Darnell HttioecBUy Uibcarg 3tl|aca, N»ui Qnrk Carpenter Estate Cornell University Library T 351.5.W74 The drawing guide :a KKlSiiiiiimmilii 3 1924 021 896 356 Cornell University Library The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924021896356 THE DRAWING GUIDE; A MANUAL OF INSTRUCTION IN INDUSTRIAL DRAWING, DESIQKED TO ACCOMPANY THE INDUSTRIAL DEAWING SERIES. WITH AN INTKODUCTOEY AETICLE ON THE PRmCIPLES AND PRACTICE OF ORNAMENTAL ART. By MARCIUS WILLSON, Auxnoa OF "sohool and family bebies or eeadebb," "manual of oe#eot lessons,*' ETC., ETO. NEW YOEK: HAEPEE & BROTHERS, PUBLISHERS, FEANKLIN SQUARE. "" 1877. Entered according to Act of Congress, in the year 1873, by Haepeb & Brothers, In the OfBce 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 Paint 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, Hadiation, and Continuity, 26 : — Prop. XI. Of Conventional Kepresentations 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. Mediaival or Gothic Ornament, 39 :— XVII. Renaissance Ornament, 41: — XVIII. Elizabethan Ornament, 42: — Modern Ornamental Art, 42. PART 11. 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, 5 1 : — 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, &1 :— 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 :— Parallelopipeds— 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, 96 : — Divisions of the Cube, 96 :— Solid Frets, 97 :— Timber Framing, 9T -.—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, 105 : — Problems,10G : — ^Post-and- Bail 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. DRAWING-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: — Kims 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 Bam, 183 :— Crown-wheel, 135 : — Eatchet Wheel, 136 :— Windlass, with Spokes, 138. DKAWING-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,MG: — Series of Platform Structures, 147 : — Pro6/em, 149. III. Cylindrical Objects In Vertical Positions 149 I. Ellipses on Diagonal Bases 1.^0 II. Ellipses on Bectangular Bases 152 Ellipses in Vertical Positions, 1 53 : — Rule, 155 : — Problems, 1 55 : — 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 169 IV. Semi-diagonal Cabinet Perspective 171-178 V. Shadows in Cabinet Perspective 179-187 APPENDIX. ISOMETRICAL JDRAWINO. 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 Platesj I. to VIII. inclusive 207-221 PREFACE. In presenting to the public the first four numbers of The Industeial Drawing Seeies, 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 PEEFACB. to the required angle. We then proceeded to prepare a somewhat elaborate work on Isometrieal 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 iis that a still more easy system of industrial drawing, more nearly approaching linear perspective in ap- pearance, and equally practical with isometrieal 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 Jo 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 Isometrieal 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. Vn portion ; and how easy it is, witli tlie 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 rnle 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 mannei", 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 o-reat 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 VI 11 PEEFACE. excellent partterns 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 Pee- sPEcnvE, as illustrated in Drawing-Books Numbers 11., 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 oi 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 ; PEEFACE. IX 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 bi'idges ; cylinders, solid or hollow, entire, or various- ly cut and divided, and in both horizontal and vertical po- sitions ; arches, both pointed and semicircular 5 braces and brackets, both plain and curvilinear ; solid quarterfoils aad 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 No. 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 PEEFACB. 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- selves — in 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. Maecitjs Willson. ViNELAND, N. J., June 5, 1873. PART I. PRINCIPLES AND PRACTICE OF ORNAMENTAL ART. I. 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 ornariientation 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 oi 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 Departm&t 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 AET. 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 Difierent Nations, and in Difierent 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 afibrded 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 Stafibrdshire 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 INDUSTEIAL DRAWING. Stated by competent judges that, through the establishment of Art Museums and Schools of Design, and the influences cxertied 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 "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 all competitors. An incident bearing upon this point is re- lated by the State Director of Art Education for Massa- chusetts, to the efiect that, " some years ago a class of thii>- teen young men spent all their leisure time in studying ORNAMENTAL ART. 11 drawing, and that now, at the time of writing, every one of them holds some impoi'tant 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 di-aughtsmen 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 : JTow 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, we commend this whole 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. 11. 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 rnle and compass ; but the ornamentation shows how far the architect was, at the same time, an artist. Peoposition I. — A Caedinal Peinciple. 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 OENAMENTAL ART. 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. — Op 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 DEAWING. ful, 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 appropiiate 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 oak, on the contrary, of angular outlines, is representative of firmness and strength. It may break, but can not bend. Peoposition III. — Of Fiem and Unbeoken, 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 pyi-amid, 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 sec Proposition XI. ORNAMENTAL AET. 21 Pkoposition IV. — Of Consteuction and Decoeation. Construction should he decorated; but 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 INDUSTEIAL 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. Peoposition 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 taten 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 Geometeioal CoNSTEtrcTiON. All surface ornamentatio?i should be based ujjon a geo- inetrical construction. Whatever the pattei'n 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 litfle 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 OENAMBNTAL AKT. 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. Peoposition Vn. — Of Methods of Subface Decoeation. 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 hy general lines ; the intermediate spaces may then be filled 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 efiect by the rep- etition of a few simple elements. " 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 I5e 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 neuti'al 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. Peoposition VIII. — Of Pkopoktion in Oknamentation. As in every perfect work 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 equal 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 proiDortions were differ- ent ; but in cnch the several parts were in just proportion the one to the other, and to the Avhole. 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 OENAMENTAL AKT. 25 the most beautiful which the uneducated eye does not readily detect. Peoposition IX. — Op Haemony and Conteast. Where great variety of form is introduced, harmony con- sists in the proper balancing and contrast of the straight, the incUtied, 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 'K — n Ys X > ^k_ -7) ( )C )C ) o O r, — )e^ O O O. O O o o ^ — y^ — r- O O O O o o 26 INDUSTEIAL DRAWING. Straight lines; so the gable contrasts admirably with the curved window-head and its perpendicular muUions. 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 eyp 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 ai'e 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. ♦ Peoposition X. — Op Disteibution, 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 aU 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. OKNAMENTAL AKT. 27 Peoposition XI. — Of Conventional Repeesentations or Natueai Objects. Flowers, or other natural objects, should 9iot be used as or- naments, but, instead thereof, we should use conventional rep- resentations founded upon them, svfficiently 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 natuealistic 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 whei-e it owes its oi'igin 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 AET. 29 might suppose they borrowed their pattei-ns 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. Tlie 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 j)archment, 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. Constructim. — 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 tof 30 INDUSTRIAL DEAWING. 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 j and they in- diiferently colored the leaves of the lotus green or blue. Iir. 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. Tet 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. OENAMENTAL AET. 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 INDTJSTEIAX 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. 19. 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 Gi-ecian 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 appi'oxima- ORNAMENTAL AET. 33 tion to nature : they were also arranged inartistically, he- 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 loaf over leaf — a prfnciple 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. Vir. 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 34 ISTDUSTEIAL DRAWING. 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. Vlir. 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- OENAMKNTAL AET. 35 pliclty 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 difiers 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 tliem ; 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 ORNAMENT. Although the Turks and the Arabians have the same re- ligion, yet, being of difierent national origin, their art rep- resentations are, as might be expected, somewhat different. The architecture of the Turks, as seen at Constantinople, 36 INDUSTKIAIi DRAWING. 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 OE MOOHISH 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 religions faith and diversities of na- tional character. The main difierences 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 alope w^s 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 OENAMENTAL AET. 37 grow 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 liad the appearance of a congeries of paintings, incrusta- tions, mosaics, gilding, and foliage ; and nothing could be more splendid and brilliant than the efiects 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. Xr. 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 calcukted 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 INDUSTEIAL DEAWING. tides in agate from Bombay, mirrors from Lahore, marble chairs from Ajmeer, embroidered shawls, scarfs, etc., from Cashmere, cai'pets 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 185]. 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, "Woe to them who dwell in a l^ouse 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 OENAMENTAL 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 OENAMENT. 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 orhament ; 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. MEDUEVAL OR GOTHIC OENAMENT. The high-pitched gable and the pointed arch, with a con- sequent slender proportion of towers, columns, and capitals, are the leading characteristics of mediaeval or Gothic archi- 40 INDUSTEIAL DRAWING. tecture, -which came into general use in Europe in the thir- teenth century. Mediseval 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 rounde(f 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 efiect 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. ORNAMENTAL AET. 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. EENAISSANCE 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 lienaissance 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 /resco. In true fresco. 42 INDUSTEIAL DEAWING. 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 AngelOj that fresco is fit for men, oil painting for women, and the luxurious and idle. XVIII. ELIZABETHAN OBNAMENT. 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 compartpients, often filled with foliage, or coats of arras, 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 MODEEN OENAMENTAL AET. 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 OEKAMENTAL 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 heen 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 OF INDUSTRIAL DRAWING. DRAWING-BOOK No. I. I. MATERIALS AND DIRECTIONS. 1. FoK Paper to draw on, use " WiUson's Cabinet Draw- ing -Paper'''' for Drawing-Books Nos. I., II., 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 semi-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 JVb. 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,HHII, and HHHH: H be;ng soft pencils, and HHHH very hard and fine. There are also what are called the jEfep^fe 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 DEAWING. [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 effect 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 circles 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 off 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 Blaclcboard Mcercises in connection with drawing on paper, every school in which industrial drawing is taught should be provided with a blackboard of convenient size, having j?we red lines painted on it, both vertically and horizontally, at right angles to one another, MATEEIALS AND DISECTIONS. 49 and two inches apart. Any careful painter can prepare a board in tliis manner. The board should not be varnished. The red lines drawn on the board will interfere very little with the iise of the board for ordinary purposes. The 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 Ko. 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. Ait'hongh free-hand drawi?iff. 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 elaliorate 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 DRAWING. [bOOK NO. I. perience. "While, therefore, we i-ecommend free-hand draw- ing in elementary exercises, and also in all portions of copies or original designs which can he 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 effect of a drawing in diagonal Cabinet Perspective (Books IL, HI., 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 ia bold relief from the paper ; and if there are any inaccura- cies in the perspective, they will be readily detected by the unnatural appearances which they will thus be made to present. 10. K 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. 1 1. 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. I^°° 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 LINES AND PLANE SUEFACES. 51 II. 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, oi- 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 INDUSTKIAL DEAWING. [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 formisd by suspending a weight hy 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 X, No. 2, is also a right angle. An acute angle (e) is an angle that is less than a right angle ; an obtuse angle (w) is an angle that is greater than a right angle. Aplane 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 are plane figures called squares. A rectilinear plane figure is a plane figure bounded by straight lines. A equare is a plane figure that has four equal sides and STRAIGHT LINES AND PLANE SUBFACES. 53 four right angles. Nos. 3, 4, and S 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 Meastcrement. 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 othei-wise 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 a 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 lai'ge is No. 5 ? (Four inches square.) How much area, or surface, does it contain ? (Sixteen square inches.) How large is No. 6 ? (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 hy the breadth, and the prod- uct will be the area. PROBLEMS FOE 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 INDUSTEIAL DRAWING. [bOOK NO. I. 3. Draw a rectangle of four by five inches. What is its area ? -jIbs.' 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 black- 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. 6. How much larger is No. 7 than No. 6 ? {Four times larger.) From the forgoing 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 five 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, 'bvit 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 are nine : if we wish to make one sixteen times as large, we quadruple the lengths of the sides ; because 4 times 4 arc sixteen: and so on to the square of any given number. %^^ Let the teacher explain more fully, if necessary, wliat is meant by the square of a number, and especially when that number represents the length of a given line. STRAIGHT LINKS AND PLANE SUEFACES. 55 PROBLEMS FOE PRACTICE. 1 . Draw a square similar to No. 3, but nine times as large. 2. Draw a square similar to No. i, 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. .■>. 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. K polygon is a plane figure having many sides and many angles. The terra, is generally applied to a plane figure of more than four angles and four sides. Freerhand 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 tlie 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 Bva&W 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 1 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 mDtrSTEIAL DRAWING. [BOOK NO. 1. I®'' In 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 tlie product will be the area, in measures of the primary erect squares. Rule A is only a special application of Rule I. {Reason for the rule. — 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 1 diagonals to a side? {Ans. 98 square inches.) What is the area of .a diagonal rectangle of 5 by Y diag- onals? (Ans. TO square inches.) Let the pupil carry out the same system on the black- board. Lesson V. — No. 1 is an erect cross, repi'esenting one thin piece, 2 inches by 8 inches, laid at right angles across another piece 2 inches by 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 horizont.-xl 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 STEAIGHT 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 a; 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 b, 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 Mackboarcl. PAGE TWO. Lesson VI. Two-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. Tliey should be copied, without the aid of a ruler, until they can be drawn with tolerable accuracy, and with facility. At b lines ai"e first drawn as at a; and then lines are drawn in- termediate between them ; c is first drawn the same as b, 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 squax'e inch. 02 58 INDUSTRIAL DRAWING. [bOOK NO. I. No. 4. What area is embraced within the dotted square ? Then how much is embraced within the portion a ? No. 5. What area is embraced within tlie 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 aeu*e-angled triangle, because each angle is less than a right angle. (See Lesson III.) The portion marked a in No. 5 is called an o5iu«e-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. 1. What area is embraced in each of the parts a of No. V ? In the central rectangle h ? 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 h (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 tliis is one of the kinds of shading very appropviate for that purpose. No. 13. What is the area of each of the rhombuses mark- ed a? (See N"o. 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? (SeeNo.9.) Of each of the star points marked 6.^ (See No. 4.) Of the central square c .? Of the whole octagon ? PROBLEMS FOE PRACTICB. 1 . Draw a rectangle similar to the dotted rectangle No. 4, but four times as large. (See Elementary Peinciple, 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. ] . 5. Draw a rhombus similar to No. 7, but containing only one fourth the area of No. 7. G. 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. 13 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 diiferent position. The area of No. 1 can easily be counted up, when it is seen that each of the 60 INDUSTRIAL DEAWING. [bOOK NO. 1. 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 hy 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. 1 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, PEOBLEMS FOE PEACTICE. 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 ontsida of the smaller ? STRAIGHT LINES AND PLANE SURFACES. Gl 2. Draw a rectangle similar to No. 4, bnt 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. 1 1. Free-hand drawing of problems similar to the foregoing on the hlackioard. Lesson VIII. 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 or, 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 i, 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, «, 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 ], the length of another side, gives the product 62 INDUSTEIA.I, DRAWING. [bOOK NO. I. ], 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 ai-ea ? 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. 1. The area of the space included within the dotted figures 1, 2, 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 a, b, c, and d? Of the inner dotted square ? Of the whole octagon ? No. 10. What is the area of the four rhombuses a, b, c,df Of the star g f Of the whole octagon ? No. 11. What is the area of the four rhombuses a, b, e, d? Of the four parts e, /, g, h ? Of the central square Ic 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. PEOBLEMS FOE PEACTICK. 1. Draw, on the drawing-paper, a three-space diagonal sqaare that shall contain 9 times the area of No. 3. 2. Draw a rhombus similar to TSo. 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 mlh 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. .'!. 4. Draw a rhombus similar to No. 8, but containing four times the area of No. 8. 5. Draw an octagon simUar to No. 9, but containing four times the area of No. 9. STRAIGHT LINES AND PLANE SUEFACES. 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 g, 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-Jiand 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. I. 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 aiid lower sides. No. 8 is a double interlacing square. First trace lightly. PEOBLEMS FOE PEACTIOE. 1 . Draw a star similar to No. 2, but nine times as large. What will be 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 a, b, c, d of No. 3 place a star like No. 2 within its own diagonal square. STEAIGHT LINES AND PLANE SUEFACBS. 65 3. Draw a pattern made up of hexagons similar to those of No. 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 shaded squares shall contain four times the area of those in No. 5. Complete thq several hexagons in a manner similar to b. 5. Draw a pattern composed wholly of figures like No. 6. The addition at a will show how the several figures are to be connected. Shade all like No. 6, or use different tints of India ink. 6. 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 X., and tJie problems, on the blackboard. Lesson XI. — 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 hasis 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 nsed 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, b, 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 ? 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 comhina- tion of dodecagons is an original pavement pattern taken from a Roman church in the Byzantine period of Roman histoiy. 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. PEOBLEMS FOE PEACTICE. 1. Draw a pattei-n formed of figures like No. 1, 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 JCI., and the problems, on the blackboard. PAGE rOUK. For convenience, we now drop the method of grouping the examples under the head oi 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 SUEPACES. 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 warp, 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 INDUSTEIAL DEAWING. [bOOK NO. I, PROBLEMS FOE PEACTICB. 1 . Draw a pattern similar to Fig. 1, but with strands of only half the width of those there represented. The rectangles shown will be of only Iiulf 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. 3, 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. b. Draw a pattern similar to Fig. 7, and of the same proportions, but containing an area i 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 hlackhoard, PAGE FIVE. Figs. 11, 12, 13, and 14 are plain border patterns; 11 and 12 being forms of the Grecian //e«, to be noticed hereafter. Pigs. 15,16, 17, 18, 19, and 20 are representations of flat braidof 3,4,5,7,9, and! 1 strands. In Fig. 16 the ne.^t move- ment is to turn the a strand upward, break it down on the dotted line 1 2, and pass it over b and under e. Then break the strand d downward on the line S 4, and pass it under a, and so on continuously. In Fig. 17 the movement is continuously from the out- side, over one andunder 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 the 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 Avhite 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 hlachhoard take Figs. 13,14, 15,16,17, and 18. They may be shaded slightly with col- ored chalks, so as to make the interlaoings jilaiu. PAGE SIX. Fig. 24 is the simple generating form of the Grecian sin- gle fret, or meandet — 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 afterwai-d 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 the 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.natiiral form in architecture, as solids. Fig. 24 requires no directions for drawing it. Fig. 1 ] , on 10 INDUSTEIAL DEAWING. [bOOK NO. I. page 5, is the same as this, with the exception that Fig. 1 1 has an interlacing band running centrally through it. The ruler may he 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 5 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 , STEAIGHT LINES AND PLANE SUEFACES. 71 been found in the ruins of Pompeii. The two bauds 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 hoiizontal 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 S, then the four inner diagonal points marked 2. Also take the eight ray points in a sim- ilar manner. Ti-ace lines very faintly from the outer to the inner points; then trace an inner set of lines equidistant 72 INDUSTRIAL DBA WING. [bOOK NO. 1. 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 Avliite, 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 difierent foTms, 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 Avhich every such decoration will impress the beholder, leaving nothing further to be desired within the scope of tlie 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. CUEYED LINES AND PLANE SURFACES. IS III. CURVED LINES AND PLANE SURFACES. PAGE EIGHT. A curved lihe 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. 1. At ^ 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 h are uniform in position an4 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 h are struck. Fig. 2 is formed of the same pattern curve used in difier- 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 T"the same form of curve is used. Let thie 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 i are, each, only half of the pattern curve, and are described from the points 1 and 2. Fig. 4. At B 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 14: INDUSTRIAL DE AWING. [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 jB, 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 B. 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 C, 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 difierent 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 difierent radii in the same pattern, with- out destroying that gracefulness of foim which is requii-ed 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 difierent forms and proportions will answer the re- quirements oi 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 a: by a cui"ved line that pleases the eye. If, now, we can draw a line exactly like it on the other side of a b, we CtlETED LINES AND PLANE SUEPACES. 15 shall have a figur^ of harmonious form, whether axhe the most graceful line, by itself, that could be drawn, or not. But if a a; should be drawn of the most graceful form pos- sible, and a y just as graceful in itself, but differing from a 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 hoi'izontal or vertical lines of the paper. Let 1, 2, S, 4, and S 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. 1 1 represents the harmonious outlines of a leaf form. Observe that the border lines of the leaf pass through the points 1, 2, S, 4, 6, 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 2, each two spaces ; the; points S, each three spaces, etc. In 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. 1 1 is equally harnipnious in proportions. The pupil 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, e, g, 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, h, . PROBLEMS FOE PEACTICE. 1. Draw the lightly dotted outlines of a cube of the same dimensions as Kg. 40 ; and then represent the cube as it would be after taking vertical sections from its four corners, and from the centres of its four sides, each four inches square at the ends. Pirst 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 6, of Book No. I., 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. 6. Draw the same frame standing on one end, with the edge of the frame fronting the spectator, Blacleboard Mcercises, — Any of the figures and problems on page 5. E 08 IND0STBIAL 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 jprojection of the moulding, whether measured horizontally by the dis- tances 1 2 and 5 6, or diagonally by the distances 2 3 and Jf 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,ybMr inches in projection, conceals from view the tioo 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 the diagonal vertical surface immediately below 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 IS 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 JABINET PEESPECTIVK — PLANE SOLIDS. 99 rigbt-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 ax-e 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. 63 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 INDUSTKIAL DEAWIKG. [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. 64 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 1.) 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 FOE PEACTICE. 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 fiimly 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 FOUU INCHES TO A SPACE. All the figures on this page, except Fig. 64, 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 thir(;y-two inches square. The base bevels upward, and the top bevels downward to the same extent. As the Jine a 6 is drawn diagonally from the corner. a, thirty-two inches, so must the line c d, drawn from the corresponding corner c, run diagou.illy in the same di- CABINET PEESPECTIVE — PLANE SOLIDS. 101 rection ; and its real length is thirty-two inches, hut 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 ff 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 cl And c and cl must be the same distance apart as n and 9n. 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 tlie 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 INDUSTKIAL 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, 4, 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 S 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. PEOBLEMS FOE PRACTICE. 1 . Draw a figure similar to Fig. 56, bat forty-four inches square at tlie bottom and top ; and with the central vertical support twenty inches high from o to X, 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 Tig. 6.5, but whose extreme side measures from corner to comer 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. I' AGE 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,h 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 m j. The centre, c, of the base of CABINET PEKSPECTIVE — PLANE SOLIDS. 103 the pyramid, must evidently be at the intersection of the central lines o p and /* 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 oi the base. The apex of the pyramid must be vertical to the point c. Fig. Ql. 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 x, 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, f, 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 a; 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, of the pyramid is drawn so as to be directly above the central diagonal line, g h, of the base. Fig. 69 is the same as Fig. 68 ; bat 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 ofi" 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 INDUSTEIAL 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 gr, 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, JD, called the shaft, and the support or base on which it rests, called the pM'estal. The pedestal is di- vided into three parts : A, the base ; JB, 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 heigh t 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 halfoia 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 as, 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 PEESPECTIVE— PLANE SOLIDS. 105 To get the full effect of the drawings on this page, view them as directed on page 50. Let the pupil view his own drawings in the same manner. PROBLEMS FOE PRACTICE. 1 . Diaw a base and pyramid similar to Kg. 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 : lase, 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 rig. 72. 4. Draw a fret like Tig. 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. Blackboard Eaercises. — 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 thiuty-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 E 2 106 IKDUSTEIAL DRAWING. [bOOK NO. 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. PEOBLEMS FOE PEACTIOE. 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 6, and we CABINET PEKSPECTIVE — PLANE SOLIDS. 107 sliall have tlie diagonal square 3 J/. G 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 ^ 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 sui'- 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 FOE PKACTICB. 1. Draw a post-and-rail fence similar to Fig. 80, but witli the square rails inserted into tlie posts diagonally. Let the posts be four inches wide ia 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 width. 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 above 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 Fig. 85, but the pickets two inches wide, twenty inches long, made of stuflf two inches thick, and placed four inches apart. 2. Draw the representation of a chest similar to Fig. 89. Suppose it to be thirty-eight inches square ; the body sixteen inches high j 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 Mcercises. — 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 12 ox S Jf, of three feet. The octagonal form is the same as that of No. 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, i, 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 DEAWING. [bOOK NO. II. ed within the No. 13 just referred to, or the same as the star figure No. 1 1 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 VIIL, page 2, of Book No. L Six inches within the series of the outer front lines is another series, whose distance from the outer lines is regulated by the points 1, 2, S, 4- 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 ^ and 5, do- sci-ibed from y, 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 Avheel, 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 PEESPECTIVE — CUEVILINEAR 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 tlie drum. A smaller shaft, or axle, T" I^ 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 cen^i-e of the end of the shaft, while h is the centre of the front face of the drum, and c the centre of the farther face or side. At ^ 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 iji 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, W, with the spokes in their places. It-is here drawn separately for the purpose of showing the methgd 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 a, on the axis, with the same radius, describe the farther circumference, only half of which is vis- ible. 2d. Now, 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 cylindi-ical surface of the drum. We therefore take a d. ] 40 INDUSTEIAL DEAWING. [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 Clearer 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 3 4 and 7 8, giving us the equidistant points f, h, j, m, o, q, 9, and 10. From/, as a starting-point, measure off any desired distance, &sfg, and take gj\.% 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 k, from m to n, 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, k, n, p, etc., measure Off three inches backward on the dotted line, as g 11, i IS, etc., for the width of the face of each spoke. Connect the points that are directly opposite, as 11 p, g IS, 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 dr.awn with a radius of twenty-four inches, giving ten inches for the length of the spokes, and passing through the points c, 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), passintr CABINET PEESrECTIVE — CUEVILINEAE SOLIDS. 141 through the points 5, S, 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 l,v w, x y, etc., which give us the side boundary-lines of the ends of the s])okes. 6th. The lines 16 17, 11 18, IS 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 A, B, and G 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 bo 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 between 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. III. width, tlie 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 i, and* the outlines of the concealed top of the farther brace. 6th. As we have the point 1 at the bottpm of the nearer brace, we now know that fhe 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, Avhile at the same time accuracy of representation may* bo adhered to, provided the very plain principles of this mode of representation are clearly understood. DRAWING-BOOK No. IV. CABINET PERSPECTIVE —MISCELLANEOUS APPLICATIONS. I. DIFJ'ERENT DIAGONAL VIEWS OF OBJECTS. In the preceding two books on Cabinet Pei-ispective, 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 oi 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 helow, 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 arc viewed. It will be observed that the bracket,/, is best 144 INDTJSTEIAL DRAWING. [cOOK NO. IV. represented in its true position from the point of view here taken. 3d. Lower left-hand view. — In division G the objects are represented as viewed from below, 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 view. — In division D the objects are represented as viewed from above, and at the the left; tlie 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 otFier 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, e, is inverted, for the purpose of showing to the best advantage the inside curves ; and the brackets f, g, 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 founda- CABINET PERSPECTIVE — MISCELLANEOUS. 145 tioiis upward to the extent of three feet. Observe that all the measures in Fig. 2 are the same, according to the scale 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 J/., 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 feet, 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--^\2i.u 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. — all, 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 iheir 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 INDUSTEIAL DEAWING. [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 4, and 1 8 the same as S 7, etc. peoblbm: fok pkactice. (This may he omitted bi) the younger pupils.^ It is required to draw, in cabinet perspective, a building similar to the ground-plan of Kg. 3, but of the following dimensions : Iiength of building {1 B), 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 6, 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 S, four feet; and let one extend from the comer above 3, 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 towai'd 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 Fig. i. 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 PBESPBCTIVE — 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 sliding-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. 1 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 1, 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 J? is a cabinet view of the fircTplace, 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 difierent 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 JT. Rising six inches from IT, we step onto the platform G or J£; four steps, each of six inches' rise, then bring us onto the platform I or JF; eight steps, each of six inches' rise, then bring us to the level of the platform D. Now it would be diflicult to ascertain whether E is high- er than D or not, if we did not know, or find out, the height of the box-like frame-work on which the flooring, £!, 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 D, where there is no projection. The frame-work (without the floor) rises four feet above the corners S 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 4, at the intersections of the 148 INDUSTEIAL DEAWING. [bOOK NO. IV. dotted lines. Now if we connect these four points, as shown 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 J). If we extend ^ 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 w, 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 S, and we shall have points through which the lower side lines of the projecting floor will 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, JS, and after- ward to put on the flooring. As it requires eight steps, of six inches each, to rise from the platform I %o D, D is four feet above I; and as the frame-worlc of JE'is 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 JS. The farther end of -Z? is even with the farther side of the frame- work of M The posts a and h 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 rigJU-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 JE 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 of^. CABINET PERSPECTIVE — MISCELLANEOUS. 149 At C are the outline walls of a Lnilding, the outer walls being two feet in thickness. At L is seen the opening into a pit, extending downward below tlie 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. PEOBLKII FOE PEACTICE. Draw a figure similar to Pig. 8, but with the following measures : Let the frame on which the flooring of E rests he twenty-seven feet square. Height of frame from corners S 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 D, 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 D, 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 Pig. 8, omitting C. 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 ISTDrSTEIAL DRAWING. [BOOK NO. IV. were directly in front of the spectator. This is the case with all the cylindrical objects represented in Book !N"o. 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- ins: illustration will show. I. ELLIPSES ON DIAGONAL BASES. PAGE FOUR.— SCALE OF TWO INCHES TO A SPACE. [N. B. — The description of Pig. 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, J/., 8, 10, H, 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 'iiagonal square within which it is drawn, and also pass- ■'ng through the corresponding points 2, Jf, 8, 10, H, 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, 2."), 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 bo accurately drawn, they will pass through the points 2, J/., 8, 10, H, 16, 20, 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 JB 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 x, and afterward at a, strike the curves inter- secting on the right at m. Then, with one point of the compasses in rn, 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 w 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 i^'will give the correspond- ing point 9. The diagonal distance from 9 to 21 is also 152 INDUSTEIAL DEAWING. [bOOK NO. IV. equal to the distance from B to a — the distance first laid off on the compasses. Ill 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 close approxi- mation to the true side curves. The end 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 ibr 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. in., or after the following plan. II. ELLIPSES ON RECTANGULAR BASES. Fig. 10. Suppose timt the cube, Fig. 9, should bo so viewed that neither tlio 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 ^ 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 di-awn on the top of a cube thus viewed would show as a perfect el- lipse, whose lesser* diameter, 13, 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 PERSPECTIVE — 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, J^, 6, 8, 10, 12, H, 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 JB C D (or, place in JB and extend to 18) ; then, still continuing the point in A, with the other point strike the central vertical line, 12 prolonged, in w. Then, with one point of the compasses in w, and the other extended to 12 above, describe the curve 9 12 15. Then, with one point in G, 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 S 21. The two side curves will thus be drawn very accurately. JEnd 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 Kg. 10 may be called upper vertical rectangular perspective. But if the spectator were horizontally to the rigfit G2 154 INDUSTRIAL DRAWING. [bOOK NO. 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 Kule, p. 85) as applicable to all drawings in cabinet perspective. Fig. 11. In this figure we have drawn the ellipse within the cabinet square AH C D,\n all respects like the ellipse of Fig. 10. The point y is the point from which we de- scribe, with the compasses, the curve S 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 JE 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 D, r, and C. FD and F C will then be the vertical side lines of the cylinder ; and through the points D, r, and G must be drawn the half of an ellipse correspond- ing in all respects to the curve F w F. As the rectangle A JB G D represents the square inclosing the top of the cylinder, so the rectangle F F Q 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,vi\ the same manner that we described the ellipse within the square A JB G D. The visible portion, Cj^-Z), of this ellipse, whose side curve is described from the point jo, we have drawn with a firm line ; the other portion, D t G, 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 G D oi Fig. 10 corresponds to the rhombus A B C 2> of Fig. 9, therefore the line G B oi Fig. 10 measures the same (in cabinet perspective) as the diagonal line (7 -B of Fig. 9; and 12 and Z> ^ of Fig. 10 the same as 12 and ^ ^ 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 eectangulae 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. PEOBLEMS FOE PEACTICE. 1. Draw a vertical solid cylinder after the plan of Kg. 12, whose diame- ter shall be thirty-six inches, and whose axis, or length, shall he 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 Kg. 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 166 INDUSTEIAL 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 cnbe 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 £ C D he 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 1 I 2 ; and x for describing the other side curve S i ^.; 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 r,kl,v p, and ij, each eight inches within the. ellipse first drawn, and complete the rectangular cabinet square E F Q height 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 g, 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 h I, and that at the points o r and V p they appeav to be double the thickness of e J and k I, CABINET PEKSPECl'IVE — MISCELLANEOUS. 157 as they would natumlly 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 B C D. A D ov B C will then represent forty-eight inches also. 2d. Describe the ellipse 1 2 3 ^ within this square, and touching the centres of its four sides. This ellipse will represent the top circle of the ring. The central line, ^ 2, 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 off a second square, E F G H, four inches below the top square, and within it describe the ellipse 6 6 7 8, 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 S 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,^ B C -Z?, of the requisite dimensions, and within it describe the ellipse, 12 34, for the upper outer circumference of the ring. 2d. Take the cabinet square, ^ J'' 1 T, which may be extended to within two inclies of the line G B with accuracy. 5th. From the point 2, the centre of the side B E, with tlie radius 2 F, describe the indefinite curve iFp. 6th. From the point F, with the radius F 1, describe the curve 1 n. Vth. From the point n, with the radius n 1, 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 about the point \?-Z 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 w, 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- »iter 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 oi Fig. 36. The thickness of the ring is two inches, and its width four inches. As the inner front ellipse is drawn within the semirdiag- onal square, C -2? 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 oif that 1V6 INDUSTEIAL DEAWING. [bOOK NO. IT. 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 4,3 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 vertical 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, te 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 S and 6 7, must be made to correspond to the inner curves ; and as i 6 or ^ 7 is a measui-c of four inches, so the point 5 must be taken four inches above the point 4- As the depth of the arch, 6 8 ov 2 10, is six inches, so the depth, wherever measured on a horizontal line, as a 6, 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 h. 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. PEOBLEMS FOE PEACTICK. ] . Draw a pointed arch, similar to Kg. 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; thiclcness of walls, 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 PEESPECTIVE — MISCELLANEOUS. lYY in semi-diagonal perspective^ust as the side view of Fig. 38 is in semi-di- agonal perspective.] 2. Draw a partial cylinder, somewhat similar to Fig. 33, 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 he 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. 89. 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 lialf the span, 8 9; or, as the vertical line 2 7 is equal to 2 8, the rafters are at an angle of forty-five degjrees. 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 « < w is a right angle, s,v is the hypothenuse; and if we add togeth- er the squares of s i and t v, and extract the square root of the sura, we shall find that the length of s u 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 INDUSTEIAL DEAWING. [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 FOK PKACTICE. [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 comer (as t) where the plate joins the post ; and the bottom of each brace, on the inside (as at f), is to be sixty-four inches below the comer (as t) 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 H. 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. 6. Draw a brace for a corner conesponding to G, and also one for a cor- ner corresponding to 7, below G. 7. 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 PEESPBCTIVE — MISCELLANEOUS. 179 V. SHADOWS IN CABINET PERSPECTIVE. As cabinet perspectLve 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 sliadows cast by objects. But where it is de- sired to give greater artistic effect to cabinet drawings, the system of perspective here adopted will 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 bo 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, E., S.,W., N. Fig. 41. This figure represents a square vertical pillar standing upon the level surface of the earth, Avhile 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, a, c, e; and that the ray a, just touching the corner j?, strikes the earth at the point 9. Then it is evident that all rays, such as g A, ij, etc., that strike the corner line 1 2, will project shad- ows upon the earth between the points 2 and 9 ; and hence 180 INDUSTRIAL DRAWING. [bOOK NO. IV. the line 2 9, 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 J).; but the shadow would become shorter and shorter as the sun rose vertically toward the zenith; and when at the zenith the shadow oi 12 would be contracted to a single point at 2. Hence two lines are required to define the shadow cast by the vertical line 1 2: the oite, a 9, called the r(iy-li?ie, giv- ing the direction of the ray of light that barely touches the corner 1; 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 hy any given point: Rule I. — Draw a rat-line, indicating the direction of the sun's 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 shadovhline 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 ^ 9 is the shadow of the vertical line 1 2, and that 9 is the shadow-point of the point i. In a similar manner it is found that the line 4 8 would be the line of shadow cast by the vertical corner line S 4, if all the pillar except its corner line S ^ were transparent ; also that 6 7 is the line of the shadow cast by the corner lino CABINET PEESPECTIVE — MISCELLANEOUS. 181 5 6. The line 8 7 must therefore he the line of shadow cast by the line 3 S. 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 surf ace, parallel to itself; and the shadoio has the same representative length as the line easting the shadow. Thus 9 * is parallel to and equal to 1 S; 8 7 to S 5; 7 11 to 6 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 a, 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- "* is the line of shadow cast by 3 J/.; m n the line of shadow cast by S 1; and w « a part of the line of shadow cast by 1 t; and that 2 n would be the shadow cast by i ^, if i ^ could cast a shadow. Also, 6 IS is the line of shadow cast by the corner line 5 6; H 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 INDUSTEIAL DRAWING. [bOOK NO. IT. on the ground by vertical lines are represented as bearing in tlie direction of northwestern three-space diagonals, as shown by the arrows a,f,h,k; 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 coi-ner 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 i 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 S 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 ^ 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 8, drawn parallel to and equal to c d, must be the line of shadow ofcd; 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 5 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. Now the shadow cast by the portion g i is evidently i w, 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 JB, in vertical position, and standing at right angles to one an- other upon the horizontal surface of the earth. The snn 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,f,u,v,x; while the shadows cast by vertical lines are in the direction of horizontal semi-diagonals, as indicated by the arrows c, d,f. Both blocks cast shadows upon the earth ; while the block JB casts a shadow, ad- ditionally, upon a part of the vertical surface of the block A. In accordance with principles already explained, the line ^^ is the shadow cast by the vertical corner line 1 2. Hav- ing g as the point of shadow cast by the corner 1, we know that the shadow cast on the earth by the line 1 7 would extend to the left from g, 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 <7 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 ^, it follows that the straight line 10 7 must cast il s shadow in a straight line from 9 to 7, from which we de- duce the following rule, applicable to all similar cases. Rule in. — 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- 6 being the shadow of S ^, and 5 6 being the shadow of 5 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 INDUSTEIAL DRAWING. [bOOK NO. IV. shadow of i r must extend from 6 to m, parallel to i r, and equal in length to i r (Rule II.). Or the point of shadow 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 G and D are similar in position to the blocks A and B of Fig. 44 ; but they here rest on a horizontal plat- form. In this figure the lines of shadow 6 a and a 1 are in all respects similar to the lines of shadow g 9 and 9 J?" in Fig. 44 — except that only that part of a i (viz., a 6), which touch- es the vertical side of C, is a shadow-line. Here b a is the shadow of 8 r ; a 6 is the shadow of r t; and 6 5 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 G, draw the horizontal two-space line of shadow 2 m ; and the intersection of these two lines, at S, will be the point of shadow of the corner 1. Draw 3 4- equal and parallel to 1 7, and S 4 will be the line of shadow of 1 7. 'Now, as 4 is the point of shadow of the corner 7, and 6 is that point in the farther vertical corner line of the block directly below 7, and level with the upper surface of G, 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 G, 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,f, 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 J" 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 £ 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 & level with the top of the third step, and through it draw the indefinite shadow-line in the direction in 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 ^ in accordance with Rule I. But 5 5 is par- allel to and equal in length to ^ 1 — from which we derive the following rule. Rule IV. — The shadow cast, hy 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 ^ 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 &. 186 INDUSTRIAL DKAWING. [bOOK KO. IV. "We are next to find the shadows cast hj the lino 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 H. And again, by Rule 11., 15 17 is the shadow of H 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 /*. 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 A 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 %i; 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 ^, therefore i r, equal and parallel to j?7^, 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. ISOMETEICAL DRAWING. I. ELEMENTARY PEINCIPLES. TsoMETEiCAx, DRAWING, 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 neai'er corner, 1, and also through the lower farther corner, the three visible faces, A, -B, (7, 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 H, 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 2, 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, 6, 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 isometvical cube, whicli 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 i, and an angle of thirty degrees with the horizontal line 8 9. It will be ob- served, also, that the lines 1 ^ and 6 6 are parallel to 2 8; 1 6 and J^ 5 parallel to 2 7; S ^ and 7 6 parallel to 2 1. In the isometrical cube, therefore, and in all isometrical draw- ing, there are only three kinds of true isometrical straight lines — vertical liaes, 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 i^ame as the vertical distance between their in- tersections. Thus, in Fig. 1, the five diagonal spaces from 1 to Jf, or 1 to 6, or 2 to 7, etc., measure the same as the five vertical spaces from 1 to 2, or ^ to 3, 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 Drawing- Paper — what is called a space in the ■one corresponding to a space in the other. From the foregoing explanation, the pupil ■vfrho is familiar with the principles of cabinet perspec- tive will have little difficulty in making every variety of plane isometrical drawings. ISOMETEICAL DKAWING. 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 corner 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. PEOBLBMS FOB PBACTICE. We would now recommend the pupil to draw, in isometrical perspective, all the figures given on the first five pages of DrawingrBook No. II. Let him talie 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 TOOT TO A SPACE. We liave 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 rectangidar moulding attached to a vertical surface obstructs the view of tJmt 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 oflf equally from the sharp edge 1 2, inasmuch as the lines 1 ^. and 1 5 intersect the base line .^ 5 at equal distances from the point 3. 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. 1 1 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 towai'd 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 Ji, 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. ISOMETEICAI, DEAWING. 193 and triangular sections cut through the pyramidal portion, sb that all except tlie 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 in.— 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 8. We here see the top, front, and left side. Fig. IV 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 sid&. Fig. 18 repi'esents the same block viewed in the direction of the diagonal that passes through the corner S. "We hero 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 the 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 "No. 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, ^, of this structure is evidently horizontal. The second portion, Ji, declines downward from the horizontal, as represented by the extent to which the line 1 S diverges downward from the diagonal horizon- I 194 APPENDIX. / tal line 1 2. If the portion H were horizontal, and of equal •width on both sides, the line 6 3 would extend to 7, and the line S 9 would coincide with 7 8. Again : if the portion C were vertical, the line 3 ^ would coincide in appearance with the line 3 6. The other por- tions of the structure require no explanation. PKOBLBMS FOE PKACTICB. The pupil ought now to find no difficulty in changing all the figures from page 6 to page 11 inclusive, and Figs. OS), 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 probkms, 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," which is applicable alike to the drawing of angles on both isometrical and rectangular bases. For inasraucli 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, isometncally 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 B, the quarter- circle J3 JD \s 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 Z>, with the distance D A, cut the curve B D at s; and from B, with the same distance, cut the curve at t. ISOMETEICAL DRAWING. 195 The curve H 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, into 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 B G and Z> C of the square. Draw a dotted lino from A to C, and the angle DAG will be half a right angle — that is, an angle of forty-five degrees, while each of the angles B A 10, 10 A 20, 20 A SO, etc., will be an angle of ten degrees. Within the larger square, A B G D,yf& 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 G and D G 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 JD 20 measures the same an- gle; and 8 r measui'es an angle of forty-five degrees, just as effectually as I) G 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 J), take the distance 8 a, and apply it to the isometric square on the line from 2 to 3, and mark the point S. A line drawn from 5 to 1 will then correspond to the line a A; and the isometric angle S 1 2 will correspond to the angle a A 8, and will represent an angle of ten degrees. If from the point ^,in the line 4 3, of the isometric square 196 APPENDIX. of eight spaces, we would lay off an angle of twenty de- grees, lay offS 7 equal to 8 p; draw a line from 7 to J^; and the angle 5^7 will be an isometric angle of twenty degrees, the same as 5 ^ jo is an angle of twenty degrees. To lay off an angle of twenty degrees from the point h in the line h d, make d c equal to8 p, and connect c h. The angle eh 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 h, form an isometric square, as g h k n, of five spaces, and from h lay off A « 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 Am 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 S a, certain angle, the angle may be laid off in the manner just illustrated. The same with any other angle which it may be I'cquired to draw on any isomctrical 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 ov 1 7 corresponds to a portion of the line A C oi the scale. The angle 7 1 10 is then an isometric angle of forty- five degrees. So also may the angle 4 3 S, if it be known, be laid off from the scale, inasmuch as the lines ^ 3 and 5 3 appear 2VisX as they would if they were in a vertical plane that coincided with 1 S* * 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 B, Fig.], page 1, of Drawing-Book No. II., which is a cube of six spaces (six inches) to a side, angles at 5 or S, up to forty-five degrees, maybe taken fiom the scale and laid off on the side G 4; ISOMETEICAL 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 cuhe 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 If and 6, up to forty-five degrees, may be laid off on the side S 3. So angles at 3 and 1, up to forty-five degrees, may be laid off on the side 2 4; and angles at S and 4i np 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 6, 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 stick proportions of 5 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 .^ or 5 should be required to be laid off on the side i .? ; or an angle at 1 or S 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 little 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. 2T, 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 ^^B C D, 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 X>, ot 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 scale ; 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 AJBGD, 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 ov 4. lOis 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 VI., 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 ^ -B CD. From one corner, as D, with a radius 3 J3, cut the side D C extend- ed, in £J, and join £IA. 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 CD, 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, P G), 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. Tlierefove, take the distance 10 A on the compasses, and, applying it to Fig. 26, lay it off on the line B D equidistant on both sides of the centre, c, and it will give the points s and t, the extremes of the major axis. Also take the distance 10 j on the compasses, and, applying it to A C, 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 JSUipse hy the Aid of the Compasses. Fig. 29, Let A Ji C D he the isometric squai-e within which the ellipse is to be drawn. Find the extreme points, s t and u v, of the major and the minor axis, as before shown. Take the point ?/ 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 G. 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, u, of the minor axis. In the same manner find the point w above C, and from it describe the side curve ^v7. 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 ghea for drawing small portions, not more than thirty degrees in extent, of the central, side, and end curves ; and the points for describing the side curves would bo a little yiir- ISOMETEICAL DEAWING. 201 The ellipse of Fig. 29 may bo considered the upper end of a vertical cylinder, having an axis, c 2, 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, JE 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 « or M 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 TOOT TO A SPACE. Fig. 35 represents a hollow cylinder, fourteen feet in ex- treme diameter, four feet in height, and having its walls one foot in thickness. It will be seen that its upper outer ellipse is di-awn within the isometric square A JB G JD, 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, c, than those we have given, while those for describing 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 B, of the isometrical square which incloses the ellipse, draw E D at right angles to A B* and equal to E A. Connect A D and B I). From Z>, with any radius, the greater the better, describe a curve, 2 3, cutting the lines B J) and A D. Mark this curve according to the divisions required in ^ c -Z>, one quarter of the ellipse, and through the points of division draw lines from D to the line A B. From the intersections of these lines with A B 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 S \s divided. Here the curve 2 3 \% 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 onAB 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 J^. (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 lire E Dha drawn from E in tlie direction of two-space diag- onals, it will be at right angles to A B. ISOMETEICAL 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 ^ to 5 and 6, from 4- to 5 and 8, from 1 to 7 and 8, and from S 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 8, 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 suiBciently 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 6, and the side J^ 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 isometrieal method of repre- sentation to all objects that are bounded by straight lines or by regular curves. Irregular surface curves may also be represented isometrieally without difficulty by first drawing them on the rectangular ruled paper, from which they may be easily transferred to the isometrieal paper, as the spaces on both measure alike. The student would do well to i-ep- resent, isometrically, all the figures and problems In Draw- ing-Books II., Ill, and IV.; and he will generally find the change quite easy from the cabinet to the isometrieal drawr ing, if he understands the former. ISOMETEICAL DRAWING. 205 TABLE FOR DRAWING CIRCLES IN ISOMETRICAL PERSPECTIVE. The figares 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. Isom. Minor Major Isom. Minor Mnjor Isom. Minor Major Diam. Axis. Axis. Diam. Axis. Axis. Diam. Axis. Axis. 1 .707 1.225 31 21.920 37.967 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 77.1,59 4 2.828 4.899 34 24.042 41.641 64 45.255 78.384 a 3.53G 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 4.9.^0 8.573 37 26.163 4,5.316 67 47.376 82.058 8 5.657 9.798 38 26.870 46.540 08 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 63.740 93.081 17 12.021 20.821 47 33.234 57.563 77 54.447 94.306 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 35.3.55 61.237 80 66.569 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 58.690 101.634 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 40.805 69.810 87 61.518 106.6,53 28 19.799 34.293 58 41.012 71.035 88 62.225 107.778 29 20.506 35.518 69 41.719 72.260 89 62.933 109.002 30 21.213 36.742 60 42.426 73.485 90 63.640 110.227 Scale of two Inches to a space. PI. I. Scale of one foot to a space. PI. II. Scale of two feet to a spaco. PI. III. Scale of one foot to a space. n. IV. n 90 80 70 60 50 ol — ■ rt ^ / / / 24 / ~ / / / / ^ ^ / / 22 A ^ / / i 1 / \ / / /l40| 20 / ,^ y / s. 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Y CO — Ll/ ,, ^ cr A — h ± lii^ 1_ __, _ li^?-| 1- 'J. u. L. J_ 1. i_ r-'Ki - Scale of one foot to a space. n. VII. Scale of ono foot to a space. PI. viir.