence in the arts of design, British 
 manufacturers could not retain, and must eventually lose, 
 their superiority in foreign markets. But the English gov- 
 ernment remained, for years, deaf to the warning ; and at 
 the great Exhibition of the Industry of all Nations, held in 
 London in 1851, England found herself almost at the bot- 
 tom of the list in respect to excellence of design in her 
 art manufactures only the United States, among the great 
 nations, being below her. This discovery aroused the En- 
 glish government to a realizing sense of the vast importance 
 of the highest and most widely diffused art education for a 
 manufacturing people ; and the result was the speedy estab- 
 lishment of an Educational Department of Science and Art, 
 from which Schools of Design have radiated all over the 
 country. In these and other schools, even ten years ago, 
 two thousand students were in training as future teachers 
 
ORNAMENTAL AKT. 15 
 
 of art, and fifteen thousand pupils were receiving an art 
 education ; while in the parish and public schools more than 
 fifty thousand children of the laboring and poorer classes 
 were receiving more or less instruction in elementary draw- 
 ing. In the higher art schools the pupils are taught not 
 only the practice, but the principles also, of ornamental de- 
 sign ; they are shown how all assemblages of ornamental 
 forms are arranged in geometrical proportions : how curves 
 must flow, the one into the other, without break or interrup- 
 tion ; and they are taught to analyze and interpret the char- 
 acteristic ideas of various styles and schools of art, such as 
 we have given a brief synopsis of under the heading of 
 " Ornamental Art among Different Nations, and in Different 
 Periods of Civilization." The wisdom of England's course 
 was very apparent at the Paris Exhibition of 1867, when 
 it was seen that England had risen, in a period of six- 
 teen years, from a position among the lowest, to one fore- 
 most among the nations in art manufactures showing the 
 effects of the art education which she had so sedulously fos- 
 tered. As humiliating as it is to our national pride, truth 
 compels us to add, in the language of another " The United 
 States still held her place at the foot of the column." In 
 England, in 1870, besides the attention given to drawing in 
 the public schools and in evening classes, there were more 
 than twenty thousand students in the art schools, and more 
 than thirty thousand in the schools of industrial science ; 
 and it is reported that, in the two following years, the num- 
 bers in both were doubled. 
 
 A notable illustration of the commercial value of the 
 beautiful in art is afforded in the colossal growth of the 
 earthenware trade in England, which started into sudden 
 notoriety when the young sculptor, Flaxman, was employed 
 to model, from fine specimens of antique sculpture, those 
 beautiful urns, vases, goblets, and other articles for table 
 service and other domestic uses, long known as the Wedge- 
 wood ware. The clay pits of Staffordshire were turned into 
 gold mines, and made a source of national wealth, when the 
 proprietors employed good artists to draw designs and se- 
 lect antique models for their workmen; and it has been 
 
16 INDUSTRIAL DBAWIXG. 
 
 stated by competent judges that, through the establishment 
 of Art Museums and Schools of Design, and the influences 
 exerted thereby, combined with popular instruction in in- 
 dustrial drawing, England has added fifty per cent, to the 
 value of her manufactured products during the past twenty- 
 five years. And, turning to the Continent, we find it is the 
 art instruction imparted in the schools and in the manufac- 
 tories of France, showing how colors are distributed, bal- 
 anced, and harmonized, both in nature and in art, that has 
 given to the silk fabrics of Lyons, the Gobelin tapestry, and 
 to other national products, their world-wide renown for har- 
 mony and beauty. In France, education in science and art 
 is now placed by law in the same rank as classical education. 
 In our own country public attention is now being turned, 
 in a very marked manner, to the subject of art education : 
 and in Massachusetts, after the subject had been agitated 
 by the leading manufacturers and merchants, laws have 
 been passed securing to pupils instruction in elementary 
 drawing in every public school in the state ; making u in- 
 dustrial or mechanical drawing" free to persons over fif- 
 teen years of age either in day or evening classes, in cities 
 and towns that have a population above ten thousand ; and 
 a State Director of Art Education has been appointed to 
 supervise the system; but, generally, throughout our schools, 
 what little imperfect instruction in art has been given has 
 thus far been confined, mostly, to the mere copying of pic- 
 tures and, where it has gone beyond that, to the education 
 of artists rather than of artisans. It is seldom addressed, 
 as it should be, to the principles and practice of ornamental 
 design ; to the harmonies of color, form, and proportion ; 
 and to such representations of objects as are most needed 
 by workingmen in the arts. This kind of art knowledge and 
 practice would not only be of interest, but of utility to all ; 
 and the mechanic who could make the best use of it in his 
 line of business would ever have a decided advantage over 
 
 O 
 
 all competitors. An incident bearing upon this point is re- 
 lated by the State Director of Art Education for Massa- 
 chusetts, to the effect that, " some years ago a class of thir- 
 teen young men spent all their leisure time in studying 
 
OEXAMENTAL AKT. 17 
 
 drawing, and that now, at the time of writing, every one of 
 them holds some important position, either as manufacturer 
 or designer." And if we would build up our manufactories 
 on a broad scale, so as to bring their products into success- 
 ful competition with those of England and France, we must 
 not rely upon a few imported draughtsmen and designers, 
 and vainly hope that uneducated artisans will work out 
 foreign patterns with taste and beauty; but we must lay 
 the foundations of art superiority broad and deep in the art 
 education of all mechanics, and in the educated tastes of the 
 people. Then draughtsmen and designers will spring up 
 wherever needed ; and the workmen in our shops and manu- 
 factories, understanding the principles of their several trades 
 and professions, will be all the more skilled in the practice 
 of them. And what we need for this is not merely a few 
 Schools of Design, and Art Museums, valuable as these may 
 be, but the introduction of the principles of design, and the 
 practice of art representation, into the education of the 
 people at large. 
 
 But here the practical question is suggested : How shall 
 we introduce Industrial Drawing into our schools, so that 
 all our youth may profit by it, when so many other impor- 
 tant studies are crowding for admission, and our teachers 
 have already quite as much as they can attend to? We 
 reply, Alternate it with the writing -lessons; and experi- 
 ence fully proves that better penmanship will be attained 
 thereby, while the drawing, and the knowledge which it 
 introduces, will be a positive gain, without any attendant 
 loss. Long ago, said that veteran educator, Horace Mann, 
 " I believe a child will learn both to draw and write sooner, 
 and with more ease, than he will learn writing alone." 
 
 In conclusion, w r e commend this w r hole subject of Indus- 
 trial Art Education as worthy the earnest consideration, 
 not only of all educators, but also of all mechanics and ar- 
 tisans, and of all who appreciate the vast proportions which 
 our manufacturing interests are destined to assume in the 
 not far distant future. 
 
II. 
 
 GENERAL PRINCIPLES OF ORNAMENTAL ART. 
 
 THERE are two kinds of beauty in Ornamental Art : the 
 one is the beauty of design and execution, arising from the 
 exhibition of skill on the part of the designer and artisan ; 
 the other is the beauty of character, which arises from the 
 expression of thought or soul in the object itself. The 
 beauty of the former is fully realized only by those who 
 are proficients in the art, and ceases to be felt when the art 
 has made a farther progress. The beauty of the latter, in- 
 asmuch as it appeals to the sensibilities of all, is universally 
 felt, although in a different degree by different individuals, 
 and is by far the most lasting ; and the former should ever 
 be subordinate to it. The difference between the two kinds 
 of beauty is best illustrated in architecture, of which orna- 
 ment is the very soul and spirit. All that utility requires 
 in the structure, skill may accomplish by the aid of mere 
 rule and compass ; but the ornamentation shows how far the 
 architect was, at the same time, an artist. 
 
 PROPOSITION I. A CARDINAL PRINCIPLE. 
 All decoration should exhibit a fitness or propriety of 
 things, just proportions, and harmony of design. 
 
 All ornaments should harmonize in expression with the 
 expression designed to be given to the objects to which 
 they are affixed. Thus there are art objects of convenience 
 and use, of sublimity, of splendor, of magnificence, of gayety, 
 of delicacy, of melancholy, etc. ; and the ornaments affixed 
 to each should fully harmonize with its character. Any 
 fabric to be ornamented should, in the first place, be suited 
 to its proposed uses ; and then, in strict keeping with the 
 main design, must be the decoration which adorns its sur- 
 face. Hence, to cover an oil-cloth, or a chair cushion, with 
 
ORNAMENTAL AET. 19 
 
 drawings of cubical blocks set on edge, as we have seen, is 
 an outrage upon the uses to which either is to be put ; and 
 alike improper is it to load a carpet, designed for the tread 
 of feet, with vases filled with fruits, and to cover it thick 
 with garlands of flowers. It is only in the richest velvet 
 carpets, elastic to the tread, and where the flowers are par- 
 tially lost in the profusion of herbage, that such excessive 
 adornment may be deemed not inappropriate. As is well 
 known, the Greek orders of architecture have manifest dif- 
 ferences of character or expression. Thus the heavy Tus- 
 can is distinguished by its severity; the manly Doric by its 
 simplicity, purity, and grandeur ; the Ionic by its grace and 
 elegance; the Corinthian by its lightness, delicacy, and 
 gayety; and the Composite by its profusion and luxury; 
 while the ornaments of the several orders fully harmonize 
 with them in expression. Thus every product of art has 
 some character of its own, and good taste demands that 
 there shall be a correspondence in the decoration given to 
 it. A degree of ornamentation that would be becoming in 
 one object, would be insipid or mean in another; as what 
 would be in good taste, and beautiful, in the robes of a 
 queen, would be inappropriate in the dress of a plain lady, 
 and tawdry in that of a peasant girl. And although wreaths 
 of flowers may alike deck the tomb and adorn the festive 
 hall, yet the variety and profusion suited to the latter would 
 not comport with the subdued feeling which is in unison 
 with the former; and the true artist will at once discern 
 the difference. The vase upon a tomb will not bear the va- 
 riety of contour that may be given to a goblet ; nor should 
 the latter have the uniformity of moulding characteristic of 
 a funeral urn. 
 
 PROPOSITION II. OF ANGULAR AND WINDING FORMS. 
 
 Angular forms denote harshness, maturity, strength, and 
 vigor. Winding forms, on the contrary, are expressive of 
 infancy, weakness, tenderness, and delicacy, as also of ease, . 
 grace, beauty, luxury, and freedom from force and restraint. 
 
 As in all objects of taste the lightest forms consistent 
 with the required strength are considered the most beauti- 
 
20 INDUSTRIAL DRAWING. 
 
 fill, so in all articles in which much strength is required 
 angular forms are generally adopted, because they require 
 a less amount of material than curvilinear forms. Hence 
 angular forms as of squares, lozenges, etc. are not only 
 best suited to such articles of furniture as chairs, tables, 
 desks, stands, etc., but also to oil-cloths, matting, plain car- 
 pets, etc., because we associate with these latter articles 
 much tread of feet and daily use ; and yet it is equally ap- 
 parent that these angular forms would not be appropriate 
 for carpets of luxurious ease, for flowing robes, curtains, 
 etc. In architecture we expect direct and angular lines, 
 because they give the impression of stability and strength ; 
 and architectural ornaments are beautiful only as they are 
 in harmony with the general character of the structure to 
 which they are affixed. An angular vase, designed for 
 holding flowers, would be exceedingly inappropriate ; while, 
 on the contrary, to make the sides of a house, or of a pyra- 
 mid, curvilinear, would none the less violate our ideas of 
 [fitness and propriety. The weeping willow, as it is appro- 
 priately named, is adapted to mournful occasions, because 
 | it bends and droops like one in affliction ; while the sturdy 
 j.oak, on the contrary, of angular outlines, is representative 
 Vof firmness and strength. It may break, but can not bend. 
 
 PROPOSITION III. OF FIRM AND UNBROKEN, AND FINE AND 
 FAINT LINES. 
 
 Firm and unbroken lines are expressive of strength and 
 boldness, with some degree of harshness. 
 
 Fine and faint lines are indicative of smoothness, fine- 
 ness, delicacy, and ease. 
 
 When the forms of objects are used to ornament articles 
 of taste or utility, they should be drawn in keeping with 
 the character of the objects themselves. Thus the visual 
 line of a column, or of a pyramid, should be bold and un- 
 broken, unless modified by distance* of view; while the 
 winding outlines of the tendrils of a vine, of a wreath, of a 
 festoon, should be exceedingly \delicate, as we say our very 
 language conforming to our ideas of the fitness of things. 
 But see Proposition XI. 
 
ORNAMENTAL ART. 21 
 
 PROPOSITION IV. OF CONSTRUCTION AND DECORATION. 
 
 Construction should be decorated ; bat decoration should 
 never be, purposely constructed. 
 
 In the weaving of lace, muslin, and other fabrics of one 
 color, in a variety of suitable patterns, and in the similar 
 braiding of mats, baskets, etc., the construction itself is, ap- 
 propriately, decorated. So may any construction as a 
 building, a robe, an article of furniture, etc. be decorated in 
 the making of it ; but to construct or plan a decoration with- 
 out regard to the application or use that is to be made of 
 it, and as if it might serve a variety of purposes, is a viola- 
 tion of the principles of true art. It is, therefore, the cor- 
 rect principle, to make the construction itself ornamental, 
 rather than to depend upon applied ornament. Hence the 
 veneering of the fronts of brick or wooden buildings with 
 marble, or articles of wooden furniture with thin layers of 
 richer wood, is a sham that gives us a feeling of disappoint- 
 ment when the cheat is known. So the painting or grain- 
 ing of wood is far less satisfactory, as a decorative agent, 
 than the bringing out and preservation of the natural grain 
 by a suitable varnish. Artistic arrangements of American 
 woods properly prepared would furnish a wonderful variety, 
 in pattern and coloring, for decorative purposes, and in far 
 better taste than most of the surface decoration that is pur- 
 posely constructed. 
 
 Every object of art production is supposed to be con- 
 structed with some definite aim, and to be designed to sub- 
 serve some purpose of utility ; or, if it be merely ornament- 
 al, it is still designed to aid in giving the true and proper 
 expression to that object to which it is affixed. In either 
 case, the style, character, and expression of the ornamental 
 are to be considered as the accessories, and to be governed 
 wholly by the character of the object of which they are 
 the appendages. A carpet, a dress, a curtain, or a chair, 
 etc., should be ornamented with reference to the circum- 
 stances and occasions of its uses ; and, evidently, it must 
 vary in decoration according as it may be designed for a 
 cottage or for a palace. So mere ornaments, as rings, brace' 
 
22 INDUSTRIAL DRAWING. 
 
 lets, brooches, etc., should be adapted to the character, per- 
 sonal appearance, and position in society of the wearer; 
 for, not all beautiful things are becoming to all places, or 
 to all persons. The proprieties of life have a very wide 
 range of application. See Proverbs xi., 22. 
 
 PROPOSITION V. OF GENERAL FORMS. 
 
 True beauty of form is produced by lines growing out of 
 one another in gradual undulations, and supported by one 
 another. There are no excrescences ; and nothing could be 
 removed and leave the design equally good or better. 
 
 These principles are best illustrated in the several orders 
 of Grecian architecture, from no one of which could any 
 portion be taken away without leaving the general form 
 defective ; and certainly no part could be enlarged without 
 giving to it the appearance of an unseemly excrescence. 
 
 PROPOSITION VI. OF GEOMETRICAL CONSTRUCTION. 
 
 All surface ornamentation should be based upon a geo- 
 metrical construction. 
 
 Whatever the pattern of the ornament, it should be 
 such that it can be traced back to a geometrical basis ; and 
 no ornament can be properly designed without such aid as a 
 groundwork. Especially is this the case in woven fabrics, 
 which are necessarily constructed on a geometrical plan. 
 
 As in the infancy of art uniformity of design was most 
 valued, as evincing the skill of the artist; and as what chil- 
 dren most admire, and, in their little attempts at art, first 
 try to execute, is uniformity and regularity, so elementary 
 drawing should begin with those simple geometrical pat- 
 terns which are the groundwork of all artistic ornamentation. 
 
 Patterns in which the geometrical arrangement is at 
 once apparent, owing to the uniformity or regularity of 
 the details, owe the first impression of beauty which they 
 give us to their expression of design on the part of the art- 
 ist; and the more intricate the pattern, and the greater 
 the number of its parts, while it still preserves its uniform- 
 ity, the higher, in the estimation of educated taste, is its de- 
 gree of beauty ; only the number of parts must not be so 
 
ORNAMENTAL ART. 23 
 
 great as to produce confusion, and thus obscure the expres- 
 sion of design. Where, however, a confused intricacy of 
 detail at first seems to prevail, nothing is more delightful 
 than to find order gradually emerging out of chaos, and a 
 consistent plan pervading the whole. When there is add- 
 ed to a beautiful design intricacy and variety of detail 
 amid uniformity, there is only needed elegance and embel- 
 lishment in the workmanship to constitute the highest de- 
 gree of ornamental art. 
 
 PROPOSITION VII. OF METHODS OF SURFACE DECORATION. 
 
 The general forms of the desired ornamentation having 
 been first drawn on some geometrical basis, consistent with 
 the character of the object to be ornamented, these forms 
 should then be subdivided and ornamented by general lines ; 
 the intermediate spaces may then be filed in ; and the sub- 
 division may be continued to any extent required, and until 
 the details can be appreciated only by close inspection. 
 
 This method of designing is adapted no less to the some- 
 times elaborate patterns of embroidered robes and tapestry 
 work, than to the more obvious geometrical arrangements 
 of squares, and parallelograms, and lozenges, and circles, 
 that are often seen in oil-cloths and carpeting. The great 
 secret of success, even in the most complicated ornamenta- 
 tion, is the production of a broad general effect by the rep- 
 etition of a few simple elements. u Variety should rather 
 be sought in the arrangement of the several portions of a 
 design, than in the multiplicity of varied forms." 
 
 In the wall or floor ornamentation of dwellings, an im- 
 portant principle to be observed is the use of modest tints 
 as a back-ground, against which the furniture can be dis- 
 played to advantage, and a due subordination to the uses 
 to which the room is to be applied as, for example, wheth- 
 er it is to express the brightness, cheerfulness, and welcome 
 of a reception-room, or the tranquillity of studious ease 
 which is adapted to the library. If to the walls be given 
 high colors, relief and roundness of ornamentation, and 
 shade and shadow, instead of flat neutral tints of one or 
 two colors, the walls are thereby apparently thrust for- 
 
24 INDUSTRIAL DRAWING. 
 
 ward, the room is made to appear smaller than it is, and 
 the furniture is dwarfed, and its natural effect destroyed. 
 So, large patterns in the carpet of a small room produce a 
 like damaging effect upon both room and furniture, and 
 destroy that feeling of satisfied repose which is ever at- 
 tendant upon true art. Vertical patterns on the walls, 
 such as columns, stripes, etc., make the walls appear higher, 
 while horizontal lines and patterns lower the ceiling. 
 
 PROPOSITION VIII. OF PROPORTION IN ORNAMENTATION. 
 As in every perfect tcorJc of Architecture a true propor- 
 tion will be found to reign between all the members which 
 compose it, so throughout the decorative Arts every assem- 
 blage of forms should be arranged on certain definite pro- 
 portions / the whole and each particular member should be 
 a multiple of some simple unit. 
 
 Thus the height of the Doric column was e'qual to six 
 times the diameter of the lower end of the shaft ; the di- 
 ameter of the upper end of the shaft was three fourths of 
 the diameter of the lower end ; and the architrave, frieze, 
 and cornice, and all other parts, had certain definite pro- 
 portions. In the other orders the proportions were differ- 
 ent ; but in each the several parts were in just proportion 
 the one to the other, and to the whole. Nor were these 
 proportions arbitrary; for they were such as were best 
 adapted to give expression to the character of the order; 
 and in no one of these orders could any important part be 
 materially changed in its proportions without doing vio- 
 lence to that harmony of design which characterized the 
 entire structure. 
 
 In the infancy of Decorative Art the proportions were 
 of the most simple kind, in accordance with the natural or- 
 der of development. It is the same with the growth of 
 art in individuals. Thus as soon as a child can draw a 
 square, its first effort is to divide it into four squares, and 
 then into a greater number; then to draw and subdivide 
 parallelograms ; then out of the squares to form lozenge- 
 shaped figures, etc., and so on, as taste and skill are devel- 
 oped. As art advances, those proportions will be deemed 
 
or.:; A MENTAL ACT. 25 
 
 the most beautiful which the uneducated eye does net 
 readily detect. 
 
 PROPOSITION IX. OF HARMONY AND CONTRAST. 
 Where great variety of form is introduced, harmony con- 
 sists in the proper balancing and contrast of the straight, 
 the inclined, and the curved. 
 
 Whether we confine our attention to structural arrange- 
 ment of edifices, or to decoration of surfaces, there can be 
 no perfect composition in which any one of the three pri- 
 mary forms is wanting. In the Greek temple the straight, 
 the angular or inclined, and the curved, are in most perfect 
 relation to one another. In the best examples of Gothic 
 architecture every tendency of lines to run vertically or 
 horizontally is immediately counteracted by the oblique or 
 the curved. Thus the capping of the buttress is exactly 
 what is required to counteract the upward tendency of the 
 
26 INDUSTRIAL DRAWING. 
 
 straight lines ; so the gable contrasts admirably with the 
 curved window-head and its perpendicular mullions. 
 
 In surface decoration any arrangement of forms, as at A, 
 consisting only of straight lines, is monotonous, and affords 
 but little pleasure. By introducing lines which tend to 
 carry the eye toward the angles, as at B, the monotony is 
 broken, and the improvement is very apparent. Then add 
 lines giving a circular tendency, as at C, and the eye reposes 
 itself within the outlines of the figure, and the harmony is 
 complete. In this case the square is the leading form or 
 tonic ; the oblique and curved forms are subordinate. 
 
 An effect similar to A, but an improvement upon it, is 
 produced by the lozenge composition, as at D. Add the 
 lines as at E, and the tendency to follow the oblique di- 
 rection is corrected ; but interpose the circles, as at F, and 
 the eye at once feels that repose which is the result of per- 
 fect harmony in the combination. 
 
 It is owing to a neglect of the principle here stated that 
 there are so many failures in wall-paper, carpets, oil-cloths, 
 and articles of clothing. The lines of wall-paper very gen- 
 erally run through the ceiling most disagreeably, because 
 the vertical is not corrected by the inclined, nor the in- 
 clined by the curved. So of carpets, the lines of which fre- 
 quently run in one direction only, carrying the eye right 
 through the walls of the apartment. Many of the checks 
 and plaids in common use are objectionable for the same 
 reason, although a great relief is sometimes found in their 
 coloring. 
 
 PROPOSITION X. OF DISTRIBUTION, RADIATION, AND CON- 
 TINUITY. 
 
 In surface decoration by curvilinear forms, all lines should 
 be harmoniously distributed, and should radiate from a par- 
 rent stem and all junctions of curved lines with curved, and 
 of curved lines with straight, should be tangential to one an- 
 other. 
 
 This is a law of the vegetable world, as seen in all plants 
 that have curvilinear forms; and Oriental practice in orna- 
 mental art is in accordance with it. 
 
ORNAMENTAL ART. 27 
 
 PROPOSITION XL OF CONVENTIONAL REPRESENTATIONS 
 OF NATURAL OBJECTS. 
 
 Flowers, or other natural objects, should not be used as or- 
 naments, but, instead thereof, we should use conventional rep- 
 resentations founded upon them, sufficiently suggestive to 
 convey the intended image to the mind without destroying 
 the unity of the object they are employed to decorate. The 
 former is called the NATURALISTIC style of ornamentation, 
 the latter the CONVENTIONAL style. 
 
 Although this rule has been universally obeyed in what 
 are deemed the classic periods of art, it has been equally 
 violated when art has been on the decline. 
 
 A fragile flower, or a delicate vine, carved in wood, stone, 
 or iron, shocks our feeling of consistency an impropriety 
 of which the Egyptians, the Greeks, and the early Romans 
 were never guilty. They made conventional representa- 
 tions of natural objects, strictly adhering to their general 
 laws of form ; and hence their ornaments, however conven- 
 tionalized (but more especially those of the ^Egyptians), 
 were always true to nature, while they never, by a too 
 servile imitation of the type, destroyed the consistency of 
 the representation. 
 
 When flowers in miniature are carved upon precious 
 stones, or even in iron, the delicacy of the workmanship may 
 overcome our sense of the unfitness of things. The flower, 
 leaf, vine, and fruit ornaments on vases and fruit-dishes are 
 certainly not beautiful except when of diminished size; 
 and even then, if carved, they should be executed in slight 
 relief, or merely etched in outline. 
 
 In contradistinction, however, to the use of color and 
 form as mere accessories in industrial art, when we come to 
 the fine art of painting, and employ it to give a representa- 
 tion of real objects or scenes in nature, or of those which 
 fancy creates, it is the naturalistic method which should 
 prevail ; for here the leading idea is a faithful portraiture 
 of what is seen or imagined ; and all other ideas must be 
 subordinate to it. Here conventionality of representation 
 would defeat the very object in view. 
 
III. 
 
 ORNAMENTAL ART AMONG DIFFERENT NA- 
 TIONS, AND IN DIFFERENT PERIODS OF CIV- 
 ILIZATION. 
 
 I. ORNAMENT OF SAVAGE TRIBES. 
 
 THE desire for ornament is universal, and it increases 
 with all people in the ratio of their progress in civilization. 
 Every where it owes its origin to man's ambition to create 
 to imitate the works of the Creator. In the tattooing of 
 the human face the savage strives to increase the expres- 
 sion by which he hopes to strike terror on his enemies or 
 rivals, or to create what appears to him a new beauty ; 
 and it is often surprising how admirably adapted are the 
 forms and colors he uses to the purposes he has in view. 
 After tattooing usually comes the formation of ornament 
 by painting or stamping patterns on the skins used for 
 clothing, or on woven cloths or braided matting. Then 
 follows the carving of ornaments on their utensils or weap- 
 ons of war. When the principal island of the Friendly 
 group was first visited, one woman was found to be the 
 designer of all the patterns on cloths, matting, etc., in use 
 there ; and for every new one, she received, as a reward, a 
 certain number of yards of cloth. 
 
 What strikes us especially in most ornamental work of 
 savages, is the adherence to that rule of art which requires 
 a skillful balancing of the masses, whether of form or color, 
 and a judicious correction of the tendency of the eye to 
 run in any one direction, by interposing lines that have an 
 opposite tendency. (See Prop, ix.) 
 
 Captain Cook, noticing the extent to which decoration 
 was carried by the Islanders of the Pacific and South Seas, 
 speaks of their cloths, their basket-work, their matting, etc., 
 as painted " in such an endless variety of figures that one 
 
ORNAMENTAL ABT. 29 
 
 might suppose they borrowed their patterns from a mer- 
 cer's shop, in which the most elegant productions of China 
 and Europe are collected, besides some original patterns 
 of their own." 
 
 II. EGYPTIAN ORNAMENT. 
 
 The origin of art among the Egyptians is unknown ; and 
 in their architecture, the more ancient the monument the 
 more nearly perfect is the art. As far back as we can 
 trace their ornamentation, a few of the more important 
 natural productions of the country formed the basis of the 
 immense variety of ornament with which the Egyptians 
 decorated their temples, their palaces, dress, utensils, arti- 
 cles of luxury, etc. 
 
 All Egyptian art was symbolic. Thus the lotus and 
 papyrus, growing so luxuriantly on the banks of the Nile, 
 the former symbolizing food for the body, and the latter, 
 being used for parchment, and thus symbolizing food for 
 the mind; certain rare feathers carried before the king, 
 and thus symbolizing royalty ; the branches of the palm, 
 and twisted cord made from its bark, etc. such natural 
 products were conventionally used in their decoration ; and 
 hence they were types of Egyptian civilization. A lotus 
 conventionally carved in stone, and forming a graceful ter- 
 mination to a column, was a fitting symbol both of plenty and 
 prosperity, and of the power of the king over countries where 
 the lotus grew ; and thus the ornament added a poetic idea 
 to what would otherwise have been but a rude support. 
 
 On this basis of symbolism, Egyptian art was of three 
 kinds : 
 
 1st. Constructive. Thus, an Egyptian column represent- 
 ed an enlarged lotus or papyrus plant, or a bundle of such 
 plants ; the base representing the root, the shaft the stalk, 
 and the capital the full-blown flower surrounded by a bou- 
 quet of smaller plants tied together by bands. And as 
 the column represented one plant, three, or a greater num- 
 ber, and so of the other parts, the variety of form to which 
 the various combinations gave rise was far beyond what 
 any other style of architecture ever attained to. 
 
30 INDUSTRIAL DRAWING. 
 
 2d. Representative. In their representation of objects, ev- 
 ery thing as a flower, for example was portrayed, not 
 as a reality, but as an ideal representation. They did not 
 attempt to portray the real flower, but something that 
 should give the idea of one ; and hence they adhered to 
 the principles of the growth of plants, in the radiation of 
 leaves, and all veins on the leaves, in graceful curves from 
 the main stalk, or the stem. (See Prop, x.) They took 
 the general form of the lotus plant, and changed it into 
 the form of the base, shaft, and capital of a column, while 
 it still retained sufficient resemblance to the lotus plant 
 to show whence the idea originated. And so on, in all their 
 art representations. (See Prop, xi.) 
 
 3d. Decorative. All the paintings of the Egyptians, pro- 
 duced by few types, are distinguished by graceful sym- 
 metry and perfect distribution of parts. They painted ev- 
 ery thing ; but using color as they did form, conventionally, 
 and to distinguish one part from another, they dealt in flat 
 tints only, using neither shade nor shadow; and they in- 
 differently colored the leaves of the lotus green or blue. 
 
 III. ASSYRIAN AND PERSIAN ORNAMENT. 
 
 The Assyrian and Persian ornaments which have been 
 discovered seem to belong to a period of decline in art, and 
 to have been borrowed, far back in the obscurity of ages, 
 from an original and more nearly perfect style perhaps 
 that from which the Egyptian itself was derived. 
 
 Assyrian ornament is represented in the same way as the 
 Egyptian, although it is not based on the same types ; and 
 indeed the natural types are very few, Yet the natural 
 laws of radiation from the parent stem, and tangential 
 curvature, are observed, although not so strictly as in 
 Egyptian art. In both styles, the carved ornaments, as 
 well as those that were painted, are mostly in the nature 
 of diagrams geometrial patterns being closely adhered to. 
 The little surface modeling that was attempted was mostly 
 in slight relief. 
 
ORNAMENTAL ART. 31 
 
 IV. GREEK ORNAMENT. 
 
 Greek art, borrowed partly from the Egyptian and part- 
 ly from the Assyrian, gave to old ideas of ornament a new 
 direction. Rising rapidly to a high state of perfection, it 
 carried the development of pure form to a degree of fitness 
 and beauty which has never since been reached ; and, from 
 the very abundant remains we have of Greek ornament, 
 we are led to believe that the presence of refined taste 
 among the people was almost universal, and that the land 
 was overflowing with artists, whose hands and minds were 
 so trained as to enable them to execute with unerring 
 truth those beautiful ornaments which to this day are the 
 great wonder of art. Greek ornament, never in profusion 
 or excess, was always strictly subordinate to the general 
 expression of the object to which it was affixed. 
 
 But Greek art was not symbolical like the Egyptian ; it 
 was meaningless, purely decorative the very embodiment 
 of beauty for beauty's sake and hence wholly aesthetic ; 
 seldom representative; and it can hardly be said to be 
 constructive. The conventional rendering of natural ob- 
 jects was so far removed from the original types as often 
 to make it difficult to recognize any attempt at imitation. 
 The ornament was no part of the construction, and was 
 thus unlike the Egyptian ; it could be removed, and the 
 structure remain unchanged. On the Corinthian capital, 
 that leading feature of Grecian ornament, the acanthus leaf, 
 is applied, not constructed as a part of the edifice. In the 
 Egyptian, the whole capital, conventionally representing 
 the full-blown flower of the lotus plant, is the ornament ; 
 and to remove any part would destroy the whole. 
 
 The three great laws which we find every where in nat- 
 ure radiation from the parent stem, proportionate distri- 
 bution of the areas, and the tangential curvature of the 
 lines are always obeyed in Greek art, and with so great a 
 degree of perfection that the attempt to reproduce Greek 
 ornament is rarely done with success. 
 
 There is now little doubt that the white marble temples 
 of the Greeks were entirely covered with painted ornament. 
 
32 INDUSTRIAL DRAWING. 
 
 This certainly was true as to the ornaments of the mouldings 
 on architrave, frieze, and cornice ; and doubtless the object 
 in these cases was to render the mouldings and carvings 
 distinct, and make the pattern visible from a distance. 
 
 V. POMPEIAN ORNAMENT. 
 
 Pompeii, a town of Italy, fifteen miles south-east from 
 Naples, was destroyed by an eruption of Mount Vesuvius 
 in the year A.D. 79. It has since been extensively exca- 
 vated, disclosing the city walls, streets, temples, theatres, 
 the forum, baths, monuments, private dwellings, domestic 
 utensils, etc., the whole conveying the impression of the 
 actual presence of a Roman town in all the circumstantial 
 reality of its existence two thousand years ago. 
 
 Pompeian ornamentation was of two kinds partly of 
 Grecian and partly of Roman origin but sufficiently dis- 
 tinct from either to require a separate notice in the history 
 of ornamental art. That derived from the Greek was com- 
 posed of conventional representations of objects in flat 
 tints, without shade or attempt at relief; the other, more 
 Roman in character, was based mainly upon the acanthus 
 scroll, and was interwoven with natural representations of 
 leaves, flowers, animals, etc. the germs of a later Italian 
 style of ornamentation. But the Pompeian style was ex- 
 ceedingly capricious, beyond the range of true art. The 
 Pompeian pavements are the types from which may be 
 traced the immense variety of Byzantine, Arabian, and Mo- 
 resque mosaics. 
 
 VI. ROMAN ORNAMENT. 
 
 The temples of the Romans were overloaded with orna- 
 ment ; and the general proportions of Roman edifices, and 
 the contours of their moulded surfaces, were entirely de- 
 stroyed by the elaborate surface modeling carved on them. 
 Nor do the Roman ornaments grow naturally from the 
 surface like the conventional forms of the Egyptian capi- 
 tal : they are merely applied to it. The acanthus leaves, 
 which by the Greeks were beautifully conventionalized, 
 were used by the Romans with too close an approxima- 
 
ORNAMENTAL ART. 33 
 
 tion to nature : they were also arranged iuartistically, be- 
 ing not even bound together by the necking at the top of 
 the shaft, but merely resting upon it. In the Egyptian 
 capital, on the contrary, the stems of the flowers round the 
 bell-shaped capital being continued through the necking, 
 at the same time represent a beauty and express a truth. 
 The introduction of the Ionic volute a Grecian feature 
 into the Roman Composite order, fails to add a beauty, but 
 rather increases the deformity. The leaf ornamentation of 
 the Romans adhered to the principle of one leaf growing 
 out of another in a continuous line, leaf within leaf, and 
 leaf over leaf a principle very limited in its application; 
 and it was only in the later Byzantine period that this 
 style began to be abandoned for the true one of a continu- 
 ous stem throwing off ornaments on either side. Then 
 pure conventional ornament began to receive a new devel- 
 opment. The true principle became common in the elev- 
 enth, twelfth, and thirteenth centuries, and is the founda- 
 tion of the Early English foliage style. 
 
 While Roman Decorative Art abounds in the most ex- 
 quisite specimens of drawing and modeling, its great de- 
 fect consists in its frequent want of adaptation to the pur- 
 poses it was required to fill as an aid to the true expres- 
 sion of architectural design. Roman decoration, like the 
 Grecian, was strictly aesthetic based on an almost rever. 
 ential regard for the beautiful, for beauty's sake alone. 
 
 VII. BYZANTINE ORNAMENT. 
 
 When in the year A.D. 328 the Emperor Constantine 
 transferred the seat of the Roman government to Byzan- 
 tium (afterward called Constantinople, from its founder), 
 Roman art was already in a state of decline, or transforma- 
 tion. Constantine employed Persian and other Oriental 
 artists, and artists from the provinces, in the decoration of 
 his capital ; and these together soon began to work a 
 change in the traditional Roman style, until at length the 
 motley mass became fused into one systematic whole dur- 
 ing the long and (for art) prosperous reign of the first Jus- 
 tinian. (A.D. 527 to 565.) 
 
 B2 
 
Si I3'DUSTIAL DliAIYI^G. 
 
 Byzantine art is characterized by elliptical curved out- 
 lines, acute-pointed and broad-toothed leaves, and thin con- 
 tinuous foliage springing from a common stem. In sculp- 
 ture the leaves are beveled at the edge, and deeply chan- 
 neled throughout, and drilled, at the several springings of 
 the teeth, with deep holes. Thin interlaced patterns are 
 preferred to geometrical designs; animal or other figures 
 are sparingly introduced in sculpture, while in color they 
 are principally confined to subjects of a holy character. 
 Rome, Syria, Persia, and other countries, all took part as 
 formative causes in the Byzantine style of art and its ac- 
 companying decoration. The character of the Byzantine 
 school is strongly impressed on all the earlier works of 
 Central and even Western Europe, which are generally 
 termed the Romanesque or Romanized style, which is con- 
 sidered a fantastic and debauched style when applied to 
 architecture. The geometrical mosaic work of Byzantine 
 art belongs particularly to the Romanesque period, espe- 
 cially in Italy. This art, which flourished principally in the 
 twelfth and thirteenth centuries, consists in the arrange- 
 ment of small diamond-shaped pieces of glass into a com- 
 plicated series of diagonal lines. Marble mosaic work dif- 
 fers from the glass only in the material used. 
 
 The influence of Byzantine art was all powerful in Eu- 
 rope from the sixth to the eleventh century, and even later ; 
 and it has served in a great degree as the basis of all the 
 modern schools of decorative art in the East and in East- 
 ern Europe. , 
 
 VIII. ARABIAN ORNAMENT. 
 
 As every distinct form or mode of civilization has been 
 characterized by its own peculiar style of art, so when the 
 religion of Mohammed spread with astonishing rapidity 
 over the East about the middle of the seventh century, and 
 over Spain in the early part of the eighth, a new style of 
 art arose, which gradually encroached, in those regions, upon 
 the already waning glories of the Byzantine period. 
 
 Some of the Arabian mosques of Cairo, erected in the 
 ninth century, remarkable alike for the grandeur and sim- 
 
OEXAMENTAL AET. 35 
 
 plicity of their general forms, and the refinement and el- 
 egance of their decoration, are among the most beautiful 
 buildings in the world. Their elegance of ornamentation 
 was probably derived primarily from the Persians, perhaps 
 modified by Byzantine influence. In their leafage orna- 
 ments we observe traces of Greek origin, especially in the 
 modified form of the acanthus leaf; but they abandoned 
 the principle of leaves growing out one from another, and 
 made the scroll continuous without break, while they re- 
 tained that universal principle of true art, the radiation of 
 lines from a parent stem, and their tangential curvature. 
 Like the Romans, they covered the floors of their public 
 buildings with mosaic patterns arranged on a geometrical 
 plan ; but it is surprising that, while the same pattern forms 
 of mosaics exist in Roman, Byzantine, Arabian, and Moor- 
 ish art, the general style of each differs widely from all the 
 others. It is like the same idea expressed in four differ- 
 ent languages. The twisted cord, the interlacing of lines 
 straight or curved, the crossing and interlacing of two 
 squares, and the equilateral triangle within a hexagon, are 
 the starting-points in each. 
 
 What is called Arabesque ornament consists of a fanciful, 
 capricious, and ideal mixture of all sorts of figures of men 
 and animals, both real and imaginary; also all sorts of 
 plants, fruit, and foliage, involved and twisted, and upon 
 which the animals and other objects rest. The Arabians 
 did not originate this style, although it is named from 
 them ; and in pure Arabesque, figures of animals are ex- 
 cluded, as they were forbidden by the Koran. 
 
 It is strange that while the Arabians have left traces of 
 fine Saracenic art all through Northern Africa, and in Spain, 
 scarcely a vestige of it can now be found in their native 
 country, Arabia. 
 
 IX. TURKISH ORXAMEXT. 
 
 Although the Turks and the Arabians have the same re- 
 ligion, yet, being of different national origin, their art rep- 
 resentations are, as might be expected, somewhat different. 
 The architecture of the Turks, as seen at Constantinople, 
 
36 INDUSTRIAL DBA WING. 
 
 is mainly based upon the early Byzantine monuments, ex- 
 cept their modern edifices, which are designed in the most 
 European style. Their system of ornamentation is of a 
 mixed character Arabian and Persian floral ornaments 
 being found side by side with debased Roman and Renais- 
 sance details. The art instinct of the Turks is quite in- 
 ferior to that of the East Indians. The only good exam- 
 ples we have of Turkish ornamentation is in Turkey car- 
 pets ; and these are chiefly executed in Asia Minor, and 
 most probably not by Turks. The designs are thoroughly 
 Arabian. The Turk is unimaginative. 
 
 X. MORESQUE OR MOORISH ORNAMENT. 
 
 In the ornamental art of the Moors, who established the 
 seat of their power in Spain during the eighth century, we 
 have another illustration of the results produced by corre- 
 sponding influences of religious faith and diversities of na- 
 tional character. The main differences between the Ara- 
 bian and Moorish edifices consist in this : that the former 
 are distinguished most for their grandeur, the latter for 
 their refinement and elegance. In ornamentation the Moors 
 were unsurpassed ; and in it they carried out the princi- 
 ples of true art, even beyond the attainments of the Greeks 
 themselves. 
 
 Arabian and Moorish art were alike wanting in symbol- 
 ism; but the Moors compensated for this want by the 
 beauty of their ornamental written inscriptions, and the 
 nobleness of the sentiments they expressed. To the artist 
 these inscriptions furnished the most exquisite lessons in 
 art; to the people they proclaimed the might, majesty, and 
 good deeds of the king ; and to the king they never ceased 
 to declare that there was none powerful but God ; that He 
 alone was conqueror, and that to Him alone was ever due 
 praise and glory. A law of the Mohammedan religion for- 
 bade the representation of animals, or of the human figure. 
 
 In the best specimens of Moorish architecture the deco- 
 ration always arises naturally from the construction ; and, 
 although every part of the surface may be decorated, there 
 is never a useless or a superfluous ornament. All lines 
 
OEXAMEXTAL AET. 37 
 
 o-row out of one another in natural undulations, and every 
 ornament can be traced to its branch or root ; and there is 
 no such thing as an ornament just jotted down to fill a 
 space, without any other reason for its existence. 
 
 The best Moorish ornamentation is found in the Alham- 
 bra, a celebrated palace of the Moorish kings, at Granada, 
 in Spain. This immense and justly famous structure, of 
 rather forbidding exterior, but gorgeous within almost be- 
 yond description, was erected in the thirteenth century; 
 and much of it remains perfect at the present day. It has 
 been said by a competent judge that "Every principle 
 which we can derive from the study of the ornamental art 
 of any other people is not only ever present here, but was 
 by the Moors more universally and truly obeyed." And 
 further, that " We find in the Alhambra the speaking art 
 of the Egyptians, the natural grace and refinement of the 
 Greeks, and the geometrical combinations of the Romans, 
 the Byzantines, and the Arabs." The walls of the Alham- 
 bra were covered with a profusion of ornamentation, which 
 had the appearance of a congeries of paintings, incrusta- 
 tions, mosaics, gilding, and foliage ; and nothing could be 
 more splendid and brilliant than the effects that resulted 
 from their combinations. The mode of piercing the domes 
 for light, by means of star-like openings, produced an al- 
 most magical effect. 
 
 XI. PERSIAN ORNAMENT. 
 
 The Mohammedan architecture of Persia, and Persian or- 
 namentation, are alike a mixed style, and are far inferior to 
 the Arabian, as exhibited in the buildings at Cairo. The 
 Persians, unlike the Arabs and the Moors, mixed up the 
 forms of natural flowers and animal life with conventional 
 ornament. 
 
 XII. EAST -INDIAN ORNAMENT. 
 
 Numerous manufactures calculated to give a high idea 
 of the ingenuity and taste of the people of British India 
 appeared in the Great Exhibition of the Industry of all Na- 
 tions, in London, in 1851. Among these were various ar- 
 
38 INDUSTRIAL DRAWING. 
 
 tides in agate from Bombay, mirrors from Lahore, marble 
 chairs from Ajmeer, embroidered shawls, scarfs, etc., from 
 Cashmere, carpets from Bangalore, and a variety of articles 
 in iron inlaid with silver. In the application of art to man- 
 ufactures the East Indians exhibit great unity of design, 
 and skill and judgment in the application, with great ele- 
 gance and refinement in the execution. In these respects 
 they seem far to surpass the Europeans, who, says Mr. 
 Owen Jones, "in a fruitless struggle after novelty, irrespec- 
 tive of fitness, base their designs upon a system of copying 
 and misapplying the received forms of beauty of every by- 
 gone style of art." All the laws of the distribution of form 
 which are observed in the Arabian and Moresque orna- 
 'ments are equally to be found in the productions of India, 
 while the coloring of the latter is said to be so perfectly 
 harmonized that it is impossible to find a discord. This, of 
 course, refers to the selected articles placed on exhibition 
 in 1851. 
 
 XIII. HINDOO ORNAMENT. 
 
 We have but little reliable information about the an- 
 cient, or Hindoo, architecture of India ; yet we know this 
 much, that the Hindoos had definite rules of architectural 
 proportion and symmetry. One of their ancient precepts, 
 quoted by a modern writer, says, u Woe to them who dwell 
 in a house not built according to the proportions of sym- 
 metry. In building an edifice, therefore, let all its parts, 
 from the basement to the roof, be duly considered." 
 
 The architectural features of Hindoo buildings consist 
 chiefly of mouldings heaped up one over the other. There 
 is very little marked character in their ornaments, which 
 are never elaborately profuse, and which show both an 
 Egyptian and a Grecian influence. 
 
 XIV. CHINESE ORNAMENT. 
 
 Notwithstanding the great antiquity of Chinese civiliza- 
 tion, and the perfection reached in their manufacturing pro- 
 cesses ages before our time, the Chinese do not appear to 
 have made much advance in the fine arts. They show very 
 
OEXAMENTAL AET. 39 
 
 little appreciation of pure form, beyond geometrical pat- 
 terns ; but they possess the happy instinct of harmonizing 
 colors. Their decoration is of a very primitive kind. The 
 Chinese are totally unimaginative ; and their ornamentation 
 is a very faithful expression of the nature of this peculiar 
 people oddness. 
 
 XV. CELTIC ORNAMENT. 
 
 The Celts the early inhabitants of the British Isles had 
 a style of ornamentation peculiarly their own, and singu- 
 larly at variance with any thing that can be found in any 
 other part of the world. Celtic ornament was doubtless of 
 independent origin, but it every where bears the impress 
 received by the early introduction of Christianity into the 
 islands. 
 
 The chief peculiarities of Celtic ornament consist, first, in 
 the entire absence of foliage or other vegetable ornament ; 
 and, secondly, in the extreme intricacy and excessive mi- 
 nuteness and elaboration of the various patterns, mostly 
 geometrical, consisting of interlaced ribbon-work ; diago- 
 nal, straight, or spiral lines ; and strange, monstrous ani- 
 mals or birds, with their tail-feathers, top-knots, and tongues 
 extended into long interlacing ribbons, which were inter- 
 twined in almost endless forms, and in the most fantastic 
 manner. Celtic manuscripts of the Gospels were often orna- 
 mented with a great profusion of these intricate designs. 
 
 What is called the Celtic ornamentation was practiced 
 throughout Great Britain and Ireland from the fourth or 
 fifth to the tenth or eleventh centuries. There was a later 
 Anglo-Saxon ornamentation, equally elaborate, employed in 
 the decoration of manuscripts of the Gospels and other holy 
 writings ; but here leaves, stems, birds, etc., were intro- 
 duced, and interwoven with gold bars, circles, squares, loz- 
 enges, quarterfoils, etc. 
 
 XVI. MEDIAEVAL OK GOTHIC ORNAMENT. 
 The high-pitched gable and the pointed arch, with a con- 
 sequent slender proportion of towers, columns, and capitals, 
 are the leading characteristics of medieval or Gothic archi- 
 
40 INDUSTRIAL DRAWING. 
 
 lecture, which came into general use in Europe in the thir- 
 teenth century. Mediaeval Gothic art, like the Egyptian, 
 was symbolic, deriving its types from the prevailing religious 
 ideas of the period. Thus the churches and the cathedrals 
 of the Middle Ages were built in the form of a cross the sign 
 and symbol of the Christian faith. The numbers three, five, 
 and seven, denoting the Trinity, the five traditional wounds 
 of the Saviour, and the seven Sacraments, were preserved 
 as emblematical in the nave and two aisles, in the trefoiled 
 arches and windows, in the foils of the tracery, and in the 
 seven leaflets of the sculptured foliage ; while the narrow- 
 pointed arches, and the numerous finger-like pinnacles, ris- 
 ing above the gloom of the dimly lighted place of worship, 
 symbolized the faith which pointed the soul upward from 
 the trials of earth to the happy homes of the redeemed. 
 The transition from the Romanesque (later Roman) or 
 rounded style to the pointed is easily traced in the numer- 
 ous buildings in which the two styles are intermingled ; but 
 the passage from Romanesque ornament to Gothic is not so 
 clear. In the latter, new combinations of ornaments and 
 tracery suddenly arise. The piercings for windows be- 
 come clustered in groups, soon to be moulded into a net- 
 work of enveloping tracery; the acanthus leaf disappears; 
 in the capitals of columns of pure Gothic style, the orna- 
 ment arises directly from the shaft, which, above the neck- 
 ing, is split into a series of stems, each terminating in a 
 conventional flower the whole being quite analogous to 
 the Egyptian mode of decorating the capital. 
 
 In the interior of the early Gothic buildings every mould- 
 ing had its color best adapted to develop its form; and 
 from the floor to the roof not an inch of space but had its 
 appropriate ornament, the whole producing an effect grand 
 almost beyond description. But so suddenly did this pro- 
 fuse style of ornament attain its perfection, that it almost 
 immediately began to decline. What is called ornamental 
 illumination, that is, the decoration of writing by means of 
 colors, and, especially, the decoration of the initial letters to 
 pages of manuscript, attained a high degree of perfection 
 under the influence of the Gothic style. 
 
OEXAMENTAL ART. 41 
 
 While Gothic ornamentation retained its conventional 
 character, there was a boundless field for development : 
 when it became a mere imitation of natural objects, and rep- 
 resented stems, flowers, insects, etc., true to life, all ideality 
 fled, and there could be no further progress in the art. 
 
 XVII. RENAISSANCE ORNAMENT. 
 
 The fact that the soil of Italy was so covered with the re- 
 mains of Roman greatness that it was impossible for the 
 Italians to forget them, however they might neglect the les- 
 sons they were calculated to teach, was probably the rea- 
 son why Gothic art took but little root in Italy, where it 
 was ever regarded as of barbarian origin. When, in the fif- 
 teenth century, classical learning revived in Italy, and the 
 art of printing disseminated its treasures, a taste for classic 
 art revived also; and the style of ornamentation to which 
 it gave rise, formed upon classic models, is called Renais- 
 sance ornament ; and the period of its glory the Restora- 
 tion, or Renaissance period. 
 
 A combination of architecture and decorative sculpture 
 was a distinguishing feature of the Renaissance style. Fig- 
 ures, foliage, and conventional ornaments were so happily 
 blended with mouldings, and other structural forms, as to 
 convey the idea that the whole sprung to life in one perfect 
 form in the mind of the artist by whom the work was ex- 
 ecuted. To Raphael (early in the sixteenth century), both 
 sculptor and painter, we owe the most splendid specimens 
 of the Arabesque style, which he dignified, and left with 
 nothing more to be desired. (See Arabian Ornament.) Ara- 
 besques lose their character when applied to large objects; 
 neither are they appropriate where gravity of style is re- 
 quired. 
 
 All the great painters of Italy were ornamental sculptors 
 also. Their sculptured ornaments were ingeniously arranged 
 on different planes, instead of on one uniform flat surface, 
 so as best to show the diversities of light and shade. Much 
 of the splendid painting done by the Italian masters, from 
 Giotto to Raphael from the year 1290 to 1520 was mu- 
 ral decoration, now generally called fresco. In true fresco. 
 
42 INDUSTRIAL DRAWING. 
 
 the artist incorporated his colors with the plaster before it 
 was dry, by which the colors became as permanent as the 
 wall itself. This kind of painting was so clear and trans- 
 parent, and reflected the light so well, as to be peculiarly 
 suited to the interior of dimly lighted buildings ; and it is 
 said that the eye which has been accustomed to look upon 
 it can scarcely be reconciled to oil pictures. It is a well- 
 known saying of Michael Angelo, that fresco is fit for men, 
 oil painting for women, and the luxurious and idle. 
 
 XVIII. ELIZABETHAN ORNAMENT. 
 
 The revival of art in Italy soon spread over France and 
 Germany, and about the year 1520 extended into England, 
 where it soon triumphed over the late Gothic style. The 
 true Elizabethan period of art embraced only about a cent- 
 ury. It is simply a modification of foreign models, and has 
 little claim to originality. 
 
 The characteristics of Elizabethan ornament may be de- 
 scribed as consisting chiefly of a grotesque and complicated 
 variety of pierced scroll-work, with curled edges; interlaced 
 bands, sometimes on a geometrical pattern, but generally 
 flowing and capricious; curved and broken outlines; fes- 
 toons, fruit, and drapery, interspersed with roughly exe- 
 cuted figures of human beings ; grotesque monsters and ani- 
 mals, with here and there large and flowing designs of nat- 
 ural branch and leaf ornament ; rustic ball and diamond 
 work ; paneled compartments, often filled with foliage, or 
 coats of arms, etc., etc. : the whole founded on exaggerated 
 models of the early Renaissance school. By the middle of 
 the seventeenth century the more marked characteristics 
 of the Elizabethan style had completely died out. 
 
 MODERN ORNAMENTAL ART. 
 
 There is, no doubt, a very decided tendency in modern 
 ornamental art to copy natural forms as faithfully as pos- 
 sible for all decorative purposes. We see this, alike, in our 
 floral carpets, floral wall-papers, floral curtains, and in the 
 
ORNAMENTAL AET. 43 
 
 floral carvings of our structures of wood, stone, and iron. 
 Yet when perfection shall have been attained in this mode 
 of ornamentation if it has not been already and which is 
 but the mere copying of nature, and devoid of all original- 
 ity of design, how little has the artist accomplished in the 
 development and application of art principles, and what fur- 
 ther can he attain to ? 
 
 But when, on the contrary, the progress of true art shall 
 be acknowledged to lie in the direction of idealizing the 
 forms of nature giving to them a conventional represen- 
 tation while adhering to the principles of natural growth, in 
 the manner in which art grew up among the Egyptians and 
 the Greeks the artist will be left free to follow the bent of 
 his genius, and to select from, and conventionally adopt, 
 whatever natural forms he may find best suited to his pur- 
 poses. Then there may be advance in art beyond the copy- 
 ing and intermingling of those olden styles, which now ex- 
 cite in us but little sympathy ; but until then we shall prob- 
 ably rest content in the idea that all available modes and 
 forms have been used by those who preceded us, and that 
 there are no untrodden domains of art left for us to explore. 
 
PART II. 
 PRINCIPLES AND PRACTICE 
 
 OP 
 
 INDUSTRIAL DRAWING. 
 
DRAWING-BOOK No. I. 
 
 I. MATERIALS AND DIRECTIONS. 
 
 1. FOB PAPER to draw on, use " Wittson's Cabinet Draio- 
 ing -Paper" for Drawing-Books Nos. I., IL, III., and IV. 
 This paper is printed in fine red or pink lines, to correspond 
 to the ruling in the Drawing-Books ; and it has a border so 
 ruled and lettered as to furnish convenient guides for the 
 accurate drawing of the diagonal and serai-diagonal lines 
 of Cabinet Perspective, as illustrated in the Second, Third, 
 and Fourth Drawing-Books. Of this drawing-paper, No. 1 
 is the same in size as the pages in the Drawing-Books, and 
 No. 2 is four times the size. There is also " Isometrical 
 Drawing- Paper No. 1," of the same size as the No. 1 Cab- 
 inet Drawing-Paper, for use in isometrical drawings, as il- 
 lustrated in the Appendix to this volume. The fine pink 
 lines of the drawing -paper do not in the least interfere 
 with the pencil drawings. 
 
 2. For PENCILS, use Faber's Nos. 1, 2, 3, and 4, which are 
 round black pencils. No. 4 being the hardest of these, is 
 used for fine, hard lines only, or very light shading ; No. 3 
 for common outline drawing and shading ; and No. 2 for 
 heavy and distinct dark lines and edges. No. 1, very dark 
 and soft, is little used. There are also very superior Fa- 
 ber pencils, of light wood, hexagonal in form, and numbered 
 by letters H, HH, HHH, and HHHH : H being soft pencils, 
 and HHHH very hard and fine. 
 
 There are also what are called the Eagle pencils the H 
 pencil for light shading and lines, and the F for common 
 shading. The common Eagle pencils marked 1, 2, and 3, 
 are of inferior grade. 
 
48 INDUSTRIAL DRAWING. [BOOK NO. I. 
 
 3. For most industrial drawings, however, India ink is 
 more convenient, and better, for shading, than the pencil. 
 A cake of good India ink, about two inches long, that will 
 go further in shading than a hundred pencils, may be bought 
 of almost any bookseller or stationer for some twenty or 
 thirty cents. Two or three camel's-hair pencils (or brush- 
 es) will also be needed. Price, three or four cents each. 
 To use the ink, put half a teaspoonful, or less, of water in a 
 small saucer (or the smallest china plate, about two inches 
 in diameter; or a small glass salt-cellar is better), and rub 
 one end of the India-ink cake in it, giving the water the 
 depth of tint that is required. With one of the brushes 
 flow the ink over those portions of the drawing that are to 
 be shaded. When the ink is dry, apply the wash a second 
 time to those portions that require a darker shade than the 
 lighter portions, and apply it a third time to those portions, 
 if any, that require a still darker shade. In this manner 
 any required depth of even shade may be given. Be care- 
 ful and not make the ink too dark at first ; and, as it dries 
 up quite rapidly in the saucer, water must be supplied from 
 time to time to keep it of a uniform tint. It produces a 
 good eifect to first wash lightly, with India ink, those por- 
 tions of a drawing that require shading ; and then, when 
 the ink is dry, to put on the line shading with the pencil. 
 
 4. For many of the curvilinear drawings, in which parts 
 or wholes of perfect circbs are used, a pair of compasses 
 adapted to receive a pencil will be needed ; or, what will 
 answer the purpose very well, a pencil may be split and 
 tied firmly to one of the legs of a pair of ordinary brass 
 compasses or dividers. 
 
 5. A ruler will also be needed for drawing long straight 
 lines. It should be beveled oif on one side to a very thin 
 edge. A ruler with one thin metallic edge is the most con- 
 venient. 
 
 6. For the purpose of Blackboard Exercises in connection 
 with drawing on paper, every school in which industrial 
 drawing is taught should be provided with a blackboard 
 of convenient size, having fine red lines painted on it, both 
 vertically and horizontally, at right angles to one another, 
 
MATERIALS AND i>I2ECTIOXS. 49 
 
 and two inches apart. Any careful painter an prepare a 
 board in this manner. The board should not be varnished. 
 The red lines drawn on the board will interfere very little 
 with the use of the board for ordinary purposes. Tho 
 school should also be provided with one pair of chalk-cray- 
 on compasses, for the drawing of regular curves on the 
 blackboard. Any ingenious carpenter can make a pair that 
 will answer very well. One of the points may be hollowed 
 out to receive the crayon, which may be tied in. 
 
 7. All the figures in a lesson, or on a page of the Drawing- 
 Books, should be first copied by the pupils on the lined 
 drawing-paper, and then the accompanying Problems should 
 be drawn, and then the free-hand blackboard exercises, 
 when such are suggested. The pupils should also explain 
 the drawings fully their measurements according to the 
 scale given on the paper, and their real measures when 
 drawn on the blackboard. But if any of the pupils are 
 too young to understand the few elementary principles of 
 surface measurement that are given in Drawing-Book No. I., 
 these principles may be passed over for the present, as 
 they will come up again in a more extended exposition of 
 the Drawing, Measurement, and Relations of Surfaces and 
 Solids. 
 
 8. Although free-hand drawing can be carried out in the 
 present series quite as extensively as in any other series, 
 and perhaps with more effect than in any other, as the 
 guide-lines at once detect all inaccuracies; and although 
 this kind of preliminary practice is important for all de- 
 signers in art, and especially for artists by profession, yet 
 we would remind teachers and pupils that it is never re- 
 lied on by architects, draughtsmen, and artisans for the 
 drawing of working-patterns or designs for industrial pur- 
 poses, and that most of the copies which are given in the 
 drawing-books for practice in free-hand drawing are there 
 executed, with elaborate care, by the aid of ruler and com- 
 pass. Even the best of artists do not hesitate to resort to 
 all possible mechanical appliances by which their work can 
 be improved ; and it would be strange, indeed, if we should 
 deny to children those aids which we allow to age and ex- 
 
 C 
 
50 INDUSTRIAL DBA WING. [BOOK NO. I. 
 
 perience. While, therefore, we recommend free-hand draw- 
 ing in elementary exercises, and also in all portions of 
 copies or original designs which can be well executed there- 
 by, we would advise advanced pupils to make use of all 
 other aids that are essential to accuracy of result. Fre- 
 quent directions are given throughout the work for free- 
 hand exercises in drawing on the blackboard. 
 
 9. For the purpose of getting the full cfiect of a drawing 
 in diagonal Cabinet Perspective (Books II., III., and IV.), 
 partially close the hand, and through the tubular opening 
 thus formed look at the drawing from a position a little 
 above and at the right of it. On thus viewing it intently 
 for half a minute, the drawing will seem to stand out in 
 bold relief from the paper; and if there are any inaccura- 
 cies in the perspective, they will be readily detected by 
 the unnatural apJ3earances which they will thus be made 
 to present. 
 
 10. If the teacher should find some few slight inaccura- 
 cies in which the diagrams in the Drawing-Books do not 
 fully come up to the descriptions of them, they must at- 
 tribute it to the occasional want of care in the artists who 
 copied them from the original drawings. The errors, how- 
 ever, are believed to be few, and of little importance ; and 
 the teacher who gets hold of the principles will easily cor- 
 rect them. 
 
 11. It should be remarked, also, that drawings in pencil 
 and India ink, if well executed, and especially if made on 
 the pink-ruled drawing-paper, will be clearer in shading, 
 more distinct in outline, and will show to better advantage 
 generally, than those in the Drawing-Books. 
 
 ElSir" 12. For convenience of adapting the explanations of 
 drawings given in the Drawing-Books to those made on 
 the blackboard, let it be understood that the lines on the 
 blackboard are in all cases (unless otherwise directed) sup- 
 posed to be drawn to the same scale as those assigned for 
 the lines of the printed drawings. 
 
STRAIGHT LIXES AND PLANE SURFACES. 51 
 
 H. STRAIGHT LINES AND PLANE SURFACES. 
 PAGE ONE. 
 
 LESSON I. Horizontal Parallel Lines. A horizontal line 
 is a line that has all its points equally high, or on a level 
 with the horizon. Parallel lines are lines that extend in the 
 same direction, and that are equally distant from one anoth- 
 er, however far they may be extended. Thus, the lines that 
 cross the paper from left to right are parallel lines, one 
 eighth of an inch apart ; and they are also horizontal lines 
 when the paper lies flat upon the table, and also when it is 
 raised to an upright position. All the lines in Lesson I. 
 may be considered horizontal and parallel. 
 
 In drawing the copies on this page, use a No. 3 or No. 2 
 pencil, rounded at the point, and not sharp. Use no ruler. 
 In figure No. 1, draw all the lines on the fine-ruled horizontal 
 red lines seen on the drawing-paper first tracing each line 
 very lightly, carrying the pencil a part of the time from left 
 to right, and a part of the time from right to left, so as to 
 acquire a free command of the hand. Finish by drawing 
 each line firm and distinct, and as true and even as possible. 
 In the first column the lines are one eighth of an inch long; 
 in the second column two eighths, or one quarter of an 
 inch ; and in the third column three eighths of an inch long. 
 The printed vertical and horizontal lines in the Drawing- 
 Book, and also on the drawing-paper, are one eighth of an 
 inch apart. 
 
 In No. I., the pencil lines are drawn on the ruled lines, one 
 eighth of an inch apart ; in No. II., they are first drawn the 
 same as in No. I., and then a line is drawn between every 
 two; in No. III., two lines are drawn equally distant between 
 every two lines first drawn as in No. I. No. III. represents 
 coarse shading. Let the pupil imitate the foregoing with 
 free-hand drawing on the red-lined blackboard, and tell the 
 lengths of the lines thus drawn as two inches four inches, 
 six inches, etc. ; and their distances apart. 
 
52 INDUSTRIAL DRAWING. [BOOK NO. I. 
 
 LESSON II. Vertical Parallel Lines. A vertical line is 
 one that is exactly upright in position such a line as that 
 which is formed by suspending a weight by a string. The 
 lines in Lesson II. represent vertical lines ; but they are 
 really vertical only when the paper is placed in an upright 
 position, and with the heading of the page upward. These 
 vertical lines are parallel, for the same reason that those in 
 Lesson I. are parallel. 
 
 Draw the lines in Lesson II. from the top downward, 
 first going over each line lightly, once or twice ; and, when 
 the line is accurately traced from point to point, finish by 
 marking it firmly. 
 
 What are the respective lengths of the lines in No. 1 ? In 
 No. 3? In No. 4? 
 
 In Nos. 2 and 3 the lines are drawn at the same distances 
 apart as in the corresponding numbers of Lesson I. In No. 4, 
 three lines are drawn equidistant between the ruled lines. 
 No. 2 represents coarse shading ; No. 3, ordinary shading ; 
 and No. 4, fine shading. 
 
 Free-hand exercises on the blackboard, similar to those di- 
 rected for Lesson I. 
 
 LESSON III. Angles, and Plane Figures. No. 1 repre- 
 sents two right angles, x, x, formed by one line meeting an- 
 other. An angle is the opening between two lines that meet. 
 
 When one straight line (a b) falls upon another straight 
 line (c d), so as to make the adjacent angles (x, x) equal, 
 the two angles thus formed are right angles. The angle at 
 a*, No. 2, is also a right angle. 
 
 An acute angle (e) is an angle that is less than a right 
 angle ; an obtuse angle (n) is an angle that is greater than 
 a right angle. 
 
 A plane is a surface, on which, if any two points be taken, 
 the straight line which joins them touches the surface in its 
 whole length. 
 
 Nos. 3, 4, and 5 tare plane figures called squares. 
 
 A rectilinear plane figure is a plane figure bounded by 
 straight lines. 
 
 A square is a plane figure that has four equal sides and 
 
STRAIGHT LINES AXD PLANE SURFACES. 53 
 
 four right angles. Nos. 3, 4, and 5 are squares. They are 
 also called erect squares, because two of the sides of each 
 are erect, or vertical. 
 
 A rectangle is a four-sided figure having only right an- 
 gles. The term is generally applied to those rectangular 
 (right-angled) figures which are not squares. Nos. 6, 7, and 
 8 are rectangles. Nos. 9 and 10 may be divided into rect- 
 angles. 
 
 Principles of Surface Measurement. 
 
 We will suppose that throughout Drawing-Book No. I. 
 the direct distance from one line to another on the ruled 
 paper is one inch, unless otherwise directed. 
 
 Then, how much space will one of the small ruled squares 
 contain ? (One square inch.) How much will four of them 
 contain ? (Four square inches.) As a standard of meas- 
 urement, each of the small squares formed by the ruling of 
 the paper is called & primary erect square. 
 
 How large is No. 3 ? (One inch square.) How much 
 area, or surface, does it contain? (One square inch.) 
 
 How large is No. 4 ? (Two inches square. That is, it 
 measures two inches on each side.) How much area, or sur- 
 face, does it contain? (Four square inches, as may be seen 
 by counting the primary squares within it.) 
 
 How large is No. 5 ? (Four inches square.) How much 
 area, or surface, does it contain ? (Sixteen square inches.) 
 
 How large is No. C ? (Two inches by three inches.) How 
 much area, or surface, does it contain ? (Six square inches.) 
 
 How large is No. Y, and what is its area? How large is 
 No. 8, and what is its area ? 
 
 Hence, 
 
 To find the area or surface measurement of any rectangle : 
 
 RULE I. Multiply the length by the breadth, and the prod- 
 uct will be the area. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw a square of three inches to a side. "What is its area? 
 
 Ans. 9 square inches. 
 
 2. Draw a square of nine inches to a side. "What is its area ? 
 
 Ans. 81 square inches. 
 
54 INDUSTRIAL DRAWING. [BOOK NO. I. 
 
 3. Draw a rectangle of four by five inches. What is its area ? 
 
 Ana. 20 square inches. 
 
 4. Draw a rectangle of six by eight inches. What is its area ? 
 
 Ans. 
 
 Let the Pupil draw the foregoing Problems on the Hlac7c- 
 board. 
 
 No. 4 has twice the length of sides of No. 3. How many 
 times larger than No. 3 is it ? (Four times larger ; because 
 No. 3 contains one square inch, and No. 4 contains four 
 square inches.) 
 
 No. 5 has four times the length of sides of No. 3. How 
 much larger than No. 3 is it ? (Sixteen times larger.) 
 
 No. 5 has twice the length of sides of No. 4. How much 
 larger than No. 4 is it ? (Four times larger.) 
 
 No. 7 has twice the length of sides of No. G. How much 
 larger is No. 7 than No. 6 ? (Four times larger.) 
 
 From the foregoing it appears that, by increasing the 
 lengths of the sides of a square or a rectangle to two times 
 their length, we form a similar figure four times as large ; 
 by increasing to three times, we form a similar figure nine 
 times as large ; by increasing to four times, we form one 
 sixteen times as large ; by increasing to Jive times, we form 
 one twenty -five times as large, etc. The same princi- 
 ple holds true with regard to a figure of any number of 
 sides. 
 
 ELEMENTARY PRINCIPLE. The areas of 'similar plane fig- 
 ures are as the squares of their similar sides. 
 
 If, therefore, we have a plane figure of any number of 
 sides, and wish to make another similar to it, lout four times 
 as large, we double the lengths of the sides ; because 2 
 times 2 are four : if we wish to make one nine times as 
 large, we treble the lengths of the sides ; because 3 times 3 
 arc nine: if we wish to make one sixteen times as large, we 
 quadrille the lengths of the sides ; because 4 times 4 are 
 sixteen: and so on to the square of any given number. 
 
 IrIP Let the teacher explain more fully, if necessary, 
 what is meant by the square of a number, and especially 
 when that number represents the length of a given line. 
 
STRAIGHT LINES AND PLANE SURFACES. 55 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw a square similar to No. 3, but nine times as large. 
 
 2. Draw a square similar to No. 4, but nine times as large. 
 
 3. Draw a square similar to No. 4, but twenty-five times as large. 
 
 4. Draw a rectangle similar to No. 6, but nine times as large. 
 
 5. Draw a rectangle similar to No. 6, but four times as large. 
 G. Draw a polygon similar to No. 10, but four times as large. 
 
 A polygon is a plane figure having many sides and miiny 
 angles. The term is generally applied to a plane figure of 
 more than four angles and four sides. 
 
 Free-hand exercises on the blackboard. Let the pupil fol- 
 low out, on the blackboard, a course of exercises similar to 
 those prescribed for the tint-lined drawing-paper. 
 
 LESSON IV. Diagonals. Diagonals are lines drawn in 
 the direction of a diagonal of a primary erect square. 
 
 A primary diagonal is a line drawn diagonally from one 
 corner to another of a primary erect square. 
 
 No. 1 is made up of primary diagonals in two directions. 
 
 No. 2 is a primary diagonal square. What is its area 
 equal to ? (Two square inches ; inasmuch as it includes 
 four halves of the small primary erect squares.) 
 
 What is the area of No. 3 ?* 
 
 What is the area of No. 4 ? No. 5 ? No. 6 ? 
 
 If No. 2 have its sides doubled in length, how much larger 
 will the figure be ? 
 
 If No. 2 have its sides trebled in length, how much larger 
 will the figure be ? 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw a diagonal square similar to No. 2, but sixteen times as large; 
 that is, containing sixteen times the area of No. 2. How long must the 
 sides be, compared with the sides of No. 2 ? 
 
 2. Draw a diagonal square similar to No. 2, but twenty-five times as 
 large. How long must the sides be, compared with the sides of No. 2 ? 
 
 3. Draw a diagonal square similar to No. 3, but nine times as large. 
 
 4. Draw a diagonal rectangle similar to No. 5, but four times as large. 
 
 * The halves of square inches included within the figures in this lesson 
 might be marked with dots, for greater facility in counting them. 
 
56 INDUSTRIAL DRAWING. [BOOK NO. 1. 
 
 Iii drawing these problems let the pupils arrange 
 them in such a manner as to economize the space on the 
 drawing-paper. 
 
 To find the area of any diagonal square, or other diagonal rectangle : 
 
 RULE A. Multiply the length in primary diagonals by 
 the breadth in primary diagonals, and TWICE the product 
 will be the area, in measures of the primary erect squares. 
 
 Kule A is only a special application of Rule I. 
 
 (Reason for the ride. The length in primary diagonals 
 multiplied by the breadth in primary diagonals will give 
 the number of primary diagonal squares ; and we then mul- 
 tiply by 2, because there are two primary erect squares in 
 each primary diagonal square.) 
 
 Thus, in No. 3, multiply 2, the length in primary diago- 
 nals of one side, by 2, the length in primary diagonals of 
 another side, and the product will be 4 ; and twice four will 
 be the area in primary erect squares, or square inches. 
 
 What is the area of a diagonal square of 7 diagonals to 
 a side ? (Ans. 98 square inches.) 
 
 What is the area of a diagonal rectangle of 5 by 7 diag- 
 onals ? (Ans. 70 square inches.) 
 
 Let the pupil carry out the same system on the black- 
 board. 
 
 LESSON Y. No. 1 is an erect cross, representing one thin 
 piece, 2 inches by 8 inches, laid at right angles across 
 another piece 2 inches by 6 inches. First draw the upper 
 piece, marked 1, and shade it lightly. The lower piece 
 might have the shading described in No. 4 of Lesson II. 
 
 No. 2. Draw the pieces in the order in which they are 
 numbered. The lower piece is first shaded with diagonal 
 lines, the same as the upper piece, and the shading is finish- 
 ed by drawing lines between the diagonals first drawn. 
 
 Nos. 3 and 4. In these, and in all similar figures, the up- 
 per pieces supposing that the pieces are in a horizontal 
 position should be drawn first. In most outline draw- 
 ings, and in lightly shaded drawings, the outline is made 
 heaviest on the side opposite to the direction from which 
 
STRAIGHT LINES AND PLANE SURFACES. 57 
 
 the light is supposed to come. Thus, in No. 4, the light 
 is supposed to come in the direction of the arrow / and 
 hence the outlines are made the heaviest where the shad- 
 ows would naturally fall. 
 
 No. 5. Observe the direction in which the light falls upon 
 this figure, as indicated by the arrow #, and the consequent 
 heavy outlines of those sides of the four pieces which would 
 be in shadow. 
 
 The shading in No. 7 should render each square distinct 
 from the others. 
 
 No. 8 is a pattern made up of only one figure, repeated 
 continuously, and so arranged as to cover the entire sur- 
 face. A very great variety of patterns, consisting wholly 
 of repetitions of one figure to each pattern, may easily be 
 designed, and drawn by the aid of the ruled paper. 
 
 What is the area of each of the squares, as they are num- 
 bered, in No. 7. 
 
 The area of the pattern figure in No. 8 ? 
 
 Free-hand exercises on the blackboard. 
 
 PAGE TWO. 
 
 LESSON VI. Tico-space Diagonals. By a two-space diag- 
 onal is meant the diagonal of a rectangle which is twice as 
 long as it is broad. It is a diagonal which passes over two 
 spaces on the ruled paper. 
 
 No. 1. The lines in No. 1 are two-space diagonals. They 
 should be copied, without the aid of a ruler, until they can 
 be drawn with tolerable accuracy, and with facility. At b 
 lines are first drawn as at a; and then lines are drawn in- 
 termediate between them; c is first drawn the same as b t 
 and is then filled in with intermediates. In this manner 
 great uniformity of shading may be attained. 
 
 No. 2 is drawn in a manner similar to No. 1. First trace 
 each line lightly, and continue to pass the pencil over it un- 
 til it is drawn with accuracy. 
 
 No. 3. As two square inches are represented in the dot- 
 ted rectangle, and as the line a b divides the rectangle into 
 two equal parts, therefore on each side of the line there is 
 an area equal to one square inch, 
 
 O 9 
 
 ~ " " THE 
 
58 INDUSTRIAL DEAWIXG. [BOOK NO. I. 
 
 No. 4. What area is embraced within the dotted square ? 
 Then how much is embraced within the portion a f 
 
 No. 5. What area is embraced within the dotted rectan- 
 gle ? Then what area is embraced within the portion a ? 
 
 The portion marked a in No. 4 is a triangle a figure of three sides and 
 three angles. It is an acute-angled triangle, because each angle is less than 
 a right angle. (See Lesson III.) The portion marked a in No. 5 is called 
 an obt use-angled triangle, because one of the angles is greater than a right 
 angle. 
 
 No. 6 is a figure called a rhombus. A rhombus is a figure 
 which has four equal sides, the opposite sides being parallel ; 
 but its angles are not right angles. What area is embraced 
 in the upper half of No. 6 ? In the whole figure ? 
 
 No. Y. What area is embraced in each of the parts a of 
 No. 7 ? In the central rectangle b f In the whole rhom- 
 bus? (16 square inches.) 
 
 No. 8. In the dotted figure No. 8 there are three of the 
 small squares ; hence the dotted figure contains an area of 
 three square inches. But the part b (as shown in No. 3 and 
 No. 5) contains an area of one square inch, and the part c an 
 area of one square inch ; hence the part a must contain an 
 area of one square inch also. 
 
 No. 9. What area is embraced in the rhombus No. 9? 
 (Let the pupil prove that each part a embraces an area of 
 one square inch, the same as a in No. 8.) 
 
 No. 10. What area is embraced in No. 10? How is it 
 shown that the upper part marked 1 contains an area equal 
 to one square inch ? 
 
 No. 11. What area is embraced in the star figure No. 11 ? 
 (Let the pupil prove that each of the points marked 1 con- 
 tains an area of one square inch.) 
 
 No. 12 is an octagonal or eight-sided figure. A regular 
 octagon has eight equal sides and eight equal angles; but 
 here, while the sides are equal, the angles are not all equal. 
 What is the area of each of the parts a of the octagon ? 
 Of the whole octagon ? 
 
 The shading of the central square of No. 12 is produced by carrying the 
 pencil from left to right with a running dotting motion. In industrial 
 drawing it is desirable to designate the different sides or surfaces of objects 
 
STRAIGHT LINES AND PLANE SURFACES. 59 
 
 very distinctly by the shading ; and this is one of the kinds of shading 
 very appropriate for that purpose. 
 
 No. 13. What is the area of each of the rhombuses mark- 
 ed a f (See No. 4 and No. 6.) What is the area of the 
 central star figure ? (See No. 11.) What is the area of the 
 whole octagon ? 
 
 No. 14. What is the area of each of the rhombuses marked 
 a ? (See No. 9.) Of each of the star points marked b f (See 
 No. 4.) Of the central square c f Of the whole octagon ? 
 
 PROBLEMS FOB PRACTICE. 
 
 1. Draw a rectangle similar to the dotted rectangle No. 4. but four times 
 as large. (See ELEMENTARY PRINCIPLE, page 54.) 
 
 2. Draw a triangle similar to a of No. 4, but nine times as large. How 
 must the sides compare in length with those of a of No. 4 ? What will 
 be the area of the triangle ? 
 
 3. Draw a triangle similar to a of No. 5, but containing sixteen times the 
 area of No. 5. 
 
 4. Draw a rhombus similar to No. 6, but containing twenty-five times the 
 area of No. 6. Shade it with two-space diagonals like 6, or c, of No. 1 . 
 
 5. Draw a rhombus similar to No. 7, but containing only one fourth the 
 area of No. 7. 
 
 6. Draw a figure similar to the a portion of No. 8, but sixteen times as 
 large. 
 
 7. Draw a rhombus similar to No. 9, but containing twenty-five times the 
 area of No. 9. 
 
 8. Draw a rhombus similar to No. 10, but having four times the area of 
 No. 10. 
 
 9. Draw a star figure similar to No. 11, but containing nine times the area 
 of No. 11. 
 
 10. Draw an octagon similar to No. 12, but containing four times the area 
 of No. 12. 
 
 11. Draw an octagon similar to No. 13, but containing nine times the area 
 of No. 13. Divide it as No. 1 3 is divided, and mark within each rhombus, 
 its area, and mark the area of the star also. 
 
 12. Draw a figure similar to No. 14, but sixteen times as large, and mark 
 within the parts a, 6, and c the area of each. 
 
 Let problems similar to the foregoing be drawn on the 
 blackboard, or selections from them, at the option of the 
 teacher. 
 
 LESSON VII. No. 1 is a two-space diagonal square ; and 
 No. 2 is the same in a different position. The area of No. 
 1 can easily be counted up, when it is seen that each of the 
 
60 INDUSTRIAL DRAWING. [BOOK NO. I. 
 
 parts marked 1 is equal to one square inch. Hence the fig- 
 ure contains five square inches. 
 
 The area of a two-space diagonal square, or of any two- 
 space diagonal rectangle, may be found by the following 
 modification of Rule I. : 
 
 To find the area of a two-space diagonal rectangle : 
 
 RULE B. Multiply the length in two-space diagonals by 
 the breadth in two-space diagonals, and FIVE times the prod- 
 uct will be the area, in measures of the primary erect squares. 
 
 Thus, in No. 1, multiply the length 1 by the breadth 1, and 
 5 times the product will be the area: 5 square inches. 
 
 What is the area of the two-space diagonal square No. 3 ? 
 
 Solution. Multiply the length 2 by the breadth 2, and 
 the product will be 4, which, multiplied by 5, will give 20 
 square inches the area. The same result will be found by 
 counting the squares, etc. 
 
 What is the area of No. 4 ? No. 5 ? No. 6 ? 
 
 No. 7 is the same form of star seen in No. 14 of Lesson 
 VI. ; and No. 8 is the same form that is seen in No. 13. In 
 drawing these figures, first trace the outlines very lightly; 
 and do not mark firmly until the positions of all the lines 
 are clearly determined. Use no ruler. 
 
 No. 9 shows two octagons intersecting each other in a 
 diagonal direction, and in such a manner that the rhombus 
 a is common to both. Any octagon may have an octagon 
 intersecting it in this manner on all of its four divisions; 
 and when the series is continued they form the pattern 
 seen in No. 10 sometimes seen in oil-cloths, carpets, etc. 
 
 No. 1 1 shows a series of octagons intersecting one another 
 vertically and horizontally, instead of diagonally as in No. 
 10. In No. 10 the rhombuses, and in No. 11 the star fig- 
 ures, are represented as shaded with a light tint of India ink. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw, on the drawing-paper, a two-space diagonal square, similar to 
 No. 1, but embracing twenty-five times the area of No. 1. Draw another 
 within the last, embracing nine times the area of No. 1. What will be the 
 lengths of the sides of each, in two-space diagonals ? The area of the small- 
 er square ? Of that portion of the larger square outside of the smaller ? 
 
STRAIGHT LINES AND PLANE SURFACES. 61 
 
 2. Draw a rectangle similar to No. 4, but embracing four times the area 
 of No. 4. 
 
 3. Draw a rectangle similar to No. 6, but embracing four times the area 
 of No. 6. 
 
 4. What area would be included in a two-space diagonal rectangle hav- 
 ing a length of eight two-space diagonal measures, and a breadth of five ? 
 (See Rule B.) 
 
 5. Draw a pattern similar to No. 10, but with the figures embracing four 
 times the area of those in No. 10. 
 
 6. Draw a pattern similar to No. 11, but with the figures embracing four 
 times the area of those in No. 11. 
 
 Free-hand drawing of problems similar to the foregoing 
 on the blackboard. 
 
 LESSON YIIL Three-space Diagonals. By a three-space 
 diagonal is meant the diagonal of a rectangle which is three 
 times as long as it is broad. Thus, the diagonal of the rect- 
 angle at cr, No. 1, passes over three spaces, and divides the 
 rectangle into two equal parts. As the rectangle includes 
 an area of three square inches, each half of it, as marked ^, 
 has an area of one and a half square inches. 
 
 No. 2. What area is included in the dotted rectangle No. 
 2 ? In each of the three parts, a, b, and c ? 
 
 No. 3 is a three -space diagonal square. Observe that 
 each of the parts marked 1 has an area of one and a half 
 square inches. Then what is the area of the whole square ? 
 (Ans. 10 square inches.) 
 
 No. 4 is a rhombus. What is its area ? The area of the 
 dotted rectangle ? 
 
 No. 5 is a three -space diagonal rectangle. Its area is 
 easily found, by counting, to be twenty square inches. But 
 the area of any three-space diagonal square, or other three- 
 space diagonal rectangle, however large, may easily be found 
 by the following rule, also a modification of Rule I. 
 
 To find the area of a three-space diagonal rectangle : 
 
 RULE C. Multiply the length in three-space diagonals by 
 the breadth in three-space diagonals, and TEN times the prod- 
 uct will be the area, in measures of the primary erect squares. 
 
 Thus, in the square No. 3, the length 1, of one side, multi- 
 plied by 1, the length of another side, gives the product 
 
G2 INDUSTRIAL DRAWING. [BOOK NO. I. 
 
 1, which, multiplied by 10, gives 10 square inches as the 
 area. 
 
 Apply the rule to No. 5, and test the result by counting. 
 What is the area ? 
 
 No. 6. What is the area of the inner dotted rectangle? 
 Of the large rectangle ? Then what is the area of the space 
 included between the two ? 
 
 "No. 7. The area of the space included within the dotted 
 figures,,./, 2 9 3 is seen, by counting, to be five square inches. 
 But the area of the part marked a is one and a half square 
 inches, and the area of c is the same, the two parts a and c 
 making three square inches. Therefore the part b embraces 
 two square inches. 
 
 No. 8. What is the area of the rhombus No. 8 ? 
 
 No. 9 is a three -space diagonal octagon. What is the 
 area of each of the parts , #, c, and df Of the inner dotted 
 square ? Of the whole octagon ? 
 
 No. 10. What is the area of the four rhombuses a, #, c, d? 
 Of the star g ? Of the whole octagon ? 
 
 No. 11. What is the area of the four rhombuses a, , c, d? 
 Of the four parts e, /, g, h ? Of the central square k f Of 
 the whole octagon ? 
 
 No. 12. What is the area of the star in No. 12 ? What is 
 the area of the star in No. 8 of Lesson VII. ? What is the 
 difference in their areas ? 
 
 Let all the foregoing be drawn on the drawing-paper. 
 
 PROBLEMS FOR PRACTICE. 
 
 1 . Draw, on the drawing-paper, a three-space diagonal square that shall 
 contain 9 times the area of No. 3. 
 
 2. Draw a rhombus similar to No. 4, but containing nine times the area 
 of No. 4. Within the rhombus thus drawn, and equidistant from its sides, 
 draw a rhombus containing four times the area of No. 4. Within this latter, 
 and equidistant from its sides, draw another equal to No. 4. Mark the 
 rhombuses thus drawn No. 1, No. 2, and No. 3, beginning with the smallest, 
 and mark the area of each. 
 
 3. Draw a rectangle similar to No. 5, but containing sixteen times the area 
 of No. 5. Draw one within this latter containing four times the area of No. 5. 
 
 4. Draw a rhombus similar to No. 8, but containing four times the area 
 of No. 8. 
 
 5. Draw an octagon similar to No. 9, but containing four times the area 
 of No. 9. 
 
STRAIGHT LINES AND PLANE SURFACES. 63 
 
 6. Draw an octagon similar to No. 10, but with other interlacing octa- 
 gons on its diagonal sides, similar to No. 10 of Lesson VII. 
 
 7. Draw an octagon similar to No. 11, but with other interlacing octa- 
 gons on its vertical and horizontal sides, similar to No. 11 of Lesson VII. 
 
 8. Draw a star similar to g of No. 10, but having four times the area of 
 ff, and inclose it with an interlacing square similar to No. 7 of Lesson VII. 
 
 Free-hand drawing of problems similar to the foregoing 
 on the blackboard. 
 
 PAGE THREE. 
 
 LESSON IX. This lesson consists of a series of net-work, 
 the finer examples of which, when used in drawing or en- 
 graving, for the purposes of shading, are called hatching. 
 
 No. 1 is a coarse diagonal net-work, in the form of squares. 
 
 No. 2 is drawn, in the first place, in the same manner as 
 No. 1 ; after which another set of lines is put in, in both 
 diagonal directions, intermediate between those first drawn. 
 
 No. 3 is first drawn the same as No. 2, after which another 
 set of lines is put in intermediate between those first drawn. 
 This kind of hatching is seen in No. 6 of the next lesson. 
 
 No. 4 is a coarse two-space diagonal net-work. 
 
 No. 5 is first drawn the same as No. 4, and is then filled 
 in with another set of lines between those first drawn. 
 
 No. 6 is a fine hatching, first drawn the same as No. 5, and 
 then filled in with another set of lines intermediate between 
 those first drawn. A sharp-pointed, hard pencil is required 
 for this shading. 
 
 No. 7 is a coarse three-space diagonal net-work. When 
 filled in with two lines intermediate between those here 
 drawn, it forms a good hatching for some kinds of shading. 
 
 All the examples in this lesson, which should be copied 
 without the aid of a ruler, will furnish good exercises in 
 drawing straight, uniform, and equidistant lines. The di- 
 rections, and the distances apart, are given in the ruling of 
 the paper. 
 
 Free-hand drawing of Lesson IX. on the blackboard. 
 
 LESSON X. No. 1 gives the outline of a star-shaped fig- 
 ure ; and No. 2 is the same divided into eight pairs of wings 
 by a vertical, a horizontal, and two diagonal lines, and then 
 
64 INDUSTRIAL DRAWING. [BOOK NO. T. 
 
 shaded. This peculiar star-shaped figure is a common form 
 of ornament in examples of Byzantine art. What is the 
 area of each of the eight pairs of wings of No. 2 ? Of the 
 whole star? 
 
 No. 3 is a star similar to No. 2, inclosed in a diagonal 
 square, but with twice the length of sides of No. 2. How, 
 then, does its area compare with that of No. 2 ? What is 
 the area of the diagonal square? (See Rule A, page 56.) 
 
 No. 4 is a hexagonal pattern covering the entire surface. 
 A hexagon is a plane figure of six sides and six angles. 
 When the sides are all equal, and the angles all equal, it is 
 a regular hexagon. What is the area of one of the hexagons 
 of No. 4? 
 
 No. 5 is a pattern composed of an elongated octagonal 
 figure and a square, the two forms combined covering the 
 whole surface. What is the area of one of the octagons ? 
 
 Nos. 4 and 5 may be varied so as to embrace a great va- 
 riety of similar patterns by changing the relative lengths 
 of the sides. Numerous oil-cloth and carpet patterns are 
 formed on this basis. Additional variety is given to Nos. 
 4 and 5 by the bordering, as indicated at a and b. Observe 
 that the exact distance of the inner lines from the outer 
 border is given by the intersections of the ruled lines. 
 
 No. 6 is an original Moorish pavement pattern, called mo- 
 saic / but it is now common, with various modifications, in 
 pavements, oil-cloths, etc. It is easily drawn on the ruled 
 paper. The hatching used in the shading is that of No. 3 
 of Lesson IX. 
 
 No. 7 is an elongated hexagonal link pattern, for borders, 
 etc. Observe the position of the heavy shaded lines on the 
 right hand and lower sides. 
 
 No. 8 is a double interlacing square. First trace lightly. 
 
 PROBLEMS FOR 
 
 1. Draw a star similar to No. 2, but nine tittles as lai'ge. What will le 
 its area ? 
 
 2. Draw a star similar to No. 3, but four times as large, and inclose it 
 with a diagonal square similar to the inclosure of No. 3. Centrally within 
 each of the corner diagonal squares similar to o, 6, c, d of No. 3 place a star 
 Ufce No, 2 within its qwi] diagqnaj square, ' ...... \ 
 
STRAIGHT LINES AND PLANE SURFACES. 65 
 
 3. Draw a pattern made up of hexagons similar to those of Xo. 4, but of 
 four times the area, and complete each in a manner similar to a. 
 
 4. Draw a pattern like No. 5, but with the exception that the shade; 1 , 
 squares shall contain four times the area of those in No. 5. Complete the 
 several hexagons in a manner similar to b. 
 
 5. Draw a pattern composed wholly of figures like No. G. The addition 
 at a will show how the several figures are to be connected. Shade all like 
 No. 6j or use different tints of India ink. 
 
 G. Draw a link pattern like No. 7, with the exception that each link shall 
 be two spaces longer than in No. 7, but of the same width. 
 
 free-hand drawing of Lesson J, and the problems, on 
 the blackboard. 
 
 LESSON XL No. 1 and No. 2 are the elements of slightly 
 different patterns formed on the basis of either an erect or 
 a diagonal square. 
 
 No. 3 is the basis of an octagonal pattern. At No. 4 the 
 short lines a a a a show the method of marking out an in- 
 ner octagon whose sides shall be uniformly distant from and 
 parallel to the sides of the larger octagon ; and No. 5 shows 
 the figure completed. 
 
 No. 6 is the pattern, as carried out, from the preceding 
 three figures. The central octagons should be shaded with 
 a light tint of India ink, and the squares with the running 
 dots in horizontal lines. The different kinds of shading 
 used in the patterns given in these books denote the variety 
 of colors employed when the pattern is used either in orna- 
 mental art, or in the designs of oil-cloths, carpets, wall-pa- 
 per, etc. 
 
 No. 7 is a dodecagon a figure of twelve sides and twelve 
 angles. When the sides are equal, and the angles equal, 
 the figure is a regular dodecagon. What is the area of 
 each of the parts a, , c, d of this figure ? Of the central 
 dotted square ? Of the whole dodecagon ? 
 
 No. 8 is a dodecagon divided into a border of squares and 
 triangles, and a central hexagon. What is the area of each 
 of the two vertical squares ? Of each of the four two-space 
 diagonal squares? Of each of the six white triangles that 
 incloses the small dark triangle? Of the central hexagon. 9 
 Of the whole figure ? 
 
66 INDUSTRIAL DRAWING. [BOOK NO. I. 
 
 No. 9 is a pattern composed wholly of intersecting dodec- 
 agons like No. 8. Each figure, it will be seen, forms a por- 
 tion .of six other like figures surrounding it. This combina- 
 tion of dodecagons is an original pavement pattern taken 
 from a Roman church in the Byzantine period of Roman 
 history. It is an admirable specimen of geometrical mosaic 
 work so common in that period; and it must have been 
 formed upon lines drawn precisely like those given on our 
 ruled paper; for in no other manner could a series of such 
 figures be drawn with accuracy. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw a pattern formed of figures like No.l, allowing the figures to 
 touch vertically and horizontally. Give to these figures the running dot 
 shading; and shade the intermediate figures formed between them with a 
 light tint of India ink. 
 
 2. Draw a pattern formed of figures like No. 2, and shade the diagonal 
 cross with a tint of India ink, and the intermediate figures with the running 
 dot shading. Leave the diagonal squares unshaded. 
 
 3. Draw a double interlacing square like No. 8 of Lesson X., and cen- 
 trally within it draw a figure like No. 8 of Lesson XI. 
 
 Patterns similar to No. 6 and No. 9 may be drawn 4, 9, 16, 25, or 3G 
 times, etc., larger than the figures here given, by increasing the lengths of 
 the sides 2, 3, 4, 5, or G times, etc. , according to the principles explained 
 on pages 53 and 54. 
 
 Free-hand drawing of Lesson XI., and the problems, on 
 the blackboard. 
 
 PAGE FOUR. 
 
 For convenience, we now drop the method of grouping 
 the examples under the head of Lessons, and here designate 
 them as separate Figures. 
 
 Fig. 1 represents an ancient Egyptian pattern of a braid- 
 ed or woven mat on which the king stood. It is formed of 
 flat strands of only two colors, each strand passing continu- 
 ously, in a diagonal direction, over two strands of different 
 colors, and then under two. The portion of a strand pre- 
 sented at one view is rectangular, and twice as long as it is 
 broad. All the lines in this figure are diagonals, and should 
 be drawn without the aid of a ruler. The shaded strands 
 
STRAIGHT LINES AND PLANE SURFACES. 67 
 
 may be gone over, first lightly, with India ink, and then 
 with pencil. 
 
 Fig. 2 is another Egyptian pattern of matting, in only 
 two colors, but presenting a view quite different from Fig. 1. 
 Here each light strand passes continuously over two dark 
 strands, and then under three dark strands. The dark 
 strands may be considered as the icarp, and are arranged 
 side by side, all running diagonally; and then the light 
 strands, being the filling, are woven in diagonally, as stated, 
 at right angles to the warp. Patterns similar to Figs. 1 and 
 2 may be formed of worsted of two colors. 
 
 Fig. 3 is the pattern of an Arabian pavement found at 
 Cairo, formed of black and of white marble, except the diag- 
 onal squares, which are of red tile. Go over the diagonal 
 squares once, and the rectangles twice, with a light tint of 
 India ink. The Arabians imitated the universal practice of 
 the Komans of covering the floors of their public buildings, 
 mosques, etc., with mosaic patterns arranged on a geomet- 
 rical system. 
 
 Fig. 4 is a decorative pattern, in different colors, from an 
 ancient Egyptian tomb. It is supposed to have suggested 
 the meander, or fret, to the Greeks. (See page 6 of draw- 
 ings.) The ruler may be used in this figure, after first indi- 
 cating the lines with the pencil alone. 
 
 Fig. 5 is an ancient Egyptian pattern, in different col- 
 ors, from the painting on a tomb. In most of these Egyp- 
 tian paintings the colors are as fresh as if put on yesterday. 
 
 Fig. 6 is an octagonal pattern forming intermediate fig- 
 ures of diagonal squares. The ruled lines furnish conven- 
 ient guides for forming the width of the octagon border. 
 
 Figs. 7 and 8 are samples of mosaic patterns based upon 
 two of the forms of the central eight-pointed star figure, so 
 common in specimens of Byzantine ornamental art. 
 
 Fig. 9 is another modification of the star figure in mosaic, 
 here inclosed by an interlacing border. 
 
 Fig. 10 represents a portion of a mosaic pavement, in dif- 
 ferent colors, from the ruins of Pompeii. Observe that the 
 running dotted shading is done very lightly, and with a 
 sharp pencil, in Figs. 7, 8, and 9 ; but much more heavily, and 
 with a blunt pencil, in a portion of Fig. 10. 
 
68 INDUSTRIAL DRAWING. [BOOK NO. I. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw a pattern similar to Fig. 1, but with strands of only half the 
 v/idth of those there represented. The rectangles shown will be of only 
 half the length of those shown in Fig. 1. 
 
 2. Draw a pattern similar to Fig. 2, but with strands of only half the 
 width of those there represented. 
 
 3. Draw a pattern similar to Fig. ?>, but make the lines of every figure 
 contained in it twice the length of those there represented. How, then, 
 will the area of each of the figures compare with the area of a similar fig- 
 ure in the copy ? 
 
 4. Draw a pattern similar to and arranged like Fig. G, but make the di- 
 agonal squares one quarter of the area of those there represented, and the 
 octagons only three spaces in height and three in width. Let the borders 
 of the figures be only straight lines. Give to the primary diagonal squares 
 the dotted shading, and leave the octagons unshaded. This will form a 
 handsome oil-cloth pattern. 
 
 5. Draw a pattern similar to Fig. 7, and of the same proportions, but 
 containing an area 4 times that of the copy. 
 
 6. Draw a pattern similar to Fig. 8, and of the same proportions, but con- 
 taining an area 4 times that of the copy. 
 
 7. Draw Fig. 9, extended upward, but make dark diagonal and smaller 
 squares in place of the dark erect- squares now shown ; then draw the same 
 pattern on the right, and also on the left, touching at the extreme angles, 
 so as thus to cover the whole paper with a harmonious pattern. 
 
 Free-hand drawing of the figures of page 4, and of all the 
 problems except the 1th, on the blackboard. 
 
 PAGE FIVE. 
 
 Figs. 11, 12, 13, and 14 are plain border patterns; 11 and 
 12 being forms of the Grecian /;<#, to be noticed hereafter. 
 
 Figs. 15, 16, 17, 18, 19, and 20 are representations of flat 
 braid of 3, 4, 5, 7, 9, and 1 1 strands. In Fig. 1 6 the next move- 
 ment is to turn the a strand upward, break it down on the 
 dotted line 1 2, and pass it over 1) and under c. Then break 
 the strand d downward on the line 3 4-, and pass it under a, 
 and so on continuously. 
 
 In Fig. 17 the movement is continuously from the out- 
 side, over one and under one / in Fig. 18, over one and under 
 two, beginning on the left ; in Fig. 19, beginning on the left, 
 over one, under two, and over one; in Fig. 20, beginning on 
 the left, over one, under two, and over two. 
 
STEAIGHT LINES AND PLANE SURFACES. 69 
 
 Fig. 21 is a pattern of interlacing diagonal net-work, em- 
 bracing diagonal squares that are distinguished by three 
 forms of shading or coloring. 
 
 Fig. 22 represents an embroidered pattern brought a few 
 years ago from the East Indies. Here \\\Q forms alone can 
 be given, as the colors can not be represented. In the orig- 
 inal pattern the four stars of each cross-shaped figure are 
 white or silver, on a black ground inclosed by a silver line ; 
 and the small dark squares and the straight lines connect- 
 ing them are golden. 
 
 Fig. 23 is the filling up of a mosaic pattern of Byzantine 
 pavement. The numerous symmetrical figures that may be 
 discerned in it show both the intricacy and at the same time 
 the harmonious simplicity of the Byzantine style. By the 
 aid of the ruled paper similar patterns of almost endless va- 
 riety may be designed. 
 
 For free-hand drawing on the blackboard take Figs. 13, 14, 
 15,16,17, and 18. They may be shaded slightly with col- 
 ored chalks, so as to make the interlacings plain. 
 
 PAGE SIX. 
 
 Fig. 24 is the simple generating form of the Grecian sin- 
 gle fret, or meander a species of architectural ornament 
 consisting of one or more small projecting fillets, or rectan- 
 gular bands, meeting, originally, in vertical and horizontal 
 directions only. Although this ornament was originated 
 by the Greeks, quite similar rudimentary forms of the fret 
 have been found among the Chinese and the Mexicans. The 
 Arabians extended the Greek fret to diagonal and curved 
 interlacing bands; and the Moors afterward extended it to 
 that infinite variety of interlaced ornaments, formed by the 
 intersection of equidistant diagonal lines, which are so con- 
 spicuous a feature in the ornamentation of the Alhambra. 
 In addition to the most important of ihz plane surface Gre- 
 cian frets, here given, and some of the Moorish that are best 
 adapted to drawing purposes, we have also shown several 
 of them in the second number of the Drawing Series, in their 
 more natural form in architecture, as solids. 
 
 Fig. 24 requires no directions for drawing it. Fig. 11, on 
 
70 INDUSTRIAL DRAWING. [i^OK NO. I. 
 
 page 5, is the same as this, with the exception that Fig. 11 
 has an interlacing band running centrally through it. The 
 ruler may be used for all the drawings on this page ; but 
 the shading of the darker parts (by India ink) should be 
 lighter than the copies. 
 
 Fig. 25 is a single fret, with the band returning upon it- 
 self at regular intervals. In drawing the frets, draw the 
 shaded portions only, and, as you proceed, trace a very faint 
 dotted line through the central part of the fret, to distin- 
 guish it from the unshaded intermediate spaces. The frets 
 are best shaded, mainly, by India ink ; but where there are 
 two interlacing bands, one of them should have the running 
 dot shading. 
 
 Fig. 26 is also a single fret, a little more complicated than 
 the former two. 
 
 Fig. 27 is a double fret, formed of two interlacing bands. 
 A single band should first be drawn throughout, tracing it 
 lightly at first ; the spaces for the other band will then be 
 readily apparent. 
 
 Fig. 28 is a double fret, formed by one single fret backing 
 upon another single fret of the same form. 
 
 Fig. 29 is an interlacing double fret. Trace one of the 
 bands throughout very lightly before beginning with the 
 other, so as not to interfere with the crossings. The ruler 
 should not be used (if at all) until the entire fret is clearly 
 but lightly marked out with the pencil alone. Observe 
 that, in all interlacing fret-work, any one band passes alter- 
 nately first over and then under another. 
 
 Fig. 30 is the same as Fig. 29, but with spaces left be- 
 tween the bands for paneling. Observe the vertical bands 
 marked a b in Fig. 29. These are separated in Fig. 30 for 
 the panels, which, in Grecian architecture, were ornamented 
 with various devices. 
 
 Fig. 31 is an interlacing double fret, similar to Fig. 30, in- 
 verted end for end,- with spaces for ornamental panels. In 
 all cases of double frets it is best to draw one of the frets 
 throughout before beginning the other. 
 
 The fret here shown, with its panels, although strictly 
 Grecian, was one of the forms of Roman pavement that has 
 
STRAIGHT LINES AND PLANE SURFACES. 71 
 
 been found in the ruins of Pompeii. The two bands com- 
 posing the fret, which are here differently shaded, were. of 
 white marble, formed of the same number of square pieces 
 as is designated by the ruling of the paper ; and the inter- 
 mediate spaces, here left unshaded, were of black marble. 
 
 Fig. 32 is an interlacing double fret with panels. 
 
 Fig. 33 is a double fret with panels, but is not interlacing. 
 Take away the panels, and the frets are doubly backed upon 
 one another. 
 
 Fig. 34 is an interlacing double fret, formed of distinct 
 portions connected by a rectangular link. 
 
 Fig. 35 is a diagonal and horizontal interlacing double 
 fret; and, as its form shows, is not Grecian. It is of Moor- 
 ish origin, and is one of the numerous kinds of complicated 
 frets, painted in various colors, and on variously colored 
 grounds, on panels of the walls of temples. 
 
 Fig. 36 is an interlacing double fret, also of Moorish ori- 
 gin. 
 
 For free-hand blackboard exercises take Figs. 28, 29, 34, and 
 36. They may be shaded lightly. 
 
 PAGE SEVEN. 
 
 Figs. 37 and 38 are borders of fret -work, formed after 
 Moorish and Arabian patterns. 
 
 Fig. 39 is an Arabian pattern of a mosaic pavement, with 
 some of the smaller subdivisions omitted. The peculiar 
 star-form of ornamentation here shown, which is of Byzan- 
 tine origin, was also used by the Arabians. 
 
 Fig. 40 is a diagonal double fret, which has been slightly 
 varied from an Arabian pattern to fit it to our purpose. In 
 copying it, either one of the bands should first be lightly 
 traced throughout. 
 
 Fig. 41 consists of two four-pointed stars interlacing, so as 
 to show an eight-rayed or eight-pointed star. In drawing 
 it, first take the centre, c, then the four inner vertical and 
 horizontal points marked 3, then the four inner diagonal 
 points marked 2. Also take the eight ray points in a sim- 
 ilar manner. Trace lines very faintly from the outer to the 
 inner points; then trace an inner set of lines equidistant 
 
72 INDUSTRIAL DRAWING. [BOOK NO. I. 
 
 from these; after which mark firmly every alternate ray 
 border across the other border lines, when, the intersections 
 being distinct, the whole can easily be finished. The rays 
 may be made either longer or shorter than those in the 
 drawing, it being considered that two diagonal spaces are 
 nearly equal to three vertical or horizontal spaces. 
 
 Fig. 42 is copied from an Arabian pattern of a mosaic 
 pavement in three colors ; white (or cream-colored), red, and 
 black. The groundwork may be said to consist of elon- 
 gated hexagons connected by interlacing diagonal squares ; 
 then there is a central interlacing fret ingeniously varied to 
 adapt it to the other portions, so as to make a perfectly har- 
 monious meander. In drawing it, first trace the three parts 
 lightly in the order here described. 
 
 Fig. 43 is also copied from an Arabian mosaic, in white, 
 red, and black. It is taken from a pavement in Cairo. It 
 will be seen that the diagonal lines here are all two-space 
 diagonals; and as the drawing conforms strictly to the orig- 
 inal, it must be true that the original pattern was formed 
 by the aid of precisely such horizontal and vertical lines as 
 we have used for guides on the ruled paper. 
 
 Observe how beautifully the nine small figures, in three 
 colors, and three diiferent forms, fill out the six -pointed 
 star-shaped figure at the intersection of the several bands. 
 The entire pattern is a fine example illustrating the fund- 
 amental principles of decoration ; that all ornament should 
 be based upon a geometrical construction, and that every 
 pattern should possess fitness, proportion, and harmony, the 
 result of all which will be a feeling of satisfied repose, with 
 which every such decoration will impress the beholder, 
 leaving nothing further to be desired within the scope of 
 the ornamentation. 
 
 For free-hand blackboard exercises take Figs. 37, 38, and 
 41. Observe the heavy shading on those sides that would 
 be in shade if the light came in the direction indicated by 
 the arrow. 
 
CURVED LINES AND PLANE SURFACES. 73 
 
 III. CURVED LINES AND PLANE SURFACES. 
 
 PAGE EIGHT. 
 
 A curved line is one which is continually changing its di- 
 rection. If the curve be uniform, it forms part of the cir- 
 cumference of a circle. 
 
 A circle is a plane bounded by a single curved line called 
 its circumference, every part of which is equally distant 
 from a point within it called the centre. The circumference 
 itself is usually called a circle. A straight line drawn from 
 the centre to any part of the circumference is called a ra- 
 dius. 
 
 Fig. I. At A are six uniform curves of five spaces' span 
 (five inches), and a depth of one space ; and at Fig. 1 this curve 
 forms part of a perfect circle. At a and b the directions 
 of the curves are changed ; but all combined form a harmo- 
 nious and equally balanced figure, because the additions a 
 and b are uniform in position and curvature. These figures 
 should be drawn with the compasses, using the pencil to 
 make the connections of the curves uniform. Let the pupil 
 find the centres from which the curves a and b are struck. 
 
 Fig. 2 is formed of the same pattern curve used in differ- 
 ent positions, but all combined to form a harmonious figure. 
 If either of the half curves, c or d, were omitted, or changed 
 in position, the harmony of the figure would be destroyed. 
 At Yihe same form of curve is used. Let the pupil find the 
 centres from which the curves are struck. 
 
 Fig. 3 is also a harmonious figure, described wholly by the 
 compasses ; but the inner border lines from e to h and from 
 g to f are described with a less radius than that used for 
 the other curves. The curves e i and g % are, each, only half 
 of the pattern curve, and are described from the points 1 
 and 2. 
 
 Fig. 4. At _Z? is another pattern curve representing a span 
 of six inches and a depth of one inch, described from the 
 centre c, with a radius of five inches. In the shaded four- 
 angled figure the pattern curve is used in four different po- 
 
 D 
 
74 INDUSTRIAL DRAWING. [BOOK NO. I. 
 
 sitions. The centres from which these curves are described 
 are easily obtained on the ruled paper. 
 
 Fig. 5 is formed wholly of combinations of the pattern 
 curve .Z?, with two half curves at the base, which, however, 
 are not described from the same centres as the curves with 
 which they unite. 
 
 Fig. 6 is also formed of combinations of the pattern curve 
 JB. Remembering that all these curves are described with 
 a radius of five spaces (or five inches), it will be easy for the 
 pupil to find their centres. 
 
 Fig. 7. We have here, at (7, a new pattern curve, of a span 
 of three spaces, and a radius of one space and a half. This 
 curve forms more than a quarter of the circumference of a 
 circle, as may be seen in the completed circle at b. 
 
 Fig. 8 is formed by very simple combinations of the en- 
 tire pattern curve C. 
 
 Fig. 9 is formed by adding, in Fig. 8, portions of the pat- 
 tern curve to the upper and lower extremities. 
 
 From the foregoing figures it will be seen that we may 
 take different portions of any one regular curve, and com- 
 bine them in a great variety of harmonious patterns. It is 
 only to a very limited extent, however, that we can combine 
 uniform curves of different radii in the same pattern, with- 
 out destroying that gracefulness of form which is required 
 to please the eye, and give to the mind a feeling of repose 
 in the contemplation. 
 
 We now come to the consideration of irregular curves, 
 such as can not be drawn, to any great extent, by the aid of 
 compasses. 
 
 Fig. 10. We have here a bell-shaped figure, drawn uni- 
 formly on both sides of the central and balancing line a b. 
 We must draw one side by the eye alone, and give to the 
 waving line as graceful a curvature as we can ; and a great 
 many different forms and proportions will answer the re- 
 quirements of graceful curvature ; but the line must, never- 
 theless, be such as will please a cultivated eye. Having, 
 therefore, the point x and the central line a b, we connect a 
 and x by a curved line that pleases the eye. If, now, we 
 can draw a line exactly like it on the other side of a #, we 
 
CURVED LINES AND PLANE SURFACES. <"5 
 
 shall have a figure of harmonious form, whether a a; be the 
 most graceful line, by itself, that could be drawn, or not. 
 But if a x should be drawn of the most graceful form pos- 
 sible, and a y just as graceful in itself, but differing from u 
 x, the combination of the two graceful forms would be dis- 
 cordant, and make an inharmonious figure, because wanting 
 in symmetry of parts. 
 
 Having, therefore, a x, we designate in it any number of 
 points in which it crosses either the horizontal or vertical 
 lines of the paper. Let 1, 2, 3, 4, and 5 be these points. 
 Then dot, lightly, the corresponding points 1, 2, 3, 4, 5 on the 
 other side of a b, and through them trace the curve, at first 
 lightly, and afterward fill it out to correspond with the line 
 a x. The figure is thus made perfectly symmetrical. The 
 top of the figure may be either pointed or circular ; and the 
 bell may be longer, or narrower, or broader, or any one out 
 of a great variety of suitable proportions ; yet if the two 
 sides are alike, the figure will not be unpleasant to the eye. 
 The two inner dotted curves give different proportions for 
 the bell, while the base remains the same. 
 
 Fig. 11 represents the harmonious outlines of a leaf form. 
 Observe that the border lines of the leaf pass through the 
 points 1, 2, <?, 4, 5, on each side of the central line c d, cor- 
 responding to one other. Thus the points 1 are, each, one 
 space from the central line ; the points , each two spaces ; 
 the points 3, each three spaces, etc. la this manner the two 
 sides of the leaf are made perfectly symmetrical. The radi- 
 ations of the veins from the midrib of the leaf are also sym- 
 metrical. 
 
 Fig. 12 is another form of leaf, and although very differ- 
 ent from Fig. 11 is equally harmonious in proportions. The 
 pupU should now have no difficulty in copying it accurately 
 on the ruled paper. 
 
 Fig. 13. Here is a group of eight uniform leaves, uniform- 
 ly radiating from a common centre. The four points a,c, 
 , (7, are, each, four spaces from the centre, and on lines at 
 right angles with one another. Hence these four points are 
 symmetrically arranged. The other four points, , d, /, A, 
 are in the diagonals, which are intermediate between the 
 
76 INDUSTRIAL DRAWING. [BOOK NO. I. 
 
 other lines, and are, each, at the distance of three diagonals 
 from the centre, a trifle greater than the distance of the 
 other four points, but sufficiently accurate to give the figure 
 all the symmetry required. Then each leaf must be drawn 
 by hand. At D the method of beginning the drawing is 
 shown ; at E it is carried still farther forward ; and in Fig. 
 13 it is completed. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw Fig. 1, and draw within it a curved line one space distant from 
 the outer border line. 
 
 2. Draw Fig. 2 in the same manner. 
 
 3. Draw Y with an outer border line one space distant from the inner 
 border line, and tint the space between the two. 
 
 4. Draw Fig. 3 inverted, with the upper part downward. 
 
 5. Draw the upper part of Fig. 5 on each of the four sides of the square 
 H, so that it shall project in four directions from the central square. 
 
 6. Draw Fig. 10 of the same length as at present, but with a base, x y, of 
 ten spaces. 
 
 7. Draw Fig. 11 eight spaces broad at 3 3. 
 
 8. Draw Fig. 12 four spaces broader than in the book, and give to the fig- 
 ure a graceful curve throughout. 
 
 Blackboard Exercises. Figs. 1 to 9 inclusive. For these 
 a pair of chalk-crayon compasses will be needed. See di- 
 rections, page 49. 
 
 PAGE NINE. 
 
 Fig. 14. The circle here drawn from the centre c has 
 longitudinal divisions, by pure curves drawn from different- 
 points on the central vertical line. 
 
 Fig. 15. The irregular curve on the left of the central line 
 a b had first to be drawn by the eye the design being to 
 form a graceful, well-proportioned curve. Its counterpart, 
 on the other side of the line, is to be drawn in the manner 
 directed for drawing the preceding figures, 10, 11, and 12. 
 
 Fig. 16 is Fig. 15 completed in outline; the two portions 
 on each side of the central line a b being made symmetrical. 
 
 Fig. 17, the completed form, is a portion of a Pompeian 
 border ornament, painted in yellow on a black ground. The 
 central figure is proportioned symmetrically also. The 
 whole is, doubtless, of Greek origin. 
 
CURVED LINES AXD PLANE SURFACES. 77 
 
 Figs. 18 and 19 are symmetrically formed leaves, the 
 drawing of which now requires no farther explanation. Ob- 
 serve how perfectly the one side corresponds to the other, 
 and how easy it is to draw such forms on the ruled paper. 
 
 Fig. 20, a common form of Grecian conventional leaf or- 
 namentation, is arranged symmetrically on each side of a 
 vertical central line. 
 
 Fig. 21, a conventional leaf form of ornamentation, can 
 now be easily drawn. 
 
 Fig. 22, also a leaf form, is (or should be) equally bal- 
 anced on each side of the line a b. Observe that the points 
 c and d are at equal distances, horizontally, from the line a 
 by so also the points e and/*, and g and h ; and each leaf di- 
 vision on one side must have a corresponding division on 
 the other side. 
 
 Fig. 23, a Grecian pattern, shows one of the many forms 
 of conventional leafage found on Grecian vases. The leaf 
 forms which the Greeks (and all the ancients) used in their 
 ornamentation were far removed from any natural types; 
 and while they are constructed on the general principles 
 which reign in all plants, they never attempt to represent 
 any particular species. The moderns, departing from the 
 true principles of ornamentation which prevail in ancient 
 art, attempt to give representations of real leaves, flowers, 
 etc., being even guilty of the anomaly of sculpturing the 
 most delicate flowers in stone ! 
 
 Observe that the two portions of each pattern figure are 
 arranged, symmetrically, on each side of a central line (as a 
 o), which equally divides not only the inclosing border, but 
 the symmetrical group of leaves also. Therefore one half of 
 each group is first to be drawn, and then the other half is to 
 be made to correspond to it, as shown in preceding figures. 
 The pupils should trace the leaf forms very lightly at first. 
 
 Fig. 24 is the form of a Grecian ornament, painted in gold 
 on a blue ground, on a sunken panel of the ceiling of a Gre- 
 cian temple. Here, also, it will be observed, the leafage is 
 purely conventional in form, and not designed to represent 
 any particular group in nature. 
 
 The pattern is arranged symmetrically around the centre c, 
 
78 INDUSTRIAL DRAWING. [BOOK NO. I. 
 
 the diagonal leablades dividing it into four equal portions. 
 Taking the upper quarter section, observe that the points 
 of the leaves b b are equidistant from the central line ; the 
 slender volutes bend over equally in graceful curves, and 
 the radiating leafage below them fills the figure with grace- 
 ful forms symmetrically disposed. Having the ruled paper, 
 the pattern can be accurately drawn without difficulty. 
 
 PROBLEMS FOB PRACTICE. 
 
 After the pupil has copied the examples on page 9, let him draw the fol- 
 lowing : 
 
 1 . Draw a figure similar to Fig. 1 7, but two spaces higher, and two spaces 
 naiTOwer. 
 
 2. Draw Figs. 18 and 19, each four spaces longer, and of the same width, 
 in the widest parts, as in the book. 
 
 3. Draw Fig. 21, symmetrically, on a smaller scale. 
 
 4. Let the pupil design a figure similar to Fig. 22, but of different propor- 
 tions. 
 
 5. Let the pupil draw the pattern figures in Fig. 23, each two spaces high- 
 er, but of the same width as at present. 
 
 Blackboard Iticercises.Figs. 16, 18, 19, 20, 21, and 22. 
 
 PAGE TEN. 
 
 All the patterns on this page are plain examples of what 
 is called Renaissance ornament; by which is understood 
 the style of ornamentation which sprung up in Italy upon 
 the revival of art after the Dark Ages. (See page 41.). The 
 examples here given are from beautiful colored specimens 
 of glazed pottery of the sixteenth century. 
 
 Fig. 25. In copying this pattern the outlines should first 
 be drawn lightly, as shown at A. First draw a single-line 
 outline of all the figures in the pattern, making each figure 
 symmetrical on each side of its central line; then finish the 
 two-line border of each figure. The minor divisions of each 
 figure can then easily be drawn. 
 
 Fig. 26. The outlines of this figure are to be drawn by 
 the compasses, as shown at B. Thus, the points a are the 
 centres of the semicircles which make up the general outline 
 of the pattern. 
 
 Fig. 27. In this pattern the outer semicircle of each fig- 
 ure is drawn from the centre a, with a radius of five spaces ; 
 
CURVED LINES AND PLANE SURFACES. 79 
 
 and the inner and smaller carves are described from the 
 points b, c y d, etc. 
 
 Fig. 28. While the lower portion of this bulb pattern is to 
 be drawn with the compasses, they can afford no aid in 
 drawing the upper portion. The outlines of the central 
 row of bulbs are first to be drawn of uniform curves on 
 both sides, for which the ruling will be a perfect guide. 
 
 PROBLEMS FOR PRACTICE. 
 
 1 . Draw and shade Fig. 25 in full, after the enlarged outline form shown 
 atZ>. 
 
 2. Draw and shade Fig. 26, according to the enlarged outline form shown 
 at (7, which is four times the size of the pattern. Why is it four times the 
 size? 
 
 3. Draw an outline of one of the series in Fig. 27, four times its present size. 
 
 4. Draw Fig. 28 lengthwise of the paper, and four times its present size. 
 
 Blackboard Exercises. The bulb-pattern Fig., and Fig. 
 25, enlarged as at D. 
 
 PAGE ELEVEN. 
 
 Fig. 29 is a Grecian pattern of a painting on vases. It is 
 still another example of conventional flower representation, 
 in which only the general principle of the pendulous flower- 
 bud is retained. It is drawn with the greatest regard to 
 symmetry. The right-hand portion, left partly unshaded, 
 shows how the forms of the pendulous buds are drawn sym- 
 metrically, and of uniform size. 
 
 Fig. 30 is a Byzantine interlaced circular pattern, sculp- 
 tured in stone, from Milan, Italy. It is very easily drawn, 
 almost wholly by the compasses. Trace the whole very 
 lightly at first. The light is supposed to come diagonally 
 from above, from the right hand. 
 
 Fig. 31 is a copy of one of the Assyrian painted ornaments 
 found among the ruins of Nineveh. The central circles, a 
 (see the end marked -S), were black ; the inclosing ring, b y 
 dark reddish brown ; the winding band, c, orange ; and the 
 other winding band, d, blue ; while the spaces within the 
 borders, outside of the winding bands, had a groundwork 
 of reddish brown. The blue winding band was separated 
 from the orange, wherever they were contiguous, by a black 
 
80 INDUSTRIAL DKAWIXG. [BOOK NO. L 
 
 line. Thus the ancient Assyrians understood the now ad- 
 mitted principle that one color should never impinge upon 
 another, and that all contiguous colors should be separated, 
 generally by either white or black lines. The entire pattern 
 may be easily and accurately drawn by the compasses. 
 The same pattern is also found differently lined and colored, 
 somewhat as shown at the end marked C. 
 
 Fig. 32 is a Byzantine pattern of interlaced ornament, to 
 be drawn wholly by the compasses. The central points 
 from which the several circles and semicircles are described 
 can easily be found by the pupil. 
 
 Fig. 33 is a partially completed pattern, showing the meth- 
 od of putting in a series of uniform divisions that radiate 
 from the centre of a circle. The circle is easily divided into 
 eight equal parts by lines radiating from the centre on the 
 vertical, horizontal, and diagonal lines, as indicated by the 
 lettering. Each eighth of the circumference is then to be 
 divided into three equal parts by the compasses, and the 
 opposite points in the circumference are connected by slight- 
 ly traced or dotted lines passing through the centre. The 
 bases of the radiating white stars can then be designated 
 with sufficient accuracy by the eye, as the points to which 
 the sides of the stars are to be drawn, on the second circle, 
 are intermediate between the radiating dotted lines. Fig- 
 ures similar to this are numerous in ancient ornamental art. 
 
 Fig. 34 is a Byzantine pattern of interlacing circles. 
 Though seemingly intricate, it is quite easily drawn, and 
 wholly by the compasses. It would, however, be impossi- 
 ble to draw it with any approach to accuracy without the 
 aid of the ruled paper ; and there can be no doubt that it 
 was originally drawn on a ground prepared with lines such 
 as those we have given. The ruling gives the exact centre 
 of every circle, and renders all measuring unnecessary. Ob- 
 serve that the heavy shadows indicate that the light comes 
 from above, and from the left. The background may be 
 shaded with a uniform tint of India ink, or by the pencil, 
 as indicated in the upper left-hand portion. 
 
 Fig. 35 is also a Byzantine interlaced ornament, a consid- 
 erable portion of which may be drawn by the compasses. 
 
CURVED LINES AND PLANE SUEFACES. 81 
 
 Thus the three circular bands, and the projections of the in- 
 terlacing loops as described from, the vertical, horizontal, 
 and diagonal centres a, a, a, etc., are all easily and accurate- 
 ly drawn. The connecting and interlacing of the loops cen- 
 trally must be done by hand, guided by the eye alone ; but 
 after having drawn one of the connections, as from b to d, 
 the others may be drawn in symmetrical conformity to it. 
 Observe how the shadows are cast, the light coming diag- 
 onally from the left, and from above. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw a pattern similar to Fig. 29, but let the semicircular black line, 
 from the end of which the flower-bud is suspended, be drawn with a radius 
 of four spaces, instead of three ; and let the flower-bud be two spaces longer 
 and one space broader at the broadest part. 
 
 2. Draw a pattern similar to Fig. 31, but describe the larger circles with 
 a radius one space greater than in the figure. 
 
 3. Draw a pattern similar to Fig. 33, but describe the outer circle with 
 a radius of eight spaces, and divide each eighth part of the circumference 
 into four equal parts, instead of three, thus giving to each eighth part one 
 ray more than in the figure. 
 
 4. Draw Fig. 34 on a larger scale, having the radius of the larger circles 
 six spaces, instead of five. 
 
 5. Draw Fig. 35 on a larger scale, at the option of the pupil. 
 
 Blackboard Exercise. So much of Fig. 35 as there is 
 room for on the board. It would require a board four feet 
 square to draw it in accordance with the enlarged ruling 
 of the board. .# 
 
 PAGE TWELVE. 
 
 We have given on this page a number of original designs, 
 for the purpose of indicating the facility with which an im- 
 mense variety of very pleasing patterns may be drawn al- 
 most wholly by the compasses alone. 
 
 Fig. 36 is an ornamental figure called a quarterfoil, in- 
 closed within a circle. The quarterfoil, often used in archi- 
 tecture, is disposed in four segments of circles, and is a con- 
 ventional representation of an expanded flower of four pe- 
 tals. The lettering shows with what ease and accuracy it 
 is drawn on the ruled paper. 
 
 Fig. 37 is an ornamental figure consisting of eight e' 
 D2 
 
82 INDUSTRIAL DRAWING. [BOOK NO. I. 
 
 ments of circles, described from the eight numbered points 
 which are at the extremities of the dotted vertical, horizon- 
 tal^ and diagonal lines. After drawing these dotted lines^, 
 of indefinite extent, but all passing through the centre c, 
 equidistant points in all of them, for describing the seg- 
 ments, may be found by cutting the lines with the circum- 
 ference of a circle described from the centre c. From the 
 points ./, #, 3, 4> etc., the segments are described with a 
 length of radius that will barely allow the inner segments 
 of circles to touch one another. 
 
 Fig. 38. The segments of circles are here drawn in a 
 manner similar to those shown in Fig. 37; and then these 
 segments are connected by ribs passing through the centre. 
 The intervening spaces are then shaded, so as to give the 
 filling -up a raised and rounded appearance, the heaviest 
 shades being found on the lower right-hand portions the 
 light being indicated by the arrow A. as coming diagonally 
 from the left hand, and from above. The spaces covered 
 by the heavy dark shade on the outside borders of the fig- 
 ure the heaviest at the right and below are marked out 
 accurately by the compasses, by moving the fi^ed point of 
 the compasses uniform distances, diagonally, from the cen- 
 tres 1 to the points 2. By looking at the figure through a 
 tubular opening, it will seem to stand out from the paper as 
 if embossed upon it in relief. 
 
 Fig. 39 is similar in its border outlines to Fig. 38; and 
 all of it, with the exception of the central portion, may be 
 easily and accurately drawn, and mostly shaded, by the 
 compasses alone. To draw the central figure accurately, 
 first trace out lightly, in the space which it covers, the ver- 
 tical, horizontal, and diagonal lines, after which the raised 
 wedge-shaped oval figures may be drawn in between 
 these lines by hand. Observe how the heavy shadows are 
 formed^ 
 
 Fig. 40, drawn on the same general plan as the preceding 
 
 two figures, requires no additional explanation. The shaded 
 
 cross-ribs are easily drawn after making the raised circles 
 
 of uniform size around their given central points. As these 
 
 t ~" 1 r>oints are determined by the tint lines on the draw- 
 
CURVED LINES AND PLANE SURFACES. 83 
 
 ing-paper, the whole figure may be drawn with the greatest 
 ease and accuracy. 
 
 Fig. 41 is a pattern of seeming intricacy, but very easily 
 drawn, and planned on the same general principles as the 
 preceding figures on the same page. Every thing is drawn 
 by the compasses from points indicated by the ruling of the 
 paper, except the raised wedge-shaped ovals, that look as if 
 embossed. 
 
 Fig. 42 is an interlaced pattern, drawn wholly by the com- 
 passes, with the exception of the central figure. The figures 
 ./, , #, 4, etc., show the eight central points from which the 
 interlacing curves are described. 
 
 blackboard Exwcises. Figures 41 and 42, omitting the 
 central eight-leaf flower patterns. 
 
 Page 12 of this book illustrates, very happily, the great 
 advantage which the ruled paper affords for drawing curvi- 
 linear patterns. It would be next to an impossibility to 
 draw these designs with accuracy without this aid; but 
 with this ruling, this kind of drawing, which is used to a 
 great extent in all the decorative arts, becomes a very sim- 
 ple matter, easy of attainment by all who can describe a 
 circle by the aid of the compasses. 
 
DRAWING-BOOK No. II. 
 
 CABINET PERSPECTIVE PLANE SOLIDS. 
 
 THE Cabinet Perspective presented in this series is a meth- 
 od of representing solids, both plane and curvilinear, in 
 such a manner that the drawings shall give, by the aid of 
 the ruled paper, the correct measurements of the objects 
 represented. The ruling on the paper is adapted to any 
 scale of measurement ; but, for the purposes of the present 
 illustration, let it be supposed that the vertical and horizon- 
 tal lines on the paper are respectively one inch apart. 
 
 In all drawings in what is called Diagonal Cabinet Per- 
 spective? objects are supposed to be viewed in a manner 
 similar to that in which the two cubesf in Fig. 1, on page 1 
 of Drawing-Book No. II., are represented. Taking the cube 
 at JB for illustration, this may be supposed to be a cube six 
 inches square, the front face of which is in a vertical posi- 
 tion. The spectator is supposed to view this cube from 
 such a point, above and at the right of the cube, that he 
 may see just as much of the upper side .of the cube as of the 
 right-hand side; so that the apparent width, 10 9 of the 
 upper face, or 12 of the side face, shall measure, in the 
 directions indicated, one half the width of the front face ; 
 and so that the diagonal corner lines, 1 2, 3 4-> and 5 6, will 
 seem to rise diagonally at an angle of forty-five degrees ; 
 while the distance from which the view is taken is supposed 
 
 * There is a beautiful practical application of Cabinet Perspective, called 
 /Semi-diagonal Cabinet Perspective, which will be found illustrated on pages 
 9-11 of Drawing-Book No. IV. See page 171. 
 
 t A cube is a regular solid body, having six equal square sides. 
 
CABINET PERSPECTIVE PLANE SOLIDS. 85 
 
 to be so great that these lines will appear to be, as they are 
 here drawn, parallel. 
 
 The front face of the cube is drawn in its real proportions 
 as a square, and as though it were seen in a vertical plane 
 directly fronting the spectator. According to the scale 
 supposed to be adopted in the ruling, the front of the cube 
 measures six inches to a side. The farther face of the cube, 
 being also vertical and parallel with the front face, and 
 therefore in a plane also directly fronting the spectator, 
 would also be drawn as a square if it could all be seen, meas- 
 uring six inches to a side. Hence each of the lines 2 4 and 
 4 6 measure six inches, the same as 1 3 and 3 5. 
 
 But each of the diagonal lines 1 2, 3 h and 5 6, being 
 corner lines of the cube, must also represent a measure of 
 six inches ; and as each of these lines extends over three di- 
 agonals of the small squares, it follows that what we call 
 one diagonal space measures twice as much as a vertical or 
 a horizontal space, whenever this diagonal space is applied to 
 the measurement of a line representing a horizontal line. 
 We may, therefore, adopt the following rule for the meas- 
 urement (and also for the drawing) of all objects in diag- 
 onal cabinet perspective. 
 
 ELEMENTARY RULE. 
 
 Drawings of surf aces that are supposed to be in a vertical 
 plane fronting the spectator are measurable, in any direc- 
 tion, according to the scale adopted for the vertical and hor- 
 izontal spaces of the ruling; while each DIAGONAL space 
 fhat measures a line in a horizontal and diagonal position is 
 to be taken as TWICE the measure of a space of the other kind. 
 
 Applying this rule to the cube at 2?, Fig. 1, we find that 
 all the horizontal lines, and also all the vertical lines that 
 cross the front face of the cube, measure each six inches in 
 length, because each extends over just six of the ruled 
 spaces, and all are in a vertical plane fronting the spectator. 
 For a similar reason each one of the vertical lines on the 
 right-hand side of the cube, and each one of the horizontal 
 lines on the top of the cube, measures six inches, because 
 each may be supposed to be in a plane like that which 
 
86 INDUSTRIAL DRAWING. [BOOK NO. II. 
 
 forms the front face of the cube directly fronting the spec- 
 tator. But such lines as 1 2, 3 4, 5 6, 8 9, 10 11, and 12 IS, 
 being seen obliquely, can not be in any plane fronting the 
 spectator; and as they lie in a diagonal direction, and rep- 
 resent horizontal lines, they are measurable by the prin- 
 ciple adopted in the latter part of the rule. Hence the line 
 a b measures five inches ; c d four inches ; g 13 four inches ; 
 m n six inches : but 8 9 measures six inches ; 10 11 nieas- 
 sures four inches ; 12 IS measures four inches, etc. The 
 cube at A measures two inches on each of its corners. 
 
 Fig. 2. Applying the scale of measurement which we have 
 adopted to the representation of the cube at D, Fig. 2, we 
 find that the front face of the cube is a square of four inches 
 to a side ; and that the diagonal horizontal distance 1 2, or 
 8 4, or 5 6, also measures four inches. Also, if we draw in- 
 termediate lines between the ruled lines on the paper, on 
 the upper face and right-hand face of the cube, so as to give 
 owe-inch diagonal measures, then sixteen one-inch squares 
 may be counted on each of the three visible faces or sides 
 of the cube. We thus have, according to the scale of one 
 inch to a space horizontally or vertically between the lines, 
 and two inches for a diagonal space, the exact measurement 
 of the three visible sides of the cube. 
 
 To find the contents of a cube: 
 
 RULE. Multiply the length of a side of the cube by itself, 
 and that product again by a side, and this last product will 
 give the contents required. (See Rule I., page 53, for the 
 measurement of surfaces.) 
 
 Thus, in the cube at D, Fig. 2, if we multiply the length 
 (1 2) of one side, which is four inches, by the length (2 4) 
 of another side, which is also four inches, we get the prod- 
 uct 16, which is the number of solid cubic inches contain- 
 ed in the upper tier of the cube as may also be seen by 
 counting them ; and as there are four of these tiers, we mul- 
 tiply the 16 by 4, and get 64, the number of cubic inches in 
 the four tiers, or in the whole cube. Or, 4 x 4 x 4 = 64 cubic 
 inches. 
 
 One cubic inch is represented at C, which, according to 
 
CABINET PERSPECTIVE PLANE SOLIDS. 87 
 
 the scale we have adopted for page 1, measures one inch on 
 each of its sides ; and at E are sixteen cubic inches, equal 
 to the upper tier in D. 
 
 In straight-line drawings in cabinet perspective, the ruler 
 may be used wherever its aid will give additional accuracy 
 to the drawing. 
 
 The contents of the cube .Z?, Fig. 1, which measures six 
 inches to a side, are found by the rule to be as follows. 
 
 Ans. 6x6x6=216 cubic inches. 
 
 The drawing at E, Fig. 2, is an example of a parallelopi- 
 ped / a figure which is defined as being a solid whose faces 
 are six rectangles,* the opposite faces being parallel, and 
 equal to each other. The drawing at F also represents a 
 parallelepiped. All squares are parallelepipeds; but all 
 parallelepipeds are not squares. 
 
 To find the contents of a rectangular (right-angled) parallelepiped : 
 
 RULE. Multiply the length by the breadth, and that prod- 
 uct by the depth, and this last product will give the contents 
 required. 
 
 The height of the rectangular solid at E, Fig. 2, is one 
 inch ; the breadth in one direction (1 2) is four inches ; and 
 the breadth in the other direction (3 4) is also four inches. 
 What are the contents? Ans. 1x4x4 = 16 cubic inches, 
 as may be verified by counting the small one-inch cubes 
 which it contains. 
 
 The length (or height) of the solid at F is three inches ; 
 the breadth or width is five inches ; and the diagonal depth 
 is two inches. What are the contents ? 
 
 Ans. 3 x5 x2 = 30 cubic inches. 
 
 The two rules just given may be combined in one, as fol- 
 lows: 
 
 To find the contents of any solid rectangular figure : 
 
 RULE. Multiply the three dimensions together, and their 
 product will be the contents required. 
 
 Fig. 3. At 6r, H, I, and e7are represented four parallelo- 
 
 * In a strict definition the sides need not, necessarily, be rectangular 
 (right-angled) ; but it is better, at present, for the pupil in drawing to con- 
 sider all parallelepipeds as of the rectangular kind. 
 
88 INDUSTRIAL DRAWING. [BOOK NO. II. 
 
 pipeds, all of the same size ; Gr being viewed in a vertical 
 position, H horizontally, and ^Tand 7 horizontally and diag- 
 onally. They may be considered pieces of timber, each two 
 inches square at the ends, and twelve inches long. What 
 are the solid contents of each? Ans. 48 cubic inches. 
 
 Observe that, according to the scale of measurement al- 
 ready explained, these pieces measure precisely the same in 
 these three different positions. 
 
 Fig. 4. According to the definition of a parallelepiped, 
 this figure, also, is one of that kind. What are its meas- 
 urements, and its contents ? 
 
 Observe that the right-hand sides of the foregoing figures 
 are represented as shaded with a deep tint of India ink, the 
 front with a lighter tint, and the top of Fig. 4 with the run- 
 ning dotted shading. The kinds of shading used in Fig. 4 
 are well adapted to all plane solids, as the object of shad- 
 ing, in cabinet perspective, is to render the several surfaces 
 as marked and distinct as possible, one from another. 
 
 Fig. 5 is a square frame composed of four pieces, each two 
 inches square; the two diagonal side-pieces each twelve 
 inches long, and the other two each eight inches long. 
 What is the size of the square which they inclose? 
 
 Fig. 6. Let the pupil describe Fig. 6 that is, tell how 
 many pieces compose the figure, their size, position, etc. 
 
 Fig. 7 is drawn, first, in the same manner as Gr of Fig. 3 ; 
 it is then divided so as to represent cubical blocks, each two 
 inches square, placed one above another. Three of these 
 blocks are represented as shaded with the hatching de- 
 Hcribed in Lesson IX. of Drawing - Book No. I., after first 
 tinting the surface with India ink. 
 
 Fig. 8 is composed of two vertical blocks, each two inches 
 square and five inches long, resting upon the ends of a piece 
 one inch by two inches, and twelve inches long; the latter 
 being viewed diagonally. 
 
 Fig. 9 is the same as No. 8 inverted, and still viewed di- 
 agonally. 
 
 Fig. 10 is also the same as Fig. 8 ; but it is here viewed hori- 
 zontally. Thus the same figure may be represented in several 
 different positions, so as to bring each side into full view. 
 
CABINET PERSPECTIVE PLANE OLIDS. 89 
 
 In these several figures observe the effect of the shading, 
 which is rapidly executed with different tints of India ink, 
 except the upper surfaces, which have the running dot-line 
 shading. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw the representation of a square frame similar to Fig. />, of the 
 same outside and inside measure as Fig. 5, but composed of pieces only one 
 inch thick instead of two inches ; and all of them two inches wide. 
 
 2. Draw a figure similar to Fig. 6, but let the upper pieces be only one 
 inch thick (or high), and let them project each one inch at both ends be- 
 yond the lower pieces. 
 
 3. Draw a figure similar to Fig. 7 in position, but four inches wide and 
 two inches thick the wide side being in front. 
 
 4. Draw a figure similar in position to Fig. 8, but composed of three pieces, 
 each twelve inches long, four inches wide, and one inch thick. 
 
 5. Draw Fig. 9 with the side in a plane fronting the spectator. 
 
 Free-hand Blackboard Exercises. Figs. 5 to 10 inclusive 
 letting a space on the blackboard represent its true meas- 
 ure, two inches. In this case only one half of a diagonal 
 measure will be required for a distance of two inches. In 
 order to render the several surfaces distinct one from an- 
 other, the dark shades may be represented by heavy ver- 
 tical chalk lines, and the lighter shades by very light ver- 
 tical lines. Or, where colored chalk crayons are accessible, 
 the dark shades may be represented by blue lines. 
 
 Fig. 11, in its complete outline, is a cube eight inches 
 square, having a small cube two inches square cut from 
 each of its four upper corners. See the form of the entire 
 cube, as represented by the dotted lines. What were the 
 contents of the entire cube ? What the contents after the 
 four corners were taken out. 
 
 Fig. 12 is a cube twelve inches square, having a piece eight 
 inches square and two inches thick taken from the centre of 
 each of its three visible sides. What were the contents of the 
 entire cube before the three pieces were taken out ? What 
 were the contents after these pieces were taken out ? Let 
 the pupil be careful, in drawing the figure, that his measures 
 shall be correct. Thus, the depth of the recess in each side 
 must measure two inches. Thus c d and m n are measures 
 
90 INDUSTRIAL DRAWING. [BOOK NO. II. 
 
 of two inches each ; and a b, being one diagonal, is also a 
 measure of two inches. 
 
 Fig. 13 represents a rectangular frame measuring ten by 
 fourteen inches, and composed of pieces two inches square 
 framed into posts two inches square and ten inches long. 
 
 CrifP' To get the full effect of the figures on this page, 
 view them from a point at the right, and above them, 
 through the opening formed by the partially closed hand. 
 The pupil should be accustomed to view his drawings in 
 the same manner. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw a figure similar to Fig. 11, but twelve inches square, having a 
 block four inches square cut from each of its four upper corners. 
 
 What would be the contents of the entire cube? What the contents after 
 the four comers were taken out? 
 
 2. Draw a figure similar to Fig. 12, but fourteen inches square, and show- 
 ing a rectangular piece ten inches square and four inches in thickness cut 
 from the centre of each of its three visible sides. 
 
 What would be the contents of the entire cube? What the contents after 
 the three rectangular pieces were taken out. 
 
 3. Draw a figure similar in all respects to Fig. 13, except that the four 
 horizontal pieces of the frame-work are to be only one inch in veriicr.l 
 thickness. 
 
 Free-hand Blackboard Exercises. Figs. 11, 12, and 13; 
 also the accompanying problems. 
 
 PAGE TWO. SCALE OF ONE INCH TO A SPACE. 
 
 Fig. 14 represents a cube eight inches square; and Fig. 
 15 represents a box formed of one-inch stuff, open at the top, 
 and of a size that will just receive the cube; so that the lat- 
 ter, when placed within the box, shall fill it even with the 
 top. At A is the cover that will just fit the top of the box. 
 
 What is the size of the cover ? Of the side J3 of the box ? 
 Of the sides C Cf Of the bottom of the box ? What arc 
 the outside measures of the box when the cover is on it ? 
 
 Fig. 16 represents a series of five blocks placed one upon 
 another, and rising in the form of stairs. The upright piece 
 in a stair (as a) is called the riser, and the part on which 
 the foot is placed (as b) is called the tread In Fig. 16 the 
 stairs are so placed that the risers front the spectator; but 
 
CABINET PERSPECTIVE PLANE SOLIDS. 91 
 
 in Fig. 17 the side of the stairway fronts the spectator, and 
 the riser is viewed diagonally. Fig. 17 measures the same 
 as Fig. 16, with the exception that in Fig. 17 the lower block 
 is omitted. 
 
 If these two figures are supposed to be drawn to a scale 
 of four inches to a space, what will be the height of each 
 riser, and the width of each tread ? What the width of the 
 stairway ? 
 
 Fig. 18 represents a cabinet frame-work formed of pieces 
 two inches square at the ends ; the whole frame measuring 
 ten* inches by twenty-six. Here the longest side fronts the 
 spectator; but in Fig. 19 the same frame-work is represent- 
 ed with the end fronting the spectator. Observe that the 
 end below c d of Fig. 18, which is there seen diagonally, is 
 not seen at all in Fig. 19 ; and that the end below a b of Fig, 
 19 is not seen at all in Fig. 18. Let the pupil describe tho 
 several pieces of which the frame-work is composed. 
 
 Fig. 20 represents a cross made of pieces measuring, at 
 the ends, two by three inches. 
 
 Fig. 21 represents a cabinet square made of one-by-two- 
 inch stuff, and placed horizontally ; but, as the spectator is 
 supposed to be above and to the right of it, one of the arms 
 seems to rise at an angle of forty-five degrees being in the 
 diagonal direction of the small ruled squares. 
 
 Fig. 22 represents the same cabinet square in a vertical 
 position, and so placed that the wide side shall front the 
 spectator. The former, as placed, is a measure for horizon- 
 tal and diagonal distances ; the latter, for horizontal and 
 vertical distances. 
 
 Fig. 23 is an upright frame-work resting on four blocks, 
 each two inches square ; having only two of its sides and 
 the top and bottom inclosed, and containing three shelves 
 in addition to the top and bottom. Let the pupil describe 
 this frame-work more fully; and tell (or write out) its meas- 
 ures in all it's parts distances apart of the shelves, etc. The 
 whole on the supposition that it is drawn to the scale of 
 one inch to the space. 
 
 The shading on this page is supposed to be done, first, 
 with India ink; some of the sides are then shaded with lines 
 
92 INDUSTRIAL DRAWING. [BOOK NO. II. 
 
 by the pencil, which, when not too heavy, give additional 
 life and spirit to the drawing. 
 
 PROBLEMS FOR PRACTICE. 
 
 As Fig. 14 represents a cube eight inches square, what are its contents in 
 cubic inches ? A ns. 8x8x8=512 cubic inches. 
 
 As it takes 231 cubic inches to make a gallon, how many gallons of water 
 would the box, Fig. 15, contain ? Ans. 2 gallons and 50 cubic inches. 
 
 1. Draw a box, similar to Fig. 15, that would contain a cube twelve inches 
 square ; and draw the cover separately. How many gallons of water would 
 such a box contain ? 
 
 2. Draw a rectangular box open at the top, the bottom of which shall 
 measure, on the inside, ten by twelve inches, and let the box be ten inches 
 deep. How many gallons of water will it contain ? 
 
 3. Draw a flight of stairs, similar to Fig. 1 7, to a scale of six inches to 
 a space, and having the risers twelve inches high, and the tread eighteen 
 inches wide. 
 
 4. Draw a frame similar to Fig. 18, but formed of stuff one by two inches 
 the one-inch being the height. 
 
 Free-hand Blackboard Exercises. Figs. 17, 18, and 20, 
 and problems 2 and 3. 
 
 PAGE THREE. SCALE OF THREE INCHES TO A SPACE. 
 
 Fig. 24. At A is represented a piece of timber six by 
 twelve inches at the end, and four feet long, broadest side 
 down, having a piece cut from the centre of the upper side 
 one foot square and three inches deep, to receive crosswise 
 nnother timber cut in like manner. At B the two pieces 
 of timber are united at right angles to one another. 
 
 The remaining figures on this page are examples of what 
 carpenters call scarfing, which means the uniting of two 
 pieces of timber longitudinally by a scarf-joint in common 
 language often called splicing ; but the latter term is more 
 properly applied to an overlapping joint that is not notched. 
 
 Fig. 25. At C and D are two pieces of timber prepared 
 for being joined lengthwise by a square scarf-joint. After 
 placing the timbers in the right position, they are pinned or 
 bolted together. Let the pupil describe the two pieces. 
 
 Fig. 26. At E is the upper piece of timber, and at F the 
 lower, to be united by a scarf-joint, which is formed of the 
 square scarf-joint combined with the splice-joint. Observe 
 
CABINET PERSPECTIVE PLANE SOLIDS. 93 
 
 that the line of union, 2 4, in both E and F, is in the direc- 
 tion of two-space diagonals ; and as the two lines are of 
 the same length, and the two timbers lie in the same rela- 
 tive positions, the cut sections must correspond with one 
 another. Moreover, E and F are redrawn at G and H; 
 and there it is seen that the diagonal line of union, 2 J^ 
 is in both cases the diagonal of the same rectangle 1 2, 
 3k 
 
 Fig. 27 shows two pieces of timber, Jand J", prepared for 
 being joined by the ordinary splice-joint. The two tim- 
 bers are to be firmly bolted together. But this is by no 
 means so firm a mode of union as is shown in Figs. 25 
 and 26. 
 
 Fig. 28 shows a still firmer mode of the scarf-joint than 
 Figs. 25 and 26. Observe that the lines of union in K meas- 
 ure precisely the same, and are in precisely the same posi- 
 tions, as in L. Thus the lines 1 2 and 3 4. are the same in 
 length and position as the lines 5 6 and 7 8, both being 
 two-space diagonals. 
 
 Fig. 29 shows, perhaps, the firmest of all modes of the 
 scarf-joint, especially for heavy timbers. JV is the upper 
 piece; and the two must be united by placing M and N 
 side by side, and driving them together. Even then, with- 
 out bolting, they can not be drawn apart lengthwise ; nor, if 
 the joints be good, can they be easily sprung in any direc- 
 tion. At Jf the two pieces are shown as they appear when 
 united. 
 
 Fig. 30 represents the same pieces that are shown in Fig. 
 29 ; but here drawn in a different position, being placed 
 side wise toward the spectator, instead of endwise toward 
 him as in Fig. 29. Observe that the measures, according to 
 the scale, are the same in the one case as in the other. That 
 mode of representation which will give the best view of 
 the object should in all cases be adopted; and in some 
 cases both modes should be used, as the one will often give 
 views of parts that are not shown in the other. 
 
 Fig. 31 shows a still different mode of scarfing, and one 
 that is very firm, and much more easily executed than Fig. 
 30. When the pieces are firmly bolted together, the inter- 
 
04 INDUSTRIAL DRAWING. [BOOK NO. II. 
 
 locking gives them great additional strength against any 
 force that might tend to wrench them sideways.* 
 
 PROBLEMS FOR PRACTICE. 
 
 1 . Draw A of Fig. 24 with the narrow side toward the spectator. 
 
 2. Draw the two parts of Fig. 25 with the narrow sides toward the spec- 
 tator. 
 
 3. Draw Tig. 28 with the dark sides of both pieces toward the spectator. 
 
 4. Design a figure similar to Fig. 30, but of different proportions in the 
 interlocking parts. 
 
 5. What are the contents of A, Fig. 24 ; the scale being three inches to a 
 space ? 
 
 6. What are the contents of the entire Figure B ? 
 
 Free-hand Blackboard Exercises. Any or all of the fig- 
 ures on page 3. 
 
 PAGE FOUR. SCALE OF TWO INCHES TO A SPACE. 
 This page shows the different methods adopted by ma- 
 sons and engineers in laying bricks. A brick of ordinary 
 size is two inches thick, four inches wide, and eight inches 
 long. Bricks which are laid lengthwise across the wall, 
 with their ends toward the face of the wall, are called head- 
 ers; and those which are laid with their lengths parallel to 
 the face of the wall, are called stretchers. The two principal 
 methods of laying bricks in walls, where much strength 
 is required, are what is called the English bond and the 
 Flemish bond. In the former, the face of the wall always 
 shows the headers and stretchers in alternate layers, or 
 courses, as they are called by bricklayers. Thus at A is a 
 
 * It may here be remarked that the only lines in Diagonal Cabinet Per- 
 spective which are susceptible of direct measurement are the horizontal, 
 vertical, and diagonal lines ; and that when any other line occurs, if it is to 
 be measured, it must be done by mathematical calculation, after first re- 
 solving it into the hypothenuse of a right-angled triangle. Thus at G the 
 lines 2 1 and 1 4 are measurable ; the former, according to the scale here 
 adopted, measures nine inches, and the latter thirty-six inches ; and the an- 
 gle 2 1 4 represents a right angle. Hence the line 2 4 is the hypothenuse 
 of a right-angled triangle ; and the teacher may show the advanced pupil 
 how to ascertain its precise length by adding together the squares of the 
 sides 2 1 and 1 4, nnd then extracting their square root. And so in all 
 similar cases of the measurement of lines that are neither diagonal, vertical, 
 nor horizontal. 
 
CABINET PERSPECTIVE PLANE SOLIDS. 95 
 
 course of headers, and at B is a course of stretchers ; and 
 in English bond the face of the wall shows alternate courses 
 of the two. For Flemish bond, however, the face of each 
 course is composed of a combination of headers and stretch- 
 ers. In both, a brick of half the ordinary width, and also 
 one of half the ordinary length, is often required ; and these 
 half bricks should be made in every brick-yard. 
 
 Fig. 32 represents a wall eight inches thick, laid in En- 
 glish bond. Observe that the first or upper course is coin- 
 posed of headers ; the second of stretchers ; the third of 
 headers ; the fourth of stretchers, etc. In drawing these ex- 
 amples (as in nearly all other cases of drawing), it is neces- 
 sary to draw the upper portions first that is, to begin at 
 the top. 
 
 Fig. 33 represents a twelve-inch wall in English bond. 
 Observe that the face of the wall shows alternate courses 
 of headers and stretchers ; while each course is composed 
 of both kinds, arranged in a peculiar order, as shown in the 
 drawing. 
 
 Fig. 34 represents a sixteen- inch wall in English bond. 
 The face is the same as in the eight and twelve inch walls. 
 The upper two courses show the arrangement. 
 
 Fig. 35 represents an eight-inch wall in Flemish bond; 
 showing that each course is composed of both headers and 
 stretchers. 
 
 Fig. 36 represents a twelve-inch wall in Flemish bond. 
 Here half bricks of both kinds are used. 
 
 Fig. 37 represents a sixteen-inch wall in Flemish bond. 
 
 Fig. 38 represents what is called English cross bond. 
 Here the face of the wall shows an arrangement in the sec- 
 ond stretcher line by which its joints come exactly below 
 the middle of the bricks in the first stretcher line ; and the 
 same arrangement of stretchers comes back every fifth line. 
 By comparing Fig. 38 with Fig. 32 the difference will be 
 manifest. 
 
 Fig. 39 represents a twenty-four-inch wall laid in Flemish 
 bond, on the farther half of which rises a twelve-inch wall in 
 English bond, both halves being capped with twelve-by- 
 twelve-inch stones two inches thick. There are also two 
 
96 INDUSTRIAL DRAWING. [BOOK ]S T O. II. 
 
 triangular abutments resting on the front half of the lower 
 wall, and abutting against the upper wall. These abut- 
 ments are twelve inches high at the back, and their bases 
 project forward twelve inches, the width of the lower half 
 of the wall. Their faces, therefore, are laid at an angle of 
 forty-five degrees ; and hence each layer of stones, of which 
 the abutment is composed, although only two inches thick, 
 presents a much greater front surface view. The teacher 
 may require of the advanced pupil the length of the face of 
 the abutment, as designated by the line 2 3 or 4 5. See 
 Note, page 94. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw an eight-inch wall in English bond, forty inches in length at the 
 base, and showing three partial courses and four full courses. 
 
 2. Draw a twelve-inch wall in English bond, forty-eight inches in length 
 at the base, and showing four partial courses and four full courses. 
 
 3. Draw a twelve-inch wall in Flemish bond, fifty-six inches in length at 
 the base, and showing four partial courses and four full courses. 
 
 4. Let the pupil design a twenty-inch wall in English bond, and also a 
 twenty-inch wall in Flemish bond, and so that the bricks will break joint in 
 the best manner possible. 
 
 JBlachboard ^Exercises. Any of the figures of page 4, 
 showing the true size of the bricks. 
 
 PAGE FIVE. SCALE OF TWO INCHES TO A SPACE. 
 
 Fig. 40 represents a cross-shaped figure formed by so cut- 
 ting a cube twenty inches square, that vertical sections, 
 eight inches square at the ends, are taken away from its 
 four corners. The form and extent of the entire cube are 
 shown by the dotted lines. Let the pupil calculate the 
 number of cubic inches taken away, and the number re- 
 maining. 
 
 Fig. 41 represents a cube of the same original size as Fig. 
 40, from which several sections are cut away, leaving what 
 is shown in the figure. Let the pupil calculate the number 
 of cubic inches taken away, and the number remaining. 
 
 Fig. 42 shows, on its front vertical face, a single fret of 
 the same form as is seen at Fig. 24, page 6, of Book No. I.; 
 but here, in Fig. 42, the fret is shown, not as a surface deco- 
 ration merely, but in the solid form, four inches in thickness. 
 
CABINET PERSPECTIVE PLANE SOLIDS. 97 
 
 Observe that the thickness is represented by one diagonal 
 space. 
 
 Fig. 43 shows the same solid fret, but with the addition 
 of a solid piece four inches square running through it cen- 
 trally, the two being framed together. 
 
 Fig. 44 represents a frame-work thirty-four inches wide 
 and fifty-six inches long, the side pieces formed of timbers 
 six inches square at the ends, and the end pieces formed of 
 timbers six by sixteen inches at the ends, the end pieces 
 being mortised into the side pieces by tenons two inches 
 thick, eight inches wide, and six inches in length. Hence 
 the mortises which receive the tenons must extend entirely 
 through the timbers. At A, B^ C, and D, the four timbers 
 which compose the frame are drawn separately, and in their 
 relative positions ; and at F they are put together. The 
 side pieces show mortises for upright posts. At E, one of 
 the end timbers is represented as standing vertically on one 
 of its tenons, with its broadest side fronting the spectator. 
 Observe that, according to the scale, it measures precisely 
 the same as C or D. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw the lightly dotted outlines of a cube of the same dimensions as 
 Fig. 40 ; and then represent the cube as it would be after taking vertical 
 sections from its four corners, anl from the centres of its four bides, each 
 four inches square at the ends. First mark out, on the upper face of the 
 dotted cube, the upper face as it would appear after the pieces were taken 
 out. What would be the solid contents of the eight pieces thus taken out ? 
 The solid contents remaining in the cube ? 
 
 2. Draw a fret similar to Fig. 34, page G, of Book No. L, representing the 
 solid form of the fret, two inches in thickness ; but draw it on an enlarged 
 plan, so that the width of the fret-line shall occupy a full space. Scale, two 
 inches to a space. 
 
 3. Draw F of Fig. 44 so that the longest side shall front the spectator. 
 
 4. Draw the same frame standing on one end, with the broad side front- 
 ing the spectator. 
 
 5. Draw the same frame standing on one end, with the edye of the frame 
 fronting the spectator. 
 
 Blackboard Zeroises. Any of the figures and problems 
 on page 5. 
 
 E 
 
98 INDUSTRIAL DKAWING. [BOOK NO. II. 
 
 PAGE SIX. SCALE OF TWO INCHES TO A SPACE. 
 
 Figs. 45 and 46 represent the same frame-work, but placed 
 in different positions in relation to the spectator ; and Fig. 
 47 represents these two frames united by string-pieces, in 
 the form of a bedstead. Let the pupil give the dimensions, 
 according to the scale adopted for the page two inches to 
 a space. 
 
 Fig. 48 represents a vertical pillar twelve inches square, 
 surrounded at the top by a moulding four inches in height 
 and projecting four inches. Observe that the projection of 
 the moulding, whether measured horizontally by the dis- 
 tances 1 2 and 5 6, or diagonally by the distances 2 3 and 
 4 5, etc., is in all parts four inches. The height of the pillar 
 is the distance 2 9, which is twenty inches. Observe the 
 following rules: 1. A horizontal moulding, or any hori- 
 zontal projection, conceals from view a portion of the front 
 vertical surface immediately below it, equal in height to one 
 half the projection. Thus the dotted line 7 8 indicates the 
 lower line of the attachment of the moulding to the front 
 face of the pillar; by which it is seen that the moulding, four 
 inches in projection, conceals from view the two inches in 
 height of the front face of the pillar immediately below the 
 attachment of the moulding. This principle is also seen in 
 the fact that the distance from the base of the pillar to the 
 moulding is measured from 9 to 7 thus making the mould- 
 ing sixteen inches above the base. 2. A horizontal projec- 
 tion conceals from view a portion of tJie DIAGONAL vertical 
 surface immediately beloio it, equal in height to the extent of 
 the projection. 
 
 Thus 8 12 is the line of the attachment of the under side 
 of the moulding to the side of the pillar; and 12 13 shows 
 the vertical extent of the side of the pillar which the mould- 
 ing conceals. 
 
 But suppose the front moulding did not project at all 
 beyond the face of the left-hand side of the column. It will 
 be seen that, in that case, the front end of the moulding 
 would terminate in the line S 10; thus having an apparent 
 projection of two inches beyond the line 7 9, the same as the 
 
CABINET PERSPECTIVE PLANE SOLIDS. 99 
 
 right-hand end of the moulding would seem to project to the 
 left of the line 8 11. All the principles of the projections 
 of rectangular mouldings may be learned from this figure. 
 
 Fig. 49 represents the same pillar, twelve inches square, 
 and the same moulding as in Fig. 48 ; but in Fig. 49 the 
 moulding is placed six inches below the top. Observe that 
 the measurements of the projection, and of the parts con- 
 cealed, are the same in both figures. 
 
 Fig. 50 represents the same pillar as in the preceding two 
 figures ; but here it is placed upon a base four inches thick, 
 that projects beyond the four faces of the pillar four inches 
 on all sides. Observe how the measurements are made 
 here also. 
 
 Fig. 51. Here we have represented eighteen inches in 
 height of the top of a pillar which is twenty inches square 
 at the ends, with a projecting moulding of four blocks on a 
 side, four inches apart, encompassing the upper end of the 
 pillar. These blocks are each four inches in height and two 
 inches in thickness, and have a projection of four inches. 
 In drawing them, first trace lightly a moulding around the 
 pillar, similar to the moulding of Fig. 48 ; then mark out, in 
 this moulding, the divisions which constitute the blocks. 
 Observe that the blocks on the two diagonal sides of the 
 pillar must necessarily be only half a diagonal in thick- 
 ness to represent two inches. Notice the portion of the 
 face of the pillar below the blocks concealed by their pro- 
 jection. 
 
 Fig. 52, which represents a pillar resting upon a block 
 that projects four inches from the face of the pillar, has a 
 narrow moulding one inch thick, with a projection of two 
 inches, encompassing the pillar near the middle of its height. 
 How far is this moulding below the top of the pillar, and 
 how far above the top of the projecting base? 
 
 Fig. 53 represents a four-legged table, placed bottom up- 
 ward, so that its several parts may be shown in the best 
 manner possible. They could not be seen so well if it were 
 represented in its natural position. Let the pupil describe 
 the size and length of the legs how they are placed in re- 
 lation to the inclosing frame ; width and thickness of this 
 
100 INDUSTRIAL DRAWING. [BOOK NO. II. 
 
 frame, length of its sides, and how joined at the corners ; 
 size of the top of the table, its projection beyond the frame, 
 thickness and number of the boards which compose it, etc. 
 
 Fig. 54 represents a diamond or lozenge-shaped figure 
 formed of two-space diagonals, and having a thickness of 
 four inches. What is its length ? Its breadth ? Its front- 
 face measure in square inches? (See page 58; No 7.) Its 
 contents in cubic inches ? 
 
 Fig. 55 is the same figure as 54, but its length is here 
 placed in a front vertical position. 
 
 PROBLEMS FOR PRACTICE. 
 
 1 . Represent a vertical piece of timber twelve inches square at the ends, 
 and twenty-four inches in height, surrounded by a moulding even with the 
 top, two inches in height, and projecting four inches; also a like moulding 
 even with the base of the timber. Be careful to get the correct height of 
 the timber, as portions of the top and bottom are concealed by the moulding. 
 
 2. Make a drawing like Fig. 47, with the exception that the posts shall 
 rest upon a frame formed of stuff four inches thick and four inches in width, 
 and that a frame formed of stuff four inches in width and one inch in thick- 
 ness (or height) shall rest upon the tops of the posts. Be careful to trace 
 the outlines very lightly at first, and only mark them firmly when the visi- 
 ble portions are correctly represented. 
 
 3. Draw a lozenge whose front face shall be like Fig. 54, but whose diago- 
 nal length shall be twelve inches. 
 
 4. Draw the same lozenge twelve inches in diagonal length, but in the po- 
 sition of Fig. 55. 
 
 Blackboard Exercises. Figures 47, 48, 49, and the fore- 
 going problems. 
 
 PAGE SEVEN. SCALE OF FOUR INCHES TO A SPACE. 
 
 All the figures on this page, except Fig. G4, contain some 
 lines that are neither true diagonals, verticals, nor horizon- 
 tals; yet their positions are as accurately defined as the 
 positions of those lines that are directly measurable. 
 
 Fig. 56 represents a kind of heavy block table, thirty-two 
 inches in height, having its top surface thirty-two inches 
 square, and resting on a base thirty-two inches square. The 
 base bevels upward, and the top bevels downward to the 
 same extent. As the line a b is drawn diagonally from the 
 corner , thirty-two inches, so must the line c d, drawn from 
 the corresponding corner c, run diagonally in the same di- 
 
CABINET PERSPECTIYE PLANE SOLIDS. 101 
 
 rection ; and its real length is thirty-two inches, but the far- 
 ther eight inches are not visible. It is best to draw the 
 top first ; then the front face ; then the diagonal lines. 
 
 Fig. 57 represents the same block table, with the excep- 
 tion that the vertical section, eight inches in length, is cut 
 out of the centres of both sides of the base. Observe that 
 as the distance if is eight inches, so must the distance g a 
 be eight inches also ; and that a b must run toward the point 
 c, which is a point on the left-hand side of the table corre- 
 sponding to d. And c and d must be the same distance 
 apart as n and m. 
 
 Fig. 58 is the same as Fig. 57, with the exception that the 
 section cut out here extends through the entire central sup- 
 port ; and Fig. 59 shows the same as Fig. 58, with the top 
 removed. 
 
 Figs. 60, 61, 62, and 63 are the same as the figures directly 
 above them, with their sides turned to the front. Observe 
 that Fig. 60 measures in all respects, according to the scale, 
 the same as Fig. 56. Thus the dotted vertical lines in the 
 base front of Fig. 56, although seen in a side view in Fig. 
 60, are in the latter, as in the former, the downward con- 
 tinuation of the columnar support, and in both cases are 
 eight inches apart, eight inches in length or height, and at 
 the same distances from the corners a and c. And if the 
 length of the line a o, in Fig. 60, should be calculated math- 
 ematically, it would be found to be of the same length as 
 the same line a o in Fig. 56. Let the pupil trace out the 
 like measurements in Figs. 61,62, and 63, with the figures 
 above them. 
 
 Fig. 64 represents a heavy frame-work sixteen inches in 
 height or thickness, and seven feet four inches square, with 
 a vertical recess of twelve by forty inches in the centre of 
 each of its four sides, and a vertical opening forty inches 
 square through the centre of the frame. 
 
 Fig. 65 represents the same frame-work as Fig. 64 ; but in 
 place of the square opening in the centre, there rises, above 
 the frame-work in Fig. 65 a four-sided pyramid, forty inches 
 square at the base, and thirty-six inches in vertical height. 
 Observe that the outlines of the base of the pyramid are 
 
102 INDUSTRIAL DRAWING. [BOOK NO. II. 
 
 the same as the upper surface outlines of the square open- 
 ing in the centre of Fig. 64, and are represented by: the 
 same figures, 1, 2, 3, J, in both cases. The centre of the 
 base of the pyramid, it will be seen, must be the point 9 
 the point of intersection of the lines 5 6 and 7 8, which 
 connect the centres of opposite sides of the base. The ver- 
 tical height of the pyramid above the base must therefore 
 be the length of the line 9 10, which represents a line drawn 
 vertically upward from the centre of the base to the apex 
 of the pyramid. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw a figure similar to Fig. 56, but forty-four inches square at the 
 bottom and top ; and with the central vertical support twenty inches high 
 from o to ar, and only four inches in thickness. 
 
 2. Draw the same with the side view brought in front. 
 
 3. Draw a figure whose base shall be similar to that of Fig. 65, but whose 
 extreme side measures from corner to corner shall be six feet, and the 
 height or thickness eight inches ; the recesses on the centres of the sides 
 sixteen by twenty-four inches ; the base of the pyramid twenty-four inches 
 square, and its vertical height fifty inches. 
 
 Blackboard Exercises. Figs. 56, 57, 58, and 62. Also 
 draw Fig. 65, giving to the frame-work one half the measures 
 denoted on the paper; but make the pyramid thirty-six 
 inches in vertical height. The dark shades may be desig- 
 nated by heavy vertical lines, with either blue or white 
 crayons. 
 
 PAGE EIGHT. SCALE OF ONE FOOT TO A SPACE. 
 
 Fig. 66 may be supposed to represent a block of stone 
 twelve feet square, and four feet in height or thickness. 
 Suppose that we wish to place, centrally, on the top of this 
 block a four-sided pyramid, eight feet square at the base, 
 and fifteen feet in vertical height from the top of the base. 
 
 Evidently the base of the pyramid will be, on all sides, 
 two feet within the outlines of the top of the base on which 
 it rests. The lines a #, b f, f d, and d a, representing the 
 outlines of the base of the pyramid, must therefore be drawn 
 two horizontal spaces within the upper side lines i m and 
 n j of the base, and one diagonal space within the front 
 and back lines i n and mj. The centre, c, of the base of 
 
CABINET PERSPECTIVE PLANE SOLIDS. 103 
 
 the pyramid, must evidently be at the intersection of the 
 central lines o p and h g, and these lines will be sufficient 
 to designate it ; but it must also be at the intersection of 
 the diagonal lines a f and d b of the base. The apex of 
 the pyramid must be vertical to the point c. 
 
 Fig. 67. The top of the base of this figure is the same as 
 the top of the base drawn in the preceding figure, and on 
 this base the pyramid, of the same dimensions as that desig- 
 nated for Fig. 66, is placed. The dotted lines and the let- 
 ters are put in to designate the same points that are given 
 in Fig. 66. Observe that the point cc, the apex or vertex of 
 the pyramid, is taken fifteen spaces vertically above the 
 centre, c, of the base ; and from x lines are drawn to the 
 corners d, /, and b. The fourth corner-line of the pyramid 
 extends from x to a, but it can not be seen because it is on 
 the side opposite to the spectator. The line c x is called 
 the axis of the pyramid. 
 
 The base of Fig. 67 is represented as ten feet in height 
 and twelve feet square ; and it has recesses in the centres of 
 its sides six feet high, eight feet wide, and two feet in depth. 
 
 Fig. 68 is a wedge-shaped pyramid, eight feet square at 
 the base, and eleven feet in vertical height. Observe how 
 the edge, a b y of the pyramid is drawn so as to be directly 
 above the central diagonal line, g A, of the base. 
 
 Fig. 69 is the same as Fig. 68 ; but the side view of the 
 wedge is here brought in front of the spectator. Observe 
 that the measurements are the same in both cases. See the 
 height, a g, in both. 
 
 Fig. 70 has the same base, eight feet square, and the same 
 height, eleven feet, as the preceding two figures; but the 
 wedge-shaped pyramid is diminished to four feet at the apex. 
 If the edge of the pyramid were the line a b, the figure 
 would be the same as Fig. 68; but the edge is diminished 
 two feet at each extremity by cutting off a r and s b. 
 
 Fig. 71 is a truncated pyramid the top being cut off 
 parallel with the base. Its vertical height is the axis line 
 c d. The lines forming the edges of the sides are drawn to- 
 ward a point in the upward extension of the line c d. 
 
 Fig. 72 represents the top of a pillar in the obelisk form 
 
104 INDUSTRIAL DRAWING. [BOOK NO. II. 
 
 the top being cut off in the form of a flat pyramid. The 
 vertex, x, must be in the line of the axis of the pyramid. 
 If the sides of the block were vertical, we might suppose its 
 base to be o, d,i,n, in the figure below it; then c x would 
 be the central line or axis. 
 
 Fig. 73 represents an obelisk, which consists of an upper 
 pyramidal part, D, called the shaft, and the support or base 
 on which it rests, called the ped'estal. The pedestal is di- 
 vided into three parts : A, the base ; 13, the die ; and C the 
 cornice. There is often a recess, or sunken part, in the die, 
 which contains the inscription, etc. The vertical height of 
 the shaft of Fig. 73 is seen to be twenty-seven feet, by count- 
 ing the spaces from n to x, which line is the axis ; and the 
 base of the shaft is eight feet square. The entire vertical 
 height of the pedestal from the bottom of the base to the 
 top of the cornice will be found to be eleven feet, as meas- 
 ured from m, the centre of the bottom of the base, to n, the 
 centre of the top of the cornice. The point m must be at 
 the intersection of the lines which connect the opposite cor- 
 ners of the bottom of the base. The face of the die is seven 
 and a half feet in height and eight feet wide ; and the recess 
 in its centre is five and a half feet in height, and six feet 
 wide, so that the recess is just one foot within the edges of 
 the die. As the cornice projects one foot, it conceals from 
 view just one half of a, foot of the upper part of the front face 
 of the die, and one foot of the upper part of the side face 
 (see Rules, page 98) ; hence the upper line of the recess on 
 the right side of the pedestal coincides in view with the bot- 
 tom line of the cornice. The recess in each of the sides of 
 the shaft is also one foot in depth. The side corner-lines of 
 the shaft are drawn toward a point directly above x, in the 
 continuation of the axis. The upper extremity of the shaft 
 is in the flat pyramidal form, similar to Fig. 72, but more 
 pointed. 
 
 Fig. 74 is a Grecian fret, the same as Fig. 25 on page 6 of 
 Book No. I., here changed into the solid form; and Fig. 75 
 is the same as Fig. 26 on the same page of Book No. I., here 
 changed into the solid form. In both cases the thickness of 
 the fret, diagonally, is the same as the width of its front face. 
 
CABINET PERSPECTIVE PLANE SOLIDS. 105 
 
 To get the full effect of the d IT. wings on this page, 
 view them as directed on page 50. Let the pupil view his 
 own drawings in the same manner. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw n base and pyramid similar to Fig. 67, but let the base be a cube 
 ten feet square ; let the pyramid be placed one foot within the outlines of 
 the top of the base, and let the vertical height of the pyramid be eighteen 
 feet. Let the recesses of the base be only one foot deep, and a foot and a 
 half from the edges of the sides. 
 
 2. Draw a wedge-shaped pyramid, similar to Fig. 68, whose base shall be 
 ten feet square, and whose vertical height shall be sixteen feet. 
 
 3. Draw an obelisk of the following dimensions : 1st. The pedestal : base, 
 fourteen feet square, and three feet in thickness ; die, ten feet square at its 
 base, and height twelve feet ; recess in die two feet from edges, and one foot 
 in depth ; cornice, a foot and a half in thickness, and projection beyond the 
 face of the die one foot. 2d. The shaft : base of shaft eight feet square, 
 placed centrally on the cornice ; vertical height thirty feet, and apex like 
 Fig. 72. 
 
 4. Draw a fret like Fig. 75, with the exception that the face of the bands, 
 as seen in front, shall be only six inches wide, and the bands eighteen inches 
 apart, while the diagonal depth or thickness of the band shall be two feet. 
 
 ItlacJcboard Exercises. Draw and shade the frets, Figs. 
 74 and 75, on the supposition that the ruled lines on the 
 blackboard are one foot apart. 
 
 PAGE NINE. SCALE OF TWO INCHES TO A SPACE. 
 
 Fig. 76 represents a post-and-board fence, its length be- 
 ing here viewed diagonally, with what is called the inside 
 of the fence exposed to view, so as to show its construction. 
 The posts are eight inches square, and twenty-six inches 
 above ground ; rails two by four inches, the lower one let 
 into the posts, and the upper one resting on the posts, both 
 flush with the front edge of the posts ; boards thirty-four 
 inches in height, one inch thick, and eight inches wide, ex- 
 cept the two end boards, which are only four inches wide. 
 In the drawing the posts are placed, for want of room on the 
 paper, much nearer together than they would be in the real 
 fence. 
 
 Fig. 77 represents the same fence that is shown in the 
 preceding figure ; but here its length is placed fronting the 
 spectator, and the end is viewed diagonally. Let the pupil 
 
 E2 
 
106 INDUSTRIAL DKAWIXG. [BOOK KO. II. 
 
 test the measurements of both figures, and see if in all re- 
 spects they fully correspond with each other. 
 
 Fig. 78 is a somewhat elaborate post-and-rail fence. Here 
 the posts are eight by twelve inches in size, and thirty-two 
 inches in height ; and the rails are two by four inches, the 
 lower three rails being let into the posts four inches. Let 
 the pupil describe the construction in full length, position, 
 distances apart, etc., of all the rails. 
 
 Fig. 79 represents a four-pointed star cut out of a plank 
 two inches in thickness. Let the pupil describe it : distance 
 of the points from the centre, etc. 
 
 PKOBLEMS FOK PRACTICE. 
 
 1. Let the pupil draw a fence similar to Fig. 76 : posts to be six inches 
 square ; rails two by three inches ; height of posts and length of boards the 
 same as in the figure. 
 
 2. Let him draw the same in the position represented in Fig. 77. 
 
 3. Let him draw Fig. 78 so that its length shall be viewed diagonally. It 
 will be found that some objects can be best represented in one position and 
 some in another. 
 
 Blackboard Exercises. Fig. 76, and problem 2. 
 
 PAGE TEN. SCALE OF TWO INCHES TO A SPACE. 
 
 Fig. 80 is another form of post-and-rail fence, in which the 
 rails, in their full size of two by four inches, are mortised 
 through the posts the top of the rail being in a horizontal 
 position. Let the pupil describe the posts, rails, etc., in full. 
 The posts are here drawn much nearer together, according 
 to the scale, than they would be placed in the real fence. 
 
 Fig. 81 will illustrate the principles of drawing the repre- 
 sentation of a fence where the square rails are let into the 
 posts diagonally. Let the upright timber here represent 
 the post, which, however, is here drawn, for convenience of 
 representation, only two inches in thickness, and twenty 
 inches in width. Suppose we wish to represent a diagonal 
 square mortise through this post, for the purpose of receiv- 
 ing a square rail of twelve inches diagonal diameter. Lay 
 off the square 1 3 5 7, of twelve inches to a side, and square 
 with the ends and sides of the post. Mark the middle points 
 in the sides of this square, and connect them, measuring 
 twelve inches from 8 to 4, and the same from 2 to , and we 
 
CABINET PERSPECTIVE PLANE SOLIDS. 107 
 
 shall have the diagonal square 2 Jf. 6 8. Represent this 
 square as cut through the post, and we shall have the ap- 
 pearance of the diagonal square mortise which is to receive 
 the rail placed in a diagonal position. Observe that the lines 
 2 8 and 4 6 are drawn in the direction of three-space diago- 
 nals. Below we have the appearance which would be pre- 
 sented by the rail passing through the post. The rail is so 
 placed, with reference to the eye of the spectator, that one 
 side of it appears very wide ; while the other side, being 
 seen very obliquely, appears very narrow. 
 
 At A we have represented the posts of the same size as 
 in Fig. 80, with the square rail passing through them diag- 
 onally. 
 
 At B we have represented the side of the post through 
 which the rail passes as fronting the spectator as it would 
 be drawn if the fence were viewed diagonally lengthwise. 
 In this case the diagonal square mortise would be well rep- 
 resented, but only one side of the rail would be seen, as in- 
 dicated by the dotted representation of it. Hence the di- 
 agonal lengthwise representation of such a fence would not 
 be a good one. 
 
 Fig. 82 is still another form of post-and-rail fence, which 
 the pupil may describe. 
 
 Fig. 83. In this figure the cross-beam, which is supposed to 
 be designed to sustain a heavy weight, is supported by two 
 braces, which are framed into the cross-beam and also into 
 the posts. The under side of the brace on the left is seen so 
 obliquely that it exposes to view only a very narrow sur- 
 face. 
 
 Fig. 84 is the same pattern of the quadruple or four-band 
 fret that is used for the setting of an Arabian mosaic in Fig. 
 43, page 7, of Book No. I. Here, in Fig.- 84, the fret alone 
 is given, and in the solid form. In the lower part of the 
 figure the half diagonal lines are marked in, to show the 
 method of representing the thickness of the fret. Each of 
 these lines, it will be seen, is drawn in a diagonal direction, 
 and the length of half a diagonal. Understanding this, the 
 whole figure is very easily executed after the original fret 
 has been drawn. 
 
108 INDUSTRIAL DRAWING. [BOOK NO. II. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw a post-and-rail fence similar to Fig. 80, but with the square rails 
 inserted into the posts diagonally. Let the posts be four inches wide in 
 front, and twenty inches in diagonal width ; and let the edges of the diago- 
 nally inserted rails be two inches from the edges of the post. Omit the up- 
 per rail, but show the mortises for it. 
 
 2. Draw Fig. 82 of the same diagonal thickness of stuff there represented, 
 but of only half the front Avidth. Outside dimensions same as in the figure. 
 
 Blackboard Exercises. Figures 80 and 82, of the same 
 real size as described. 
 
 PAGE ELEVEN. SCALE OF TWO INCHES TO A SPACE. 
 
 Fig. 85 represents a section of a plank picket-fence, ac- 
 cording to the designated scale. The pickets are made of 
 stuff four inches wide and two inches thick. The bottom 
 rail to which they are spiked is four inches by five inches at 
 the ends ; and the upper rail is three inches by four inches. 
 Observe that if the spikes are driven in horizontally, and 
 perpendicular to the face of the pickets, they must have a 
 seemingly upward diagonal direction, as indicated by the 
 line c a. Hence, if they are driven into the central line of 
 the rail, their heads must be below that line, as indicated in 
 the drawing. 
 
 Fig. 86 represents a horizontal box eight inches square at 
 its two open ends, and twenty inches in length, having its 
 ends framed into and resting upon two vertical pieces of 
 plank, each sixteen inches square and two inches thick. 
 
 Fig. 87 is the Grecian double fret represented in Fig. 30, 
 page 6, of Drawing-Book No. I. ; but here drawn to a larger 
 scale, and put into the solid form. Observe that the bands 
 are two inches in thickness, the same in width, and the same 
 distance apart. Observe, also, that the short corner diag- 
 onal lines are all drawn to the centres of the small squares ; 
 and that they thereby measure the required thickness of the 
 fret, and also give the correct diagonal direction for the 
 solid. 
 
 Fig. 88 is also a Grecian double fret, of the kind seen on 
 page 5, of Book No. I., Fig. 12. It is here put into the solid 
 form, and is used for the bordering of a tablet, which is sup- 
 posed to be ornamented. 
 
CABINET PERSPECTIVE PLANE SOLIDS. 109 
 
 Fig. 89 represents a heavy plank chest, thirty-six inches 
 square on the bottom, and twenty-one inches in height when 
 the lid is closed. In the top of the chest is placed a tray, 
 having in it nine partitions. This tray rises three inches 
 ahove the body of the chest ; but its partitions are only two 
 inches deep ; and when the lid shuts down it incloses within 
 it the three inches' elevation of the tray. Let the pupil ex- 
 amine and test all the measurements, and see if the lid will 
 accurately fit over the tray, and also be even with the out- 
 side of the chest. What is the thickness of the top of the 
 lid? 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw the representation of a board-picket fence : rails the same as in 
 ITig. 85, but the pickets two inches wide, twenty inches long, made of stuff 
 two inches thick, and placed four inches apart. 
 
 2. Draw the representation of a chest similar to Fig. 80. Suppose it to 
 be thirty-eight inches square ; the body sixteen inches high ; top five inches 
 high, made of stuff two inches thick ; but the tray made of one inch stuff, 
 and rising three inches above the body. Divide the tray into sixteen square 
 divisions, each eight inches square (including the partitions) ; and let the 
 partitions be three inches deep. Place the cover on the farther side, oppo- 
 site the front. 
 
 Blackboard Exercises. Figures 85 and 87, and problem 1. 
 
 PAGE TWELVE. SCALE OF SIX INCHES TO A SPACE. 
 
 Fig. 90 represents a solid octagonal block, five feet in 
 length, with a face diameter, on the line 1 2 or 3 4, of three 
 feet. The octagonal form is the same as that of Xo. 13, 
 page 2, of Book No. I. Observe that, in drawing the visible 
 sides of the block, we draw diagonal lines from the points 
 5, 1, 6, 4, and 7, and in all cases a distance of five diagonals, 
 representing five feet. 
 
 It will be easy to find the solid contents, in cubic inches 
 or feet, of such a block, after the directions for measuring 
 surfaces given in the preceding book. Thus, on the scale 
 of six inches to a space, the front face of Fig. 90 measures 
 six square feet ; and as the length of the block is five feet, 
 the contents of the block must be five times six, or thirty 
 cubic feet. 
 
 Fig. 91 has the same front face as the star figure contain- 
 
110 INDUSTRIAL DRAWING. [BOOK NO. II. 
 
 ed within the No. 13 just referred to, or the same as the 
 star figure No. 11 of Lesson VI., on the same page. As the 
 face of the star form, Fig. 91, contains an area, according to 
 the scale, of two square feet, and as the length of the solid 
 is six feet, the solid contents of Fig. 91 would be twelve 
 cubic feet. 
 
 Fig. 92 is a hexagonal, or six-sided solid, six feet in length, 
 its two parallel ends being formed of hexagons. The pupil 
 can now, doubtless, easily calculate the cubic contents of 
 this figure. 
 
 Fig. 93 is an octagonal figure, of the same front outline as 
 No. 10 of Lesson VIII., page 2, of Book No. I. Six inches 
 within the series of the outer front lines is another series, 
 whose distance from the outer lines is regulated by the 
 points -7, 2, 3, J h Both the outer and the inner lines are 
 drawn in the direction of three-space diagonals. This com- 
 pleted figure forms a hollow octagonal tube, one foot in 
 length, whose sides are six inches in thickness, with an ex- 
 treme diameter, both vertical and horizontal, of four feet. 
 The points 5 and 6, one diagonal space distant from 4 and #, 
 regulate the drawing of the inner boundary-lines of the far- 
 ther end. These lines also are drawn in the direction of 
 three-space diagonals, and are hence parallel to their corre- 
 sponding front lines. 
 
 Fig. 94 is a solid octagonal block, of four feet vertical and 
 horizontal diameter, and six feet in length. The pupil, re- 
 ferring back to Book No. I., should now be able to calculate 
 its cubical contents. Observe that the shading is such as 
 most clearly to distinguish the several visible sides of the 
 block. 
 
 Fig. 95. The outline of the front face of the block here 
 represented forms a dodecagon, or figure of twelve sides, the 
 same as is shown on page 2 of Book No. I. ; while the hol- 
 low or opening through the block is hexagonal. It is easy 
 to see how the lines bounding the several sides of the block 
 lengthwise, both on the outside and inside, are to be drawn 
 all in the direction of diagonals. The outside lines must 
 all, of course, be of equal length ; and the lines forming the 
 boundary of the farther face of the dodecagon must be par- 
 
CABINET PERSPECTIVE PLANE SOLIDS. Ill 
 
 allel to those forming the boundary of the front face. But 
 they are made parallel with perfect ease, because their posi- 
 tion and limits are definitely designated by the ruling of 
 the paper. 
 
 Let the pupil, referring back to the outlines of the same 
 dodecagon on page 2 of Book Xo. I., calculate the solid con- 
 tents of Fig. 95. 
 
 Fig. 96 has the same front face as Fig. 95, but the solid 
 formed upon it is only one foot in length. It is thus drawn 
 in order to show a portion of the inner outlines of the far- 
 ther end of the block. What are the cubical contents of 
 this figure? 
 
 Fig. 97, having the same outline front form as Fig. 90, is 
 drawn to inclose an area four times as large as Fig. 90. 
 (See ELEMENTARY PRINCIPLE, page 54.) This figure is a 
 hollow octagon, two feet in length, and having its sides six 
 inches in thickness. Observe that the front outline is formed 
 by drawing two-space diagonals throughout ; observe, also, 
 that the inner boundary-lines of the octagon, at both ends, 
 are the same in direction, but necessarily of less length. 
 
 Fig. 98 has the same outline front form as Fig. 93, but is 
 drawn to inclose an area four times as large as Fig. 93. 
 (See ELEMENTARY PRINCIPLE, page 54.) It is drawn of 
 three-space diagonals instead of two. Observe how easily 
 and accurately in all these figures the inside lines of the 
 front face of the octagon are drawn, so as to be exactly 
 parallel to the outside lines. Thus, after taking the point 
 1, in Fig. 98, the line 1 7 is drawn in the direction of a 
 three-space diagonal until it intersects the diagonal line 7 10; 
 then the line 2 7 is drawn as a three-space diagonal, and it 
 will intersect the diagonal 7 10 in the point 7. In like 
 manner all the front inner-face lines are drawn. Observe, 
 also, that the points 6 and 5 are designated by being three 
 diagonal spaces distant from 3 and 4- Then 6 12 and 5 12 
 are drawn in the direction of three-space diagonals, meeting 
 in the same point, 12. 
 
 Fig. 99 is a frame-work which requires little explanation. 
 Observe that the braces must start at equal distances from 
 the vertical post ; and that the distance 1 2 must be the 
 
112 INDUSTRIAL DRAWING. [BOOK NO. II. 
 
 same as 3 4* The width of the braces, as measured by the 
 spaces 1 6 and 4 5, must also be the same ; although, as the 
 side edge of the nearer brace is seen most obliquely, it ap- 
 pears narrower than the farther brace. The pupil should 
 give the measurements throughout, and imitate the shading 
 with India ink and the running dot-line. 
 
 Fig. 100 represents a foot-bridge resting on piers laid in 
 cut stone, or in brick three inches thick, six inches wide, and 
 twelve inches long. These piers are three feet high, laid 
 up vertically on three sides, four feet by five feet at the base, 
 and two feet by five feet at the top. The extreme width 
 of the bridge is five feet; length, thirteen feet; top of rail 
 from top of pier, three feet nine inches. The timbers (called 
 sleepers) on which the floor is laid are twelve inches vertical 
 height, and nine inches wide; the flooring is represented as 
 of about t wo and a half inch plank ; five of the planks being 
 each two feet wide, and the two end planks each eighteen 
 inches wide. The posts are six by twelve inches, cut out 
 so as to let half their thickness (three inches) rest on the 
 sleepers. The planks come out even with the outer edges 
 of the posts, and hence they project three inches beyond 
 the sleepers. The top rails are three by six inches. 
 
 Figs. 101 and 102 represent patterns of cubical blocks, 
 such as are sometimes worked in worsteds of three colors. 
 We have also seen a carpet of the same pattern, although it 
 is a very poor pattern for such a purpose. No further di- 
 rections are required for drawing or shading these figures. 
 
 Blackboard Exercises. Figures 90, 93, 96, and 99. Ob- 
 serve that the farther braces of Fig. 99 run downward in 
 the direction of diagonals, and the front braces in the direc- 
 tion of three-space diagonals. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw an octagonal solid similar to Fig. 90, but whose front face is 
 only one fourth as large. Be careful to draw the lines of the front face in 
 the direction of two-space diagonals, and half the length of those in the 
 figure. 
 
 2. Draw a solid similar to Fig. 91, but whose front face shall be four 
 times as large. 
 
 3. Draw a solid similar to Fig. 92, which shall contain four times the 
 
CABINET PERSPECTIVE PLANE SOLIDS. 113 
 
 cubic contents of Fig. 92, and whose front face shall contain four times the 
 area of Fig. 92. 
 
 4. Draw a dodecagon hollow body, the extreme outline of whose front 
 face shall be the same as Fig. 95 : let the sides of the hollow body be of 
 the same length, but only half the thickness of the sides of Fig. 95. 
 
 5. Draw a figure the same as Fig. 99, with the exception that the side 
 shall front the spectator, and the platform base shall be one foot wider than 
 there represented. 
 
 G. Draw Fig. 100 of the same dimensions as given, but with the side 
 fronting the spectator. 
 
DRAWING-BOOK No. III. 
 
 CABINET PERSPECTIVE CURVILINEAR SOLIDS. 
 
 IT has been observed that, in all the representations of 
 objects given in the preceding book, the side of the object 
 fronting the spectator is drawn in its natural form and pro- 
 portions this front side being supposed to be in a vertical 
 position. For example, the front side of the cube at -K, 
 Fig. 1 of Book No. II., is drawn an exact square. So in 
 curvilinear solids, that side fronting the spectator is drawn 
 in its natural form, and is supposed to be in a vertical po- 
 sition. Therefore, if we wish to represent a cylinder in cab- 
 inet perspective, and to draw it with the end fronting the 
 spectator, that end must be represented in its natural form 
 as a perfect circle. 
 
 PAGE ONE. SCALE OF ONE INCH TO A SPACE. 
 
 Fig. 1 is a circle drawn from the centre, c, with a radius, 
 c a, of five inches. It may therefore represent the end of 
 a cylinder of ten inches' diameter. 
 
 Fig. 2. Here c may be taken as the centre of the end of a 
 cylinder of ten inches' diameter. A cylinder is a solid whose 
 bases or ends are equal parallel circles, at right angles to the 
 axis of the cylinder. The axis is the line passing through 
 the centre of the cylinder at right angles to the ends of the 
 cylinder. The circumference of a circle is often called its 
 pe-riph'-er-y; hence, also, we speak of the periphery of a cyl- 
 inder. 
 
 In Fig. 2, if we suppose the cylinder to be ten inches in 
 length, the line c ct, drawn diagonally from the front centre, 
 
CABINET PERSPECTIVE CURVILINEAR SOLIDS. 115 
 
 c, will represent the axis of the cylinder; and c will be the 
 central point of the near end, and d the central point of the 
 farther end. If, therefore, we describe a circle from the 
 point d, with a radius of five inches, this circle, only one 
 half of which could be seen in the solid cylinder, would givo 
 the true form and position of the farther end of the cylinder. 
 Connect the extreme edges of these circles by the lines p r 
 and in n, and we have all the visible outlines of the cyl- 
 inder. 
 
 Fig. 3 is the same as Fig. 2, with the cylinder shaded, 
 so as to represent it in its solid form. . Observe here that 
 
 d, the same as in the preceding figure, five diagonal spaces 
 from c (ten inches), is the centre from which the visible 
 boundary of the farther end of the cylinder is described. 
 From points in the line c d are to be described the several 
 circular bands which encompass the cylinder; and these 
 bands will be the same distances apart as are the centres 
 from which they are described. 
 
 Fig. 4. Here a longitudinal right-angled section, embrac- 
 ing one quarter of the cylinder, is taken away. This shows 
 the position of the axis, c d, better than it can be seen in the 
 preceding two figures. From the centre, d, are described, 
 with the compasses, the visible portions of the farther end 
 of the cylinder. The sides are drawn in the same manner 
 as in Fig. 3. 
 
 Fig. 5 represents a cylinder of the same size, originally, 
 as in the preceding three figures, but having, first, one quar- 
 ter cut away, as in Fig. 4, and then, in addition, all but the 
 lower left-hand quarter of the middle section is taken away. 
 Measurements of the several sections may be made on the 
 vertical, horizontal, and diagonal lines in the same manner 
 as in all plane solids. 
 
 Fig. 6. Here, also, the cylinder is ten inches long, as meas- 
 ured on the line of its axis, c d. The upper half of the front 
 portion, four inches in length, is first taken away ; then a 
 portion of the farther part, but less than a quarter, is re- 
 moved. 
 
 Fig. 7 is the same as Fig. 6, but shaded so as to show the 
 several parts more distinctly. 
 
116 INDUSTRIAL DRAWING. [BOOK NO. III. 
 
 Fig. 8 represents three quarter-sections of a cylinder five 
 inches in diameter and four inches long. 
 
 Fig. 9 represents a cylinder four inches in length and ten 
 inches in diameter, with an opening four inches square ex- 
 tending through it longitudinally and centrally. Here c is 
 the centre of the front end of the cylinder ; and at two di- 
 agonal spaces from it will be the centre from which the 
 boundary of the farther end is described. Circles running 
 around the cylinder are described by the compasses from 
 points on the axis. Observe that the length of the cylinder 
 is not only measurable on the axis, but also on the other di- 
 agonal lines 3^56, etc., which must all be equal each 
 being the measure of two diagonals. 
 
 Fig. 10 represents the outlines of a hollow cylinder four 
 inches in length and ten inches in diameter, with walls one 
 inch in thickness. As c is the centre of the front end, we 
 first describe from it the outer circle, or periphery, of the 
 cylinder, with a radius of five inches, and then the inner cir- 
 cle, with a radius of four inches, which leaves one inch be- 
 tween them for the thickness of the walls. From r7, the 
 centre of the farther end, we next describe two like circles, 
 or parts of circles, the visible portion of the smaller circle 
 being 1 2 3 y which is the visible portion of the farther inside 
 boundary. 
 
 Fig. 11 is the same as Fig. 10 in outline, but fully shaded. 
 
 Fig. 12 represents a cylinder four inches in length and 
 ten inches in diameter, with a solid cylinder, or axle, four 
 inches in diameter and twenty inches in length, passing cen- 
 trally through it the axis of the longer cylinder coinciding 
 with the axis of the shorter one. Here c d represents the 
 axis of the short cylinder, and a b the axis of the long one. 
 The length of either, or both, may be measured in any one 
 of the several ways mentioned for measuring Fig. 9. 
 
 PEOBLEMS FOE PEACTICE. 
 
 1. Draw a solid cylinder six inches in diameter and twelve inches in 
 length. 
 
 2. Draw the same, but represent one quarter taken from it, similar to 
 Fig. 4. 
 
CABINET PERSPECTIVE CURVILINEAR SOLIDS. 117 
 
 3. Draw the lower left-hand quarter of a cylinder whose entire dimen- 
 sions would be eight inches in diameter and ten inches in length. 
 
 4. Draw the lower right-hand quarter of a cylinder of the same dimen- 
 sions. 
 
 5. Draw the lower horizontal half of a cylinder whose entire dimensions 
 would be eight inches in diameter and twelve inches in length. 
 
 6. Draw a hollow cylinder of fourteen inches in diameter, six inches in 
 length, and one-inch thickness of walls. 
 
 7. Draw the lower half only of a hollow cylinder of twelve inches in di- 
 ameter, 12 inches in length, and one-inch thickness of wall. 
 
 Blackboard Exercises. The chalk crayon compasses will 
 be required here. See page 49. Draw and shade all the 
 figures on page 1, and also the problems. 
 
 PAGE T \YO.-SCALE OF OXE FOOT TO A SPACE. 
 
 Fig. 13 represents the outlines of a solid, four feet by ten 
 feet at the base, and eleven feet entire height : but the upper 
 part is a semicircle. The front semicircular outline is de- 
 scribed from the point c, and the farther one from the point 
 #, distant, diagonally, from c, four feet. Observe that the 
 side line, a 5, is drawn so as to touch the semicircles at the 
 extreme limit of our view of the semicircular top ; and that 
 the points a and b are necessarily at the intersections of di- 
 agonals, from c and ce, with the semicircles. Let it be no- 
 ticed that, as the measurements 1 , 3 4, 5 6 9 7 , 9 10, a b, 
 and c x, are ail equal in the real object, so they must be 
 equal in the drawing. 
 
 Fig. 14 is the same as the preceding figure shaded. 
 
 Fig. 15 is similar to Fig. 14 ; but it rests on a base which 
 projects laterally one foot; and the semicircular top also 
 projects laterally one foot, equal to the projection of the 
 base. 
 
 Fig. 16 is the same as the preceding figure fully shaded. 
 
 Fig. 17 represents the outlines of an archway, formed of 
 two parallel walls, each two feet thick, six feet high, and 
 six feet wide, placed eight feet apart, and surmounted by a 
 semicircular arch of the same thickness as the walls on which 
 it rests. 
 
 Fig. 18 is the same archway represented as reversed in 
 position, and placed arch downward. This shows the out- 
 
118 INDUSTRIAL DRAWING. [BOOK NO. III. 
 
 lines of the bases of the walls in full, and a part of the un- 
 der side of the arch, which could not be shown in the draw- 
 ing above it. Figs. 19 and 20 are the same as 1Y and 18, 
 shaded. In this way an object may be represented in dif- 
 ferent positions, when it is desirable to show as fully as pos- 
 sible the construction of its several parts. 
 
 Fig. 21 represents a semicircular arch resting on four pil- 
 lars, two on a side. The pillars are four feet square and 
 fourteen feet high, each pair resting on a base one foot in 
 thickness, six feet wide, and ten feet in length, and sur- 
 mounted by a cap (capital) of the same dimensions. The 
 width of the archway, as measured from a to b, is twelve 
 feet ; and the height, from c to 2, twenty-one feet. From the 
 point x the front semicircles are described ; and from y the 
 semicircles which bound the farther end of the arch are de- 
 scribed. 
 
 The divisions of the front of the arch, 2, 3, 4, &> etc., are 
 laid oif by the compasses; and from the points of division 
 straight lines are drawn toward the centre, a?, just as the 
 lines would tend if the real arch were laid with cut stone. 
 Let it be observed, also, that the lines 5 7, 6 , 9 10, etc. r 
 drawn in the direction of diagonals, must all be of precisely 
 the same length in the drawing, according to the scale, a? 
 they would be in the real arch. In fine, all diagonal, ver- 
 tical, and horizontal lines in the drawing give, according to 
 the scale, perfectly accurate measurements of the structure. 
 Parts of perfect circles may then be measured or calculated 
 with the same accuracy as in the real object. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw a figure similar to Fig. 1 4, but with a width of eight feet in front, 
 depth of two feet, and total vertical height of ten feet. 
 
 2. Draw one like Fig. 19 in all respects, except that a central vertical 
 section, two feet in depth, is represented as taken out, from right to left, 
 throughout the entire archway thus leaving the front and the rear of 
 the archway perfect, and taking out the third part that lies between 
 them. 
 
 3. Eepresent the same inverted that is, by placing the arch downward. 
 
 4. Draw a figure similar to Fig. 21, but of the following dimensions: 
 Place two columns on a side, each two feet square and twelve feet high, ten 
 feet apart in front, and each pair six feet apart diagonally. Let each pair 
 
CABINET PERSPECTIVE CURVILINEAR SOLIDS. 119 
 
 have a base and cap similar to those in the figure, and also complete the 
 arch in a manner similar to that in the figure. 
 
 Blackboard Exercises. Figures 14, 19, and 20; also the 
 foregoing problems. 
 
 PAGE THREE. SCALE OF TWO INCHES TO A SPACE. 
 
 Fig. 22. We have here represented a pair of braces rest- 
 ing upon a platform two inches thick, eight inches wide, 
 and twenty-eight inches long, and so placed as to be equally 
 inclined toward a central line, 2 5. 
 
 Fig. 23 represents a platform the same as in the preced- 
 ing figure, but with curvilinear braces placed upon it, and 
 equally inclined toward a central line, 2 5. In this case, 
 the bases of the braces, and the height 5 2, being the same 
 as in the preceding figure, we take any point, as y or c, from 
 which we can describe, with the compasses, a curve that 
 will pass through the points 1 and 2. This point, y or c, 
 may be farther from the left-hand brace, or nearer, according 
 to the kind of curve that we desire for the brace : but, hav- 
 ing taken it, we must describe, from the same point, the cor- 
 responding curve 7 9. Diagonally from y or c we must 
 take the point z or x, making y z or c x equal to 7 8 ; and 
 from z or x describe the curve 3 4- This latter curve would 
 terminate at the point 10, the vertex of the farther face of 
 the arch, on the under side. For describing the curves of 
 the right-hand brace, we must take corresponding points at 
 the left of the platform. 
 
 Fig. 24 shows the left-hand brace of the preceding figure, 
 as drawn by itself alone. Here the point 10 is visible, and 
 is seen to be in continuation of the curve 3 4> 
 
 Fig. 25. This figure represents a semicircular arch eight 
 inches (y i) in width, two inches' thickness of wall, described 
 with an extreme radius (c d or y g) of twenty-two inches, 
 and resting upon a base eight inches wide, two inches thick, 
 and fifty-two inches in length the whole divided vertically 
 in the drawing, and at right angles to the face of the arch, so 
 as to separate arch and platform into two equal halves, six 
 inches apart. Suppose the two halves to be brought togeth- 
 er centrally : the arch will thea form a perfect semicircle. 
 
120 INDUSTRIAL DRAWING. [BOOK NO. III. 
 
 Here all the curvilinear lines on the left are described 
 from the points c and JB, and those on the right from the 
 points y and z. Let the pupil observe, that while the line 
 from 10 to b is a curve struck by the compasses from the 
 point a;, that b a is a straight line, drawn according to the 
 principles that were explained in relation to Figs. 13 and 
 14. When the curve from 10 to b has reached the point , 
 its continuation is below the visible surface of the arch; and 
 the straight line b a is the surface of the arch, just as the 
 line b a in Fig. 13 is the surface of the arch there repre- 
 sented. 
 
 Fig. 26 is the same as the preceding figure inverted. The 
 upper sides of the platform and arch here become the under 
 sides, and the points x c and y 2, consequently, follow the in- 
 version. 
 
 Fig. 27 represents a hollow cylinder of forty-eight inches' 
 outside diameter, as measured from 1 to 2 ; eight inches in 
 length, as measured from 2 to $, 4 to 5, 6 to 7, or any where, 
 diagonally, on the circumference ; and walls of two inches' 
 thickness. The two circles of the front end of the cylinder 
 are described from the point t; and the visible parts of the 
 two circles which bound the farther end of the cylinder are 
 described from the point a?, which must be two diagonal 
 spaces from , in order to make the length of the cylinder 
 eight inches. The two front circles should be described 
 first. 
 
 Fig. 28 is what is called a quarter/oil in architecture; 
 which is an ornamental figure disposed in four segments of 
 circles, the front face of which is a conventional resemblance 
 of an expanded flower of four petals. (See Fig. 36, page 81.) 
 The front segments of circles, forming the face of the orna- 
 ment, are described from the points a, b, c, c?, the four angles 
 of a diagonal square; and as the ornament has a depth of 
 eight inches, the segments of circles forming the farther 
 side of the figure are described from the points ic 9 x, y, z, two 
 diagonal spaces from the other centres. The shading lines 
 of the segments of circles are described from points between 
 the front and rear centre?, on the dotted lines. In drawing 
 the figure, first describe the front segments from the points 
 
CABINET PERSPECTIVE CURVILINEAR SOLIDS. 121 
 
 , &, c, c7, then from the other points describe such portions 
 of the other segments as could be seen. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw a figure similar to Fig. 22, but make the plank base four inches 
 longer, and place the braces four inches farther apart, while they shall be 
 of the same height, 5 2. Then on the lower side of the plank base place 
 a like pair of braces, same distances apart, etc., and meeting at a point the 
 same distance below the plank that 2 is above it. 
 
 2. Draw Fig. 23 with changes similar to those that were required to be 
 made in Fig. 22 for the preceding problem, putting a pair of curvilinear 
 braces below as well as above the platform. 
 
 3. Draw Figs. 25 and 26 united in one figure, and on the same base, 
 but on opposite sides of it, and without any separation in the base or in 
 the semicircular segments. 
 
 4. Draw a cylinder similar to Fig. 27, but with an extreme diameter of 
 forty inches ; walls four inches in thickness, and length of cylinder sixteen 
 inches. 
 
 5. Draw a quarterfoil similar to Fig. 28 ; but take the centres of the 
 front segments in the four comers of a diagonal square of only four diag- 
 onal spaces to a side : draw the outer segments with a radius of eight 
 inches, and the inner with a radius of six inches ; and make the quarterfoil 
 only four inches in diagonal thickness. The measure for the radius must 
 be taken on vertical or horizontal lines. 
 
 After drawing the foregoing problems, examine them carefully through 
 the opening formed by the partially closed hand. 
 
 Blackboard Exercises. Figs. 22, 23, 25; and any of the 
 problems. 
 
 PAGE FOUR. SCALE OF TWO INCHES TO A SPACE. 
 Fig. 29 represents a bracket of forty-four inches' height, 
 thirty inches' projection, and eight inches' thickness. The 
 front of the bracket (here the diagonal view) embraces two 
 segments of circles; the upper one described with a radius, 
 3 a or 3 b, of twenty-two inches ; and the lower one with 
 a radius, 1 m or 1 rc, of six inches. Observe that from the 
 point 3 the curves a b and/.? are described; and that from 
 the point 4 the curve c d is described. The parts cp and 
 a n are not portions of the curves, but are straight lines. 
 From the point 1 the curve m n is described ; and from the 
 point 2 is described the hidden curve op. In selecting the 
 points 1 2 and 3 4 we are influenced wholly by the kind 
 
 F 
 
122 INDUSTRIAL DBA WING. j_BOOK NO. III. 
 
 and extent of the curve which we wish to describe, just as 
 in describing the curve of the real bracket; but the diagonal 
 distance from 1 to 2 and from 8 to 4 is determined by the 
 thickness of the bracket. 
 
 Fig. 30 is the same as Fig. 29, shaded in full. 
 
 Fig. 31 is a bracket of precisely the same dimensions as 
 Fig. 30, but drawn in an inverted position, with the top of 
 the bracket downward, by which means the hollows, or 
 curved portions, are represented more fully than in the 
 preceding figure. 
 
 Let the pupil observe that, wherever the thickness of the 
 bracket is measured, it measures in all parts the same. 
 Thus the diagonal lines a c, d b,g h,p n, o m, s r, etc., all 
 measure the same. 
 
 Fig. 32 shows the outlines and mode of drawing a plain 
 and regular trefoil, which is an architectural ornament dis- 
 posed in three segments of circles, being a conventional 
 representation of three-leafed clover which is, botanically, 
 a trefoil plant. 
 
 Observe that the points -?,#,#, which are the centres from 
 which the front curves are drawn, are at the three corners 
 of an equilateral triangle ; and that the points 4, &, 6, from 
 which the visible portions of the curves of the farther side 
 are drawn, are also at the corners of an equilateral triangle, 
 one diagonal space distant from the front corners. 
 
 Fig. 33 shows the same trefoil as Fig. 32, but it is here 
 cut into and through a cylindrical plank four inches in 
 thickness, and thirty-six inches in diameter. Observe that 
 a and b are the central points for describing the outlines of 
 the cylinder. 
 
 Fig. 34. Here we have a quarterfoil of the same dimen- 
 sions as in Fig. 28, but here placed within a cylinder of the 
 same dimensions as that shown in Fig. 27. Observe that 
 the central point of the frOnt end of the cylinder is tj and 
 that from this point, with a radius of eleven inches, we de- 
 scribe a circle which touches the convex front edges of the 
 quarterfoil; from the same point t, but with a radius two 
 inches greater, we describe another circle ; and thus get 
 the front face of the cylinder. In this manner the inclosing 
 
CABINET PERSPECTIVE CURVILINEAR SOLIDS. 123 
 
 cylinder should first be drawn; then the four central points, 
 a, , <?, d, of the front face of the quarterfoil, should be desig- 
 nated at the four corners of a diagonal square. These points 
 must be taken according to the size of the quarterfoil that 
 can be drawn; but it will be observed that they are at 
 equal distances, vertically and horizontally, from the central 
 point t. The point x is taken the same as in Fig. 27; and 
 the points tc,ic,y,2, the same as in Fig. 28. To get the full 
 effect of this drawing, view it through the opening of the 
 partially closed hand. (See page 50.) 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw a bracket similar to Fig. 30, but of the following dimensions: 
 height of bracket, forty inches ; projection, 2G inches ; thickness, four 
 inches ; the form of curves, etc. , according to judgment. 
 
 2. Draw the same inverted, similar to Fig. 31. 
 
 3. Draw a trefoil on the basis of a triangle of thirty-two inches to a side; 
 make it eight inches in diagonal depth or thickness ; and let it have walls, 
 which shall be two inches thick, similar to the walls of the quarterfoil in 
 Fig. 28. 
 
 Blackboard Exercises. The foregoing problems. 
 
 PAGE FIVE. SCALE OF TWO INCHES TO A SPACE. 
 
 Fig. 35 is a conventional leaf pattern, drawn so as to 
 represent four leaves whose central surface lines form the 
 diagonals of a square; then these leaves are represented as 
 being four inches in thickness, the under sides of the leaves 
 being of the same form and dimensions as the upper sides. 
 
 First : the outline of one side of the upper or front sur- 
 face of one of the leaves, say A^ is drawn according to judg- 
 ment. Then it will be easy, as shown in Book No. I., to 
 draw the other side symmetrical with it by the aid of the 
 ruled paper. Thus the curvilinear outlines of the leaf^l, 
 on both sides of the central line, are a little more than one 
 space from the point a; nearly three spaces, on both sides, 
 from the point b; four spaces from the point c; three spaces 
 from the point c?, etc. In this manner the outlines of the 
 upper surfaces may all be accurately drawn by the hand 
 alone. 
 
 Next : we have to draw the visible portions of the out- 
 
124 INDUSTRIAL DEAWING. [BOOK NO. III. 
 
 lines of the under or farther sides of the leaves. If we take 
 any point, as $, in the leaf A^ the point directly opposite 
 to it, on the farther side of the leaf, must be just one diag- 
 onal space from it that is, its opposite point must be seen 
 at 4j the point opposite 5 must be seen at 6 ; the point 
 opposite r, the end of the upper side of the leaf, must be at 
 jt?, which latter point is the end of the lower side of the 
 leaf. In this manner we may designate as many of the 
 visible points of the lower edges of the leaf as we may de- 
 sire; and then we have only to draw the curves through 
 them to have the outlines of the under side of the leaf, so far 
 as they can be seen from our point of view. In the leaf 
 marked A, the extreme point of the under side of the leaf is 
 seen at p, one diagonal space from r; in the leaf B, the 
 extreme point of the under side of the leaf is seen at m, one 
 diagonal space from / in the leaf D, the extreme point 
 of the lower side is seen at 7i, one diagonal space from g ; 
 and in the leaf (7, the extreme point of the lower side of the 
 leaf is at x, one diagonal space from o ; and therefore it 
 could not be seen unless the leaf were transparent. Thus 
 the pupil will see what portions of the lower edges of these 
 four leaves must be visible, and how he must draw them ; 
 and if he will remember that any point on the farther side 
 of a vertical object put in cabinet perspective must be in a 
 diagonal direction from its corresponding point on the 
 nearer or front side, and will regulate the distance on the 
 diagonal according to the scale adopted, he will find little 
 difficulty in drawing any solid curvilinear object. 
 
 Fig. 36 is the same as Fig. 35, only shaded to show the 
 form distinctly. 
 
 Fig. 37 is a solid isosceles* triangle, eight inches in thick- 
 ness, as represented by the diagonal lines 1 2, 7 S, and 9 10. 
 Let it be understood that the face, 1 7 P, of the triangle is in 
 a vertical position, fronting the spectator. This face, then, 
 presents to our view its natural proportions, and is meas- 
 ured in any direction by the measurement given to the ver- 
 tical and horizontal lines of the ruling two inches to a space. 
 
 * An isosceles triangle is one which has only two sides equal. 
 
CABINET PERSPECTIVE CURVILINEAR SOLIDS. 125 
 
 Taking the length, a 7, with the compasses, and applying it 
 to the ruled lines, we find that a 7 is a little more than twen- 
 ty-eight inches long ; and we find in the same manner that 
 the base, 1 9, is about seventeen inches long, as nearly as it 
 can be measured by the compasses.* Let it be noticed here 
 that the rule of diagonal measures does not apply to any 
 lines whatever on the surface, 1 7 9; because 1 7 9 is here 
 supposed to be a vertical surface fronting the spectator, 
 and not a horizontal surface. See Rule, page 85. Observe 
 that the point on the under side of the triangle, opposite to 
 7, is seen at 2; the point opposite to 3 is seen at 4 > the 
 point opposite to 5 is seen at 6; the point opposite to 7 is 
 seen at #, etc. all in the direction of diagonals. The farther 
 face of this solid triangle, being also in a vertical position, 
 must measure the same as the front face. 
 
 Fig. 38 is the same in form and measurement as the pre- 
 ceding figure, but inverted in position. We here see the 
 base, 1 2 10 9, quite fully, but no portion of the sides. 
 
 Fig. 39 is a curvilinear quadrangular solid, supposed to 
 be placed with the front side in a vertical position. It. 
 measures thirty-four inches between the opposite angles 
 as from 1 to 2 and from 3 to 4, and is four inches in thick- 
 ness. The curvilinear sides of the front surface are described 
 with a radius of six diagonals, or about seventeen inches, 
 from the four angles of an erect square ; and the curvilinear 
 sides of the farther face are described from the four angles 
 of another erect square of the same size, whose corners are 
 one diagonal space from the corners of the first-mentioned 
 square. Observe that no part of the lower left-hand side or 
 edge of the solid is visible ; but the dotted line, s ?, shows 
 its position on the farther face of the solid. 
 
 Fig. 40 is the same curvilinear solid that is represented in 
 the preceding figure, but the sides of the front face are here 
 described from the four angles of a diagonal square. The 
 
 * The exact lengths of the lines a 7 and 1 9 can be found, inasmuch as 
 each forms the hypothenuse of a right-angled triangle, which the ruling on 
 the paper will show. In calculating their lengths, it will he found that a 7 
 is a little more than twenty-eight inches long, and that 1 9 is a mere frac- 
 tion less than seventeen inches. 
 
126 INDUSTRIAL DRAWING. [BOOK NO. III. 
 
 points for describing the curves of the farther edges are 
 f 9 A, #, and d, which are each one diagonal space from the 
 other points. 
 
 Fig. 41 is a solid semicircular and rectangular architect- 
 ural band, the front vertical face of which is two inches in 
 width, and the thickness or depth of which is four inches. 
 
 Taking the curvilinear section A, it will be seen that the 
 two curves of the front face are described from the point #, 
 with a radius of three and of four inches ; and that the two 
 curves of the farther face (a part only of each being visible) 
 are described from the point 4, one diagonal space from #, 
 with the same radii as the front curves. The points 5 and 6 
 are, in like manner, the centres for describing the curves of 
 the section J5. The points for the other curves are also 
 given. The front face is shaded with a light wash of India 
 ink, and the other parts by darker washes and the pencil. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw the leaf forms B, (7, and D at short distances apart, but in their 
 present relative positions, and give to each a thickness of eight inches. 
 'Shade them as in Fig. 3G. 
 
 2. Draw Fig. 37 of the same dimensions as described ; but let the cen- 
 tral line, a 7, be in a horizontal position, twenty-eight inches in length 
 tue point 7 being at the right hand. 
 
 3. Draw Fig. 38 of the same dimensions as Fig. 37, but having the cen- 
 tral line, 4 5, m a horizontal position the point 5 being at the left hand. 
 
 4. Draw an erect square frame of forty inches to a side, in the clear that 
 is, having a clear measure of forty inches on the inside ; let the depth be 
 eight inches, and the thickness of the material one inch. Within this frame 
 place a solid similar to Fig. 39, having its curvilinear sides described from 
 the inner corners of the frame with a radius of twenty inches ; and let the 
 solid also be eight inches in thickness. Shade the frame and the inclosed 
 solid properly. 
 
 Blackboard Exercises. Fig. 41, as much of it as can be 
 put on the board, and problem 4. 
 
 PAGE SIX. SCALE OF TWO INCHES TO A SPACE. 
 Fig. 42 represents five quarter-sections of the rims of 
 wheels, the front faces or edges of which are all in the same 
 vertical plane, fronting the spectator. If the rims were en- 
 tire, they would represent what is called a nest, lying one 
 
CABIXET PERSPECTIVE CURVILINEAR SOLIDS. 127 
 
 within another. As the front edges all lie in the same 
 plane, they are all described from the same centre, a, each 
 rim being made two inches in thickness. The farther edges 
 of the rim are likewise in one and the same plane ; and as 
 the rims are four inches wide, they must be described from 
 a point, , one diagonal space from a. The lines crossing the 
 front edges of the rims all radiate from the point a. 
 
 Fig. 43 represents the side walls of a beveled tub, whose 
 vertical height, when placed on its bottom, is sixteen inches ; 
 extreme diameter across the top, twenty-eight inches ; and 
 extreme diameter across the bottom, forty inches. 
 
 The top of the tub being supposed to be in a vertical plane 
 fronting the spectator, and c being its central point, the 
 outer circle of the top is described from c, with a radius, 
 c #, of fourteen inches ; while the inner circle is described 
 with the radius, c g, of twelve inches. 
 
 Now, as the tub is to be sixteen inches in extreme height 
 (or depth), we take the point x, four diagonal spaces from c; 
 and the point x is then the central point of the bottom of 
 the tub. Hence, with x as a centre, and with a radius, x b, 
 of twenty inches, we describe a circle for the outer circum- 
 ference of the bottom of the tub, thus making its diameter 
 forty inches. With a radius two inches less, we describe 
 from x the circle for the inner circumference of the bottom 
 of the tub, which gives us the visible part, dsf, of this inner 
 circumference of the bottom. The lines r p and m n are 
 drawn tangent to (that is, touching) the outer circumfer- 
 ence of the top of the tub, and the outer circumference of 
 the bottom of the tub. 
 
 Suppose, now, that we wish to put circumference lines 
 around the tub, and passing through certain points, , 4, 
 and 6, of the surface. In the same proportions that the line 
 i k is divided by the points , 4, and 6>, divide the axis line 
 c x by the points .?, , and 5 ; then, with one point of the 
 compasses in the point 1, and the other extended to the 
 point 2, describe the first curve ; with one point in <<?, and 
 the other in 4-> describe the second curve ; with one point in 
 5, and the other in 6, describe the third curve. The visible 
 portion of the inside of the tub is viewed so obliquely that 
 
128 INDUSTRIAL DRAWING. [BOOK NO. III. 
 
 its widest portion, t s, is hardly one third of the apparent 
 extent of i k / although both are, in reality, of the same ex- 
 tent. This figure should be viewed from above, and on the 
 right, through the partly closed hand, when the effect will 
 be very striking. 
 
 Fig. 44 represents nine cylinders, each fourteen inches in 
 extreme diameter, four inches in depth, and walls one inch 
 in thickness, placed in three rows forming a square, so that 
 the cylinders touch one another. 
 
 Fig. 45 represents one of the stones from a carved mould- 
 ing. It measures thirty-eight inches in height, twenty-six 
 inches in breadth, and twenty-four inches in thickness as 
 measured any where diagonally. In making a working 
 drawing of such a figure, certain measurements can be given 
 definitely for directions to the stone-cutter as the height, 
 the breadth, the distance from b to 11, from 11 to 9, from 9 
 to d; the size of the semicircle described from c, etc. ; and all 
 these measurements will appear accurately in the drawing. 
 Then the curve, 3 5 7 d, may be drawn so as best to please 
 the eye. Having this drawn, the corresponding curve on the 
 opposite side must be draw r n, in all its parts, at a diagonal 
 distance from the first curve representing twenty-four inches. 
 Thus all the diagonal lines, such as 5 6, 7 8, etc., that can 
 be drawn across the face of the carving, must measure pre- 
 cisely the same as the lines c x, 3 4, 9 10, 11 12, etc. ; for 
 all alike measure the thickness of the stone. 
 
 Fig. 46 is a conventional heart-shaped solid, sometimes 
 used in architectural ornamentation. 
 
 The points a and b being taken vertically above c and d, 
 a suitable distance, according to judgment, the face of one 
 half of the figure must be drawn by the guidance of the eye 
 alone, but making the width of the face the same through- 
 out, except where it varies slightly at or near the two ex- 
 tremities. It will then be easy to make the face of the oth- 
 er half symmetrical with it, according to directions given 
 in Book No. II. Then, if the thickness of the ornament is 
 to be four inches, or one diagonal, the diagonal width of the 
 drawing is determined in the same manner as in the pre- 
 ceding figure. 
 
CABINET PERSPECTIVE CUBVILIXEAE SOLIDS. 129 
 
 PROBLEMS FOB PBACTICE. 
 
 1 . Draw five quarter-sections of the rims of wheels, similar in all re- 
 spects to those in Fig. 42, but placed in the upper right-hand quarter of the 
 circle. 
 
 2. Draw the outlines of a bottomless tub similar to Fig. 43, but of the fol- 
 lowing dimensions. Extreme upper (front) diameter, twenty-four inches ; 
 vertical height, measured on the axis of the tub, twenty-four inches ; ex- 
 treme diameter of bottom of tub, thirty-four inches ; thickness of walls, one 
 inch. 
 
 Blackboard Exercises. The foregoing problems. 
 
 PAGE SEVEN. SCALE OF TWO INCHES TO A SPACE. 
 
 Fig. 47 represents a cubical block, twenty -eight inches 
 square, having semicircular grooves of four inches' radius 
 running diagonally through the centres of its upper, lower, 
 and right and left hand sides, with a circular aperture of 
 four inches' radius running from the front face diagonally 
 through the centre of the block. 
 
 Observe that the centres a, #, c, d, <?,/", from which the 
 semicircles on the front and on the farther side are de- 
 scribed, are situated at the middle point of the edges of the 
 cube, just where they would be placed for marking out the 
 semicircular grooves on the real cube. 
 
 Figs. 48 may be supposed to be cut out of a block twen- 
 ty-eight inches in height (m n or e g) ; twenty-eight inches 
 in depth (m o or e/); and thirty-two inches in width (m e 
 or n (/). The figure consists of a frame-work having four 
 corner posts of two by four inches, supporting a half-cylin- 
 der above and one below, with their concave surfaces out- 
 ward, the whole cut out of the block. The thickness of the 
 walls of the half-cylinders is four inches. Their semicircular 
 ends are described from the points tf, , c, and d. The point 
 d t for describing the farther end of the lower half-cylinder, 
 must evidently be at the middle of the farther lower edge 
 of the block, just as c is at the middle of the front lower 
 edge. But c is eight spaces, horizontally, to the left of the 
 corner #/ so must d be eight spaces, horizontally, to the left 
 of the corner h. If the pupil will carry out all the meas- 
 urements, just as they are in the real object, he will find 
 
 F2 
 
130 INDUSTRIAL DRAWING. [BOOK NO. III. 
 
 but little difficulty in the accurate representation of the 
 most intricate plans and patterns. 
 
 Fig. 49 represents two cylinders, lying, horizontally and 
 diagonally, side by side on a frame-work of twenty-four by 
 thirty- two inches. As the centres of the farther ends of 
 these cylinders are at the points b and c7, the length of their 
 axes, a b and c d, and, consequently, the length of the cylin- 
 ders, is twenty-four inches. The cylinders are placed, at 
 both extremities, four inches from the ends of the frame. 
 Complete the circle of two-inch radius, of which d is the 
 centre, and the circle will strike the frame at the same dis- 
 tance from the farther end that the circle described from c 
 strikes it from the front end. 
 
 Fig. 50. This figure consists of two portions, A and B 
 A being the same as B inverted. 
 
 Taking, first, the portion marked A: the two semicircu- 
 lar curves a and #, of the front face, are described from their 
 common centre, y ; while the two marked c and d, of the 
 farther face, are described from their common centre, z. In 
 a similar manner, the two semicircular curves a and , of 
 the front face of B, are described from w; and the curves 
 c and d, of the farther face, are described from a*. 
 
 Fig. 51 consists of a series of the same forms as A, of the 
 preceding figure, here framed together. The pupil should 
 now be able to describe this frame-work in full, and to des- 
 ignate the respective centres from which the several pairs 
 of curves (front curves and rear curves) are described. 
 
 Fig. 52 consists of a series of the same forms asJB, of Fig. 
 50, here framed together; or the same as Fig. 51 inverted. 
 The pupil should now be able to describe it in full. In 
 making a drawing of such figures either in copying, or in 
 drawing from a description the front portions should be 
 drawn first. 
 
 Fig. 53 is a combination of the two figures 51 and 52; 
 requiring merely the placing of the two figures in proper 
 position, and the completing of the vertical corner posts, 
 which are left short in Figs. 51 and 52. 
 
 In making a drawing of Fig. 53, the front face should first 
 be drawn, then the right-hand side; then the other parts 
 
CABINET PERSPECTIVE CURVTLIXEAR SOLIDS. 131 
 
 snoald first be traced lightly, and only filled in firmly when 
 the positions of all the parts are distinctly seen. 
 
 PROBLEMS FOE PRACTICE. 
 
 1. Draw a block the same in size as Fig. 47, but with semicircular grooves 
 of eight inches' radius taken out centrally and diagonally through four of 
 the sides, as in Fig. 47. 
 
 2. Draw a figure similar to Fig. 48 ; but let the height of the front be 
 twenty-eight inches, its width twenty-four inches, and the diagonal depth of 
 the block twenty inches ; let the greater curves of the semicircular grooves 
 be struck with a radius of ten inches, and the lesser curves with a radius of 
 eight inches. 
 
 3. Draw a solid cylinder of six inches' radius, and twenty-four inches' di- 
 agonal length, and let it rest centrally upon a diagonal platform twelve 
 inches wide in front, four inches thick, and twenty-four inches' diagonal 
 length, so that the ends of the cylinder shall be even with the ends of the 
 platform. 
 
 Blackboard Exercises. Figs. 48 and 50, and problems 1 
 and 3. 
 
 PAGE EIGHT. SCALE OF THREE INCHES TO A SPACE. 
 
 Fig. 54 represents a wheel eleven feet and a half in ex- 
 treme diameter; rim, thirty inches wide and six inches 
 thick ; the cylindrical nave or hub, four feet in diameter 
 and thirty inches in length ; and having eight flat spokes 
 or radii, each six inches thick, and the same width as the 
 rim and hub. 
 
 The foregoing dimensions being given, we proceed in the 
 following manner to make a drawing of the wheeL 
 
 1st. Taking x as the centre of the vertical front face of 
 the wheel, with a radius, x <r, of five feet nine inches, de- 
 scribe the circle abed for the circumference of the wheel, 
 being eleven feet and a half in diameter. 
 
 2d. Six inches within the circumference, with the radius 
 x e, describe a circle, which, with the former, will give the 
 thickness of the rim six inches. 
 
 3d. With the radius ie/, of two feet, describe a circle for the 
 circumference of the hub, four feet ; and with the radius x g, 
 of one foot, describe a circle for the cylindrical opening which 
 is to receive the axle, which is to be two feet in diameter. 
 
 4th. Trace lightly the vertical, horizontal, and diagonal 
 
132 INDUSTRIAL DRAWING. [BOOK NO. III. 
 
 lines through the centre, x, hereby dividing the rim into eight 
 equal parts ; and on each side of these several lines lay off a 
 space of three inches, and draw the lines representing the 
 thickness of the spokes, terminating them, in one direction, 
 by the inner circle of the rim, and in the other direction by 
 the outer circle of the hub. This completes the outlines of 
 the front vertical face of the wheel. 
 
 5th. As the rim is to be thirty inches wideband the hub 
 and spokes of the same diagonal extent, we take x y, thirty 
 inches, for the axis, and also for the thickness or width of 
 the wheel. Then y will be the centre of the farther verti- 
 cal face of the wheel. 
 
 6th. From the centre, y, and with the radius y z, of five feet 
 nine inches, describe the outer circumference of the farther 
 face of the wheel; and from the same point y, with a radius 
 six inches less, that is, with a radiuses, describe a circle (only 
 half of which, h ij, will be visible) for the inside circumfer- 
 ence of the farther face of the rim ; then, with a radius, y r, 
 of two feet, describe a circle (of which k Us a part) for the 
 outer circumference of the farther face of the hub; and 
 finally, with a radius, y p, of one foot, describe a circle (of 
 which m n is the only visible part) for the inner circumfer- 
 ence of the farther face of the hub. All the necessary cir- 
 cles will thus be completed drawn in the very same man- 
 ner, from the two centres, #, y, of the axis, that they would 
 be drawn from on the real wheel. 
 
 7th. The positions and correct width of the front faces of 
 the spokes having been marked out under the fourth divi- 
 sion, we have only to draw from their extremities the diag- 
 onal lines 1 2, 3 4-, 5 6, 7 8, etc., across the inside of the rim 
 and the outside of the hub, and the lines for the farther vis- 
 ible edges of the spokes, and the outlines of the spokes, so 
 far as they are visible, will be completed. 
 
 8th. Shade the several parts in such a manner as to make 
 each part distinct from the others, but at the same time 
 paying as much attention as you can, consistent with dis- 
 tinctness, to the laws of light and shade. The light in this 
 drawing, as in most of the preceding, comes diagonally from 
 the upper left-hand corner. 
 
CABINET PERSPECTIVE CURVILINEAR SOLIDS. 133 
 
 Observe with what accuracy all the parts, in accordance 
 with the scale laid down, and with the principles of repre- 
 sentation adopted, maintain their correct measurements, the 
 same as in the real object. Thus all measurements at right 
 angles across the outside or the inside of the rim of the real 
 wheel, and across the outside or the inside of the hub, meas- 
 ure the same, thirty inches. So, also, in the drawing, all the 
 diagonal lines across the rim or the hub as 1 2,3 ^5 6, 7 8, 
 v w, etc. measure precisely the same, five diagonal spaces, 
 or thirty inches. 
 
 Let this wheel be viewed as directed on page 50. The 
 pupil should not only draw this wheel, but also one of a dif- 
 ferent size, and with the spokes arranged differently. 
 
 PROBLEM: FOR PRACTICE. 
 
 1. Draw a wheel, similar to Fig. 54, of the following dimensions: Ex- 
 treme diameter, ten feet and a half; rim, thirty-six inches wide and three 
 inches thick ; the cylindrical nave, or hub, thirty-six inches in diameter, 
 three inches in thickness, and thirty-six inches in length ; and having eight 
 flat spokes or radii, each three inches thick, and the same width as the rim 
 and hub. 
 
 Blackboard Exercise. Let the pupil plan and draw a 
 wheel similar to that described in the problem, and of such 
 size as the board will permit. 
 
 TAGE NINE. SCALE OF THREE INCHES TO A SPACE. 
 
 Fig. 55 represents a wheel eleven feet and a half in ex- 
 treme diameter; extreme width of double rim and double 
 hub, four feet; thickness of rim, three inches; extreme di- 
 ameter of the hub, four feet, and diagonal extent of the two 
 parts, four feet ; thickness of rim of the hub, three inches ; 
 thickness of the broad paddle-like spokes, three inches, and 
 width four feet. 
 
 This figure consists of two similar wheels, the second being 
 placed immediately back of the first, and each having its 
 separate rim and hub, but the two being united in one wheel 
 by the broad paddle-like spokes, which are common to both. 
 
 1st. In drawing this figure, draw the front wheel first, in 
 the same manner that Fig. 54 was drawn w being the cen- 
 tre of the front face of the wheel, and x the centre of the 
 
134 INDUSTRIAL DRAWING. [BOOK NO. III. 
 
 opposite side. Draw the circles lightly with the compasses ; 
 then draw lightly, in like manner, the front and rear circles 
 of the farther wheel, from their centres y and z. 
 
 2d. Observe that the eight broad spokes are arranged in 
 a manner different from those in Fig. 54, there being no 
 spokes on the diagonal lines 3 12 and 7 a ; and we have 
 placed none on these lines, because, in these positions, there 
 could be no side view of the spokes that are in the direction 
 of 3 12, as there is none in Fig. 54. This is, therefore, a bet- 
 ter arrangement of these portions of the wheel. 
 
 3d. In order to arrange the eight spokes equidistant from 
 one another, and intermediate between the centrally verti- 
 cal, horizontal, and diagonal lines of the wheel, first draw 
 these lines lightly, and mark their points, as 1, 3, 5, 7, 9, 11, 
 etc., on the circumference of the front face of the wheel ; 
 then take the points 2, ^, 6, 8, 10, etc., intermediate between 
 them, and connect the opposite points by light lines passing 
 through the centre, w. On each side of these latter lines 
 lay off half a space (an inch and a half), and draw lightly 
 the lines which determine the front width of the spokes. 
 Each pair of these lines passes equidistant on both sides of 
 the centre, w. The extreme ends of the spokes are then all 
 drawn diagonally across both parts of the rim, and the oth- 
 er ends of the spokes are drawn in like manner across both 
 parts of the hub. The dotted continuations of the lines show 
 where the lines would appear if the parts which obstruct 
 the view were transparent. 
 
 Observe how the broad spokes are let in, centrally, be- 
 tween the two rims which compose the hub, and that they 
 are let in the exact depth of the rims, three inches. The 
 portions that are let in are, therefore, continuous with the 
 inner sides of the rims of the hub. Every part of this wheel 
 will show beautifully when viewed as directed on page 50. 
 
 If the pupil will be careful to have all the measures cor- 
 respond with the measures of the real wheel of the given di- 
 mensions, and will connect those points, and those only, that 
 are connected in the real wheel, he will have no difficulty 
 in drawing this, and all similar representations of real ob- 
 jects, with mathematical accuracy. 
 
CABINET PERSPECTIVE CURVILINEAR SOLIDS. 135 
 
 PAGE TEN. SCALE OF THREE INCHES TO A SPACE. 
 
 Fig. 56 is a drawing of what is called a croicn-wheel, 
 which is a wheel with cogs or teeth cut out of the rim, and 
 get at right angles to the plane of the wheel. See a differ- 
 ent representation of this same wheel in Fig. 23 of Book 
 No. IV. 
 
 In this figure the ends of the projecting cogs or teeth, 
 thirty-two in number, form the vertical front face of the 
 wheel, which has w for its centre ; and the extreme diameter 
 of the wheel 1 5 or 3 7 is eleven feet. The rim, including 
 the teeth, is thirty inches wide and -three inches in thick- 
 ness, and the cogs are one foot in length ; the four support- 
 ing arms (or spokes) of the wheel, which are six by twelve 
 inches, are set back even with the farther face or side of the 
 wheel, and are back eighteen inches from the ends of the 
 cogs : x is the centre of the farther vertical side, and hence 
 to x, the proper axis, is thirty inches in length ; but v x, the 
 axis of the short hub, is only twelve inches in length. The 
 hub is eighteen inches in diameter, as measured from m to w, 
 and its rim is three inches in thickness. Through the hub 
 passes an axle on which the wheel turns, twelve inches in 
 diameter and sixty inches in length, having s t for its axis. 
 
 1st. In drawing a figure with the dimensions given for 
 this wheel, first take some point, as w, for the centre of the 
 front face, and with a radius, w 3, of five and a half feet, de- 
 scribe the outer front circumference of the circle of cogs ; 
 and then, with a radius three inches less, describe the inner 
 front circle, which gives the thickness, three inches, of the 
 cogs. From x describe two like circles for the farther edge 
 of the rim. 
 
 2d. Divide the front circumference into eight equal parts, 
 by the dotted vertical, horizontal, and diagonal lines, radi- 
 ating from the centre, w; then, with the compasses, make 
 eight equal divisions in each of these parts, which will give 
 to each part four equal cogs, with spaces between them of 
 the same width as the cogs. 
 
 3d. As the cogs are to be twelve inches in length, take 
 the point y, twelve inches diagonally from ?, and with the 
 
136 INDUSTRIAL DRAWING. [BOOK NO. III. 
 
 same radius as w 3, describe a circle; and also describe one 
 with a radius three inches less; and these two circles will 
 give the true measurable length of the teeth, both on the 
 inside and on the outside of the rim. 
 
 4th. Now if, in the real wheel, the cogs bevel outward 
 from the centre, w, they must be so drawn here, in the follow- 
 ing manner: The lines at the ends of the cogs that limit 
 their width (not thickness) must all be directed toward 
 their common centre, w as c ?o, a w, 3 w, etc. ; but the cor- 
 responding lines at the base of the cogs must be directed 
 toward their common centre, y as I y, 9 y, g y, p y, etc. 
 just as they are in the real wheel. 
 
 5th. All lines that bound the teeth lengthwise as h i,j I; 
 o r, etc. must be drawn in the direction of diagonals. These 
 lines, however, are drawn next after the end lines that limit 
 the width of the cogs; and, when drawn, they fix the posi- 
 tion of the base lines of the cogs. 
 
 6th. From the point v, the centre of the front face of the 
 short hub, describe the two circles, one with a radius of nine 
 inches, and the other with a radius of six inches, for the 
 front face of the hub ; and from a;, with a radius of nine 
 inches, describe a circle (only a small part of which is seen) 
 for the outer circumference of the farther face of the hub. 
 From s and t are described the circles for the ends of the 
 axle. 
 
 7th. As v is the centre of the vertical circle which limits 
 the front ends of the spokes, from v, with a radius of five 
 feet and three inches, describe a circle for that purpose. The 
 central lines of the front faces of the spokes must then pass 
 through v, and the width of the spokes is laid out at equal 
 distances on each side of these central lines. The shading 
 is similar to that of the preceding two figures. 
 
 The dimensions of the wheel are well seen in the dotted 
 squares AB CD, E F Q II, which bound its front and rear 
 vertical faces. 
 
 PAGE ELEVEN. SCALE OF THREE INCHES TO A SPACE. 
 
 Fig. 57 represents the face or side of a ratchet wheel, a 
 wheel which has teeth like those of a circular saw. The 
 
CABINET PERSPECTIVE CURVILINEAR SOLIDS. 137 
 
 teeth may be either long and sharp-pointed, or short and 
 more blunt in appearance; but in either case the inner 
 edges of the teeth are not directed toward the centre of the 
 wheel. 
 
 1st. This figure represents the face or side of a wheel fifty- 
 four inches in extreme diameter, having x for its centre. 
 After describing the outer circumference, take any required 
 distance on a diameter, as 3 d, and from x, with the radius 
 x d, describe the circle d ef, which is to limit the length of 
 the teeth. 
 
 2d. Divide each quarter of the circumference into as many 
 equal spaces as there are to be teeth in it. From the centre, 
 x, describe an inner circle, a b c, large or small, according to 
 the amount of bevel that is to be given to the teeth. 
 
 3d. From the points of division in the circumference, as 
 #,#,4, etc., draw lines tangent to (touching) the inner circle, 
 r.<5 seen in the drawing such as 3 a, 8 c, etc. ; and also, 
 from the same points of division in the circumference, draw 
 the tooth lines 3 g, h o, etc., intersecting the former tangen- 
 tial lines on the second circle. But the drawing itself will 
 explain every thing that is needed to be understood. 
 
 Fig. 58 represents not only the side view of a ratchet 
 wheel, but the dimensions, also, of all its parts, in length, 
 breadth, and thickness. 
 
 Fig. 58 represents a ratchet wheel twelve feet in extreme 
 diameter, having a rim fifteen inches wide and six inches 
 thick, into which thirty-two teeth are cut, to a depth of nine 
 inches. It has a hub forty-two inches in diameter, with a 
 central cylindrical opening for a shaft, or axle, twelve inches 
 in diameter : and twelve spokes, of three by six inches, and 
 thirty-six inches long, set at equal distances apart, connect 
 the rim and the hub. 
 
 1st. Here x is the centre of the vertical front face or side 
 of the wheel, and y the centre of the farther side. As the 
 circle 1111, described from x, with a radius of six feet, 
 limits the points of the teeth on their front face, so the circle 
 222%, described from y, with the same radius, limits the 
 points of the teeth on the farther side of the w T heel. There- 
 fore, if we divide the outer circumference, 1111, into thirty- 
 
138 INDUSTRIAL DRAWING. [BOOK NO. III. 
 
 two equal parts, as in Fig. 57, and from all the points of di- 
 vision draw diagonal lines, like d g,f%, m n, etc., intersect- 
 ing the circle 2222, we shall have the lines of the edges 
 of all the teeth. 
 
 2d. We get the required bevel for the front edges of the 
 teeth, in the same manner as in Fig. 57, by describing the 
 circle a a a from the centre, a?, and drawing all lines, such as 
 d CjfSj m p, etc., tangent to that circle. But these latter 
 lines are limited by the circle 3 3 8, which we describe from 
 x, limiting the length of the front edges of the teeth. A 
 circle, 4 4 4 9 described from ?/, with the same radius, limits 
 the edges of the teeth on the farther side ; but all these 
 edges, like g A, 2 4, n'r, etc., whether concealed from view 
 or not, must be drawn tangent to the rear circle, b b b, de- 
 scribed from ?/, and corresponding to the front circle, a a a. 
 
 By following these simple principles, describing all the 
 circles, and drawing all the lines, just as they would be in 
 the real wheel, the dimensions of every tooth are accurately 
 represented, and all measure precisely the same. 
 
 3d. The twelve spokes, each three inches wide in front, 
 and six inches in thickness, are laid out by dividing each 
 quarter of the inner circumference of the rim into three 
 equal parts, and then laying out the width equally on each 
 side of these divisions. By this arrangement of the spokes 
 all show to good advantage. 
 
 The pupil should not only draw this wheel, but also one 
 of a different size, and with a greater number of teeth. 
 
 Fig. 59 represents a chain-pulley wheel, the links of the 
 chain being adapted to fit over the projecting triangular 
 cogs. The design is sufficiently illustrated by the drawing, 
 without the necessity of any description; but the pupil 
 should fully describe the wheel. 
 
 PAGE TWELVE. SCALE OF TWO INCHES TO A SPACE. 
 
 Fig. 60 represents a windlass, viewed diagonally length- 
 wise. It consists of a cylindrical shaft, XX, twelve inches 
 in diameter and eighty-eight inches in length ; four inches 
 from the two ends of which are two drums, W W, each 
 twenty-eight inches in diameter and twelve inches in thick- 
 
CABINET PERSPECTIVE CURVILINEAR SOLIDS. 139 
 
 ness. Projecting from the centre of the circumference of 
 each drum, at right angles to the axis, are eight spokes or 
 levers, three inches by four inches in size, and with a pro- 
 jection of ten inches beyond the drum. 
 
 A smaller shaft, or axle, Y y, six inches in diameter, pro- 
 jects, centrally, sixteen inches beyond the ends of the larger 
 shaft, and, immediately beyond the ends of this larger shaft, 
 rests in two bracing supports, each eight inches wide, fitted 
 to receive the axle. 
 
 At A the front drum is drawn separately, together with 
 the end of the shaft and the projecting axle, but without 
 the spokes or levers, and without the support in which the 
 axle rests. Here a is the centre of the axle, where it con- 
 nects with the shaft, and also the centre of the end of the 
 shaft, while b is the centre of the front face of the drum, and 
 c the centre of the farther face or side. 
 
 At J? is a separate drawing of the support in which the 
 front axle rests. The timber forming the base in which the 
 braces rest is forty inches in length, four inches in vertical 
 thickness, and eight inches in width. The supporting braces 
 are two by eight inches, and about twenty-three inches in 
 length above the base. The tie-brace at the top is cut out 
 in a semicircular form, six inches in diameter, to receive the 
 axle. 
 
 At C is a representation of the drum, TFJ with the spokes 
 in their places. It is here drawn separately for the purpose 
 of showing the method of placing the spokes accurately in 
 the centre of the cylindrical surface of the drum. This draw- 
 ing may be made in the following order: 
 
 1st. Take some point, a, as the centre, from which, with a 
 radius of fourteen inches, describe the circumference of the 
 nearer and visible end of the drum ; and from d, twelve 
 inches diagonally from , on the axis, with the same radius, 
 describe the farther circumference, only half of which is vis- 
 ible. 
 
 2d. Xow, as the spokes to be inserted are three inches 
 wide on their face or front side, and four inches diagonally, 
 they must be inserted in the central third portion of the 
 cylindrical surface of the drum. We therefore take -a d, 
 
140 INDUSTRIAL DRAWING. [BOOK NO. III. 
 
 representing the line of the axis of the drum, and divide it 
 into three equal parts by the points b and c; and from b 
 and c as centres, with a radius of fourteen inches, we de- 
 scribe the two dotted circles encompassing the drum. These 
 two circles mark out the central third part of the cylindric- 
 al surface of the drum, as they would appear to the eye if 
 the drum were transparent. The spokes must therefore 
 start out from the drum between these circles, and touching 
 them. 
 
 3d. These spokes we may consider four in number, equi- 
 distant from one another, passing at right angles through 
 the axis of the drum, and presenting eight projecting arms 
 to our view. For convenience, we will suppose the circum- 
 ference of the nearer dotted circle to be divided into eight 
 equal portions by the vertical line 1 2, the horizontal line 
 5 6, and the two diagonal lines 8 4 and 7 8, giving us the 
 equidistant points /, /i, J, m, o, q, 9, and 10. From /, as a 
 starting-point, measure off any desired distance, as/<7, and 
 take g as the point for the intersection of a corner of the 
 spoke with the dotted line. From h measure off the same 
 space to i, and the same from^' to A?, from m to ??, from o to 
 p, etc., for points at which the corners of spokes pass through 
 the circumference of the drum. 
 
 4th. From each of the several points, g, i, 7j, ??, p, etc., 
 measure off three inches backward on the dotted line, as 
 g 11, i 12, etc., for the width of the face of each spoke. 
 Connect the points that are directly opposite, as Up, g 13, 
 etc., by very light lines extended indefinitely, every pair of 
 which will pass equidistant on each side of the central point, 
 b, and will represent the front faces of the spokes as passing 
 through the axis of the drum. 
 
 5th. The length of the projecting arms, or spokes, may be 
 determined by circumscribing a circle around the centre, b. 
 Here the circle, which may be called the nearer face circle 
 of the spokes, is drawn with a radius of twenty-four inches, 
 giving ten inches for the length of the spokes, and passing 
 through the points e, 8, x, etc. Now, as the spokes are four 
 inches in diagonal width, we describe another circle with the 
 same radius from c (four inches diagonally from b), passing 
 
CABINET PERSPECTIVE CURVILINEAR SOLIDS. Ill 
 
 through the points 5, 3, 1, 8, w, y, etc., thus limiting the di- 
 agonal width of the spokes. Suppose, now, that the corner 
 face lines of the spokes, as g e, 16 r, etc., extend to the nearer 
 face circle e 8 x. From the intersections of these corner 
 lines with this nearer face circle, giving the points t, r, v, z, x, 
 etc., we draw lines diagonally, as t u, r s, e I, v w, x y, etc., 
 which give us the side boundary-lines of the ends of the 
 spokes. 
 
 6th. The lines 16 17, 11 18, 12 19, etc., showing the in- 
 tersections of the sides of the spokes with the surface of the 
 drum, we also draw diagonally for all lines that are par- 
 allel in the real wheel must be parallel in the drawing. . 
 
 The outlines of the spokes are thus completed all accu- 
 rately drawn in their proper places, as projecting from the 
 central third part of the surface of the drum. 
 
 At D the several parts described in the sections ^1, B, and 
 C are put together, resting on a timber platform. Let the 
 pupil verify the following measures according to the scale. 
 
 1st. The distance, x y, between the drums, is fifty-six 
 inches. The distance, c d, on the axis, between the centres 
 of the inner faces of the two drums, must give the same 
 measure. 
 
 2d. As each drum is twelve inches in thickness, the dis- 
 tance, r z, between the outer faces of the drums, must be 
 eighty inches. The distance, bg, between the points on the 
 axis, from which the circumference of these outer faces are 
 described, must give the same measure. 
 
 3d. As the large shaft extends four inches beyond the 
 extreme faces of the two drums, the length of the shaft, 
 whether measured diagonally on its surface or on its axis, 
 must be eighty-eight inches. The farther end of the large 
 shaft is concealed from view, but its position can be easily 
 determined by counting twenty-two spaces diagonally from 
 the point 5. 
 
 4th. As the tops of the bracing supports of the axle are 
 in immediate contact with the ends of the shaft, the dis- 
 tance beticeen the inner faces of the tops of these supports 
 must also measure eighty-eight inches. 
 
 5th. As each of the bracing supports is eight inches in 
 
142 INDUSTRIAL DRAWING. [BOOK NO. Ill, 
 
 width, the distance, s t, between the extreme faces of the 
 tops of these supports, must be one hundred and four inches. 
 We thus get the point t, and the outlines of the concealed 
 top of the farther brace. 
 
 Cth. As we have the point 1 at the bottom of the nearer 
 brace, we now know that the corresponding point 2 of the 
 farther brace must be one hundred and four inches from it 
 diagonally that is, twenty-six diagonal spaces. The dis- 
 tance from 3 to 4 must measure the same. 
 
 We have been thus particular in describing the drawing 
 of this figure for the purpose of showing that every part 
 may be drawn with the most perfect accuracy, and so as to 
 show its exact measurement. With the aid of the ruled 
 drawing-paper all such drawings may be executed with 
 great rapidity; and any required degree of modification, 
 also, from the plain angular forms, may be given to them. 
 Thus the spokes may be round, or even tapering, and light- 
 er than here shown, and all the parts may be modified, 
 while at the same time accuracy of representation may be 
 adhered to, provided the very plain principles of this mode 
 of representation are clearly understood. 
 
DRAWING-BOOK No. IV. 
 
 CABINET PERSPECTIVE MISCELLANEOUS 
 APPLICATIONS. 
 
 I. DIFFERENT DIAGONAL VIEWS OF OBJECTS. 
 
 Ix the preceding two books on Cabinet Perspective, ob- 
 jects have been represented not only as viewed diagonally, 
 with their principal surfaces in vertical and horizontal posi- 
 tions, but as viewed diagonally from one particular point, 
 which point is supposed to be diagonally at the right of the 
 object, and above it. But although, for the purpose of 
 avoiding confusion, it is best to represent all objects from 
 this one point of view whenever it can be done to advan- 
 tage, yet objects may be represented diagonally in cabinet 
 perspective equally well from any one of four points of 
 view, as will be seen from the drawing on the first page of 
 Book No. IV. 
 
 PAGE ONE. 
 
 1st. Upper right-hand view. In division A the objects 
 are represented as viewed diagonally from above, and at the 
 right, the same as in all the drawings in the preceding two 
 books. The arrow, m, is represented as coming from the po- 
 sition, at an infinite distance away, from which the objects 
 in A are viewed. 
 
 2d. Lower right-hand view. In division B the objects are 
 represented as viewed from below, and at the right ; and 
 here the arrow, n, is represented as coming from the posi- 
 tion, at an infinite distance away, from which the objects 
 are viewed. It will be observed that the bracket,/, is best 
 
144 INDUSTRIAL DRAWING. [iJOOK NO. IV. 
 
 represented in its true position from the point of view here 
 taken. 
 
 3d. Lower left-hand view. In division C the objects are 
 represented as viewed from beloio, and at the left; and here 
 the arrow, o, is represented as coming from the point from 
 which the objects are viewed. 
 
 4th. Upper left-hand mew. In division D the objects 
 are represented as viewed from above, and at the the left / 
 the arrow, p, being represented as coming from the position, 
 at an infinite distance away, from which the view is taken. 
 
 Thus two of the views are represented as taken from 
 above, one from the right and the other from the left ; and 
 the other two views are represented as taken from below, in 
 like manner. In this way we get inside views of the curves 
 of the four brackets ; but all of the brackets might have 
 been drawn equally well according to the plan of division A, 
 if they had been placed in proper positions for the purpose. 
 In A the bracket, c, is inverted, for the purpose of showing 
 to the best advantage the inside curves ; and the brackets 
 /*, y, and h might have been drawn in similar positions. 
 
 II. GROUND-PLANS AND CABINET-PLANS OF 
 BUILDINGS. 
 
 PAGE TWO. SCALE OF ONE FOOT TO A SPACE. 
 
 Fig. 1 is the ground-plan of a building, whose outer walls, 
 one foot in thickness, embrace an extent of eighteen by twen- 
 ty-six feet. It will be seen that the lined paper is admira- 
 bly adapted to ground-plans or surface representations, as 
 it is only necessary to adopt any convenient scale, when 
 the plan may be laid down with perfect accuracy without 
 any measurement, by simply counting the spaces. In a 
 ground-plan, like Fig. 1, we can show the position and thick- 
 ness of the walls and partitions, and the width and position 
 of doors and windows, etc. 
 
 Fig. 2 is a cabinet-plan of the walls, doors, windows, etc., 
 of Fig. 1 ; the whole being represented from the fotmda- 
 
CABINET PERSPECTIVE MISCELLANEOUS. 145 
 
 tions upward to the extent of three feet. Observe that all 
 the measures in Fig. 2 are the same, according to the st-:ik' 
 adopted, and the principles laid down for cabinet drawing, 
 as in Fig. 1. Thus, in Fig. 2, the length of the side wall, 1 2, 
 is twenty-six feet ; and the distance, 1 4, is eighteen feet 
 the same, in both cases, as in Fig. 1. But in Fig. 2 it is 
 shown that the windows in the front room come down with- 
 in six inches of the bottom of the wall, while those in the 
 back room come down to within two feet and a half of the 
 bottom of the wall. These are features which can not be 
 shown in a ground-plan. 
 
 Fig. 3 is another ground-plan, representing a building 
 with two rooms, one fourteen by sixteen feetj and the other 
 thirteen by sixteen. The position for a stair-way is given in 
 a corner of the large room, and the position for a shelf is 
 given in the other. Rectangular openings for fire-places 
 are also given in the partition wall; but we can not show 
 any thing more than a mere ground -plan in this kind of 
 drawing, and must look to the next figure for any thing like 
 details. 
 
 Fig. 4 is a cabinet view of the plan of Fig. 3, embracing 
 not only the ground-plan, but the walls, bench, stair-way, 
 platform to entrance, etc. a//, in fine, that belongs to the 
 plan of the building, to the extent of three feet above the 
 stone foundations, while the height of the stone wall above 
 the earth is shown also. 
 
 In making a drawing of this kind it is best to begin at 
 the top, and work downward, tracing every thing lightly at 
 first. The walls may be made of any desired height; and 
 the only objection to making them then-full height is that 
 the front wall will then obscure the rear wall. It is well, 
 sometimes, to draw the front wall and the right-hand wall 
 only two or three feet high, and represent the other two of 
 their full height, with beams, joists, etc., projecting a short 
 distance from them. Or any section of the building may 
 be represented separately, and a scale larger than that used 
 for the walls may be adopted for portions that require a 
 minute representation. In this manner the most elaborate 
 building may be fully represented, even to the most minute 
 
 G 
 
146 INDUSTRIAL DRAWING. [BOOK NO. IV. 
 
 details, and with far greater accuracy than could be attain- 
 ed by description alone. 
 
 Fig. 5. At A are represented a stair-way and platform, 
 with the side to the front ; and at B the same are repre- 
 sented with the stairs presenting the front view. Observe 
 that the measurements in the one case are the same as in 
 the other the distance 5 2 being the same as 6 ., an ^ * $ 
 the same as 3 7, etc. 
 
 PROBLEM FOR PRACTICE. 
 
 (This may be omitted by the younger pupils.} 
 
 It is required to draw, in cabinet perspective, a building similar to the 
 ground-plan of Fig. 3, but of the following dimensions : Length of building 
 (1 2\ forty feet ; width (1 3\ twenty-four feet ; thickness of walls, eighteen 
 inches ; room on the left, twenty by twenty-two feet in the clear ; room on 
 the right, seventeen by twenty-two feet in the clear. Draw the front wall, 
 division wall, and wall on the right-hand side, only three feet high, but the 
 wall on the rear, and left-hand side, eleven feet high. Put in the openings 
 for the windows, a and b, in the centres of the ends of the rooms, three feet 
 from the floor, four feet wide, and seven feet high ; the door, c, two feet 
 from the partition wall, four feet wide, and seven feet high ; let a beam one 
 foot square at the end rest upon the rear wall and left-hand side wall ; let 
 a beam of the same size project from the rear beam four feet over the par- 
 tition wall ; and let one project forward from the upper corner above 2, 
 four feet; and let one extend from the corner above <?, four feet to the 
 right, over the front wall. Carry a tier of four shelves, two and a half feet 
 apart, twelve feet long, and two feet wide, across the window, up against 
 the rear wall and right-hand corner of the room on the right, the top of the 
 lower shelf being two and a half feet from the floor. Carry a flight of eight 
 stairs, four feet wide, with one-foot risers and one-foot tread, up against the 
 rear wall of the room on the left, onto a platform four feet square in the cor- 
 ner ; and then have four stairs running from the platform toward the front, 
 up to the top of the wall. Let there be one supporting post, six inches 
 square, at the angle where the stairs turn, and one to support the end of the 
 last stair, three feet six inches from the wall, and coming out flush with the 
 end of the stair. 
 
 Let the other windows, doors, etc., be similar, in size and position, to those 
 in Tig. 4. 
 
 PAGE THREE. SCALE OF ONE FOOT TO A SPACE. 
 Fig. 6 is the ground-plan of a building twelve feet in ex- 
 treme width, and thirty-four feet in length, with a platform 
 entrance to the front of two by six feet, and three steps, 
 
CABINET PERSPECTIVE MISCELLANEOUS. 147 
 
 each a foot wide, ascending to it. There are two rooms, 
 each ten by twelve feet, with a hall between of six by ten 
 feet, and openings four feet wide in the partition walls, for 
 slid ing-doors, leading from the hall into the two rooms. The 
 jambs of the fire-places bevel outward. At A the ground- 
 plan of the fire-places is drawn on a larger scale, showing 
 the exact bevel of the jambs. 
 
 Fig. 7 is the same as the preceding figure, here changed 
 to the cabinet-plan, and drawn to the same scale as Fig. 6. 
 The ground-plan of the walls represented in Fig. 6 is, in Fig. 
 7, transferred to a stone foundation, which is raised one foot 
 above the ground at the corner .7, and two feet at the corners 
 2 and 3. Hence the front steps, which have six-inch risers 
 and one-foot tread, must rise a distance of two feet. 
 
 At B is a cabinet view of the fire-place, drawn to the 
 same scale as the ground-plan in A. Observe how the fig- 
 ures of the one correspond to the figures of the other in rel- 
 ative position, and that the measures are the same in both. 
 
 Fig. 8 represents a series of platform structures of different 
 elevations. Thus, if we begin with the level at J, we as- 
 cend toward the left an inclined or sloping plane until we 
 rise three feet, to the level of H. Rising six inches from Jf, 
 we step onto the platform Gr or J5f/ four steps, each of six 
 inches' rise, then bring us onto the platform I or F; eight 
 steps, each of six inches' rise, then bring us to the level of 
 the platform Z>. 
 
 Now it would be difficult to ascertain whether JEis high- 
 er than D or not, if we did not know, ovfind out, the height 
 of the box-like frame-work on which the flooring, E, rests, 
 and likewise its projection beyond the sides of the frame. 
 The size of this frame-work is sixteen feet square ; and the 
 floor, here represented as six inches in thickness, projects 
 one foot on all sides, except toward Z>, where there is no 
 projection. The frame-work (without the floor) rises four 
 feet above the corners 5 and 7, and six feet above the corner 
 6. Hence the upper corners of the frame-work would be 
 seen at the points 1 and #, the former four feet vertically 
 above 5, and the latter six feet vertically above 6; and the 
 other corners at the points 3 and J h at the intersections of the 
 
148 INDUSTRIAL DBA WING. [BOOK NO. IV. 
 
 dotted lines. Now if we connect these four points, as shoxvn 
 in the drawing, we have the outlines of the top of the frame- 
 work. 
 
 We have now to put a floor six inches thick on the top 
 of this frame-work, and projecting one foot beyond all its 
 sides except toward D. If we extend 4 1 one half a diag- 
 onal space, that is, one foot, to o, the latter point will be the 
 lower left-hand corner of the projecting floor ; and the point 
 7i, six inches above it, will be the upper left-hand corner of 
 the floor. Extend the side lines of the top of the frame one 
 foot beyond their intersections at 2 and 5, and we shall have 
 points through which the lower side lines of the projecting 
 floor w r ill pass, to make the other lower visible corners t and 
 v. Draw from t and v lines six inches vertically upward, 
 and we shall have the upper right-hand corners of this pro- 
 jecting floor. In order to see fully the truth of this de- 
 scription, it would be well, first, to draw the outlines of the 
 top of the frame lightly, without the flooring, E, and after- 
 ward to put on the flooring. 
 
 As it requires eight steps, of six inches each, to rise from 
 the platform I to Z>, D is four feet above I; and as the 
 frame-work of E\$ four feet high above the level of the cor- 
 ner 5, and as on this frame-work is placed a flooring six 
 inches thick, it follows that D is six inches below E. The 
 farther end of D is even with the farther side of the frame- 
 work of E. 
 
 The posts a and b are flush, on both their outer sides, with 
 the corners of the frame-work above which they are placed ; 
 the posts d and r are flush with the right-hand side of the 
 frame-work, and consequently are one foot from the right- 
 hand edge of the flooring ; while the posts p and c are 
 flush with the front and rear sides of the frame-work, and 
 are one foot each from the front and rear edges of the 
 flooring. 
 
 The posts on the platform E are three feet high above 
 the floor. The posts on the platform D are also three feet 
 high ; but on the latter posts is a rail six inches thick, so 
 that the top of the rail is even with the top of the railing 
 ofM 
 
CABINET PERSPECTIVE MISCELLANEOUS. 149 
 
 At C are the outline walls of a building, the outer walls 
 being two feet in thickness. At L is seen the opening into 
 a pit, extending downward below the flooring. 
 
 Observe that the window openings in the frame of E are 
 the same in relative position, number, size, etc., on the right- 
 hand side as on the front, and that there is the same real 
 width of window-sill represented on the right-hand side as 
 on the front. 
 
 To master the drawing of the platforms in Fig. 8 with 
 the projection of E on three sides, and no projection to- 
 ward D with the arrangement of the posts, etc., the whole 
 should be drawn on a larger scale. 
 
 PROBLEM FOR PRACTICE. 
 
 Draw a figure similar to Fig. 8, but with the following measures : Let 
 the frame on which the flooring of E rests be twenty-seven feet square. 
 Height of frame from corners 5 and 7 six feet, and from corner 6 eight feet. 
 On this frame place a floor one foot in thickness, and one foot projec- 
 tion on all sides except toward Z>, where there is to be no projection. 
 Posts of platform E four feet high above floor, six by twelve inches in size, 
 and arranged as in Fig. 8. Platforms I and F six feet wide. Platform Z>, 
 and steps ascending to it, eight feet wide ; and a sufficient number of steps, 
 each six inches' rise and twelve inches' tread, for D to be six inches below 
 E. Top of railing of D to be even with top of railing of E. Let there be 
 four window openings, or recesses, in front, and the same number on side 
 of E, each three feet wide and four feet high, a foot and a half from the 
 top of the frame ; and let the depth of the recesses be one foot. These 
 window openings to have three feet space between them, and those nearest 
 the corners to be three feet from corners. In other respects make the 
 structure similar to Fig. 8, omitting (7. 
 
 III. CYLINDRICAL OBJECTS IN VERTICAL 
 POSITIONS. 
 
 The representations of circles, wheels, cylinders, etc., in 
 cabinet perspective, have thus far been made on the suppo- 
 sition that the axes of the cylinders, etc., lie in a horizontal 
 position, although placed diagonally with reference to the 
 point from which the view is taken ; and the vertical cylin- 
 drical ends have been drawn perfect circles, as though they 
 
150 INDUSTRIAL DRAWING. [BOOK NO. IY. 
 
 were directly in front of the spectator. This is the case 
 with all the cylindrical objects represented in Book No. III. 
 Thus, referring to the cabinet cube for illustration, we have 
 hitherto represented the circle as drawn on the front ver- 
 tical face of the cube. But if the circle were to be drawn on 
 the obliquely viewed top or side of the cube, the circle would 
 have the form of a particular kind of ellipse* as the follow- 
 ing illustration will show. 
 
 I. ELLIPSES ON DIAGONAL BASES. 
 
 PAGE POUR. SCALE OF TWO INCHES TO A SPACE. 
 
 [N. B. The description of Fig. 9 should be read over, but the figure need 
 not be drawn.] 
 
 Fig. 9. We have here drawn a cabinet cube of twenty 
 spaces (forty inches) to the side, the nearer vertical face of 
 the cube fronting the spectator ; and on this front face of 
 the cube we have described a circle touching the centres of 
 the four sides of the square. As this circle is drawn with 
 a radius of ten spaces, the circumference will pass through 
 the intersections of the ruled lines of the paper marked 2, 4-, 
 8, 10, 14, 16, and 20, 22, in addition to the points 0, 6, 12, and 
 18, making twelve points in the intersections of the ruled 
 lines through which it will pass.f On the top, and also on 
 the right-hand side of the cube, we have also represented a 
 circle of the same size as that in front the circumference 
 ia each case touching the centres of the four sides of the 
 aiagonal square within which it is drawn, and also pass- 
 ' ng through the corresponding points 2, 4, 8, 10, IJf., 16, and 
 20, 22. The representations on the top and right-hand side 
 of the cube are ellipses and a sufficient number of points 
 in their curves may be known to enable one to draw the 
 curves with great accuracy. 
 
 Thus, take the construction of the upper ellipse for illus- 
 
 * An ellipse is an oval or oblong figure, which corresponds to an oblique 
 view of a circle. 
 
 t A circle drawn with a radius of 5, 10, 15, 20, 25, or 30 spaces, etc.. 
 will pass through twelve points in the intersections of the ruled lines. This 
 is susceptible of geometrical proof. 
 
CABINET PERSPECTIVE MISCELLANEOUS. 151 
 
 tration : First, rule the upper surface of the cube to corre- 
 spond to the ruling on the front face. Then all the lines, 
 and their intersections, on the upper surface, will correspond 
 to those on the front face. Mark any required number of 
 points on the front face, through which the circle passes, 
 and mark the corresponding points on the upper surface, 
 and then through these latter points draw the ellipse, and 
 it will correspond to the circle. Thus, if the ellipses be 
 accurately drawn, they will pass through the points 2, 4, 
 8, 10, 14, 16, 0, 22, etc., of the top and right-hand side of the 
 cube. More points may be taken, if required, and thus the 
 ellipses may be drawn quite accurately by hand. This 
 mode must give an accurate cabinet representation of a cir- 
 cle drawn on the top and side of a cube. View the whole 
 figure through the opening of the partly closed hand for 
 a half- minute or so, and the ellipses will gradually appear 
 as perfect circles. 
 
 But the two side curves of each of these ellipses may be 
 quite accurately drawn in the following easy manner. Sup- 
 pose we wish to draw the ellipse in the cabinet square 
 B C F E. Place one point of the compasses in the point 
 B, and extend the other to the point a, the centre of the 
 opposite long side ; and with the compasses thus extended 
 strike a curve across B F, the diagonal of the square, and 
 dot the point b. Then, with one point of the compasses in 
 F) and with the same stretch, dot the point d, corresponding 
 to b. With the compasses still spread as before, first with 
 the point at a;, and afterward at a, strike the curves inter- 
 secting on the right at m. Then, with one point of the 
 compasses in m, strike the side curve a a; of the ellipse. It 
 may be prolonged in the direction of a to 7. Then, with 
 the points of the compasses respectively in 18 and 12, the 
 centres of the other two sides, with the same stretch as be- 
 fore, strike the curves that intersect at n; and from n as a 
 centre describe the other side curve 12 18, prolonging the 
 curve to about the point 19. Half the distance from d to 
 B will give the point 21 of the ellipse, on the diagonal ; 
 and half the distance from b to J^will give the correspond- 
 ing point 9. The diagonal distance from 9 to 21 is also 
 
152 INDUSTRIAL DRAWING. [BOOK NO. IY. 
 
 equal to the distance from B to a the distance first laid 
 off on the compasses. 
 
 In a similar manner the side curves of the upper ellipse 
 may be described, by laying off the same distance as be- 
 fore, from B to o, and describing the intersecting curves 
 from x and o, etc. This method will give a dose approxi- 
 mation to the true side curves. 
 
 The ei.d curves of the ellipse should be drawn by the eye, 
 after first marking the points of the curve on the diagonal, 
 as before directed. Fig. 9 should be referred to as a guide 
 for drawing the forms of all similar end curves in ellipses 
 thus situated ; but yet it will seldom be necessary, and not 
 often desirable, to draw cabinet ellipses in those positions. 
 It will generally be found most convenient to draw the 
 curves either in the positions and in the manner shown in 
 Book No. III., or after the following plan. 
 
 II. ELLIPSES ON RECTANGULAR BASES. 
 
 Fig. 10. Suppose that the cube, Fig. 9, should be so 
 viewed that neither the right-hand side nor the left-hand 
 side could be seen at all, but that the top of the cube should 
 appear directly above the front face, and of the same width 
 as in Fig. 9. The whole would then be seen as in Fig. 10, 
 in which the front face is the same in all respects as in Fig. 
 9 ; and the top has the same width, 12, as o A in Fig. 9 
 but the right-hand side of the cube is not visible. Now 
 the top of the cube, supposed to be viewed from an infinite 
 distance, is rectangular; and it is of the same length, from 
 left to right, as the front face ; and its apparent width, 12, 
 is just half the width or height of the front face. Hence 
 the top or upper side, A B C D, as thus viewed, has a 
 length twice its width ; and the circle that should be drawn 
 on the top of a cube thus viewed would show as a perfect el- 
 lipse, whose lesser* diameter, 12, is just one half the length 
 of the greater* diameter, 6 18. 
 
 This ellipse, representing a circle forming the end of a 
 
 * The lesser diameter of an ellipse is called its conjugate diameter, and 
 the greater is called its transverse diameter. 
 
CABINET rEr.SPECTIYE [MISCELLANEOUS. 153 
 
 vertical cylinder, may be drawn after the manner first indi- 
 cated in Fig. 9, by dotting points to correspond, in relative 
 position, to the points marked on the circle below. The el- 
 lipse is here drawn in this manner, by dotting the points, 
 and then drawing the curve through them by hand. All 
 the points in the ellipse that are accurately marked in this 
 way must be correct. As the circle, described with a ra- 
 dius often spaces, passes through the twelve points 0, 2,4, 
 8, 10, 12, 14, 16, 18, 20, 22, so must the ellipse above it, if accu- 
 rately drawn, pass through the corresponding twelve points, 
 numbered in like manner. 
 
 Side Curves. But the side curves, at least, of this and of 
 all similar ellipses, may be very accurately drawn by the 
 compasses in the following manner. Place one point of the 
 compasses in A, and extend the other to 6, the middle of 
 right-hand side of the cabinet square A B C D (or place 
 in .Z> and extend to 18) ; then, still continuing the point in 
 A, with the other point strike the central vertical line, 12 
 prolonged, in 10. Then, with one point of the compasses in 
 ic, and the other extended to 12 above, describe the curve 
 9 12 15. Then, with one point in C, and the other extended 
 to 18 (half way between A and D), strike the line 12 pro- 
 longed above 12, for the point on the opposite side to cor- 
 respond to the point w ; from which point, thus found, and 
 with the other point of the compasses extended to 0, de- 
 scribe the curve 3 21. The two side curves will thus be 
 drawn very accurately. 
 
 End Curves. The end curves may be best drawn by 
 hand, in the following manner, by the aid of guide circles. 
 Thus : 
 
 On the line 6 18 take any point, v, so that v 6 shall be 
 equal to v 9, and from v describe a circle passing through 
 the points 9,6,3: this circle will then serve as a guide for 
 drawing the end curve of the ellipse, which must pass a lit- 
 tle within the circle, and at the same time be a natural and 
 graceful continuation of the side curves. The other end 
 curve is drawn in a similar manner. 
 
 Note. The representation given in Fig. 10 may be called upper vertical 
 rectangular perspective. But if the spectator were horizontally to the rifjht 
 
 G2 
 
154 INDUSTRIAL DRAWING. [BOOK XO. IV. 
 
 of the centre of the front circle, at the proper distance, he would not see the 
 top of the cube, but the right-hand side would be visible ; and on that side 
 might be described an ellipse like the one now seen at the top of Fig. 1 ; 
 only the longest diameter of the ellipse would then be in a vertical position. 
 This might be called right horizontal rectangular perspective. In the same 
 manner, if the spectator were horizontally to the left of the centre of the 
 front circle, the left side of the cube might be seen, and on that side might 
 be described an ellipse, and this might be called left horizontal rectangular 
 perspective. In the same way the spectator might be supposed to be ver- 
 tically below the centre of the front circle, so as to see the lower side of the 
 cube, on which might be described an ellipse. But this latter would be a 
 position so unusual that we would not recommend objects to be thus drawn. 
 The ellipse is drawn in precisely the same manner in the several positions 
 here mentioned. 
 
 The position of the horizontal cylinder in Fig. 13 is different, as regards 
 the spectator, from any of the positions above described, but the Eule on the 
 next page applies equally to all of them ; and even this does not differ in 
 principle from the general rule (ELEMENTARY RULE, p. 85) as applicable 
 to all drawings in cabinet perspective. 
 
 Fig. 11. Iii this figure we have drawn the ellipse within 
 the cabinet square A J3 C _Z), in all respects like the ellipse 
 of Fig. 10. The point y is the point from which we de- 
 scribe, with the compasses, the curve 3 w 21 ; and from the 
 point z we describe the curve 9 12 15. These points are 
 found in the same manner as the corresponding points for 
 describing the ellipse of Fig. 10. Suppose this ellipse to 
 be the upper horizontal end of a vertical solid cylinder 
 forty inches in diameter. The distances w 12 and E F 
 therefore alike represent forty inches. Suppose the cylin- 
 der to be ten inches in height, and that we wish to draw 
 the outline of the visible side of it toward the spectator. 
 The cylinder will then extend downward from E, w, and F^ 
 five spaces, to ./>, r, and C. ED and F C will then be the 
 vertical side lines of the cylinder; and through the points 
 J9, r, and C must be drawn the half of an ellipse correspond- 
 ing in all respects to the curve E w F. As the rectangle 
 A B C D represents the square inclosing the top of the 
 cylinder, so the rectangle E F Gr H represents the square 
 embracing the bottom of the cylinder; and if we describe 
 a cabinet ellipse within this lower cabinet square, it will 
 give the outlines of the bottom of the cylinder. Therefore 
 
CABINET PERSPECTIVE MISCELLANEOUS. 155 
 
 we describe an ellipse within the square E F G H, in the 
 same manner that we described the ellipse within the square 
 ABCD. The visible portion, C r D, of this ellipse, whose 
 side curve is described from the point p, we have drawn 
 with a firm line ; the other portion, Z> t C, which would 
 be invisible unless the cylinder were transparent, we have 
 drawn with a slightly dotted line. 
 
 Fig. 12. Here the solid cylinder, the method of drawing 
 which has been fully described in Figs. 10 and 11, is shown 
 as completed and shaded. Any vertical cylinder, of any 
 given dimensions, may be easily drawn by the method here 
 described, and with such accuracy that all its parts may be 
 readily measured. 
 
 Let it be observed that as the rectangle A B C D of 
 Fig. 10 corresponds to the rhombus A B C D of Fig. 9, 
 therefore the line C B of Fig. 10 measures the same 
 (in cabinet perspective) as the diagonal line C B of 
 Fig. 9; and 12 and D A of Fig. 10 the same as 12 
 and D A of Fig. 9, etc. Therefore we have the follow- 
 ing rule for representing objects in rectangular cabinet per- 
 spective : 
 
 RULE. When, in drawing objects in RECTANGULAR cabinet 
 perspective, a diagonal space is changed to a vertical space, 
 the latter, when thus used to represent a horizontal distance, 
 has the same measure as the former. 
 
 Fig. 13. The upper part of this figure is drawn on the 
 plan of Fig. 10 and Fig. 12. The lower part is a horizontal 
 cylinder whose end directly fronts the spectator. Observe 
 that the length of the horizontal cylinder, as measured from 
 1 to 2, is sixteen inches ; length of the short vertical cylin- 
 der only four inches. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw a vertical solid cylinder after the plan of Fig. 12, whose diame- 
 ter shall be thirty-six inches, and whose -axis, or length, shall be eighteen 
 inches. 
 
 2. Draw a horizontal cylinder whose diameter shall be twenty inches, 
 and length twenty-four inches, with one end directly fronting the spectator, 
 as in Fig. 13 ; and at the farther end of this cylinder, and starting even 
 with the top of it, draw a vertical cylinder of the same dimensions as the 
 
156 INDUSTRIAL DRAWING. [BOOK NO. IV. 
 
 horizontal cylinder. The horizontal cylinder is to project forward from the 
 front vertical side of a cube twenty inches square; and the vertical cylinder 
 is to rise from the upper side of the same cube. 
 
 3. Draw a cube twenty-four inches square, in right horizontal rectangular 
 perspective. Centrally placed on the right-hand side of it, extend out 
 horizontally to the right another cube sixteen inches square ; and on the 
 right-hand side of .this draw a horizontal cylinder twelve inches in length 
 and sixteen inches in diameter. Shade all the front surfaces light, and 
 the right-hand side surfaces dark. Then view the drawing as directed on 
 page 50, except that the eye of the spectator should be centrally to the 
 right of it. Also place the length vertically, and view it in that position 
 until it seems to stand out boldly from the paper. 
 
 PAGE FIVE. SCALE OF TWO INCHES TO A SPACE. 
 
 Fig. 14. It is required to make a drawing of the top of 
 a hollow vertical cylinder eighty inches in extreme diame- 
 ter, and whose sides are eight inches in thickness. 
 
 1st. Let A B C D be a rectangular cabinet square of 
 eighty inches to a side. Within it describe an ellipse 
 touching the centres of its four sides, after the manner 
 shown in Figs. 10 and 11. Here w will be the point for 
 describing the side curve 112; and x for describing the 
 other side curve 8 i 4; while t and s are the points for de- 
 scribing the guiding circles for the outer end curves. 
 
 2d. Now, as the walls of this hollow cylinder are to be 
 eight inches thick, the inner circle that bounds the walls on 
 the inside must be eight inches within the outer circle; so we 
 take o r, k I, v p, and ij, each eight inches within the ellipse 
 first drawn, and complete the rectangular cabinet square 
 E F G II eight inches, on all sides, within the outer square. 
 Within this smaller square describe an ellipse touching the 
 centres of its four sides, and we shall have the outlines of 
 the top of the hollow cylinder, as required. 
 
 3d. This inner ellipse is drawn in all respects like the 
 outer one. Thus y and z, found as before shown, are the 
 points from which its side curves are described; and m and 
 n are the points from which the guiding circles for drawing 
 the ends of the ellipse are described. 
 
 4th. Observe that the walls of the cylinder appear the 
 thinnest at the points ij and k I, and that at the points o r 
 and v p they appear to be double the thickness of ij and k /, 
 
CABINET PERSPECTIVE MISCELLANEOUS. 157 
 
 as they would naturally appear when viewed from the given 
 point above and in front of the cylinder. 
 
 Fig. 15. It is required to make a drawing of a thin flat 
 ring, forty - eight inches in diameter and four inches in 
 width. 
 
 1st. Take the horizontal line A B, of twenty-four spaces, 
 representing forty-eight inches, and on it construct the rect- 
 angular cabinet square A JB C D. A D or JB C will then 
 represent forty-eight inches also. 
 
 2d. Describe the ellipse 1234 within this square, and 
 touching the centres of its four sides. This ellipse w r ill 
 represent the top circle of the ring. The central line, 4 , 
 must be extended upward, so as to get the point from which 
 the lower side curve is to be described. 
 
 3d. As the bottom circle of the ring is four inches below 
 the top circle, it is to be described within a square four 
 inches below the square which contains the top circle. 
 Therefore lay on a second square, E F G- J7, four inches 
 below the top square, and within it describe the ellipse 
 5678, touching the centres of its four sides at the points 
 5, 6, 7, 8. The outlines of the drawing, which may now be 
 shaded, will thus be completed. 
 
 4th. Notice that 1 5 and 3 7 are straight vertical lines, 
 representing the width of the ring, four inches; and that 
 the curve 9 6 10 would touch the points 5 and 7 if it could 
 be seen throughout. 
 
 5th. Observe that this ring is drawn as having no appar- 
 ent or measurable thickness. 
 
 Fig. 16. It is required to make a drawing of a ring forty- 
 eight inches in diameter, four inches in width (or height, as 
 here viewed), and four inches in thickness. 
 
 1st. Draw the cabinet square, A. IB C D, of the requisite 
 dimensions, and within it describe the ellipse, 1 2 3 4> for the 
 upper outer circumference of the ring. 
 
 2d. Take the cabinet square, EFGr H, also of forty-eight 
 inches to a side, four inches below the first-mentioned square, 
 and within it draw an ellipse in all respects like the ellipse 
 5 6 7 8 of Fig. 15, with the exception that only the front 
 half of the ellipse in Fig. 16, as denoted by the figures , 8, 7, 
 
158 INDUSTRIAL DRAWING. [BOOK NO. IV. 
 
 is visible; the farther half being concealed by the thickness 
 of the ring. 
 
 3d. In order to get the lateral thickness of the ring, take 
 a cabinet square, I J I X, four inches within the square 
 A BCD, and describe the ellipse, 9 10 11 1#, touching the 
 centres of the four sides of this inner square. The thickness 
 of the ring, as seen on its upper horizontal edge, will then 
 be represented by the space between the two ellipses 1234 
 and 9 10 11 12. 
 
 4th. It now remains to describe the visible portion of the 
 lower inner circle of the ring. As this must be drawn four 
 inches below the upper inner circle, take the cabinet square, 
 U V W JT, four inches below the square I J K X, and 
 within it describe an ellipse touching the centres of its 
 sides. Of this ellipse, a little less than the farther half, 
 op r, will be visible. 
 
 Observe the quadrangle (four sided figure) at each corner 
 of the drawing. The two outer corners of each of the four 
 quadrangles are the corners of the two cabinet squares 
 within which the two outer ellipses are drawn ; and the two 
 inner corners are the corners of the two squares within 
 which the two inner ellipses are drawn. Similar quadran- 
 gles may be found in the drawings of all cylinders on a 
 rectangular basis ; and these quadrangles are excellent 
 guides for the formation of the squares within which the 
 several ellipses are to be drawn. Observe that, in all simi- 
 lar drawings of hollow cylinders, the distance A E, or B F, 
 or C 6r, or D H, represents the height of the cylinder. 
 
 The four ellipses described in this figure, representing 
 the four circles that form the outlines of the real hollow 
 cylinder, are all the guides that are required to draw any 
 hollow cylinder, of any definite .height, diameter, and thick- 
 ness of walls, in the vertical position represented by a rect- 
 angular cabinet square, as distinguished from a diagonal 
 cabinet square. A little practice will enable one to draw 
 these ellipses with great facility, while the ruling of the 
 drawing-paper insures accuracy. These four ellipses an- 
 swer to the four circles used in drawing a hollow cylinder 
 in diagonal cabinet perspective. 
 
CABINET PERSPECTIVE MISCELLANEOUS. 159 
 
 Fig. 17 is the same in outline as Fig. 16, but is here shaded, 
 the better to show the effect. Observe the four corner 
 quadrangles, which are here retained as guides for the four 
 cabinet squares within which the ellipses are to be drawn. 
 The points from which the upper side curves are drawn are 
 below the border of the page. Observe that the point 1 is 
 the centre of the upper two squares and ellipses; and that 
 the point 2 is the centre of the lower two squares and el- 
 lipses ; while the line 1 2 represents the axis of the short 
 cylinder or ring. 
 
 Fig. 18 represents the same ring as in Fig. 17, but with a 
 portion of the front of the ring cut out vertically, and at 
 right angles to its inner and outer surfaces. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw a vertical hollow cylinder whose extreme diameter shall be 
 fifty-two inches ; length of axis (height of cylinder), twenty inches, and 
 thickness of walls, six inches. 
 
 2. Draw two hollow cylinders after the plan of Fig. 13, page 4. Let each 
 of the cylinders be thirty-two inches in diameter, sixteen inches in length, 
 and the walls four inches in thickness. Remember that it is to be drawn 
 on a scale of "two inches to a space." This measure applies to both ver- 
 tical and horizontal lines on the front end of the horizontal cylinder, while 
 it means four inches to a space on lines representing horizontal lines run- 
 ning directly from the spectator. Thus, in Fig. 13, each of the lines 1 3, 
 and 4 $i an( i 8 9 measures twelve inches ; and 2 6 measures four inches ; 
 but G 7 measures twelve inches, and 1 2 measures sixteen inches. 
 
 PAGE SIX. SCALE OF TWO INCHES TO A SPACE. 
 
 Fig. 19. It is required to draw, in rectangular cabinet per- 
 spective, a hollow vertical cylinder, forty inches in extreme 
 diameter, fourteen inches in height, the walls four inches 
 thick, and to be encompassed by three bands, each two 
 inches wide, one even with the top, one even with the bot- 
 tom, and the third intermediate between the other two. 
 
 1st. Draw the hollow cylinder after the manner described 
 in preceding figures. Referring to Fig. 19, let the pupil ob- 
 serve that it is the cabinet square A J3 C D that incloses 
 the top of the cylinder, and the square E F ' G JET that in- 
 closes the bottom. Hence E and .F are fourteen inches 
 vertically below A and 7?, and H and G are fourteen inches 
 
160 INDUSTRIAL DBA WING. [BOOK NO. IV. 
 
 vertically below D and C, thus measuring two inches to a 
 vertical space. And yet, when the line A D is taken as the 
 short visible side of a rectangular cabinet square, and is con- 
 sidered as running horizontally from the spectator, being 
 what is called fore-shortened, it is measured as four inches 
 to a space, making the line forty inches in length. This 
 principle must be kept carefully in mind, being equivalent 
 to the principle of the measurement of "diagonal spaces" 
 in Books No. II. and No. III. 
 
 2d. To describe the encircling bands. The principle is the 
 same as explained in describing them in diagonal cabinet 
 perspective; but we will show the application here in detail. 
 
 As w is the point from which, with the distance w 4> we 
 describe the upper and outer side curve embraced in 1 4 8, 
 so, to describe the side curve of the lower side of the first 
 band, we move the point of the compasses one space (two 
 inches) downward, from w to r, and with the same stretch 
 of the compasses as before describe the curve. Then, as 
 the next band is four inches (two spaces) lower, we move 
 the point of the compasses two spaces (four inches) down- 
 ward, from r to s, and describe the upper side curve of the 
 second band. So continue until all the bands are drawn. 
 The end portions of the curves are drawn without the com- 
 passes, each to its given point on the side lines 1 5 and 3 7. 
 
 We have also described six curves (portions of ellipses) 
 on the visible inner side of the cylinder, between m and 9 9 
 the better to show the curvature of the inner side of the 
 cylinder. These curves are two inches apart; and the up- 
 per and the lower one are each two inches from the top 
 and bottom inner curves of the cylinder. These curves are 
 the same in form (so far as seen) as the upper curve I m n; 
 and as the side portion of I m n is described from the point 
 z, so, to describe the side portions of these six curves, carry 
 the point of the compasses, at each remove, one space (two 
 inches) downward below z. 
 
 Fig. 20 is the same cylinder as represented in Fig. 19, but 
 is here drawn in diagonal cabinet perspective, after the 
 manner shown in Book No. III. Observe that both cylin- 
 ders measure the same, according to the principles of meas- 
 
CABINET PERSPECTIVE MISCELLANEOUS. 161 
 
 urement applicable to each. Observe, also, that the two 
 cabinet squares, AB CD, E F G II, of Fig. 20, correspond 
 to the similarly lettered cabinet squares of Fig. 19. 
 
 Fig. 21. It is required to draw a horizontal wheel of the 
 following dimensions in rectangular cabinet perspective: 
 Wheel to be fifty-six inches in extreme diameter; horizontal 
 thickness of rim, four inches ; width of rim (or height, as 
 it lies horizontally), two inches ; a hub at the centre six- 
 teen inches in extreme diameter, same vertical thickness as 
 the width of the rim, and with a central eight-inch circular 
 opening through it for the axle. There are to be eight 
 spokes between the hub and rim, each four inches in hori- 
 zontal thickness, and same vertical width as the rim, radia- 
 ting from the centre of the hub, and at equal distances apart. 
 
 1st. Draw the upper and lower ellipses which form the 
 boundaries of the rim in the manner described for drawing 
 hollow cylinders in preceding figures. 
 
 2d. Draw the boundaries of the hub in the same manner. 
 
 3d. From the centre, q, with a radius of twenty-eight 
 inches, describe a circle passing through the extreme points 
 o and t of the wheel. Mark the points w, s,>, and n for the 
 diagonals of the circle ; then these points, together with the 
 centrally dividing points m, t, r, and o, will divide the circle 
 into eight equal parts. From each of the eight given points, 
 beginning at m and moving to the left around the circle, 
 lay off with the compasses any required distance, as m 1 
 and n 2, and mark accordingly the points 1, 2, 3, 4-> etc., for 
 the positions in the wheel in which the spokes are to be rep- 
 resented. Then, as the spokes are to be each four inches 
 in horizontal thickness, take with the compasses a width of 
 two spaces, and lay off that distance on the circle from the 
 points /, 2, 3, 4, 5, etc., and we shall have the distances 1 x, 
 2 y, S 2, 4 -10, etc., to represent the horizontal thickness of the 
 spokes. From the given points draw the dotted lines per- 
 pendicularly toward the horizontally dividing line o t, and 
 from the points where any pair of these dotted lines strike 
 the outer upper ellipse of the wheel, draw lines to the corre- 
 sponding opposite points of the ellipse, and the apparent upper 
 surface width of the spokes will be given. Observe that the 
 
162 INDUSTRIAL DRAWING. [BOOK NO. IV. 
 
 nearer these spokes approach, in lengthwise position, a line 
 running directly from the spectator, the wider their upper 
 surfaces appear, and the narrower their side views appear. 
 
 To get the apparent vertical thickness of the spokes, and 
 the lines of their junction with the inside of the rim and the 
 outside of the hub, we have only to draw lines vertically 
 downward from the points where the upper lines of the 
 spokes intersect the ellipses of the rim and hub, and then 
 draw the lower lines of the spokes as shown in the drawing. 
 The same result would be attained by supposing the spokes 
 to extend through the rim, as shown by the dotted lines. 
 
 Fig. 22 is another illustration of the principles of dividing 
 the ellipse into any number of equal parts, as set forth in 
 Fig. 21. 
 
 It is required to make a drawing of a vertical tub, of the 
 following dimensions, in rectangular cabinet perspective: 
 The tub is to be twelve inches in outside height; forty inches 
 in extreme diameter ; having a bottom one inch thick from 
 the level of the chimes below ; its sides to be vertical, and 
 composed of twenty-four staves of uniform width, and two 
 inches thick; and the tub is to be encompassed by two 
 hoops, each two inches wide, the upper one one inch from 
 the top of the tub, and the other even with the bottom. 
 
 1st. Draw the hollow cylinder for the walls of the tub 
 the same as in preceding figures, with the exception of the 
 inside lower ellipse. 
 
 2d. Describe the semicircle, 6 6, and with the compasses 
 divide it into twelve equal parts. This is easiest done by 
 commencing at 0, and dividing each quarter as from to 
 8, then from 3 to 6, etc. into three equal parts. This will 
 give the corresponding points, 1, 2, 3, 4, & 9 6-> o n each side of 
 0. From these points draw dotted lines vertically down- 
 ward until they intersect the outer upper ellipse ; also sup- 
 pose these dotted lines to be extended until they intersect 
 the nearer half of this outer upper ellipse. These twenty- 
 four points of intersection with the outer ellipse will be the 
 points of division for the apparent outer surface width of the 
 twenty-four staves. 
 
 3d. The top dividing lines of these staves must all be di- 
 
CABINET PERSPECTIVE MISCELLANEOUS. 163 
 
 rected toward the centre, <?, just as they would be in the 
 real tub, and the intersections of these lines with the inner 
 upper ellipse will give the points for the inside vertical di- 
 visions of the staves. It will be noticed that the staves di- 
 minish in apparent width from the vertical centre to the 
 right and left sides, as their surfaces are seen more and 
 more obliquely, but that they increase in apparent thickness, 
 just as they would appear to do in viewing the real tub. 
 
 4th. Observe that the inside height of the tub, as meas- 
 ured from 7 to 8, is only eleven inches, as required to be, 
 while the outside height is twelve inches. Therefore the 
 farther lower inner ellipse must be drawn one inch higher 
 than it would be drawn if the tub had no bottom in it. 
 
 5th. Observe, also, that A B C D is the square within 
 which the upper outer ellipse is drawn, and that E F Gr H 
 is the square within which the lower outer ellipse is drawn. 
 The distance apart from A to E, etc. depends upon the 
 height of the cylinder. In Fig. 21 the distance is two inch- 
 es, while here it is twelve inches. To prevent confusion of 
 lines, the lower quadrangles only are here dotted. 
 
 6th. The hoops are to be drawn according to the method 
 explained in Fig. 19. At the sides they should project, to 
 the extent of their thickness, beyond the vertical side lines 
 of the tub. 
 
 PROBLEMS FOR PRACTICE. 
 
 1 Draw a wheel similar to Fig. 21 , but of the following dimensions : 1"o 
 be sixty-eight inches in extreme diameter ; horizontal thickness of rim, two 
 inches ; width, or vertical height, of rim, six inches ; a hub at the centre 
 twenty inches in extreme diameter, same vertical height as rim, and with a 
 sixteen-inch central opening through it for the axle. There are to be eight 
 spokes between the hub and rim, each two inches in horizontal thickness, 
 and same width vertically as the rim, radiating from the centre of the hub, 
 and at equal distances apart. 
 
 2. Draw a tub similar to Fig. 22, but of the following dimensions : To be 
 fifty-six inches in extreme diameter ; sixteen inches in outside height ; a 
 bottom that comes up two inches from the level of the chimes below ; its 
 sides to be vertical, and composed of thirty-two staves of uniform width, 
 and two inches thick ; and the tub is to be encompassed by three hoops, 
 each two inches wide the upper one, one inch below the tops of the chimes; 
 the lower one, even with the lower ends of the chimes ; and the other, four 
 inches above the lower hoop. 
 
164 INDUSTRIAL DEAWING. [BOOK NO. IV. 
 
 PAGE SEVEN. SCALE OF THREE INCHES TO A SPACE. 
 
 Fig. 23. The same crown-wJieel that is shown in Fig. 56, 
 page 10, of Book No. III., is here drawn with its axis, t s, 
 in a vertical position. The rectangles A B C D and 
 E F G II represent the squares A B C D and E F G II 
 of Fig. 56. 
 
 Here w is the centre of the tipper horizontal surface of the 
 wheel, across the ends of the cogs ; and the wheel, as in Fig. 
 56, measures eleven feet in extreme diameter from 16 to 16. 
 From the centre, w, we therefore describe the two ellipses 
 which give the thickness of the ends of the cogs, as in Fig. 
 56 we described the two circles from the corresponding cen- 
 tre, 10. 
 
 Twelve inches below w we take the point y, from which, 
 as a centre, we describe the two ellipses which determine 
 the length of the cogs, and also the thickness of the wheel 
 between their bases. Six inches below y we take the point 
 u, which must be on the level of the tipper surfaces of the 
 four arms which support the rim ; and from v as a centre 
 we describe the ellipse which gives the limit for the ends 
 of these arms at their upper extremities. As these arms are 
 to be twelve inches in vertical height, or thickness, we take 
 a point a;, twelve inches below v, from which we describe 
 the ellipse which limits the lower extremities of the arms, 
 and also gives the inside lower boundary of the wheel. 
 
 As the axle on which the wheel turns extends fifteen 
 inches beyond both the upper and the lower face of the 
 wheel, the point s must be the centre of one end of the axle, 
 and the point t of the other; and as the diameter of the axle 
 is twelve inches, we have the dimensions of the ellipse de- 
 scribed around s for one end of the axle. The other end, t, 
 of the axle can not be seen. 
 
 The point v is the centre of the upper end of the hub, 
 which is eighteen inches in diameter. 
 
 As the wheel is to contain thirty-two cogs, at uniform 
 distances apart, and of the same width as the spaces be- 
 tween them, we divide each quarter of the semicircle, 16 16, 
 into eight equal parts; and from the points of division on 
 
CABINET PERSPECTIVE MISCELLANEOUS. 365 
 
 the semicircle draw lines vertically downward until they in- 
 tersect the outer ellipse 16 17 16 ; and the points of inter- 
 section will give the points for the divisions of the outer el- 
 lipse for the cogs and the spaces between them. From these 
 points of division, lines, as d w, a w, etc., drawn to the centre, 
 w, give the boundary-lines for the sides of the cogs at their 
 ends. In a similar manner, lines drawn like b y, g y, p y, 
 etc., give the boundary -lines for the sides of the cogs at 
 their bases. 
 
 The four arms, or spokes, which support the rim are to be 
 six inches in thickness, and twelve inches in vertical height, 
 the same as in Fig. 56. Here we have placed these arms in 
 diagonal positions, because they thus show to the best ad- 
 vantage; and the only remaining point now is to give the 
 true representation for their thickness six inches. 
 
 For this purpose, from v, the centre of the ellipse which 
 passes through the extremities of the arms, we describe a 
 circle (of which uj is a part) with a radius of sixty-three 
 inches (the length of each arm from the centre, v). From 
 the point i, where the diagonal v i intersects the circle, draw 
 the vertical line i r to intersect the before-mentioned ellipse, 
 and it will intersect it at r, on the diagonal of the ellipse. 
 From the point i measure off, on the circle, with the com- 
 passes, * h and ij 9 each equal to one ruled space, and there- 
 fore each representing three inches; and from h andj draw 
 lines vertically downward to intersect the ellipse. The space 
 included bet ween the points of the intersections of these lines 
 with the ellipse will then represent six inches ; and if lines 
 be drawn from the points of intersection parallel to r v, we 
 shall have the true apparent width of the upper surface of 
 this one arm or spoke of the wheel. The true width of the 
 others may be found in a similar manner; or it may be 
 taken by the compasses, and applied to the other arms. 
 
 At A the method of drawing the hub, together with the 
 arms radiating diagonally from it, may be more distinctly 
 seen. 
 
 Let it be observed, here, with what perfect accuracy a line 
 drawn vertically downward from the intersection of the cir- 
 cumference of a circle with its diagonal must intersect the 
 
166 INDUSTRIAL DRAWING. [BOOK NO. IV. 
 
 ellipse drawn below it on the same diameter on Us diagonal. 
 Owing to this principle, any division, and any measurement, 
 made on the circumference of the circle, according to what- 
 ever scale may be adopted, may be accurately represented 
 on the ellipse. 
 
 Thus, in B, we have a circle, and within it a cabinet el- 
 lipse, both drawn on the same diameter, m n. Here D J? is 
 drawn on a diagonal of the circle, and II F on a diagonal 
 of the ellipse. Observe, also, that every diagonal of the 
 cabinet ellipse drawn like this, in a rectangle, is wbat we 
 have called a two-space diagonal line, or a semi-diagonal. 
 From the point -?, where the circumference of the circle in- 
 tersects its diagonal, draw a line vertically downward, and 
 it will intersect the ellipse on its diagonal at the point 5. 
 Now the measure 1 4 is represented on the ellipse by 5 8, 
 and 1 2 by 5 6 ; and if 1 4 is equal to two spaces, or six 
 inches, and 1 2 equal to the same, then 6 8 will represent 
 a distance of twelve inches ; and the shaded band, 6 t s 8, 
 which extends across the ellipse, will be twelve inches wide. 
 Remembering that the cabinet ellipse represents the circle 
 viewed obliquely, we see why all measures on the circle 
 must have their accurate representations on the ellipse. 
 Also, we see from the three representations, Fig. 23, and A^ 
 and -Z?, that the diagonal of a cabinet ellipse must be drawn 
 on a two-space diagonal line that passes through the centre 
 of the ellipse. 
 
 PROBLEM FOR PRACTICE. 
 
 Let the pupil, af.ev drawing Fig. 23, draw another crown-wheel of any 
 given dimensions that may be assigned. 
 
 PAGE EIGHT. SCALE OF TWO INCHES TO A SPACE. 
 Fig. 24. It is required to make a drawing, in rectangular 
 cabinet perspective, of a tub twenty-six inches in vertical 
 height, forty-eight inches' extreme measure across the top, 
 thirty-two inches' extreme measure across the bottom ; the 
 sides formed of thirty-two staves of equal width, and two 
 inches in thickness, encompassed by two hoops, each two 
 inches wide one of them two inches from the top of the 
 tub, and the other one inch from the bottom ; the tub to 
 
CABINET PERSPECTIVE MISCELLANEOUS. 167 
 
 have a false bottom of one-inch stuff, seven inches in ver- 
 tical height from the level of the top of the tub ; and in the 
 centre of this bottom an opening twelve inches square. 
 
 1st. Mark off the cabinet square, A B G D, forty-eight 
 inches to a side, and within it, and touching the centres of 
 its four sides, describe an ellipse for the upper outer circum- 
 ference of the tub. 
 
 2d. Two inches within this outer ellipse describe another, 
 and we shall then have the outlines of the top of the tub. 
 
 3d. From m, the centre of the top diameter, mark off m n, 
 twenty-six inches, for the vertical height, or axis, of the tub. 
 The point n will then be the centre of the bottom. 
 
 4th. With n for the centre, mark off the square, EF G H, 
 of thirty-two inches to the side, and within this square de- 
 scribe an ellipse, only the nearer half of which, 123, need 
 be firmly drawn ; and this part of the ellipse will give the 
 visible portion of the outside circumference of the bottom 
 of the tub. 
 
 5th. By connecting the extreme points p 1 and r S of the 
 diameters of the upper and lower ellipses, the lines thus 
 formed will give the side outlines of the tub ; and if these 
 lines be continued downward, they will meet in a point, w, 
 on the downward continuation of the axis of the tub. All 
 the outside dividing lines of the staves will tend toward this 
 point, w, while the inside dividing lines will tend toward 
 the point z, above w. The point z is found by drawing 
 lines from 8 and 9 parallel to p 1 and r #, until they meet 
 at a point in the downward continuation of the axis. 
 
 6th. Take the point 7, seven inches, on the axis, below my 
 and 7 will be the centre of the circle (or ellipse, in the draw- 
 ing) which bounds the outer upper edge of the false bottom 
 of the tub. 
 
 If the sides of the tub were vertical, we should measure 
 off, from the inside top of the tub at the farthest point, 4 &> 
 equal to m 7y but as the line 4 5 extends downward to- 
 ward the spectator, and is in reality longer than m 7, it 
 must appear so in the drawing. The point 5 (when accu- 
 rately found) must be in the middle point of the farther 
 side of the cabinet square which bounds the ellipse that 
 
168 INDUSTRIAL DRAWING. [BOOK NO. IV. 
 
 limits the false bottom of the tub, and which ellipse has 7 
 for its centre. To obtain, with accuracy, the point 5, and 
 the square within which this ellipse is to be drawn, draw 
 the horizontal line from 7 toward 10, and it will intersect a 
 line drawn from 9 to z at a point a. The distance, 7 a, will 
 then be one half of the longer diameter of the ellipse which 
 bounds this false bottom of the tub ; and if 7 a be twenty 
 inches, we must so place the point 5 that the horizontal dis- 
 tance, 7 5, shall also be twenty inches. We may thus eas- 
 ily construct the square within which the ellipse must be 
 drawn. Centrally within the ellipse mark out the opening, 
 twelve inches square, and give to the false bottom, on its 
 farther side, a depth or thickness of one inch. 
 
 Vth. The divisions showing the apparent width of the 
 staves are obtained in the same manner as in Fio-. 23. 
 
 O 
 
 Fig. 25. It is required to draw an octagonal tub (of eight 
 equal sides), having an extreme upper diameter of forty- 
 eight inches, a vertical height of twelve inches, a lower ex- 
 treme diameter of forty inches, the sides two inches in thick- 
 ness, and the whole resting on a two-inch thick octagonal 
 bottom that projects two inches beyond the sides. 
 
 1st. Mark off the square, A B C D, of forty-eight inches 
 to a side ; and within it, and touching the centres of its 
 four sides, draw lightly an ellipse, of which ij k is the far- 
 ther half. As the central vertical line j n (vertical as seen 
 on the paper, although it represents a horizontal line), the 
 central horizontal line i /, and the two diagonals A C and 
 B Z>, divide the circle represented by this ellipse into eight 
 equal parts, so the intersections of these lines with the el- 
 lipse give us the eight corners of an octagon. Connect the 
 points or corners thus found, and two inches within this 
 octagon draw another in like manner, and we shall have rep- 
 resented the thickness of the walls of the octagon. This 
 inner octagon may be drawn without first drawing its el- 
 lipse, simply by drawing its lines parallel to the surround- 
 ing octagon, and two inches within it. 
 
 2d. From m, the centre of the two ellipses and of the oc- 
 tagon just described, measure twelve inches vertically down- 
 ward to n. Take n 2 and n 3, each twenty inches, and on 2 3, 
 
CABINET PERSPECTIVE. MISCELLANEOUS. 169 
 
 as the central horizontal line, mark off the cabinet square 
 E F G HE II or F G being half the length of E F or of 
 H G. Within this square draw an ellipse touching the cen- 
 tres, 1, 3, 4, 2, of its four sides. Divide the nearer half of 
 this ellipse so as to give the four nearer corners of an octa- 
 gon, in the same manner as the upper octagon M-as formed. 
 Connect the corners of this lower octagon with the corners 
 of the upper octagon by the lines i 2, n 4> k 3, etc. 
 
 3d. Two inches within the lower square, E F G If, inscribe 
 an ellipse, in the same manner that the inner upper ellipse 
 was drawn, and find the corners <5, #, 7 in the same manner 
 that the corners 8, 9, 10 were found. Connect these lower 
 corners with the upper corners. This inner octagon may 
 be drawn by simply drawing its lines parallel to the outer 
 octagon. 
 
 4th. Take 2 p and 3 r, each one space (two inches), and 
 from p and r draw lines parallel to the border octagonal 
 lines, and we shall have the projection of the bottom two 
 inches beyond the walls of the octagon, thus forming an 
 octagon two inches beyond the other; or this octagon may 
 be formed by first drawing an ellipse on p r as a central base 
 line. The two inches' thickness of the bottom, on which the 
 octagonal tub rests, is easily designated by simply marking 
 points one space down from the corners above. 
 
 Observe that the thickness of the walls of the octagonal 
 tub, and the extension of the platform bottom, naturally ap- 
 pear the greatest on the extreme right and left sides of the 
 drawing. Observe, also, that the farther inner side of the 
 tub, as measured by the line 9 6, appears deeper than the 
 nearer side as measured by the line n 4. The cause of this 
 is that the line 9 6 is more nearly at right angles to the 
 line of vision than the line n 4- 
 
 III. ARCHES IN DIAGONAL PERSPECTIVE. 
 
 Figs. 26 and 27 are drawn in diagonal cabinet perspec- 
 ve ; but it is the upper left-hand view of them that is here 
 given, as explained on page 144, and illustrated at J9, page 1, 
 of Book No. IV. 
 
 H 
 
170 INDUSTRIAL DRAWING. [BOOK NO. IV. 
 
 Fig. 26. The lightly shaded portion of Fig. 26 represents 
 the face of a pointed archway fronting the spectator, and is 
 therefore drawn in its true relative proportions: the base 
 line of the archway, a d, being thirty-two inches, and the 
 height, e/, thirty-eight inches. 
 
 The outer curve line, af 9 of the arch, is here drawn by the 
 compasses, from a point twenty-six spaces fifty-two inches 
 to the left of a; or it may be drawn from any other point 
 that will give a graceful curve connecting a and f* but 
 having drawn this curve, the opposite curve, d f 9 must be 
 made to correspond to it. Then the inner curves, p h and 
 g h, must be drawn from the same points from which the two 
 outer curves were drawn, but each with a radius of twenty- 
 four spaces. The curve v h must be drawn from a point 
 twenty-four spaces to the left of the corner v. 
 
 As Fig. 26 is a front view of the arch, it is very easily 
 drawn. 
 
 Fig. 27 is a diagonal view of the front of Fig. 26. Here 
 the diagonal figure, a b c d, must measure the same as the 
 corresponding rectangle of the preceding figure ; and the 
 curve a f must be drawn by hand, and be made to pass 
 through points corresponding to those through which the 
 curve a f of Fig. 26 passes. Thus the points n and m of 
 Fig. 26 are respectively four and eight inches from certain 
 points, 10 and 1^ in the corner line ab ; and the correspond- 
 ing points n and m of Fig. 27 must be similarly situated. 
 So any point in any one of the curves of Fig. 26 should, 
 if accurately drawn, have its corresponding point, by meas- 
 ure, in a corresponding curve of Fig. 27. This principle 
 should apply accurately throughout the two figures the 
 measures from point to point in the one being the same 
 (according to the principles of measurement) as the meas- 
 ures between corresponding points in the other. Thus the 
 two points n, n, in Fig. 26 are each twelve inches horizontally 
 distant from the point w ; the corresponding points n, n in 
 Fig. 27 are also each twelve inches (three diagonal spaces) 
 distant from their intermediate point 10. The under side 
 of the arch must measure four inches, horizontally, in any 
 part of it. So the outside of the arch, from a r tofj, must 
 measure, in any part, four inches horizontally. 
 
CABINET PERSPECTIVE MISCELLANEOUS. 171 
 
 Having Fig. 26 as the pattern, and marking any required 
 number of points in its curves, it is easy to mark corre- 
 sponding points in Fig. 27, and draw the curves through 
 them. 
 
 Fig. 28, drawn in diagonal perspective, consists of a rect- 
 angular box, open at top and bottom, thirty-two inches in 
 outside width, sixty-four inches in length, thirty-four inches 
 in height, and walls four inches thick, having an archway 
 opening through both ends, and two archways on a side. 
 The side archways are the same in size as those at the ends; 
 and the face of the archway in the front end, being drawn 
 in its true relative proportions, is the guiding pattern for 
 drawing the others. Let the pupil describe the dimensions 
 of the arch, method of drawing the side arches, etc. If the 
 pupil will follow the directions given for Fig. 27, he will 
 probably draw Fig. 28 with more accuracy than it is given 
 in the book. 
 
 The principles here illustrated will be a sufficient guide 
 for the drawing of all arches in diagonal perspective. First 
 draw an arch of the required dimensions in its true relative 
 proportions, with its face fronting the spectator: it will 
 then be easy to make a diagonal view of the same arch 
 transferring the measures of the one to the other. 
 
 IY. SEMI-DIAGONAL CABINET PERSPECTIVE. 
 PAGE NINE. SCALE OF TWO INCHES TO A SPACE. 
 
 In what has been called Diagonal Cabinet Perspective, a 
 cubical block is supposed to be viewed in such a manner 
 that the horizontal corner lines of the top and bottom of the 
 cube that run from the spectator seem to rise diagonally 
 at r.n angle of forty-five degrees that is, half way from the 
 horizontal to the vertical. Thus in Fig. 29, if the cube were 
 drawn in diagonal cabinet perspective, its upper horizontal 
 face would be bounded by the lines 1 , 2 3 t 3 7, and 7 1; 
 and 5 4 would be its visible lower side line. 
 
 Now suppose that the eye of the spectator should be 
 lowered so that the line 2 3 should be brought down and 
 
172 INDUSTRIAL DRAWING. [BOOK NO. IV. 
 
 made to coincide with the line 89: then 1 2 would be 
 brought down to 1 8, and 7 8 to 7 9, and 5 4 to 5 10; and 
 thus the lines 1 8, 7 9, and 5 10 would be in the direction of 
 semi-diagonals, or two-space diagonals, as shown in Book 
 No. I., page 2. The front face of the cube would remain 
 the same as in diagonal perspective ; but the top and. side 
 would be changed as shown in Fig. 29. This is what is 
 called " $e/m-Diagonal Cabinet Perspective;" and it is spe- 
 cially adapted to the drawing of buildings or other large 
 structures in full elevation, as hereafter shown. 
 
 In this modification of Cabinet Perspective, the principles 
 of measurement remain the same as in the general system 
 before explained. Thus, in diagonal perspective, the line 
 1 2, according to the scale for the page, measures sixteen 
 inches: but the line 18 must also measure sixteen inches : the 
 line 1 2 extends from the point 1, diagonally, to the fourth ver- 
 tical line at 2; and the line 1 8 extends from the point 1, 
 semi-diagonally, to the fourth vertical line at 8 that is, the 
 line 1 2 passes over four spaces, and the line 1 8 also passes 
 over four spaces ; and in each case a space whether a diag- 
 onal or a semi-diagonal measures four inches. If, therefore, 
 we consider that each sem /-diagonal (that is, the distance on 
 the line 1 8 from one vertical line to another) measures the 
 same as a diagonal, the rule of measurement is unchanged. 
 The semi-diagonal cube, Fig. 29, therefore measures sixteen 
 inches to a side, the same as the dotted diagonal cube. 
 
 Fig. 30 represents a cubical block, of twenty-four inches 
 to a side, from each upper corner of which is taken away a 
 cubical block eight inches square. There is an eight-inch 
 square opening cut through the block, centrally, from the 
 lower part of the front side ; and also an opening of the 
 same dimensions on the right-hand side. 
 
 Fig. 31 is a frame -work in semi -diagonal perspective. 
 The central post measures eight inches square at the ends; 
 and each half of it is thirty-six inches in length on each 
 side of the cross-beam. The cross-beam is four by eight 
 inches, and sixty -four inches in length, while the braces 
 would measure four inches square at the ends if the ends 
 were squared. Inasmuch as the side of the frame fronting 
 
CABINET PERSPECTIVE MISCELLANEOUS. 173 
 
 the spectator is drawn in its true relative proportions, the 
 same as all front views in the cabinet perspective of plane 
 solids, therefore all lines, drawn in any direction whatever, 
 on any portion of this front view, are measurable by the 
 scale of two inches to a space. Hence the line a >, drawn at 
 right angles across the brace, must be the true measure of 
 the width of the brace ; and, according to the scale, it is 
 found to measure two spaces, or four inches. The line c cl, 
 beino- one semi-diasronal, measures four inches. We thus 
 
 O O 7 
 
 obtain the size of the timber used for the brace. 
 
 The extreme length of a brace, 1 3, or its equal in length, 
 4 , may be found by actual measurement ; for, as the line 
 1 3 is in a plane at right angles to the line of vision, it may 
 be measured by the scale. Thus measured, it will be found 
 that the line 1 S is equal to twenty spaces equal to forty 
 inches. Or it may be measured by the rule for the extrac- 
 tion of the square root. Thus 1 2 3 is a right-angled tri- 
 angle; and therefore, if we add together the squares of 
 the sides 1 2 and 1 3, and extract the square root of their 
 sum, we shall get forty inches for the length of the line 1 3. 
 Try it. 
 
 Observe that the braces are flush with the farther side of 
 the central post, and with the farther side of the cross-beam. 
 The dotted continuations of the lines of the braces show the 
 attachment of the braces to those sides of the post and cross- 
 beam which are invisible to the eye. 
 
 Fig. 32 is the same as Fig. 31, but so placed that the front 
 view of Fig. 31 is made the semi-diagonal view in Fig. 32. 
 Let the pupil test all the measurements, and see that they 
 are the same in both figures. Thus the line 6 7 in Fig. 32 
 measures the same, according to the principles of measure- 
 ment adopted, as the line 6 7 in Fig. 31. Fig. 32, however, 
 does not give so good a view of the frame as is shown in 
 Fig. 31. 
 
 Fig. 33 is the same as Fig. 95 of Book No. II., but is here 
 changed from diagonal to se??n'-diagonal perspective. Ob- 
 serve that, if the same scale be adopted, the measurements 
 will be the same in both cases. 
 
 Fig. 34 is a hollow cylinder sixteen inches in length and 
 
174 INDUSTRIAL DRAWING. [BOOK NO. IV. 
 
 eight inches in extreme diameter, having its walls two 
 inches in thickness. The length of the cylinder may either 
 be measured on its axis, a b, or any where semi-diagonally 
 on its circumference as on the line c d. 
 
 Fig. 35 is a cylinder eight inches in length and sixteen 
 inches in diameter, having an opening eight inches square 
 through it centrally lengthwise. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw, in semi-diagonal perspective, a cube thirty-two inches square, 
 and take from the centres of each of its three visible sides a piece twenty- 
 four inches square on the face, and four inches in thickness. 
 
 2. Draw a frame the same as Fig. 31, with the exception that the braces 
 are to be placed flush with the front side of the central post and cross-beam. 
 
 PAGE TEX. SCALE OF TWO INCHES TO A SPACE. 
 
 Fig. 36 represents a cube having a circle described on its 
 top, and also one on each of its two visible sides, the whole 
 being drawn in semi-diagonal perspective. The representa- 
 tive ellipse on the right-hand side of the cube is of the same 
 width as in diagonal perspective, but not of the same pro- 
 portions; but the ellipse on the top has only half the width 
 that it would have in diagonal perspective. The front side 
 being a square, the circle is there drawn as a circle, in its 
 true proportions. See Fig. 29, and explanation, for the rel- 
 ative proportions of the three visible sides of the cube. 
 
 As the upper side of the cube is divided into the same 
 ' inmber of semi-diagonal squares that the front side contains 
 'cgular squares, the former representing the latter, therefore 
 the ellipse drawn on the top must pass through the same 
 corresponding points in the semi-diagonal squares that the 
 circle passes through in the squares of the front side, on the 
 principle that was explained in the drawing of Fig. 9. The 
 ellipse on the top would be best drawn by first dotting the 
 principal points through which it must pass, and then drawing 
 the curve carefully by hand with a sharp-pointed hard pencil. 
 
 The ellipse on the side might be drawn by the same meth- 
 od, on the principle explained with reference to Fig. 9 ; but 
 the side curves may be drawn with great accuracy by the 
 compasses in the following manner: 
 
CABINET PERSPECTIVE MISCELLANEOUS. 175 
 
 1st. From the point 1, the centre of the side C f] with the 
 radius 1 B, describe the indefinite curve B 5. 2d. From 
 the point B, with the radius B 2, describe the curve 2 3, in- 
 tersecting the curve B 5 at the point 3. 3d. From the point 
 3, with the radius 3 2, describe the curve 2 4. 4th. From 
 the point 4, with the radius 4 !>> describe the side curve 
 617, which may be extended to within two inches of the 
 line C B with accuracy. 
 
 5th. From the point 2, the centre of the side B E, with 
 the radius 2 F, describe the indefinite curve Fp. 6th. From 
 the point JF\ with the radius F 1, describe the curve 1 n. 
 7th. From the point n, with the radius n -/, describe the 
 curve 1 m. 8th. From the point m, with the radius m 2, 
 describe the side curve 9 2 10 ; which may be extended to 
 s bout the point 11 with accuracy. 
 
 Thus the two side curves of the ellipse will be drawn, to- 
 gether with portions of the two end curves. The remaining 
 portions of the end curves must be drawn by hand to the 
 points v and ?, taking care that they pass through points 
 in the small semi-diagonal squares corresponding to the 
 points through which the circle on the front passes. Let 
 this figure be viewed intently through the opening formed 
 by the partially closed hand, and the ellipses will soon take 
 the appearance of perfect circles. 
 
 Fig. 37 represents a ring drawn in semi-diagonal perspec- 
 tive, and in the same position as shown by the ellipse on 
 the right-hand side of the cube, Fig. 36. The inner diam- 
 eter of the ring, as measured either by the line v w or 1 2. 
 is forty-eight inches, the same as the ellipse 1 v 2 w of Fig. 
 3G. The thickness of the ring is two inches, and its width 
 four inches. 
 
 As the inner front ellipse is drawn within the semi-diag- 
 onal square, (7 B E F, so the outer elliptic circumference of 
 the front face of the ring must be drawn within a semi-di- 
 agonal square two inches larger, in every direction, than 
 the inner square, as shown by the surrounding dotted semi- 
 diagonal square. 
 
 The true width of the ring is marked out by extending 
 the points of the compasses two spaces, and laying off that 
 
176 INDUSTRIAL DRAWING. [BOOK NO. IV. 
 
 distance horizontally to the left, from the right-hand por- 
 tion of the inner front ellipse, and also from the left-hand 
 portion of the outer front ellipse. Thus the horizontal 
 measures 3 4, 5 6, 7 8, 9 10, etc., must each be equal to two 
 spaces, if the ellipse be accurately drawn. 
 
 Fig. 38 represents a pointed arch drawn in semi-diagonal 
 perspective. The opening, or span of the arch, 1 2, is forty 
 inches; and the ^ 7 ertical height, 3 4, is also forty inches 
 although the arch proper is fourteen inches less in height, 
 as the walls do not begin to converge until they reach a 
 height of fourteen inches from the base. 
 
 If this arch were to be drawn from the given measure- 
 ments, it should first be drawn within a rectangular square 
 fronting the spectator. It could then be changed with ac- 
 curacy to a semi-diagonal view, according to the method ex- 
 plained for transferring the measures of Fig. 26 to Fig. 27. 
 
 Or, if the nearer inner curve, 1 4-, he drawn by the eye, ac- 
 cording to the judgment, the farther inner curve may be 
 drawn to correspond to it. Then the outer curves, 6 5 and 
 5 7, must be made to correspond to the inner curves ; and 
 as 1 6 or 2 7 is a measure of four inches, so the point 5 must 
 be taken four inches above the point 4- -As the depth of 
 the arch, 6 8 or 2 10, is six inches, so the depth, wherever 
 measured on a horizontal line, as a b, c d, 11 12, 9 5, etc., 
 must be six inches. 
 
 Fig. 39 is a cylinder one foot in diameter, and seven feet 
 four inches, or eighty-eight inches, in length. Its axis is a b. 
 At each end of the cylinder is a projecting tenon twelve 
 inches long, eight inches wide, and four inches in thickness ; 
 and longitudinally through the cylinder is a mortise three 
 inches wide, extending to within four inches of the ends, 
 and coinciding in direction with the width of the tenons. 
 
 PROBLEMS FOR PRACTICE. 
 
 1. Draw a pointed arch, similar to Fig. 38, first in rectangular perspec- 
 tive, and then in semi-diagonal perspective, of the following dimensions : 
 Span of arch, forty-eight inches ; height of opening, forty-eight inches ; 
 thickness of \valls, eight inches ; depth of arch, twelve inches. [In the 
 first drawing, while the front view of the arch is to be on a rectangular basis, 
 and therefore in its relative proportions throughout, the side view is to be 
 
CABINET PERSPECTIVE MISCELLANEOUS. 177 
 
 in semi-diagonal perspective just as the side view of Fig. 33 is in semi-di- 
 agonal perspective.] 
 
 2. Draw a partial cylinder, somewhat similar to Fig. 30, but of the fol- 
 lowing dimensions : Length of cylinder, eight feet ; diameter, sixteen 
 inches ; tenons, sixteen inches in length, otherwise the same as in Fig. 39 ; 
 upper side of the cylinder to be taken off horizontally, even with the top of 
 the tenons ; and a mortise, four inches wide and eighty inches in length, to 
 extend longitudinally through the cylinder, equidistant from the two ends, 
 and coinciding in direction with the width of the tenons, as in Fig. 39. 
 
 PAGE ELEVEN. SCALE OF FOUR INCHES TO A SPACE. 
 
 Fig. 40. According to the scale of measurement adopted, 
 Fig. 40 represents a small building, fifteen feet long, and 
 ten feet eight inches wide, with posts nine feet four inches 
 high. The four corner posts and sills are eight inches square 
 at the ends ; the two middle posts are four inches square, 
 and the central cross-sill is four by eight inches. The plates 
 on which the rafters rest are four by eight inches, while the 
 rafters are four inches square. The rafters are placed at 
 what is called half pitch the height, 2 7, being equal to 
 half the span, 8 9; or, as the vertical line 2 7 is equal to 
 2 #, the rafters are at an angle of forty-five degrees. 
 
 Observe that the lower ends of the rafters come down at 
 equal distances on both sides below the plate, as indicated 
 by the line 5 6; and that the ends are sawed off horizon- 
 tally. The rafters are twelve inches apart, or sixteen inches 
 between their central lines. 
 
 Observe that the braces are all of the same length forty 
 inches, inside measure and that they are placed flush with 
 the outside of the frame. Thus the inside of the top of the 
 extreme brace at the right, at s, is twenty-four inches from 
 the corner t; and the inside of the bottom of the brace, at 
 v, is thirty-two inches below the corner t. Now, as s t v is 
 a right angle, s v is the hypothenuse ; and if we add togeth- 
 er the squares of s t and t v, and extract the square root of 
 the sum, we shall find that the length of s v is forty inches. 
 All the braces are arranged in like manner. In all cabinet 
 work, and in buildings, braces are generally arranged in the 
 proportions of three measures for one side of the triangle 
 and four for the other; and then five will be the measure 
 
 H2 
 
178 INDUSTRIAL DBA WING. ^BOOK NO. IV. 
 
 for the hypothenuse, whether the measure be in inches or 
 in feet. 
 
 The only difficulty in drawings similar to Fig. 40 con- 
 sists in placing the braces in their correct positions, accord- 
 ing to the measurements assigned to them ; and, in order to 
 arrange them accurately, it is evident that we must first 
 find those corners which are hidden from view, as shown 
 in Figs. 31 and 32. The following problems will aid in 
 elucidating the principles which govern the drawing of 
 braces in the various positions in which they usually occur 
 in a building. 
 
 PROBLEMS FOR PRACTICE. 
 
 [In these problems the posts are to be sixteen inches square at the ends; 
 and the plates or cross-beams are to be four by sixteen inches at the ends. 
 Referring for illustration to the brace at A, the top of each brace (as s) is 
 to be forty-eight inches, on the inside, from the corner (as t) where the 
 plate joins the post ; and the bottom of each brace, on the inside (as at v\ 
 is to be sixty-four inches below the corner (as /) where the plate joins the 
 post. The inside length of each brace will then be eighty inches. The 
 braces are to be twice the size of timber shown in Fig. 40, and to be placed 
 flush with the outside of the frame. In each case, let the outlines of the 
 concealed end of the brace be dotted, as in Figs. 31 and 32.] 
 
 1 . Draw a brace of the given dimensions connecting a post and plate, as 
 in the corner A. Only a sufficient length of post and plate may be drawn 
 to show the plate to advantage. 
 
 2. Draw a brace for a corner corresponding to B. 
 
 3. Draw a brace for a corner corresponding to C. 
 
 4. Draw braces for corners corresponding to D and //. 
 
 5. Draw a square frame-work of two upright posts, and plate, and sill, 
 corresponding to the left-hand end of the building, Fig. 40, and put a brace 
 of the given dimensions, and flush with the outside, in each of the four cor- 
 ners of the frame. 
 
 G. Draw a brace for a corner corresponding to G, and also one for a cor- 
 ner corresponding to 7, below G. 
 
 1. Draw braces of the given dimensions for corners corresponding to the 
 four corners embraced by the sills of the building, and let the braces be 
 flush with the tops of the sills. . Let the sills be sixteen inches square at 
 the ends. 
 
CABINET PERSPECTIVE MISCELLANEOUS. 179 
 
 V. SHADOWS IN CABINET PERSPECTIVE. 
 As cabinet perspective is designed for the artisan rather 
 than the artist, we have thus far striven for no effect in our 
 drawings beyond what is requisite to convey, in as simple a 
 manner as possible, correct ideas of the forms and dimen- 
 sions of objects. To this end such simple methods of plain 
 shading have been introduced as will most readily distin- 
 guish one surface from another, while no attention has been 
 given to the shadows cast by objects. But where it is de- 
 sired to give greater artistic effect to cabinet drawings, 
 the system of perspective here adopted w r ill enable the 
 draughtsman to define the outlines of shadows with the 
 greatest ease, and with a degree of mathematical accuracy 
 hitherto unattainable.' On page 12 we give a few illustra- 
 tions of the principles which are applicable to this subject. 
 
 PAGE TWELVE. SCALE OF SIX INCHES TO A SPACE. 
 
 The objects here represented are supposed to stand on a 
 horizontal plane, which may be understood to be the level 
 surface of the earth ; and the spectator is supposed to be 
 looking down upon them from above, and at the right, and 
 from a northerly direction. Hence the left-hand side of the 
 paper is east, the upper side is south, the right-hand west, 
 and the lower side is north as indicated by the large cap- 
 itals, K, S., TV., N. 
 
 Fig. 41. This figure represents a square vertical pillar 
 standing upon the level surface of the earth, while the sun, 
 elevated at some distance above the horizon, and shining 
 upon the pillar from a southeasterly direction, causes the pil- 
 lar to cast a shadow on the earth, as shown in the drawing. 
 
 The sun is so far distant from the earth that the rays of 
 light coming from it may be regarded as parallel. Suppose 
 that the rays of light come from the southeast in a semi- 
 diagonal direction, as indicated by the arrows, or, c, e; and 
 that the ray #, just touching the corner 1, strikes the earth 
 at the point 9. Then it is evident that all rays, such as 
 g h, ij, etc., that strike the corner line 1 2, will project shad- 
 ows upon the earth between the points and 9; and hence 
 
ISO INDUSTRIAL DRAWING. [BOOK NO. IY. 
 
 the line 2 P, which indicates a line of shadow drawn on the 
 level surface of the earth, will be the shadow of the vertical 
 corner line 1 2. This line of shadow, 2 9, although hori- 
 zontal, will diverge away from the line 2 4, and also be 
 lengthened, just in proportion to the southerly direction of 
 the sun, and its nearness to the eastern horizon. If the sun 
 were directly east of the line 1 2, and on the horizon, its 
 shadow would be extended indefinitely from 2 in the direc- 
 tion of 2 4> but the shadow would become shorter and 
 shorter as the sun rose vertically toward the zenith; and 
 when at the zenith the shadow of 1 2 would be contracted 
 to a single point at 2. 
 
 Hence two lines are required to define the shadow cast 
 by the vertical line 12: the one, a 9, called the ray-lute, giv- 
 ing the direction of the ray of light that barely touches 
 the corner ! and the other, called the shadow-line, drawn 
 through the corner 2, and intersecting the ray-line at 9. 
 
 Hence, to find the shadow cast on a horizontal plane by 
 any given vertical line, or by any given point: 
 
 RULE I. Draw a BAY-LINE, indicating the direction of 
 the surfs rays, from the top of the given line to the horizontal 
 plane which is on a level with the lower end of the given line; 
 then draw a SHADOW-LINE from the lower end of the given 
 line to the point where the ray-line strikes the horizontal 
 plane. The shadow-line thus drawn will be the shadow of 
 the given line. 
 
 The shadow cast by any given POINT on a horizontal plane 
 may be found by first drawing a vertical line from it to the 
 horizontal plane, and then finding the shadow of the given 
 line, as before. The shadow cast by any required point in the 
 given line may thus be obtained. 
 
 Following out the principles of light and shade in con- 
 nection with this rule, it will be seen, as before shown, that 
 2 9 is the shadow of the vertical line 1 2, and that 9 is the 
 shadow-point of the point 1. 
 
 In a similar manner it is found that the line 4 8 would bo 
 the line of shadow cast by the vertical corner line 3 J>, if all 
 the pillar except its corner line 3 4 were transparent; also 
 that 6 7 is the line of the shadow cast bv the corner line 
 
CABINET PERSPECTIVE MISCELLANEOUS. 181 
 
 5 6. The line 8 7 must therefore be the line of shadow cast 
 by the line 35. If 5 10 could cast a shadow, its shadow 
 would be the line 7 11, parallel to and equal in length to 
 5 10; and if 10 1 could cast a shadow, its shadow would be 
 the line 11 9, parallel to and equal in length to 10 1. 
 
 The following Rule may also be deduced from Fig. 41. 
 
 RULE II. Every horizontal line casts a shadow, on a 
 horizontal surface, parallel to itself; and the shadow has the 
 same representative length as the line casting the shadow. 
 Thus 9 8 is parallel to and equal to 1 3; 8 7 to 3 5; 7 11 
 to 5 10; 11 9 to 10 1, etc. 
 
 Fig. 42. In this figure the horizontal lines of shadow cast 
 on the ground by vertical lines are represented as bearing 
 in the direction of southwestern four -space diagonals, as 
 shown by the course of the arrows , b, c, d,e; and the rays 
 of light as coming diagonally downward, really from the 
 northeast, as indicated by the arrows f,g, h, i,j, although 
 they seem to come from the southeast. Here the north 
 and east sides of the objects are in the light, and the south 
 and west sides in shadow, as they would be if the objects 
 were south of the equator, and the sun had risen midway 
 toward the zenith. 
 
 Observe, here, that 4 m is the line of shadow cast by 3 4; 
 m n the line of shadow cast by 3 1 ; and n v a part of the 
 line of shadow cast by 1 t; and that 2 n would be the 
 shadow cast by 1 2, if 1 2 could cast a shadow. Also, 6 13 
 is the line of shadow cast by the corner line 5 6; 14 15, the 
 shadow cast by 8 7 ; 15 16, the shadow cast by 7 9; and 
 
 16 17, the shadow cast by 9 11. The farther vertical corner 
 line of the pillar, represented by the dotted line 11 12, also 
 casts a shadow on the top of the platform, which shadow is 
 represented by the line 12 17; but only a small portion, 
 
 17 x, of this line of shadow is visible. Observe, here, the 
 strict application of Rule II., as to the direction and length 
 of the shadows cast by horizontal lines. Let the pupil ex- 
 plain the method of finding the shadow cast by any given 
 line in Fig. 42, or by any given point in the structure. 
 
 Fig. 43. In this figure the north and west sides of the 
 objects are in shadow; the horizontal lines of shadow cast 
 
182 INDUSTRIAL DRAWING. [BOOK NO. IV. 
 
 on the ground by vertical lines are represented as bearing 
 in the direction of northwestern three-space diagonals, as 
 shown by the arrows #,/",/*,&/ and the rays of light as 
 coming semi-diagonally downward from the southeast, as 
 indicated by the arrows r, s, t,v, etc. The important point 
 to be noticed in this figure is that the square pillar casts 
 its shadow beyond the platform, and beyond the shadow 
 of the platform also. 
 
 The shadows cast by the two visible sides of the plat- 
 form are easily obtained. The shadow 2 3, of the vertical 
 corner line 1 2, is obtained in the same manner as the shadow 
 of the corresponding corner line of Fig. 41. The shadow 
 that would be cast by 1 c on the horizontal surface of the 
 earth must be parallel to and equal in apparent length to 
 1 c. (See Rule II.) Now, if the platform did not intercept 
 a portion of this shadow, the shadow of 1 c would be the 
 line 5 7; but the platform intercepts a portion equal to 3 4 
 that is, the shadow of that part of the line included be- 
 tween the points 1 and b; and it is the shadow of the part 
 b c only that passes beyond the platform, and shows itself 
 on the ground in the line 6 7. The two shadow-lines 3 4 
 and 6 7. must therefore be equal to 1 c. 
 
 Having now the point 7 as the point of shadow cast by 
 the corner c, we know (Rule II.) that 7 #, drawn parallel 
 to and equal to c d, must be the line of shadow of c d ; and 
 hence 8 will be the point of shadow of the corner d. But 
 this point of shadow may also be found by the general 
 rule, in the following manner (see Rule I.) : Extend the 
 vertical corner line d i down to m that is, to the level of 
 the earth on which the platform rests ; and then the 
 problem becomes one to find the shadow cast on the earth 
 by the line d m. From m draw a three-space diagonal line 
 of shadow in the direction m 8; and through d draw a two- 
 space diagonal ray-line, which, at its intersection with the 
 former line, will give the point 8. 
 
 We next wish to find the shadow cast by the vertical 
 corner line d i. It is evident that the shadow that would 
 be cast on the ground by the whole line d m would be the 
 line m 8; but 9 8 is the only part of this shadow-line that 
 
CABINET PERSPECTIVE MISCELLANEOUS. 183 
 
 is not intercepted by the platform ; and 9 8 is the shadow 
 cast by g d. Kow the shadow cast by the portion g i is 
 evidently i n, which latter line is found by drawing a three- 
 space diagonal line of shadow through the corner , and in- 
 tersecting it by the two-space diagonal ray-line g n. This 
 completes the outlines of shadow cast by the platform and 
 the pillar. 
 
 Fig. 44 represents two rectangular blocks, A and B, in 
 vertical position, and standing at right angles to one an- 
 other upon the horizontal surface of the earth. The sun is 
 in the southwest, and at such an elevation that its rays pass 
 downward in the direction of semi-diagonals, as indicated 
 by the arrows s,t y u,v,x; while the shadows cast by vertical 
 lines are in the direction of horizontal semi-diagonals, as 
 indicated by the arrows c,c?,/. Both blocks cast shadows 
 upon the earth; while the block B casts a shadow, ad- 
 ditionally, upon a part of the vertical surface of the block A. 
 
 In accordance with principles already explained, the line 
 2 g is the shadow cast by the vertical corner line 1 2. Hav- 
 ing g as the point of shadow cast by the comer .7, we know 
 that the shadow cast on the earth by the line 1 7 would 
 extend to the left from <?, parallel to 1 7, and of a length 
 equal to 1 7. But this line of shadow is intercepted at 9 
 by the vertical surface of the block A ; and it is evident 
 that g 9 is the shadow of 1 10 only. Where, then, does the 
 part 10 7 cast its shadow ? As the point 10 casts its shadow 
 at 9, and as the shadow of 7 is at 7 itself, on the vertical 
 surface of A, it follows that the straight line 10 7 must cast 
 its shadow in a straight line from 9 to 7, from which we de- 
 duce the following rule, applicable to all similar cases. 
 
 RULE HI. The two points of shadow cast on a plane 
 surface by the extremities of a straight line being given, the 
 shadow of the line will be in a straight line connecting the 
 two given points of shadow. 
 
 The additional lines of shadow in Fig. 44 present no diffi- 
 culties the line 4 5 being the shadow of 3 4, and 5 6 being 
 the shadow of 3 i. The line i r casts a shadow, and a part 
 of it is out of view, 'on the farther or eastern side of the 
 block A. But the shadow of the corner i is at 6; and the 
 
184 INDUSTRIAL DRAWING. [BOOK NO. IV. 
 
 shadow of i r must extend from 6 to m, parallel to i r, find 
 equal in length to i r (Rule II.). Or the point of shadov.- 
 cast by the corner r may be found by the regular method, 
 as follows: As n is the farther but invisible corner of the 
 block, directly below r, if we draw a horizontal two-space 
 line of shadow through n, and a two-space diagonal ray-line 
 through r, their point of intersection at m will be the point 
 of shadow cast by the corner r. 
 
 Fig. 45. In Fig. 45 the ray-lines, and the shadows of verti- 
 cal lines, are the same as in the preceding figure ; and the 
 two blocks C and D are similar in position to the blocks A 
 and JB of Fig. 44 ; but they here rest on a horizontal plat- 
 form. 
 
 In this figure the lines of shadow b a and a 1 are in all 
 respects similar to the lines of shadow g 9 and 9 7 in Fi<r. 
 44 except that only that part ofal (viz., a ), which touch- 
 es the vertical side of (7, is a shadow-line. Here b a is the 
 shadow of 8 r; a 6 is the shadow of r t; and 6 3 is the 
 shadow of t 1. 
 
 To find, independently, the point of shadow cast by the 
 corner 1, through 1 draw the ray-line 1 n; and through 2, 
 the point where the corner line 1 10 comes in contact with 
 the upper surface of (7, draw the horizontal two-space line 
 of shadow 2 fn ; and the intersection of these two lines, at 
 8, will be the point of shadow of the corner 1. Draw 3 4 
 equal and parallel to 1 7, and 3 4 will be the line of shadow 
 of 1 7. Now, as 4 is the point of shadow of the corner 7, 
 and 5 is that point in the farther vertical corner line of the 
 block directly below 7, and level with the upper surface of 
 C, it follows that ^ 5 is the line of shadow of 7 5 that part 
 of the farther vertical corner line that is above the block C. 
 Only a small part of this shadow-line, 4 5, is visible. 
 
 The shadows cast on the ground by the platform, and by 
 the end of the block (7, require no farther explanation than 
 may be obtained from the drawing itself. 
 
 Fig. 46 is drawn in semi-diagonal cabinet perspective; 
 and it will be seen, from this figure, that the same principles 
 and rules of shadow apply to semi-diagonal as to diagonal 
 cabinet perspective. 
 
CABINET PERSPECTIVE MISCELLANEOUS. 185 
 
 In this drawing the sun is supposed to be southwest from 
 the spectator, half way between the zenith and the horizon ; 
 and hence the ray-lines are in the direction of diagonals, as 
 indicated by the arrows a, b, c, etc. ; while we assume that 
 the lines of shadow cast by the vertical lines on horizontal 
 surfaces are in the direction of three-space diagonals, as in- 
 dicated by the arrows e,/, g, etc. 
 
 From the principles already explained, it is easy to find 
 the shadows cast on the ground by the lines forming the 
 ends of the steps; and the shadows cast by the block I^on 
 the top of the first step, and on the riser of the second, are 
 in all respects like the shadows cast, in Fig. 44, by the block 
 IB on the ground, and on the vertical side of the block A. 
 It is evident, also, that the line 1 2 (a part of the vertical 
 corner line 6 2) casts its shadow on the top of the second 
 step, from 2 to 3. 
 
 Now the point 1 casts its shadow at the point 3 ; and we 
 find, in the following manner, that the point 4 casts its shad- 
 ow at the point 5. Take the point m in the line 6 2, on a 
 level with the top of the third step, and through it draw the 
 indefinite shadow-line in the direction m 5. Through 4 draw 
 a diagonal ray-line, and the point 5, at which it intersects 
 the line drawn through m, will be the point of shadow cast 
 by the point 4-> in accordance with Rule I. But 5 3 is par- 
 allel to and equal in length to 4 1 from which we derive 
 the following rule. 
 
 RULE IV. The shadow cast, by a vertical line, on a ver- 
 tical plane surface, is parallel to the vertical line itself; and, 
 if the vertical plane surface have sufficient extent, the shadow 
 will have the same length as the line casting the shadow. 
 
 We next wish to find the shadow cast by the part 4 6 of 
 the line 6 2; and to this end we first find, in accordance 
 with Rule I., the point of shadow cast by the corner 6. 
 Thus : we take the point m, in the line 6 2, on a level with 
 the top of the third step, and through m draw a line in the 
 direction (m 7) of the shadows for vertical lines ; and then 
 through 6 draw a ray-line, which we find intersects the for- 
 mer line at 7, thus showing that 7 is the point of shadow 
 cast by the corner 6. 
 
186 INDUSTRIAL DRAWING. [BOOK NO. IV. 
 
 We are next to find the shadows cast by the line 6 h. 
 Referring to Rule II., we find, as 7 is the shadow-point of 6, 
 so 7 9 is the shadow of 6 8. 
 
 Also, referring to Fig. 44, we find, on the principles there 
 explained, that as 7 9, in Fig. 44, is the shadow cast by 
 7 10; so, in Fig. 46, 9 11 is the shadow cast by 8 10. Or, by 
 taking the point s, vertically below 10, and on a level with 
 the top of the fourth step, we may show by the triangle, 
 10 s 11, that the point 11 is the shadow of the point 10 
 (Rule I). 
 
 Again, by Rule II, 11 13 is the shadow of 10 12. Also, 
 according to the principle just referred to in Fig. 44, 13 15 
 is the shadow of 12 14. And again, by Rule II., 15 17 is the 
 shadow of 14 16. 
 
 We have next to find the shadow cast by that portion of 
 the line embraced between 16 and h. As 16 is the farthest 
 point in the line 6 h that casts its shadow on the steps, 
 therefore 16 h must evidently cast its shadow beyond the 
 steps, on the ground. Let us, then, find the point of shadow 
 cast by the corner h. 
 
 Take the point v, fifteen spaces vertically below h, on 
 the ground-level ; draw the three-space diagonal line of 
 shadow through v, and the diagonal ray-line through h; 
 and their point of intersection, at j, will be the point of 
 shadow cast on the level ground by the corner h. Then 
 j i, drawn parallel and equal to h 16, will be the line of 
 shadow of h 16. The other portion of h 6, at the right of 
 16, casts its shadow on the steps, as already shown ; and 
 at the point i its shadow is lost in the shadow of the 
 steps. 
 
 The line j x, equal and parallel to h t, is the shadow of 
 h t; and x u would be the shadow of the farther vertical 
 but invisible corner line t u; but only the portion x z, of 
 this shadow-line, can be seen. Observe, also, that as the 
 ray-line passing through 16 just touches the point 17, and 
 strikes the ground at i, and as r is the point of shadow cast 
 by p, therefore i r, equal and parallel to 17 p, must be the 
 shadow cast on the ground by 17 p. So accurately do all 
 portions of the shadows harmonize with one another, and 
 
CABINET PERSPECTIVE MISCELLANEOUS. 187 
 
 beautifully illustrate the principles of shadows, as deduced 
 from this system of drawing. 
 
 The shadows cast by curvilinear objects may be defined 
 with equal accuracy, on the general principles already ex- 
 plained, with some modifications ; but we have not space to 
 illustrate the subject here. 
 
APPENDIX. 
 
 ISOMETRICAL DRAWING. 
 I. ELEMENTARY PRINCIPLES. 
 
 TSOMETRICAL DsAwixG, or Isometrical Perspective, is based 
 upon the following principles : If a cubical block, as shown 
 in Fig. 1, Plate I., and as seen shaded in Fig. 2, be supposed 
 to be viewed from an infinite distance, and from such a po- 
 sition that the line of vision shall pass through the upper 
 and nearer corner, 1, and also through the lower farther 
 corner, the three visible faces, A, B, G T , of the cube will ap- 
 pear to be equal in measure, the one to the other. Any 
 boundary-line of the upper surface, A, will measure the 
 same as any boundary-line of the face .Z?, or of the face C. 
 Thus the line 1 4 will measure the same as the line 1 6, or 
 4 5, or 6 5, or 1 ,?, or 2 3, etc. And any measure taken on 
 any one line will give the same relative distance when ap- 
 plied to any other line. Hence the appropriateness of the 
 term Isometrical, which is formed from two Greek words 
 signifying equal in measurement. 
 
 The isometrical cube is based upon the geometrical prin- 
 ciple for inscribing a hexagon in a circle. Thus, to in- 
 scribe a hexagon in the circle, Fig. 1, take the radius, 1 4, 
 and, beginning at 2, apply it six times to the circumference, 
 and it will give the points 2, 3, 4-> 5 9 #> 7. Join these points 
 by straight lines, and we shall have the six equal sides of 
 a regular hexagon. Connect the alternate corners of the 
 hexagon with the central point, 1, and retain the circumfer- 
 ence of the circle, and we shall have the isometrical cube in- 
 scribed in a circle. 
 
190 APPENDIX. 
 
 In the isoraetrical cube, which is supposed to stand upon 
 a horizontal surface, while the spectator looks down upon it 
 diagonally, each of the lines 2 S and 2 7 forms an angle of 
 sixty degrees with the vertical line 2 1, and an angle of 
 thirty degrees with the horizontal line 8 9. It will be ob- 
 served, also, that the lines 1 4 and 6 5 are parallel to 2 3 ; 
 1 6 and 4 5 parallel to 27; 34 and 7 6 parallel to 21. In 
 the isometrical cube, therefore, and in all isoraetrical draw- 
 ing, there are only three kinds of true isometrical straight 
 lines vertical Imes, and the two kinds of diagonals as seen 
 in Fig. 1. But unless the diagonal lines form exact angles 
 of sixty degrees with the vertical lines, the drawings made 
 on them will be distorted ; and as these lines can not be 
 made with sufficient accuracy with the pen or pencil, we 
 have had them accurately engraved, and printed in red ink 
 on drawing-paper. By the aid of such paper all difficulty 
 in making accurate isometrical drawings is now removed, 
 as the ruling is a perfect guide for the direction of all the 
 diagonal lines ; and the vertical lines, as will be seen, follow 
 the intersections of the diagonals. 
 
 Let it be observed, also, that the diagonal distance from 
 point to point in the intersections of the diagonal lines is 
 precisely the same as the vertical distance between their in- 
 tersections. Thus, in Fig. 1, the five diagonal spaces from 
 1 to 4-> or 1 to 6, or 2 to 7, etc., measure the same as the five 
 vertical spaces from 1 to 2, or 4 to #, or 6 to 7, etc. More- 
 over, the Isometrical Drawing-Paper is so ruled as to cor- 
 respond, in measure, to the ruling of the Cabinet Drawiny- 
 Paper what is called a space in the one corresponding to 
 a space in the other. From the foregoing explanation, the 
 pupil who is familiar with the principles of cabinet perspec- 
 tive will have little difficulty in making every variety of 
 plane isoraetrical drawings. 
 
ISO1IETRICAL DRAWING. 191 
 
 II. FIGURES HAVING PLANE ANGLES. 
 PLATE I. SCALE OF TWO INCHES TO A SPACE. 
 
 According to the scale here adopted, Fig. 1 represents the 
 outlines of a cube of ten inches to a side, and Fig. 2 is the 
 same, shaded. The student will notice the difference be- 
 tween the mode of measurement here adopted and that 
 used in cabinet perspective. In the latter, also, one face of 
 the cube the front vertical face would be drawn in its 
 natural proportions, as a perfect square. 
 
 Fig. 3 represents a cube of ten inches to a side, having 
 rectangular pieces six inches square and two inches in thick- 
 ness cut from the centres of its three visible faces. Let the 
 pupil compare this drawing with that of Fig. 12, page 1, of 
 Drawing-Book No. II. The cube shows to excellent advan- 
 tage in isometrical drawing. 
 
 Fig. 4 represents an inverted frame sixteen inches square, 
 with comer posts two inches square and six inches in length. 
 
 Fig. 5 is" the same as the English cross bond shown in 
 Fig. 38, page 4, of Drawing-Book No. II. The two figures 
 illustrate, very happily, the two methods of representation 
 cabinet and isometrical. The scale adopted being the 
 same in both cases, the bricks measure the same in both. 
 
 Fig. 6 is the same as the upper part of Fig. 88, page 11, 
 of Drawing-Book No. II. A figure of this kind, evidently, 
 does not show to so good advantage in isometrical as in 
 cabinet drawing. The former is best adapted to the rep- 
 resentation of objects whose side views are nearly equal in 
 proportion. 
 
 PROBLEMS FOR PRACTICE. 
 
 "We would no\v recommend the pupil to draw, in isometrical perspective, 
 all the figures given on the first five pages of Drawing-Book No. II. Let 
 him take the measures as there indicated by the scale, but let him remem- 
 ber that a diagonal space is there to be taken as twice the length of a ver- 
 tical or horizontal space, while in isometrical drawing a diagonal space 
 and a vertical space measure the same, and are to be considered of equal 
 length. 
 
 The pupil would do well to draw all the problems, also, connected with 
 these first five pages. 
 
192 APPENDIX. 
 
 PLATE II. SCALE OF ONE FOOT TO A SPACE. 
 
 "We have here adopted a scale of one foot to a space, al- 
 though any scale whatever, that is most convenient, maybe 
 used. 
 
 Fig. 7 is intended to represent the upper part of a pillar 
 five feet square, around which a moulding of one foot pro- 
 jection and one foot in height is to be placed, even with the 
 top. The shaded portion shows the attachment of the 
 moulding to the pillar. 
 
 Fig. 8 shows the moulding as attached, and concealing 
 from view a portion of the pillar down to the line 8 9 10. 
 Hence the following rule : 
 
 RULE. Any horizontal rectangular moulding attached 
 to a vertical surface obstructs the mew of that surface below 
 the moulding to an extent equal to the extent of the projection 
 of the moulding. 
 
 Fig. 9. The dotted outline represents a cubical block four 
 feet square, while the shaded portion shows a "Wedge cut 
 from it. The sides of the wedge bevel off equally from the 
 sharp edge 1 2, inasmuch as the lines 1 4 and 1 5 intersect 
 the base line 4 & a * equal distances from the point 8. 
 
 Fig. 10. The pillar in this case is of the same size as that 
 seen in Fig. 8 ; but in Fig. 10 the moulding is cut up into 
 three cubical blocks on a side, each one foot square. 
 
 Fig. 11 shows how triangular blocks attached to the top 
 of a column may be represented. The dotted continuations 
 of the lines of the farther two blocks show the concealed 
 points on the column toward which the lines are to be 
 drawn. 
 
 Fig. 12 represents a truncated pyramid, the base of which 
 is surrounded by rectangular mouldings. Observe that the 
 side lines of the pyramid are drawn toward the point x. 
 
 Fig. 13. Very tall four-sided pillars, gradually tapering, 
 and having a flat pyramid at the summit, as at A and .#, 
 are called obelisks. Observe that the apex, in these two 
 obelisks, is in the central vertical line of the pyramid. 
 
 Fig. 14. This pyramid has a rectangular section, one foot 
 in depth, cut from each of the two visible sides of the base. 
 
ISOMETRICAL DRAWING. 193 
 
 and triangular sections cut through the pyramidal portion, 
 so that all except the four edges of the pyramid, and one 
 foot in thickness of its base proper, are cut away. The far- 
 ther edge of the pyramid is concealed by the front edge. 
 
 PLATE III. SCALE OF TWO FEET TO A SPACE. 
 
 Although any object drawn isometrically is supposed to 
 be viewed in the direction of the diagonal of a cube, yet we 
 may view any one face of a cube, or any one side of any 
 rectangular object, from four different positions, and at the 
 same time view it in the direction of some one of the diag- 
 onals of a cube. Thus : 
 
 Fig. 15 represents a block viewed in the direction of the 
 diagonal that passes through the corner 1. We here see 
 the top, front, and right side. 
 
 Fig. 16 represents the same block viewed in the direction 
 of the diagonal that passes through the corner 2. We here 
 see the top, front, and left side. 
 
 Fig. 17 represents the same block viewed in the direction 
 of the diagonal that passes through the corner 4. We here 
 see the bottom, front, and right side. 
 
 Fig. 18 represents the same block viewed in the direction 
 of the diagonal that passes through the corner 3. We here 
 see the bottom, front, and left side. 
 
 Figs. 15 and 16 are viewed from above, and 17 and 18 
 from below. These are similar to tl.e different views of ob- 
 jects in cabinet drawing, as represented on page 1 of Draw- 
 ing-Book No. IV. 
 
 Fig. 19 is the same as Fig. 56, of page 7, in Drawing-Book 
 Xo. II., although the designated scales are different. By 
 adopting the same scale, the figures will measure alike. 
 
 Fig. 20 represents two flights of steps, ascending in dif- 
 ferent directions, and leading to a platform seven feet in 
 height. As each step rises half a space that is, one foot, 
 seven steps are required to reach the platform. 
 
 Fig. 21. The upper roof, A, of this structure is evidently 
 horizontal. The second portion, .Z?, declines downward 
 from the horizontal, as represented by the extent to which 
 the line 1 8 diverges downward from the diagonal horizon- 
 
 I 
 
194 APPENDIX. 
 
 tal line 1 2. If the portion J5 \Vere horizontal, and of equal 
 width on both sides, the line 6 3 would extend to 7, and the 
 line 3 9 would coincide with 7 8. 
 
 Again: if the portion C were vertical, the line 3 4 would 
 coincide in appearance with the line 3 5. The other por- 
 tions of the structure require no explanation. 
 
 PROBLEMS FOE PRACTICE. 
 
 The pupil ought now to find no difficulty in changing all the figures from 
 page 6 to page 11 inclusive, and Figs. 09, 100, 101, and 102, of page 12, 
 of Drawing-Book No. II., into isometrical drawings. If he think this 
 would require too much labor, he would do well to work out the problems, 
 at least, isometrically. 
 
 III. THE DRAWING OF ISOMETRICAL ANGLES. 
 
 The rectangular ruling on the upper part of Plate IV. is 
 designed to correspond precisely in the measure of its 
 spaces that is, in the distance from line to line, measured 
 on the lines to the isometrical ruling on the lower part of 
 the plate. So, also, the ruling on the " Isometrical Drawing- 
 Paper" corresponds, in like manner, to the ruling on the 
 " Cabinet Drawing - Paper." On this basis we have con- 
 structed a " Scale of Angles," w^hich is applicable alike to 
 the drawing of angles on both isometrical and rectangular 
 bases. For inasmuch as any one of the four angles of a 
 rectangular square may be divided into ninety equal de- 
 grees, so also may any one of the four angles of a corre- 
 sponding isometrical square be divided into equivalent iso- 
 metrical degrees, isometrically representing the angles of 
 the rectangular square. Thus : 
 
 Fig. 22. Scale of one foot to a space. In the geomet- 
 rical square A ~B C Z>, the quarter -circle B D is divided, 
 by the full lines which diverge from A, into nine equal 
 parts, each part representing ten degrees at the corner A. 
 The division is best made by the compasses, in the follow- 
 ing manner : 
 
 From D, with the distance D A, cut the curve I> D at sy 
 and from J5, with the same distance, cut the curve at t, 
 
ISOilETKICAL DEAWIXG. 195 
 
 The curve IB D will thus be divided into three equal parts, 
 representing angles of thirty degrees each at the point A. 
 Next divide each of these parts, by the compasses, in to three 
 equal portions, and the entire curve will then be divided 
 into nine equal parts, of ten degrees each. Through these 
 points of division draw lines from A, and extend them to 
 the sides C and D C of the square. Draw a dotted line 
 from A to C, and the angle D A C will be half a right 
 angle that is, an angle of forty-five degrees, while each of 
 the angles D A 10, 10 A 20, 20 A 30, etc., will be an angle 
 often degrees. 
 
 Within the larger square, A B CD, we may count twen- 
 ty-five different squares, each having one of its angles at A; 
 and on the two sides of each of these squares, opposite A, 
 we have the same degrees marked off, by the lines diverg- 
 ing from A, that we have on the sides B C and D C of 
 the larger square. Thus the measure 8 p, on the side of a 
 square of eight spaces, measures an angle of twenty degrees 
 at A, as truly as the measure D 20 measures the same an- 
 gle; and 8 r measures an angle of forty-five degrees, just 
 as effectually as D C measures the same angle. 
 
 Now, inasmuch as any one of the twenty-six rectangular 
 squares that may here be designated exactly measures an 
 isometric square of the same number of spaces to a side, 
 the measures of angles on any one of these rectangular 
 squares may be used to lay off like angles on a correspond- 
 ing isometric square. Thus : 
 
 Fig. 23. It is required to lay off, from the point 1 in the 
 line 1 2, an angle of ten degrees. As the lines 1 2 and 2 3 
 are two sides of an eight-space isometric square, they corre- 
 spond to the two lines A 8 and 8 r (in Fig. 22), two sides 
 of an eight-space rectangular square, and measure the same. 
 From the point 8, on the line A D, take the distance 8 a, 
 and apply it to the isometric square on the line from 2 to 3, 
 and mark the point 5. A line drawn from 5 to 1 will then 
 correspond to the line a A ; and the isometric angle 512 
 will correspond to the angle a A 8, and will represent an 
 angle of ten degrees. 
 
 If from the point 4, in the line 4 3, of the isometric square 
 
196 APPEXDIX. 
 
 of eight spaces, we would lay off an angle of twenty de- 
 grees, lay off $ 7 equal to 8 p; draw a line from 7 to 4; 
 and the angle S 4 7 will be an isometric angle of twenty 
 degrees, the same as 8 A p is an angle of twenty degrees. 
 
 To lay off an angle of twenty degrees from the point b 
 in the line b d, make d c equal to S p, and connect c b. The 
 angle c b d will then be an angle of twenty degrees. 
 
 To lay off an angle of forty degrees at the point g in the 
 line g A, form an isometric square, as g h k n, of five spaces, 
 and from h lay off h i equal to 5 m of the five-space rectan- 
 gular square, and connect g i. Then h g i will be an iso- 
 metric angle of forty degrees, the same as the angle 5 A m 
 is an angle of forty degrees. 
 
 The angles laid off in Figures 24 and 25 may now be eas- 
 ily drawn. It is not necessary, in any case, to lay off a full 
 isometric square to correspond to the rectangular square. 
 It is sufficient to have one side of the isometric square, and 
 enough of the other side to receive the measure from the 
 rectangular square. 
 
 If, in Fig. 19, Plate III., it be required to make 2 1 3 a 
 certain angle, the angle may be laid off in the manner 
 just illustrated. The same with any other angle which it 
 may be required to draw on any isometrical square. So 
 also, in Fig. 21, if it be known what angle the line 3 1 forms, 
 in the real object, with the horizontal line 1 2, or 1 7, the 
 angle may be laid off from the scale, by considering that 
 1 2 or 1 7 corresponds to a portion of the line A C of the 
 scale. The angle 7 1 10 is then an isometric angle of forty- 
 five degrees. So also may the angle 4 & &> if it be known, 
 be laid off frdm the scale, inasmuch as the lines 4 $ and 5 3 
 appear just as they would if they were in a vertical plane 
 that coincided with 1 * 
 
 * Note. The scale shown in Fig. 22 may be applied to the drawing of 
 definite angles in cabinet perspective, when the measures of angles can be 
 taken on that edge of a cabinet square which measures the same number of 
 spaces as the edge of a corresponding rectangular square. 
 
 Thus in the cabinet cube 7?, Fig. 1, page 1, of Drawing-Book Xo. II., 
 which is a cube of six spaces (six inches) to a side, angles at 5 or 3, up to 
 forty -five degrees, maybe taken from the scale and laid ofFon the side 6 4l 
 
ISOMETRICAL DRAWING. 197 
 
 IV. THE ISOMETRIC ELLIPSE AND ITS APPLICATIONS. 
 
 The Isometric Ellipse is the ellipse which is drawn within 
 an isometric square, touching the middle points of its sides, 
 as the three ellipses in Fig. 26, Plate V. The isometric 
 ellipse represents a circle viewed in the position of a side 
 of an isometric cube.* 
 
 Fig. 26. Plate V. Here is represented a cube which meas- 
 ures ten spaces to a side, and on each of its three visible 
 faces is an isometric ellipse which represents a circle drawn 
 touching the middle of the sides of the inclosing square. 
 
 and angles at 4 ant l &> U P to forty-five degrees, may be laid off on the side 
 53. So angles at 3 and 1, up to forty-five degrees, may be laid off on the 
 side 24; and angles at 8 and ^, up to forty-five degrees, may be laid off on 
 the side 1 3. Angles for the front face may be laid off on all the sides of 
 that face. But to lay off an angle at , on the line 3 4 although the meas- 
 ure 3 4 would make the angle 3 6 4 one of forty-five degrees, yet for lesser 
 angles we should be obliged to take such proportions of 3 4 as the measures 
 for angles, on the scale, bear to the entire side of the rectangular square 
 from which the measures are taken. It would be the same when an angle 
 at 4 r 3 should be required to be laid off on the side 12; or an angle at 
 1 or 2 should be required to be laid off on the side 3 4- 
 
 The same principles apply to the laying off of angles in semi-diagonal 
 cabinet perspective. See pages 10 and 11 of Drawing-book No. IV. Yet, 
 for practical purposes in all working drawings, the true angles, or inclina- 
 tions of lines, can generally best be laid off by some known measurements 
 on the objects themselves. 
 
 * Note. In the isometric ellipse, what is called the major axis (greater 
 diameter) is a liule more than once and seven tenths the length of the minor 
 axis ; and it is of the same length as the diameter of the circle which the 
 ellipse represents. Thus, in Fig. 26, the upper ellipse represents a circle 
 whose diameter is s t that is, it represents the outer circle of Fig. 27, Plate 
 VI., while the inner circle of Fig. 27 is the one we are obliged to compare 
 it with in prescribing the rules of practical isometrical drawing. The rea- 
 son of this is that the square within which the circle is drawn is dimin- 
 ished in apparent length of sides by an isometrical view of it; and we 
 adapt the scale of our drawing to the apparent size, and not to the real 
 size. Hence we draw a rectangular square, as A B CD, of Fig. 27, hav- 
 ing the same real length of sides as the apparent length of the sides of the 
 isometric square, A B C D, of Fig. 26 ; and then any lines, divisions, or 
 points of the one may have corresponding lines, divisions, and points in the 
 other. That is, both may be drawn to the same s^ah ; and one may be 
 used to illustrate the other. Thus the two kinds, cabinet and isometrical 
 drawing, perfectly harmonize in measurement. 
 
198 APPENDIX. 
 
 Taking, first, the upper face of the cube for illustration, 
 we see that it is an isometrical square of ten spaces (ten 
 feet) to a side, and crossed by equidistant isometrical lines 
 parallel to the sides. In Fig. 27, Plate VI., we have the rect- 
 angular square A I> CD, of ten spaces (ten feet) to a side, 
 and also crossed by the same number of equidistant lines 
 parallel to the sides. A circle is also drawn within this 
 rectangular square touching the middle points of its four 
 sides, which circle is represented by the ellipse of Fig. 26. 
 Now, as the inner circle of Fig. 27 is a circle of five spaces' 
 radius, the circumference passes through the twelve num- 
 bered points of the intersections of the ruled lines, as there 
 designated from 1 to 12 inclusive. (See page 150.) The el- 
 lipse of Fig. 26 must therefore pass through the same num- 
 ber of corresponding points in the ruling, so that we thus 
 have twelve definite points through which the ellipse must 
 be drawn. The ellipse may therefore, by these aids, be 
 drawn quite accurately by the hand alone, by tracing a 
 symmetrical curve through these twelve points. The same 
 holds good as to the ellipses on the other two visible faces 
 of the cube. 
 
 Any isometric ellipse that represents a circle of ten, twen- 
 ty, thirty, forty, etc., spaces' diameter, may thus have twelve 
 of its points given. But when the ellipses represent circles 
 of other proportions, they must be drawn by the aid of the 
 following principles and methods : 
 
 Scale of Diameters and Axes of Isometric Ellipses. 
 In every isometric ellipse there is. in addition to the ma- 
 jor and the minor axis, what is called the isometric diam- 
 eter. Thus, in the upper ellipse of Fig. 26, 1 7 or 4 10 is the 
 isometric diameter of the ellipse its position being central- 
 ly equidistant from, and parallel to, the sides of the inclos- 
 ing isometric square. The isometric diameter is equal to a 
 side of the isometric square. Hence, when an isometric 
 square is laid down on isometrically ruled paper, the iso- 
 metric diameter of the ellipse that may be drawn within it 
 is also known, and may be located by merely counting the 
 spaces on either of the side lines of the square. 
 
ISOMETEICAL DRAWING. 199 
 
 The proportions which the minor axis, the major axis, and 
 the isometric diameter bear to one another are also known ; 
 and a table of these relative proportions is given on page 
 205. We have also prepared, in Fig. 28, Plate YL, a diagram 
 scale, in which the proportions are given, by measure, for el- 
 lipses of any size up to one whose isometric diameter is not 
 more than thirty-five spaces of the isometric ruling given 
 on the isometric drawing-paper. The scale, however, may 
 easily be extended to any required size. 
 
 Fig. 28. Plate VI. To illustrate. The scale is made in 
 the following manner: Take a rectangular square of any 
 equal number of spaces to a side, as A B CD. From one 
 corner, as D, with a radius D B, cut the side D C extend- 
 ed, in E, and join E A. From the points where the vertical 
 ruled lines from above intersect the line A E, draw hori- 
 zontal lines to the line A D. These thirty-five horizontal 
 lines, thus drawn, measuring from one space up to thirty- 
 five spaces, represent the isometric diameters of that number 
 of ellipses, while to each isometric diameter is assigned its 
 proper major axis and minor axis. 
 
 Thus, if the isometric diameter of the ellipse be the line 
 D E, its major axis will be A E, and its minor axis A D. 
 Again: if the isometric diameter of a required ellipse be 
 twenty spaces, its measure will be the horizontal line from 
 the point 20, on the diagonal line A E, to the line AD; 
 its major axis will be the measure from 20 to A; and its 
 minor axis will be the measure from the. point A, down to 
 the intersection of the line A D, with the horizontal line 
 drawn from 20. Or, what is the same thing, the minor axis 
 will be the measure from the point 20 up to the point t on 
 the line A B. 
 
 Fig. 26. Plate V. Application. Suppose that, in Fig. 
 26, we have merely the isometric square A B C D, of ten 
 spaces to a side, and wish to draw within it an isometric el- 
 lipse touching the middle points of its four sides. The iso- 
 metric diameter being 7 1 (or its equal, B C), we observe 
 that it is represented by the horizontal line on the scale, 
 Fig. 28, from 10, on the diagonal line A E, to the vertical 
 line A D. The line 10 A is, then, the major axis of the el- 
 
200 APPENDIX. 
 
 lipse, and 10 j the minor axis. Therefore, take the distance 
 10 A on the compasses, and, applying it to Fig. 26, lay it 
 off on the line 13 J) equidistant on both sides of the centre, 
 c, and it will give the points 5 and t, the extremes of the 
 major axis. Also take the distance 10 j on the compasses, 
 and, applying it to A 6 r , Fig. 26, lay it off equidistant on 
 both sides of the centre, c, and it will give the points u and 
 v, the extremes of the minor axis. 
 
 In this same manner may the major and the minor axes 
 of the ellipses on the sides of the cube be laid off. And as 
 the diagram scale (Fig. 28) may be easily and accurately 
 made of any required size, on the isometrical drawing-pa- 
 per, the extreme points of the major axis, the minor axis, 
 and the two isometrical diameters eight points in all 
 may be found for any required isometrical ellipse. Through 
 these points the curve may be traced by hand ; or it may be 
 better drawn by the compasses, with great approximate 
 accuracy, in the following manner. 
 
 To Draw the Ellipse by the Aid of the Compasses. 
 
 Fig. 29. Let A IB C D be the isometric square within 
 which the ellipse is to be drawn. Find the extreme points, 
 s t and u v 9 of the major and the minor axis, as before 
 shown. Take the point y so as to make t y equal to t D, 
 and from y describe a curve passing through t and barely 
 touching the sides D C and D A. The other end curve of 
 the ellipse is to be drawn in like manner. 
 
 Make c x equal to c C. Take the distance, t x, by the 
 compasses, and lay it off from A to z. With one point of 
 the compasses in z, and the other extended to 1 or 10, de- 
 scribe the side curve 1 u 10. It should pass through the 
 extremity, it, of the minor axis. In the same manner find 
 the point w above G", and from it describe the side curve 4 v 7. 
 In this manner the ellipse will be so accurately drawn that 
 even in large ellipses the eye can scarcely detect a variation 
 from the true outline.* 
 
 * Note. A more accurate method might be given for drawing smrtll 
 portions, not more than thirty degrees in extent, of the central, side, and end 
 curves ; and the points for describing the side curves would be a \\tt\efar- 
 
ISCirilTBICAL DEAWIXG. 201 
 
 The ellipse of Fig. 29 may be considered the npper end 
 of a vertical cylinder, having an axis, c #, of thirteen feet, 
 and a diameter, 1 7, or 4 10, of ten feet. The side lines, t 3 
 and s 5, are drawn the same as the side lines of vertical cyl- 
 inders in cabinet perspective, while the ellipse for the bot- 
 tom only half of which is visible is drawn within the 
 isometric square, E F G H, in the same manner as the up- 
 per ellipse. The side curves on the cylinder are described 
 from points below w^ by continued removals of two spaces 
 downward, and all with the same stretch of the compasses, 
 w v or w 4; while the end curves are described in a similar 
 manner, from points vertically below y and n. 
 
 Fig. 30 is an ellipse representing a circle of only five spaces' 
 (five feet) diameter, described on the top of a vertically 
 placed block; and Fig. 31 represents one of the same di- 
 mensions on the end of a horizontally placed block. Fig. 
 32 represents a cylinder, six feet in diameter and three feet 
 in length, placed horizontally, the end of it being in the 
 same position as the ellipse on the left-hand side of Fig. 26. 
 Fig. 33 is a cylinder of the same dimensions as Fig. 32, but 
 the visible end of it is in the position of the ellipse on the 
 right-hand side of Fig. 26. The upper part of Fig. 34 rep- 
 resents a vertical cylinder, two feet in diameter and four feet 
 in length, cut from a block two feet square at the end. 
 
 PLATE VII. SCALE OF ONE FOOT TO A SPACE. 
 Fig. 35 represents a hollow cylinder, fourteen feet in ex- 
 treme diameter, four feet in height, and having its w r alls one 
 foot in thickness. It will be seen that its upper outer ellipse 
 is drawn within the isometric square A J2 CD, and that the 
 inner ellipse is drawn within a square one foot within the 
 outer square. The farther inner bottom curve "must, evi- 
 dently, be drawn within a square of the same dimensions as 
 the upper inner square. 
 
 ther from the centre, r, than those we have given, while those for describiig 
 the end curves would be a trifle nearer the centre, c, than those we have 
 given. But this method would require one half of the side curves of the 
 ellipse to be drawn without the aid of compasses ; and the result would sel- 
 dom be as accurate as by the method we have given. 
 
 12 
 
202 APPENDIX. 
 
 Fig. 36 illustrates a method of dividing the ellipse into 
 any number- of equal parts, or of making in it any required 
 divisions. 
 
 From E, the centre of a side, A 13, of the isometrical 
 square which incloses the ellipse, draw ED at right angles 
 to A It* and equal to E A. Connect A D and JB D. 
 From .Z), with any radius, the greater the better, describe a 
 curve, 2 3, cutting the lines B D and A D. Mark this 
 curve according to the divisions required in A c D, one 
 quarter of the ellipse, and through the points of division 
 draw lines from D to the line A 1>. From the intersections 
 of these lines with A J3 draw lines to the centre, c, of the 
 ellipse, and the quarter part of the ellipse will be divided in 
 the same proportions as the curve 2 3 is divided. Here 
 the curve 2 3 is designed to be divided into eight equal 
 parts four parts on each side of the central point E; and 
 hence one quarter of the ellipse is divided into the same num- 
 ber of equal parts. If the same divisions are to be continued 
 throughout the ellipse, transfer the points of division on A J3 
 to the other sides of the isometric square, and from them 
 draw lines to the centre, c, etc. 
 
 Or the method given in Figs. 21, 22, 23, and 24, of 
 Drawing-Book No. IV., may be adopted for all isometrical 
 ellipses. 
 
 V. MISCELLANEOUS APPLICATIONS. 
 
 Fig. 37. To draw an isometrical octagon within an iso- 
 metrical square. And, 1st, when one of the sides of the oc- 
 tagon is to coincide, in part, with the sides of the isometrical 
 square : 
 
 Within the isometrical square lay down the points 1 2 
 and 3 4 (taken from diagram, Fig. 28) for the extremities 
 of the major and the minor axis of the ellipse to be drawn 
 within the square. Then through these points draw lines 
 at right angles to the two axes, and the lines thus drawn 
 
 * If the line E D be drawn from E in the direction of two-space diag- 
 onals, it will be at right angles to A B. 
 
ISOilETKICAL DRAWING. 203 
 
 will be four of the sides of the required octagon. The other 
 four sides will be those portions of the sides of the isometric 
 square lying between the intersections of the first four lines. 
 It will be seen that the ruling of the paper is a perfect guide 
 for drawing lines at right angles to the major and the mi- 
 nor axis. 
 
 2d. When the centre of each of the four sides of the iso- 
 metric square is to be touched by an octagonal corner. 
 
 Draw two lines from each extremity of the major and 
 minor axes to the centres of the two sides adjacent each ex- 
 tremity, and the octagon will be completed. Thus draw 
 lines from 2 to 5 and 6, from 4 to 5 and , from 1 to 7 and 
 8, and from 3 to 6 and 7. Inner lines may easily be drawn 
 parallel to the outer lines. 
 
 Fig. 38 is an octagonal tub or box nine and a half feet in 
 vertical height ; and the sides, one foot in thickness, bevel 
 outward from the top downward. The top is inclosed by 
 an isometrical square of ten feet to a side, and the bottom 
 by a square of thirteen feet to a side. The figure itself will 
 sufficiently explain the method of drawing it. 
 
 PLATE VIII. SCALE OF ONE FOOT TO A SPACE. 
 
 Fig. 39 shows that the half of a cylinder, of three-feet ra- 
 dius, cut longitudinally and vertically, has been taken from 
 a piece of timber measuring at the upper end four feet by 
 six feet. The hollow in the timber is semicircular, but 
 shows here as the half of an isometric ellipse, viewed in the 
 position of the ellipse on the right-hand side of the cube in 
 Fig. 26. 
 
 Fig. 40 represents a semicircular arch of twelve-feet span 
 in its extreme measure from 2 to 3, five feet in length, and 
 with walls two feet in thickness. The arch shows the up- 
 per halves of two isometric ellipses, each viewed in the po- 
 sition of the upper half of the ellipse seen on the right-hand 
 side of the cube in Fig. 26. The outer ellipse is drawn 
 within an isometrical square of twelve feet to a side, and 
 the inner ellipse within one of eight feet to a side. The 
 farther curve is part of an ellipse like the outer front ellipse. 
 The method of drawing the lines for the uniform layers of 
 
204 APPENDIX. 
 
 stones that form the arch is sufficiently illustrated by the 
 drawing itself. 
 
 Fig. 41 is the drawing of a small building, and, according 
 to the scale here given, it is only six by eight feet on the 
 ground, with corner posts only three feet high, the ridge 
 rising two feet above the level of the tops of the posts, and 
 the chimney two feet above the ridge. Observe how the 
 ridge runs centrally over the building, and how the chimney 
 is placed centrally on the ridge, and also equidistant from 
 the two extremes of the ridge. 
 
 Fig. 42 represents a structure having a ground-plan in 
 the form of a cross. The four roofs have sloping ends as 
 well as sloping sides, and are what are called hip-roofed. 
 Moreover, the slope of the ends is the same as the slope of 
 the sides. Thus the point 3 is two feet above the level of 
 the tops of the posts ; and if the end 4 , and the side 4 5, 
 were extended upward, the horizontal distance to the side 
 would be the same as the horizontal distance to the end, 
 being three feet in both cases. 
 
 Fig. 43 is a clustered column, formed of four pieces, each 
 one foot square at the upper end, but each beveling outward 
 below. The moulding around it is beveled also, to corre- 
 spond to the sides of the column. 
 
 With the aid of the foregoing illustrations and the iso- 
 metrical drawing-paper, the student ought now to meet with 
 little difficulty in applying the isometrical method of repre- 
 sentation to all objects that are bounded by straight lines 
 or by regular curves. Irregular surface curves may also be 
 represented isometrically without difficulty by first drawing 
 them on the rectangular ruled paper, from which they may 
 be easily transferred to the isometrical paper, as the spaces 
 on both measure alike. The student would do well to rep- 
 resent, isometrically, all the figures and problems in Draw- 
 ing-Books II., III., and IV. ; and he will generally find the 
 change quite easy from the cabinet to the isometrical draw- 
 ing, if he understands the former. 
 
ISOMETKICAL DRAWING. 
 
 205 
 
 TABLE FOR DRAWING CIRCLES Ds ISOMETRICAL 
 PERSPECTIVE. 
 
 The figures in the columns of Isometrical Diameters denote the lengths 
 of isometric diameters (or sides of isometric squares) ; and the figures in 
 the other two columns denote the corresponding lengths of the minor and 
 major axes. Thus, if an ellipse is to be drawn in an isometric square of 1 
 spaces to a side, the isometric diameter will be 10 spaces in length, the 
 minor axis will be 7.071 spaces in length, and the major axis 12.247 spaces 
 in length. The Table gives the relative proportions of the isometric diam- 
 eters, minor axes, and major axes for all isometric ellipses drawn in iso- 
 metric squares of from 1 to 90 spaces in diameter. The principle holds 
 good whatever measure of length the figures in the columns of Isometrical 
 Diameters represent. 
 
 Ipom. 
 Diam. 
 
 Minor 
 Axis. 
 
 Major 
 Axis. 
 
 L*ora. 
 Diam. 
 
 Minor 1 Major 
 Axis, j Axis. 
 
 Isoin. 
 Diam. 
 
 Minor 
 Axis. 
 
 Major 
 Axis. 
 
 1 
 
 .707 
 
 1.225 
 
 31 
 
 21.920 
 
 37.9G7 
 
 61 
 
 43.134 
 
 74.709 
 
 2 
 
 1.414 
 
 2.449 
 
 32 
 
 22.627 
 
 39.192 
 
 62 
 
 43.841 
 
 75.934 
 
 3 
 
 2.121 
 
 3.674 
 
 33 
 
 23.335 
 
 40.417 
 
 63 
 
 44.548 
 
 77.159 
 
 4 
 
 2.828 
 
 4.899 
 
 34 
 
 24.042 
 
 41.641 
 
 64 
 
 45.255 
 
 78.384 
 
 5 
 
 3.536 
 
 6.124 
 
 35 
 
 24.749 
 
 42.866 
 
 65 
 
 45.962 
 
 79.608 
 
 6 
 
 4.243 
 
 7.348 
 
 36 
 
 25.456 
 
 44.091 
 
 66 
 
 46.669 
 
 80.833 
 
 7 
 
 4.950 
 
 8.573 
 
 37 
 
 26.163 
 
 45.316 
 
 67 
 
 47.376 
 
 82.058 
 
 8 
 
 5.657 
 
 9.798 
 
 38 
 
 26.870 
 
 46.540 
 
 68 
 
 48.083 
 
 83.283 
 
 9 
 
 6.364 
 
 11.023 
 
 39 
 
 27.577 
 
 47.765 
 
 69 
 
 48.790 
 
 84.507 
 
 10 
 
 7.071 
 
 12.247 
 
 40 
 
 28.284 
 
 48.990 
 
 70 
 
 49.497 
 
 85.732 
 
 11 
 
 7.778 
 
 13.472 
 
 41 
 
 28.991 
 
 50.215 
 
 71 
 
 50.205 
 
 86.957 
 
 12 
 
 8.485 
 
 14.697 
 
 42 
 
 29.698 
 
 51.439 
 
 72 
 
 50.912 
 
 88.182 
 
 13 
 
 9.192 
 
 15.922 
 
 43 
 
 30.406 
 
 52.664 
 
 73 
 
 51.619 
 
 89.406 
 
 14 
 
 9.899 
 
 17.146 
 
 44 
 
 31.113 
 
 53.889 
 
 74 
 
 52.326 
 
 90.631 
 
 15 
 
 10.607 
 
 18.371 
 
 45 
 
 31.820 
 
 55.114 
 
 75 
 
 53.033 
 
 91.856 
 
 16 
 
 11.314 
 
 19.596 
 
 46 
 
 32.527 
 
 56.338 
 
 76 
 
 53.740 
 
 93.081 
 
 17 
 
 12.021 
 
 20.821 
 
 47 
 
 33.234 
 
 57.563 
 
 77 
 
 54.447 
 
 94.300 
 
 18 
 
 12.728 
 
 22.045 
 
 48 
 
 33.941 
 
 58.783 
 
 78 
 
 55.154 
 
 95.530 
 
 19 
 
 13.435 
 
 23.270 49 
 
 34.648 
 
 60.012 
 
 79 
 
 55.861 
 
 96.755 
 
 20 
 
 14.142 
 
 24.495 
 
 50 
 
 35.355 61.237 
 
 80 
 
 56.569 
 
 97.980 
 
 21 
 
 14.849 
 
 25.720 
 
 51 
 
 36.062 
 
 62.462 
 
 81 
 
 57.276 
 
 99.204 
 
 22 
 
 15.556 
 
 26.944 
 
 52 
 
 36.770 
 
 63.687 
 
 82 
 
 57.983 
 
 100.429 
 
 23 
 
 16.263 
 
 28.169 
 
 53 
 
 37.477 
 
 64.911 
 
 83 
 
 58.690 
 
 101.654 
 
 24 
 
 16.971 
 
 29.394 
 
 54 
 
 38.184 
 
 66.136 
 
 84 
 
 59.397 
 
 102.879 
 
 25 
 
 17.678 
 
 30.619 
 
 55 
 
 38.891 
 
 67.361 
 
 85 
 
 60.104 
 
 104.103 
 
 26 
 
 18.385 
 
 31.843 
 
 56 
 
 39.598 
 
 68.586 
 
 86 
 
 60.811 
 
 105.328 
 
 27 
 
 19.092 
 
 33.068 
 
 57 
 
 40.305 
 
 69.810 
 
 87 
 
 61.518 
 
 106.553 
 
 28 
 
 19.799 
 
 34.293 
 
 58 
 
 41.012 
 
 71.035 
 
 88 
 
 62.225 
 
 107.778 
 
 29 
 
 20.506 
 
 35.518 
 
 59 
 
 41.719 
 
 72.260 
 
 89 
 
 62.933 
 
 109.002 
 
 30 
 
 21.213 
 
 36.742 
 
 GO 
 
 42.426 
 
 73.485 
 
 90 
 
 63.640 
 
 110.227 
 
TTIIVEKSITY 
 
Scale of two inches to a space. 
 
 PI. I. 
 
Scale of one foot to a space. 
 
 PI. II. 
 
UNIVERSITY 
 
Scale of two feet to a space. 
 
 PI. in. 
 
Scale of one foot to a space. 
 
 PI. IV. 
 
 J80 
 
 150 
 
 7 
 
 I'igj. 2 
 
 Vr. /V 
 
 FT 
 
 14 
 
 1! 
 
 is 
 
 2ta 
 
UHIVERSIT7 
 
Scale of one foot to a space. 
 
 PJ. V 
 
Scale of one foot to a space. 
 
 PL VI. 
 
 \ 
 
 -^T 
 
 s 
 
 * 
 
 \/ 
 
 -/ 
 
 \, 
 
 T/ 
 
 7 
 
 
 K 
 
Scale of one foot to a space. 
 
 PI. VII. 
 
OP THE 
 
 UHIVEKSIT7 
 
Scale of one foot to a space. 
 
 PI. VITT. 
 
UNIVEESITY OF CALIFOENIA LIBEAEY, 
 BEEKELEY 
 
 THIS BOOK IS DUE ON THE LAST DATE 
 STAMPED BELOW 
 
 Books not returned, on time are subject to a fine of 
 50c per volume after the third day overdue, increasing 
 to $1.00 per volume after the sixth day. Books not in 
 demand may be renewed if application is made before 
 expiration of loan period. 
 
 FEB 28 192t 
 SENT ON ILL 
 
 NOV 1 2 2002 
 
 U. C. BERKELEY 
 
 20m-ll,'20 
 
B 15869