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LIBRARY OF CONGRESS. 
 
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 UNITED STATES OF AMERICA. 
 
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WHITE'S INDUSTRIAL DRAWING 
 
 THE SCIENCE AND ART 
 
 MODEL AND OBJECT DRAWING 
 
 AND FOR SELF-INSTRUCTION OF TEACHERS AND ART-STUDENTS 
 
 IN THE THEORY AND PRACTICE OF DRAWING 
 
 FROM OBJECTS 
 
 \ 
 
 ^^4 BY 
 
 tf' LUCAS BAKER 
 
 n 
 •art^fHastfr 
 
 FORMERLY SUPERVISOR OF DRAWING IN THE PUBLIC SCHOOLS OF THE 
 CITY OF BOSTON 
 
 ILLUSTRATED ^ ^,^ , ,333 1 
 
 Copyright, 1883, by 
 IVISON, BLAKEMAN, TAYLOR, AND COMPANY 
 
 PUBLISHERS 
 
 NEW YORK AND CHICAGO 
 

CONTENTS. 
 
 PAGE 
 
 NTRODUCTION . . . . . . . .5 
 
 Terms and Definitions 11 
 
 Of Limits 12 
 
 Of Extension 12 
 
 Quantities of the First Degree. — Lines . . .12 
 Quantities of the Second Degree. — Surfaces . . . . 13 
 
 Quantities of the Third Degree. — Volumes 16 
 
 Quantities of the Fourth Degree. — Inclination . . . 17 
 
 Words Denoting Position and Relation 17 
 
 Orthographic Projections . . 18 
 
 How TO Read x\pparent Forms .26 
 
 The Diascope 29 
 
 Analysis of Apparent Forms .30 
 
 The Drawing of the Rectangle or of the Square . . . 32 
 
 The Apparent Forms of Angles 34 
 
 The Drawing of the Cube 36 
 
 Method of Drawing the Hexagon and the Hexagonal Prism . 39 
 
 The Circle 44 
 
 3 
 
4 CONTENTS. 
 
 PAGE 
 
 Position of the Apparent Diameter. — Illustration . . .48 
 Apparent Form of Circle seen Obliquely . . . . . 49, 50 
 The Recession of the Apparent Diameter. — Illustration . .52 
 Apparent Forms of Parts of Circles. — Illustration . . . 55 
 
 Method of Drawing Circular Objects 57 
 
 Rules for Drawing the Cylinder 57? 58 
 
 Apparent Widths of the Bases 59 
 
 The Position of the Major Axis of the Ellipse ... 60 
 
 Bands and Rims 66-68 
 
 The Law of Rims Demonstrated 70 
 
 The Drawing of Ellipses 71 
 
 Drawing the Triangle and Triangular Frames . . , . 72 
 
 The Frame-Cube . 74 
 
 Drawing the Single Cross 75 
 
 Drawing the Double Cross 76 
 
 Drawing the Frame-Square . . . , . . . . 80 
 
 The Use of Diagonals 81 
 
 The Cube 82 
 
 Groups of Rectangular Solids and Triangular Prisms . . ^ '^Z 
 
 Groups with Hexagonal Prism. — Vases 84 
 
 Light, Shade, Refleci^ed Light, Cast Shadow, and Reflections . 86 
 
 Light and Shade on the Cube %^ 
 
 Light and Shade on the Cylinder 90 
 
 Shading the Sphere, Cone, Etc 91, 92 
 
 Methods of Shading 93 
 
 Reflections 95 
 
 APPENDIX 99 
 
INTRODUCTION. 
 
 HE tendency of the American people to study art 
 marks an era in our intellectual life. Students 
 ^ of art multiply rapidly : art-schools are well 
 filled, and private teachers are in great de- 
 mand. All branches of art are receiving atten- 
 
 tion, and especially the industrial department. 
 
 There are two sources of art-instruction, — the 
 teacher, and nature. There are also two methods of 
 practice, — working from copies, and working from 
 nature. Multitudes of private pupils do nothing but 
 ^Qx copy the work of others, and consequently they never 
 
 acquire the power to produce original work themselves. The two 
 methods may be combined, but nature must always be regarded as 
 the great instructor. We can do no greater service to our pupils 
 than to prepare them to learn from nature, to open their eyes and 
 minds to the harmonies and melodies which she has in ample store 
 for them. 
 
 There is no department of public instruction better adapted to 
 the development of the powers of observation than drawing from 
 objects. 
 
 5 
 
6 MODEL AND OBJECT DRAWING. 
 
 The art-student, in progressing through the various branches of 
 his study, is soon confronted with the necessity of making for himself 
 original drawings from objects. He can not long follow copies, and 
 depend upon them for guidance : he must read forms independently, as 
 he would read a book ; and he must give his own rendering of them. 
 
 At this stage he is presumed to have acquired a ready hand in 
 drawing from the copy, and to be in possession of some knowledge 
 of Plane Geometry. Thus prepared he enters upon a tour of investi- 
 gation, not unlike the explorer of a new country. He must note all 
 the facts presented to his observation, and deduce all the laws dis- 
 coverable by his understanding. 
 
 To the student it is emphatically a field of discovery. His eyes 
 must be opened to new facts, which have been hitherto unnoticed by 
 him. His method of seeing is to be changed from the casual and 
 accidental to the accurate and discriminating method which penetrates 
 and comprehends the subtleties of the apparent forms of objects, and 
 of light, shade, shadow, reflections, and color. Every teacher of art 
 knows that the principal part of his work is teaching his pupils to 
 see and how to see. The pupil begins with little knowledge of the 
 apparent forms of objects, and with no habit of observing them. 
 This knowledge must be acquired, and the habit of seeing must be 
 formed. This is the only foundation for true progress. In this 
 respect, to draw is to knotv ; and not to know, is not to be able to draw. 
 
 The subject of Object-Drawing has a basis of fact throughout. 
 There is no guess-work ; mathematical precision pervades the whole ; 
 every question can be settled by reference to fundamental prin- 
 ciples. 
 
 Model-drawing is the best possible preparation for sketching from 
 
INTRODUCTION. 7 
 
 nature. The student graduating from the study of models goes fully- 
 equipped to the delineation of natural scenery or of architectural 
 objects. Without this preparation the results of his efforts would 
 be uncertain, and accurate only by accident. It furnishes the scien- 
 tific basis for free sketching ; and without it, and an understanding of 
 its principles, no artist can count himself secure in his work. 
 
 The first part of model-drawing, viz., that relating to apparent 
 forms, is closely related to Descriptive Geometry ; while the second 
 part, viz., light, shade, shadow, and reflection, falls within the prov- 
 ince of the fixed laws of light. The third division, viz., color, has 
 also its fixed limitations and conditions : hence the whole field of our 
 subject falls within the domain of science, and only partially within 
 that of taste. 
 
 The models used in this department are geometrical forms, and 
 objects based on these, as the sphere, cylinder, cone, cube, prism, 
 pyramid, plinths, vases, rings, etc., supplemented by numerous 
 objects of utility and beauty, whose forms bear close relationship 
 to geometrical types. To become thoroughly familiar with the prin- 
 ciples of the whole subject should be the aim of every student of pic- 
 torial or industrial art; for thus only will the way become clear for 
 any future advancement. 
 
 Model-drawing also possesses an educational value that ought to 
 commend it to every true teacher. The general tendency of the 
 course of instruction in the public schools, aside from drawing, is 
 toward the development of the world of ideas, and not toward the 
 development of the power of observation. Indeed, so strongly is this 
 the case, that the mind is drawn away from the real, visible, and tangi- 
 ble, to the contemplation of the unseen and ideal. Thus our pupils 
 
8 MODEL AND OBJECT DRAWING. 
 
 come to belong to the class, that, ''having eyes, see noty We say, 
 then, that the discipline derived from the practice of this subject 
 tends to put the pupil in full possession of his faculties. 
 
 Emerson says, "The study of art is of high value to the growth 
 of the intellect ; " and Goethe called drawing " That most moral of 
 all accomplishments," saying, '' It unfolds and necessitates attention, 
 and that is the highest of all skills and virtues." 
 
 Attention makes the scholar, the want of it the dunce. 
 
 It is said that the artist knows what to look for, and what he sees ; 
 and it is almost equally true, that the untrained in model and object 
 drawing do not know what to look for, or what they see. It is for 
 these reaso7is that our subject has a high educational utility over and 
 above all considerations of its industrial or commercial value. Model- 
 drawing in particular, and drawing in general, should be well taught in 
 our public schools, in order to secure a more complete development of 
 the mental powers. 
 
 Moreover, this subject opens to the pupil new sources of enjoy- 
 ment ; as it unfolds new powers,, and extends the area of his mental 
 vision, while it increases the value of his labor in life. The power 
 he derives from it enters into all skills and labors, and adds another 
 segment to the arc of his being. 
 
 The student has presented to his mind, for his comprehension, a 
 multitudinous series of facts relating to form, light and shade, shadow 
 and reflection. The whole series must be appropriated and digested, 
 and made a part of the student : he must assimilate the whole 
 if he would attain to a complete mastery of the subject. The best 
 method for the teacher to follow, is to place before his pupils a single 
 model, and then, — first, to lead them carefully to recognize the several 
 
INTRODUCTION. 9 
 
 facts, relations, and principles involved in its apparent form ; secondly, 
 to note the distribution of light, shade, shadow, and reflection on the 
 same ; and, thirdly, to deduce the general principles which the observa- 
 tion and comparison of these appearances are found to establish. 
 
 It is not enough merely to set the pupil to work on the models. 
 His powers of observation are undeveloped, and need directing. At 
 the same time, the rules should be deduced by the pupil, and not 
 furnished ready-made by the teacher. The pupil should be taken 
 into partnership with the teacher in the analysis of the subject, and 
 taught to write down his own conclusions. He will thus appropriate 
 and assimilate the facts for his own use, so that he will feel he is in 
 full possession of them. 
 
 The practice in all branches of our school instruction should be 
 to lead and direct the pupil's minds in all their investigations, rather 
 than to impose upon them a burden of arbitrary dogmatism without 
 regard to their power of assimilation. 
 
 In the practice of model or object drawing we place the objects 
 before us in suitable positions, and proceed to draw them with pencil, 
 brush, or crayon, in line, light, and shade, or in color, as we may 
 choose. The method is wholly a freehand process throughout : we 
 use no instruments but the pencil, brush, stump, and^ rubber ; and 
 we proceed upon certain general and fundamental principles which 
 are to be noticed hereafter, to make the representation upon whatever 
 surfaces we may have chosen for that purpose. Model and Object 
 Drawing, then, is a study for the artist as well as for the mechanic. 
 
 In Perspective Drawing, which is really a branch of Descriptive 
 Geometry applied to the representation of objects as they appear, we 
 make a drawing of an object or objects wholly or mainly with instru- 
 
lO MODEL AND OBJECT DRAWING, 
 
 ments for measurement and execution, following certain fixed and 
 determined laws of intersection of lines and planes, from certain 
 assumed or fixed data or measurements, upon whatever plane surface 
 we may have selected for that purpose. 
 
 It is a mechanical and not a freehand process : hence it is not 
 the ordinary method followed by the artist in securing his *' views," 
 but it is generally the method employed by the architect to render 
 apparent the results of his inventions and combinations. 
 
 It will be seen, therefore, that, in practice. Object Drawing and 
 Perspective Drawing are essentially different. But, however differ- 
 ent the practice in these two departments may be, there are certain 
 fundamental principles common to both ; and they are in complete 
 harmony, the one with the other. If there seem to be contradic- 
 tions, they are apparent only, and not real, and are owing to a want 
 of understanding of the subjects under consideration. 
 
Model and Object Drawing. 
 
 TERMS AND DEFINITIONS. 
 
 HE terms used in drawing, so far as they relate 
 to mathematical quantities, should be identical 
 with those used in Geometry ; and they should 
 be given the same value. 
 
 It may be useful, therefore, to insert here 
 a partial analysis of geometrical quantities, with 
 their definitions, for the use of those who are not other- 
 wise familiar with the same. 
 
 A class of beginners should be taught to distinguish 
 and to define geometrical quantities as a preparation for 
 model or perspective drawing. Let them begin with the four kinds 
 of geometrical quantities, and learn to refer any quantity to its own 
 class : this is the first step in getting at the correct definition. 
 
 In Geometry there are four different kinds of quantities, some- 
 times called quantities of different degrees. 
 
 First, Quantities of Length : all lines belong to this degree. 
 Second, Quantities of Surface : all surfaces belong to this degree. 
 Third, Quantities of Volume : all solids belong to this degree. 
 Fourth, Quantities of Inclination : all angles belong to this degree. 
 
12 MODEL AND OBJECT DRAWING. 
 
 The degree, or kind, to which any quantity belongs determines the 
 first word or words of the definition of that quantity. The last part 
 of the definition refers to the manner of limitation or boundary. 
 
 OF LIMITS. 
 
 Points limit lines, lines limit surfaces, surfaces limit volumes ; or, 
 to reverse the statement, we should have these limitations in the 
 following order : volumes are limited by surfaces, surfaces are limited 
 by lines, and lines are limited by points. 
 
 Or again : quantities of the first degree, or kind, are limited by 
 points ; quantities of the second degree are limited by quantities of 
 the first degree ; and quantities of the third degree are limited by 
 quantities of the second degree. 
 
 Quantities of the fourth degree are limited by lines or planes. 
 
 OF EXTENSION. 
 
 Extension is ultimately the occupation of space. Extension has 
 three dimensions, — length (lines), breadth (surface), thickness (limited 
 space or volume). 
 
 A POINT is the zero of extension, as it possesses neither of the 
 three elements of extension : hence it is position only. 
 
 QUANTITIES OF THE FIRST DEGREE.-LINES. 
 
 There are straight, curved, broken, and mixed lines. 
 A STRAIGHT LINE is the direct distance between two points. A 
 straight line is one without change of direction. 
 
SURFACES. 13 
 
 A CURVED LINE is one in which the direction is constantly chan- 
 ging. The change of direction is constant, or constantly increasing 
 or diminishing by a certain law of ratio ; or it may be irregular. A 
 curved line may lie wholly in a plane, or in a regularly curved surface, 
 or in an irregularly curved surface. 
 
 QUANTITIES OF THE SECOND DEGREE. -SURFACES. 
 
 Surfaces are of several kinds, such as regularly curved surfaces, — 
 those of the sphere, cylinder, cones, etc. ; rolling and wrinkled sur- 
 faces ; broken and warped surfaces ; and surfaces which are neither 
 warped, broken, nor curved in any direction, but are straight in all 
 directions : these last are called Planes. 
 
 A PLANE, therefore, is any straight surface. Planes are considered 
 infinite if not limited ; and hues Hmit planes, as stated above. A 
 plane takes its name from the manner of its limitation. Thus, when 
 a plane is limited by a curved line, every point of which is equally 
 distant from a point within the plane, the plane is called a Circle. 
 
 A CIRCLE, then, is a plane limited by a curved line, every point of 
 which is equally distant from a certain point within the plane called 
 the center. (It will be observed here, that the distinction between the 
 plane of the circle and its limiting line is kept clearly in view.) 
 
 Again, a plane limited by three straight lines is called a Triangle: 
 therefore, a TRIANGLE is a plane limited by three 
 straight lines. Triangles are of five kinds. Right- 
 angle triangles (Fig. A), having one right angle ; 
 Right-angle Isosceles triangles, having a right angle 
 and two equal sides (Fig. B) ; Equilateral triangles, having the three 
 
14 MODEL AND OBJECT DRAWING, 
 
 sides equal (Fig. C) ; Isosceles, having two sides equal (Fig. D) ; and 
 Scalene, having the three sides and angles unequal (Fig. E). 
 
 From the same analogy we should have the following definitions 
 of planes. 
 
 A SQUARE is a plane hmited by four equal straight lines, which 
 make four right angles one with another. 
 
 A RECTANGLE is a plane limited by four straight lines, the opposite 
 lines being equal, and forming four right angles. 
 
 A RHOMBUS is a plane limited by four equal straight lines, having 
 only its opposite angles equal. ■ 
 
 A RHOMBOID is a plane limited by four straight lines, only the 
 opposite lines being equal, and forming equal opposite angles. 
 
 A REGULAR PENTAGON is a plane limited by five equal straight lines 
 forming five equal angles. 
 
 A REGULAR HEXAGON is a plane limited by six equal straight lines 
 forming six equal angles. 
 
 A REGULAR HEPTAGON is a plane limited by seven equal straight 
 lines forming seven equal angles. 
 
 A REGULAR OCTAGON is a plane limited by eight equal straight lines 
 forming eight equal angles. 
 
 A REGULAR NONAGON is a plane limited by nine equal straight lines 
 forming nine equal angles. 
 
SURFACES, 15 
 
 A REGULAR DECAGON is a plane limited by ten equal straight lines 
 forming ten equal angles. 
 
 An ELLIPSE is a plane limited by a curved line, every point of which 
 is equal in the sum of its distances from two points within the plane 
 called the foci. An ellipse is said to have two axes, or diameters : 
 they are at right angles to each other; and they are called the major 
 and minor axis, or, in common language, the longer and the shorter 
 diameters. 
 
 Returning to the circle and its different parts and their limitations, 
 the definition of each part is dependent upon the kind of quantity to 
 which it belongs. Thus, the CIRCUMFERENCE is the line of limitation ; 
 and the CIRCLE is the plane limited. The circumference becomes the 
 figure of the circle. A part of the cir- 
 cumference of a circle is called an Arc 
 (Fig. I). 
 
 The SEMICIRCLE is the half-plane of the 
 circle limited by the semi-circumference 
 and the subtending diameter. 
 
 A SECTOR is a part of the plane of a 
 circle limited by two radii and the in- 
 cluded arc. 
 
 A SEGMENT is a part of the plane of a 
 circle limited by an arc and its chord. It will be observed, that, in the 
 foregoing definitions of the several limited planes, the word ''figure'' 
 is not used. It seems that this word tends to confusion; preventing, 
 in some cases, the mind from seizing at once the idea. We may say 
 that every limited plane has a figure, but the figure is not the plane : 
 the circle has a figure ; yet the figure of a circle is not the circle, but 
 
1 6 MODEL AND OBJECT DRAWING. 
 
 the perimeter, or circumference, of the circle. We can never find the 
 area of a figure ; because the figure is only outline, and not area at all. 
 All figures, as such, belong to quantities of the first degree. 
 
 QUANTITIES OF THE THIRD DEGREE.-VOLUMES. 
 
 Extending on all sides of us, above and below, is the infinite space 
 of the universe in which all worlds and beings have their existence. 
 Whenever any portion of this infinite unlimited space becomes lim- 
 ited in any manner, such portion of space becomes a volume ; there- 
 fore, — 
 
 A VOLUME is any limited portion of space, and the volume takes its 
 name from the method of its limitation. 
 
 A SPHERE is a volume limited by a curved surface, every point of 
 which is equally distant from the center of the sphere. 
 
 A CUBE is a volume limited by six equal squares. 
 
 A PYRAMID is a volume limited by a polygon and as many equal 
 isosceles triangles as the polygon has sides. 
 
 A CONE is a volume limited, both by a circle as a base, and a curved 
 surface which is straight in the directions of all lines drawn from 
 the circumference of the base to a point in a line perpendicular to the 
 center of the circle, called the Apex; or a cone would be limited as 
 described by the revolution of a right-angle triangle about one of its 
 sides adjacent to the right angle. 
 
 A CYLINDER is a volume limited by two opposite equal and parallel 
 circles, and by a surface curved in the direction of the circumferences 
 of the circles, and straight at right angles to this direction. 
 
 A PRISM is a volume limited by two equal, opposite, and parallel 
 
POSITION AND RELATION. IJ 
 
 polygons, and as many equal rectangles as either of the polygons has 
 sides. 
 
 QUANTITIES OF THE FOURTH DEGREE. - INCLINATION. 
 
 When two lines in the same plane incline to each other, the inclina- 
 tion is called an Angle. Angles are of three kinds, Right Angles 
 (Fig. F), Acute Angles (Fig. G), and Obtuse Angles (Fig. H). When 
 
 one line meets another line, forming two equal angles on the same side 
 of the line met, both angles are Right Angles. The point of intersec- 
 tion of two lines forming an angle is called the Vertex of the angle. 
 There may be four right angles in the same plane having their vertices 
 in the same point. An Acute Angle is less than a right angle. An 
 Obtuse Angle is greater than a right angle. The inclination of two 
 planes also forms an angle. The inclination and intersection of three 
 or more planes, at one point, form a Solid Angle. 
 
 WORDS DENOTING POSITION AND RELATION. 
 
 Two other classes of definitions are important to the student ; viz., 
 those of words which denote position, and those of words which denote 
 relation. 
 
 First, Words denoting position ; namely, vertical, horizontal, level, 
 flat, inclined. All these terms signify position, without relation to 
 any other object save the earth itself. That is to say, a line in any 
 
1 8 MODEL AND OBJECT DRAWING. 
 
 of these positions is so of itself alone, without the aid of any other 
 line. 
 
 Second, Words denoting relation ; namely, parallel, perpendicular, 
 tangent, secant, etc. A line in any of these positions bears a certain 
 definite relation to some other line, and changes position with such 
 line. A PARALLEL LINE is one which is everywhere equally distant 
 from another line ; while a vertical line is vertical alone, and of itself, 
 from its position only. 
 
 A VERTICAL LINE is one in an upright position, pointing to the center 
 of the earth. 
 
 A HORIZONTAL LINE is one, all points of which are on the same level. 
 A horizontal line drawn through any point is perpendicular to a 
 vertical line drawn through the same point, and the vertical is perpen- 
 dicular to the horizontal line. * 
 
 An INCLINED LINE is one, all points of which are at different elevations. 
 
 A line is perpendicular to another line when it makes a right angle 
 with it. 
 
 A line is tangent to another line when it touches it at a single 
 point, and would not cut it if both were produced. 
 
 ORTHOGRAPHIC PROJECTIONS. 
 
 In order to understand clearly some of the illustrations and 
 descriptions which follow in this book, we think it advisable to ask 
 the attention of the student to a brief preliminary statement of the 
 leading principles and methods of Orthographic Projection. The 
 object of these projections is, to show the real forms of objects, or 
 combinations of objects ; so that any one understanding these methods 
 
ORTHOGRAPHIC PROJECTIONS, 
 
 19 
 
 of representation can construct from such drawings the things repre- 
 sented. These methods are generally used by architects, machinists, 
 ship-builders, and inventors, to represent in detail the forms, dimen- 
 sions, combinations, and methods of action, of whatever they may 
 invent or design. They are also useful in demonstrating many geo- 
 metrical principles, with reference to perspective, forms of shadows, 
 intersections of solids, etc. 
 
 Two planes of projection at right angles to each other are em 
 ployed. One of these is named the Vertical plane of Projection, 
 the projection itself on this plane being generally called the Eleva- 
 tion : the other plane is named the Hoidzontal plane of Projection, and 
 the projection on it is called the Plan. The plan and elevation of a 
 building or machine, drawn to dimensions, gives an idea of its form, 
 size, and method of construction. Two or more vertical or horizontal 
 projections may be drawn where they are required to determine addi- 
 tional details. By these means the most complicated combinations 
 can be made apparent. 
 
 The use we shall make of these 
 methods will be to show the apparent 
 forms of some objects, and to demon- 
 strate certain mathematical principles. 
 
 Let us suppose that we have, as in 
 Fig. 2, two planes represented by sheets of 
 paper at right angles to each other, — one 
 in a vertical, and the other in a horizontal, 
 
 position, intersecting or touching each other in the line G L. These 
 planes are represented in a perspective view, and we will say they are 
 each one foot square. Let us suppose, further, that the sun is in the 
 
 X'iS.2 
 
20 
 
 MODEL AND OBJECT DRAWING, 
 
 west. Place the vertical plane so the sun's rays will strike the plane 
 
 at right angles to its surface, while they pass parallel to the horizontal 
 
 plane. 
 
 Now, if we hold a four-inch square plane, or piece of paper, parallel 
 
 to the vertical plane, at a little distance from it, with two of its sides 
 
 vertical, the paper will throw upon the vertical plane a shadow which 
 
 will have the precise form and dimensions of the four-inch square. 
 
 We may call this shadow the vertical projection of the square. 
 
 With the square in the same position, suppose the sun directly 
 
 over-head: the horizontal projection of the square will be cast down 
 
 upon the horizontal plane. 
 
 This projection is a straight line, four inches long. In the figure 
 
 the vertical projection of the square is the square A' B' C D', and its 
 
 horizontal projection in the same position is the straight line A B. 
 
 It is not customary to represent, as above, 
 these planes of projection in a perspective view, 
 but simply to draw a horizontal line on the paper, 
 representing the intersection of the vertical and 
 the horizontal planes, and to regard that part of 
 the paper above the line as the vertical plane, and 
 that part below the line as the horizontal plane. 
 This line is called the ground-line, and it is 
 marked with the letters G L. 
 
 Let us analyze the case above described (Fig. 
 3). Rays of light moving in horizontal parallel 
 lines, perpendicular to the vertical plane above 
 
 the line G L, cause the shadow of the square to fall upon that plane ; 
 
 and rays of light moving vertically downward, in parallel lines, cause 
 
 mg.s 
 
ORTHOGRAPHIC PROJECTIONS. 21 
 
 the shadow of the square to be cast on the horizontal plane. The 
 first shadow is a square, and the second is a line. Hence, when we 
 see the two elements projected, we know of what form they are the 
 projections: since the horizontal projection is only a line, we see that 
 the object which is the origin of projection must be merely a plane, 
 because it possesses no appreciable thickness ; and, since we have a 
 square for the vertical projection, we know that the plane is in the 
 form of a square. Thus we are able to understand the form of an 
 object from its projections. 
 
 By observing still further the two projections, we should also see 
 what position the object occupies with reference to both planes. Since 
 A B is parallel to the ground-line, we know that the square is parallel 
 to the vertical plane ; and, when we see that A' B' is parallel to the 
 ground-line, we know that the lower and upper edges are parallel to 
 the horizontal plane. 
 
 In Figs. 4 and 5 the horizontal and vertical projections of several 
 solids are shown. 
 
 First, we have the sphere at A ; having, for its horizontal and ver- 
 tical projections, a circle. It is, of course, the same in both : but it 
 should be observed that two circles at right angles to each other, and 
 intersecting at the horizontal diameter of each, would give the same 
 projections ; but, if these planes were revolved into different positions, 
 as in Figs. C E H and K, the projections would show that they were 
 planes, and not a sphere. 
 
 At B we have the projections of a cube. Two squares at right 
 angles to each other would give the same projections. At C we have 
 the cube revolved on the horizontal plane, so as to bring one diagonal 
 of the upper and lower sides perpendicular to the vertical plane. 
 
22 
 
 MODEL AND OBJECT DRAWING. 
 
 In this position, two square planes would not give the horizontal and 
 % vertical projections of the solid, as at C. In this figure we observe 
 that the horizontal projection gives the true form and dimensions 
 of a side of the cube, and that the vertical projection does neither. 
 
 At D we have the horizontal and vertical projections of a cone, — 
 the horizontal being a circle equal to the base of the cone ; and the 
 vertical projection, a triangle equal to a vertical section through the 
 axis of the cone. 
 
 At E we have the same tipped up, with its base and axis oblique 
 
 Tig. A 
 
 to the horizontal plane. This projection is made by revolving, on the 
 point b' as a center, a! b' in its horizontal position on the vertical 
 plane, to the position d' b' ; on this as a base constructing the triangle. 
 The horizontal projection of the same is made by carrying forward to 
 the right, from D to E, the diameter c d; letting fall the dotted ver- 
 ticals from a!' b' to' determine a b. It is evident that this iatter diam- 
 eter, a b, will be foreshortened. Upon these two diameters, the 
 horizontal projection of the circle must be drawn : it will be an ellipse. 
 The dotted vertical, let fall from the apex e\ will give the place of the 
 
ORTHOGRAPHIC PROJECTIONS, 
 
 23 
 
 vertex e in the horizontal projection. From this point draw tangents 
 to the ellipse, and the figure will be complete. 
 
 At F we have the projections of a four-sided pyramid : the vertical 
 projection is a triangle, equal to a vertical section through the axis 
 and diameter of the base. 
 
 The horizontal jfig. 5 
 shows the projec- 
 tion of the base : and 
 the four isosceles 
 triangles, in their 
 oblique positions, 
 forming the sides of 
 the pyramid, are 
 projected at a be, 
 bee, c e d, d e a ; 
 
 each having the common point e. The projections of its oblique 
 position at H are obtained similarly to those of the cone, after first 
 constructing its projections at G, where it has been revolved on the 
 horizontal plane through a quarter circumference. 
 
 At I, J, K (Fig. 6), we have, in succession, the projections of a four- 
 sided prism in several positions. At I the sides of the prism are per- 
 pendicular, and parallel to the vertical plane ; at J the prism has been 
 revolved so as to bring the sides at an angle of 45° to the vertical 
 plane ; and, at K, it is tipped up so that the bases and sides make 
 angles with the horizontal plane. The method of drawing these pro- 
 jections will be readily understood by what has preceded. 
 
 The reader will further observe, that the projection of any particu- 
 lar line or plane may be studied from these projections of solids. 
 
24 
 
 MODEL AND OBJECT DRAWING. 
 
 For instance, at I the edge of the prism, represented by the line e' a! in 
 the vertical projection, has its horizontal projection in the point a; and 
 in the same way the remaining edges, represented by the other ver- 
 tical lines in the vertical projection, have their horizontal projections 
 in their corresponding points. We conclude, therefore, from what was 
 found in the case of the four-inch square, and in the present investi- 
 gation, that the vertical projection of a vertical line is a vertical line 
 
 Fig, 6 
 
 of the same length, and that the horizontal projection of a vertical 
 line is a point. 
 
 If we take the lines, ad, be, at I, in the horizontal projection, 
 which are the projections of the two opposite sides of both bases of 
 the prism, the bases being perpendicular to the vertical plane, we see 
 that their vertical projections are found in the points a' , e' , b' , f. 
 Therefore, we conclude that the horizontal projectiofi of a horizontal 
 line is a straight line of the same length ; and, if the line is perpen- 
 dicular to the vertical plane, its vertical projection will be a point. By 
 
ORTHOGRAPHIC PROJECTIONS. 25 
 
 examination of the several planes bounding this solid, we see that 
 the horizontal bases are projected on the horizontal plane in squares 
 of the same size, since, in all these projections, the rays are assumed 
 to be parallel ; and that also the two side planes, which are parallel 
 to the vertical plane, are projected in rectangles of the same magni- 
 tude. We may say, then, that, zvhen a plane is parallel to either plane 
 of projection, its projection on that pla7ie will be equal to the plane 
 itself. 
 
 If we examine the front face of the prism, as projected in a' b' f e', 
 we see that it has its horizontal projection in the line ab, and the 
 other three sides of the prism have their horizontal projections in the 
 lines b c, c d, da; the two bases have their horizontal projection in 
 abed, and their vertical projection in the lines a! b' and e' f: hence, 
 whenever a plane is pe7pe7tdicular to either plane of projection^ its projec- 
 tion on that plane will be a straight line. 
 
 In J we have the vertical planes of the prism in their oblique 
 positions projected on the vertical plane ; and we see, comparing them 
 with the vertical projections in I, that neither projection is of the 
 same size as the plane itself : but, in the horizontal projection, we 
 have the two bases of the prism projected in their true form and 
 dimensions. Compa.re with a b c d '\x\. \. 
 
 By comparing K with I and J, we see that none of the planes 
 limiting the solid are shown in their true dimensions. The analysis 
 of this subject might be carried on to any extent, and deductions 
 made, and processes developed, for showing various combinations 
 and forms, intersections of solids, projections of shadows, principles 
 of construction, etc. ; but we have given enough of the principles of 
 Orthographic Projections to enable the attentive student to under- 
 
26 MODEL AND OBJECT DRAWING. 
 
 stand the illustrations given in the body of the book. This is all that 
 is necessary for our present purpose. 
 
 HOW TO READ APPARENT FORMS. 
 
 If one had the faculty, when looking at a house, for example, of 
 making it appear like a flat spot of a certain shape, disregarding the 
 
 fact that certain surfaces are retreating, thus 
 reducing the whole to one vertical plane, he 
 would have the most complete qualification 
 for rapid sketching (Fig. 7). Indeed, this is 
 just what the artist endeavors, as far as possi- 
 ble, to do in order to read forms. It is not 
 difficult to read off rapidly the outline after the whole complex arrange- 
 ment of planes, constituting the house, or the group of buildings, has 
 been reduced to one plane. But our knowledge of the retreating of 
 the planes, and of their many combinations, makes it very hard to 
 secure the apparent form of the whole group. 
 
 Herein our knowledge of real forms and directions seems to stand 
 in the way of our appreciation of other facts relating to appearances ; 
 so that it always happens that the beginner draws the forms as he 
 knows them to exist, instead of representing them only as they appear 
 to his eye. To draw what y 07 l see, to paint what yon see, and not what 
 yonr knowledge leads yon to imagine you see, must be the constant 
 admonition of the teacher. Works of imagination may be excellent, 
 and greatly to be prized ; but, at this stage, neither the imagination 
 nor the knowledge of the pupil is of any avail. He must depend only 
 upon his eyes. Seeing with the eyes, and knowing from data in the 
 
BOW TO READ APPARENT FORMS. 
 
 27 
 
 Fiff.8 
 
 mind, are very different acts ; and the province of each is separate 
 
 from that of the other. 
 
 Taking the cube with three faces visible, if we can make the whole 
 
 block appear like a flat spot on a vertical plane 
 
 when seen horizontally, we can then draw the 
 
 various lines with accuracy by referring each to an 
 
 imaginary horizontal or vertical, passing through 
 
 one end of the same, and by noting the angle 
 
 (Fig. 8). The inclination of all lines may be 
 
 determined by reference to the vertical or to the horizontal. 
 To sum up these suggestions, we say that all attempts at com- 
 parison of lengths and 
 positions of lines must 
 be made on a plane per- 
 pendicular to the axis 
 of sight, or, in other 
 words, perpendicular to 
 the central ray from the 
 object to be drawn. A 
 common way is, to hold 
 out the pencil at arm's- 
 length, in such a posi- 
 tion that one end is 
 as near to the eye as 
 the other, and then to 
 
 compare two lines as to their apparent lengths, or their positions 
 
 with regard to each other, or to a horizontal or a vertical line. 
 
 Thus, relative apparent lengths, and relative apparent positions, 
 
28 
 
 MODEL AND OBJECT DRAWING. 
 
 Tig. 9 
 
 may be determined. See cut of hands showing the positions of the 
 pencil. 
 
 A very satisfactory and conclusive method of testing the accuracy 
 of a drawing of a simple object, after it is made, is to cut out the 
 drawing with a pen-knife, running the point around the outside, or the 
 outer lines of the whole figure, and folding back the different planes 
 on certain lines. Thus, in the case of the cube. Fig. 9, run the knife 
 along the full lines, and fold back the several squares on the dotted 
 
 lines, and then hold the paper at such a 
 
 /"~ ■/ distance from the eye that the model from 
 
 which the drawing w^as made will appear to 
 
 just fill the opening. Any error in the work 
 
 will be seen at once. In the same way the 
 
 drawing of any separate plane may be tested 
 
 by putting in place all the other planes, leav- 
 
 \ ing the one to be determined folded back. 
 
 / \ \ Care must be taken to hold the paper in a 
 
 / / \ \ position perpendicular to the central ray 
 
 ""^^---./ V'-'''''' from the object to the eye. 
 
 A very simple method of finding the ap- 
 parent position of a line, when neither horizontal nor vertical, is to hold 
 out the pencil as above directed, so as to coincide with the line to be 
 determined, and, with the other hand holding up the paper, bring the 
 pencil against it in a position corresponding to that of the line. The 
 direction of the line on the paper will thus be readily determined. 
 The pupil may also put up in front of the eye a plate of glass, and, 
 holding the head fixed in one position, may trace upon it the outline 
 of the object. 
 
THE niASCOFE. 
 
 29 
 
 THE DIASCOPE. 
 
 The DIASCOPE is a simple contrivance for testing apparent forms. 
 
 This instrument is simply'a frame, across which are drawn fine 
 wires or threads, at equal distances, in two opposite directions, divid- 
 ing the space inclosed into a number of equal squares. A frame four 
 inches square, inside measure, is a convenient size. The frame should 
 be made of some thin material, and provided with a handle. The 
 
 'Fig. 10 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 inner lines of the frame may then be divided into half-inch spaces, 
 and small holes should be made near the inner edges at the points 
 of division. Small wires or threads may be drawn through these 
 holes from opposite sides, dividing the whole space, for instance, into 
 sixty-four equal squares. 
 
 When completed, the Diascope may be held up in a vertical posi- 
 tion, between the object to be drawn and the eye, so that the central 
 ray of light from the object will pass through the Diascope at right 
 angles to its plane. With it in this position, the observer will be 
 
30 MODEL AND OBJECT DRAWING. 
 
 enabled to read off without difficulty many of the apparent inclinations 
 and magnitudes. 
 
 The side of a cigar-box, and two or three yards of fine iron or copper 
 wire, is all the material required in the construction of this instrument 
 (Fig. 10). 
 
 ANALYSIS OF APPARENT FORMS. 
 
 Every visible object transmits to the eye of the observer rays of 
 light from every part of its visible surface. The rays of light move 
 in straight lines and converge as they approach the eye ; so that the 
 whole bundle of rays from an object is able to enter the eye through 
 the small opening called the pupil, and, traversing the body of the eye, 
 is received on the inner side of the posterior-wall, called the retina. 
 On it the image of the object is formed, exactly similar to the appar- 
 ent form of the object itself, only greatly reduced in size and reversed 
 in position. In order to understand the explanations which follow, it 
 is important to consider attentively this bundle of converging rays 
 which the eye receives from every object upon which it is turned. 
 Every object seems to be charged with the luminous quality we call 
 light, which is profusely diffused abroad in all directions. Whenever 
 the eye is directed to any object, it receives a shower of these lumin- 
 ous vibrations. It suits our present purpose to regard these vibrations 
 of light as moving in straight lines ; that is, a bundle of lines from 
 an object converging to the eye. The form of the bundle of rays 
 depends upon the form of the object. 
 
 Thus, if a square be placed directly in front, so that the eye is 
 equally distant from each of the four corners, it is plain that the rays 
 of light from this square, converging to the eye, will form a true right 
 
ANALYSIS OF APPARENT FORMS, 31 
 
 pyramid, having four sides, with the square for its base, and its apex 
 in the eye as in Fig. 11. 
 
 In this case the sides of the pyramid of rays would be bounded 
 by four equal isosceles triangles ; and the central ray of light Cy from 
 the square, would be the axis of the pyramid of rays. If, now, this 
 pyramid of rays is cut by a plane 
 perpendicular to the axis or central f^s- " 
 
 ray, and parallel to the base, the 
 section will be geometrically simi- 
 lar to the base, that is a square. 
 The section will, therefore, be a 
 
 true picture of the square, and will correspond in form to the little 
 spot in the eye formed by the square. 
 
 If the square is turned obliquely to the eye, so that the rays of 
 light are thrown off obliquely to the surface of the square, and a cross- 
 section of the rays is made perpendicular to the central ray, the sec- 
 tion will present a true picture of the apparent form of the square 
 in its oblique position; and it will be exactly similar to the image 
 formed in the eye by the rays from the square in its oblique position. 
 There are several ways of making these facts apparent. One method 
 is, by employment of models in a conical or pyramidal form built 
 obliquely on several bases, showing cross-sections. The only objec- 
 tion to this mode of experiment and proof is in the cost of the models, 
 which are difficult of construction. 
 
 An easier method is, to set up a plate of glass perpendicular to the 
 central ray, and, looking through it at right angles to its surface upon 
 any object, to trace upon the glass with a common pencil, or one made 
 of soap, the outline of the object, with the head in a fixed position. 
 
32 
 
 MODEL AND OBJECT DRAWING, 
 
 The outline on the glass will be a true picture of the object. The 
 glass will be a cross-section of the bundle of rays from the object (Fig. 
 
 12). 
 
 Thus, the picture of the plane abed will be accurately traced on 
 
 the transparent plane interposed 
 
 — — -/r^- at T P. Hence, we may state 
 
 this general principle : A true pie- 
 
 tw'e of an objeet may be obtained 
 
 by tracing its apparent form on a 
 
 transparent plane perpendicular to the central ray from the object, or by 
 
 a cross-section of the rays from the object perpendicular to the central ray. 
 
 Tig, 13 
 
 THE DRAWING OF THE RECTANGLE OR OF THE SQUARE. 
 
 The drawing of the rectangle or the square presents a few points of 
 special interest, which the stu- 
 dent would do well to consider, 
 and to master completely, in 
 order to make the drawing of 
 all rectangles easy and sure. 
 
 First, when two sides, ab 
 and c d (in this case, the upper 
 and lower sides), of a square or 
 a rectangle are perpendicular to 
 the central ray, but one of 
 them, dc, 2it a greater distance 
 from the eye than the other, 
 as in Fig. 13, then the two lines which are perpendicular to c.r.. 
 
THE RECTANGLE OR THE SQUARE. 
 
 33 
 
 i.e., a b and dc^ are seen to be parallel ; but, since they are un- 
 equally distant from the eye, the nearer line, ah, will appear to be 
 longer than dc. 
 
 Thus, in Fig. 14, which is the plan of the above, the rays from c d 
 will be seen to cross ab 2X c' d' ; so that, relatively to ab, c d will 
 appear to be only as long as c" d" on the transparent plane. If we 
 examine the image formed on T P, we find that it consists of the 
 
 following elements : viz., apparent height, i to 2 
 
 Fig.is^n — ! — ^" (Fig. 15) ; the apparent length of ^r^ is c' d'\ and of 
 
 / \ ab \^ d b' ; the apparent length of a d \s a! d'\ and 
 
 J- 2 \' of be, b' c" ; thus, the figure of the rectangle will be 
 
 given in the figure 0! b' c" d'' . It will be seen, there- 
 fore, that the lines a d and b c will appear to be convergent lines, seem- 
 ing to approach each other as they recede from the eye. By assuming 
 four points on any two receding lines, we could construct a rectangle 
 as above, and proceed in the same method to show the convergence. 
 
 In the same way it may be proved that all receding parallel lines, 
 in whatever position, seem to converge or incline to each other as 
 they recede, and would, therefore, if extended sufficiently, meet in 
 the same point. In all 
 cases this will appear from „_.. ^„ t 
 the fact that the distance 
 between them, which is a 
 line of a certain length, 
 seems to diminish in length 
 as it becomes more distant. 
 
 Thus, in Fig. 16 let E represent the position of the eye, and i, 2, 3, 
 the positions of three equal lines in the same plane with the eye and 
 
 msi 16 
 
34 MODEL AND OBJECT DRAWING, 
 
 with each other. Let the line i i' be at a certain distance, 2 2' at 
 twice, and 3 3' at three times, the distance of 11' from the eye. 
 Draw lines from the extremities of each of these lines to E, and, 
 at their intersection with T P, we shall have their relative apparent 
 lengths. Thus, 2 2' will appear to be one-half as long as i \\ because 
 it is twice the distance from the eye ; and 3 3' will appear to be one- 
 third as long as 11', because it is three times as far from the eye. 
 Hence it follows that the apparent length of a line is inversely pro- 
 portional to its distance from the eye. If 2 2' and 3 3' were moved 
 up to the position of i \\ they would appear to be of the same 
 length. 
 
 We have thus obtained these additional general principles : viz.. 
 First, Equal magnitudes appear equal at equal distances ; Second, 
 Equal magnitudes appear unequal at unequal distances ; and, Third, 
 Equal magnitudes appear inversely proportional to their distances. 
 
 These principles determine the convergence of all parallel lines as 
 they recede from the eye. 
 
 THE APPARENT FORMS OF ANGLES, 
 
 Place a square plane in such a position that all the angles are 
 equally distant from the eye, as in Fig. ij, abed. It is evident, 
 that, in this position, all the angles will appear to be right angles, 
 as they really are ; but if the plane is revolved about c d into the 
 position a b' c d' , so as to bring a b into the position of a' b' , the 
 appearance will be at once changed, and all the right angles will 
 have been apparently destroyed. Thus, the angles at a' and b' will 
 appear to have been opened, while those at c' and d will appear to 
 
THE APPARENT FORMS OF ANGLES. 
 
 35 
 
 have been partly closed. If the revolution of the plane about the line 
 c' d' were continued, the process of opening one set and closing the 
 other set would go on until all the angles would appear to be extin- 
 guished ; the points a! and b' coming into the same line with the 
 eye, and the whole plane assuming the appearance of a straight 
 line. 
 
 Now, since the angles at d and 
 b' in the oblique position appear to 
 be opened more than right angles, 
 and since rays from the angle ^are 
 more oblique than at b' , and since 
 the angles at c' and d! appear partly 
 closed, considering what was shown 
 on p. 30 we may deduce the follow- 
 ing general statements : — 
 
 Whenever a rectangular plane is 
 seen obliquely, the nearest and the 
 
 farthest angles appear obtuse, the latter being the more obtuse ; and 
 the tzvo intermediate angles appear always acute. 
 
 This rule will apply to every possible position of the rectangle and 
 of the square, which is only a particular case of the rectangle. As 
 rectangular (solids) volumes are drawn by representing their separate 
 faces, and as each face must be solved or read by itself, as well as 
 with reference to the others, the principles above stated go far to 
 enable the student to represent accurately rectangular solids. 
 
 There is, however, one other deduction which may be noticed. If 
 we have three rectangular planes in an oblique position, as, for 
 instance, the three sides of a cube forming one solid angle, there will 
 
36 MODEL AND OBJECT DRAWING. 
 
 appear to be three obtuse angles about that point. This will always 
 be the case when three sides are visible : there 
 
 Fig. 19 
 
 can never be a combination of one right and two 
 obtuse angles, or of one acute and two obtuse angles ; 
 but the three angles about that nearest point of the 
 cube must always be obtuse, as in Fig. 19. 
 
 The advantage of this rule will be appreciated 
 by every teacher, as it offers at once a test for many 
 doubtful points where the eye alone might not be 
 able to detect the error. 
 
 THE DRAWING OF THE CUBE. 
 
 Definition : The CUBE is a volume bounded by six equal squares. 
 First, place the cube on a horizontal plane directly in front, with the 
 two side-lines of the front square equally distant from the eye ; the 
 top of the square being a little nearer than the bottom, so that only 
 the front and the top of the cube will be seen (Fig. 20). In this posi- 
 tion the front face of the cube is usually drawn as a 
 square, with the side-lines vertical, for the same reason 
 that we should draw the sides of a house vertical, and 
 not converging as they recede upward. We should 
 then ascertain by observation, on the pencil held at 
 arm's length in a vertical position, corresponding with 
 ac, the measurement of the apparent height of the 
 upper face of the cube. Let us suppose it to be one-fourth as high 
 as the front face. Divide one vertical side of the front face into four 
 equal parts, and place one of these parts above the line a b, and 
 
THE DRAWING OF THE CUBE, 37 
 
 draw ef of indefinite length, parallel to ab. Next observe how 
 much shorter ef appears to be than a b, and mark its apparent length 
 on a b, and draw dotted vertical lines from these points to e and 
 f: the lines a e and bfmdiy now be drawn, and the figure is complete. 
 
 Next place the cube so that three sides will be visible ; the model 
 still resting on a horizontal plane, showing the front, right side, and 
 top (Fig. 21). 
 
 The first line to be drawn is the nearest vertical, a b. This line is 
 the measure of every other line. The second line, a Cy 
 must be placed by observing its position in the model, 
 its degree of inclination to an imaginary horizontal 
 line through a, and its length compared with the 
 standard line a b. 
 
 Then the third line, af^ should be read from the 
 model, as to position, inclination, and length, in a 
 similar manner. We have now one line in each of the three sets 
 of parallels to be drawn. 
 
 Since every other line in the model is parallel to one of these 
 three, therefore the three lines are the ruling lines of the drawing. 
 We should next observe if ^<3f is shorter than ab^ and, if so, how much, 
 representing it in its true proportion : then draw b d. Compare f e 
 with a b, and draw it. Connect b with e. By drawing b d and b e, 
 the convergence of the lines fg and eg has been determined ; so that 
 it is only necessary that they should have the same degree of conver- 
 gence, as the lines are respectively parallel to each. These lines 
 complete the drawing of the model. If correctly drawn, there will be, 
 first, three obtuse angles about the point a; second, the angles at 
 d, g, and e will also appear obtuse, and more obtuse than the angles 
 
38 MODEL AND OBJECT DRAWING, 
 
 in their respective planes at a ; third, the remaining angles will be 
 acute. 
 
 The third position of the cube is one in which the three faces will 
 appear about equal. Place the cube on an inclined plane, or put some- 
 thing under the back corner, so that there will be no vertical lines in 
 the model. 
 
 In this position let a be the nearest point : draw first the line a by 
 which seems to be nearest vertical ; then the line ^ ^ to the left ; and 
 third, a dj comparing the last two lines with the first to obtain their 
 
 different lengths (Fig. 22). Having ob- 
 Fig, 22 ^""~;;^S<^' tained the positions and lengths of these 
 
 three lines, it only remains to draw the 
 other six lines with the proper convergence, 
 which must be noted from the model itself. 
 There will be, when complete, three sets of 
 lines ; each set converging to a different 
 point. 
 
 Let us observe, again, that about the 
 point a we have three obtuse angles, and that the opposite angle on 
 each face is more obtuse than the angle in the same plane at a, and 
 that the angles at c, d, and b are all acute. There is one other rule 
 very useful in the criticism of drawings by pupils deducible from this 
 case; viz.. Take the two faces A and B, and call c e^ df, and ab 
 the side-lines of the two faces, a b being the dividing line : then these 
 side-lines will converge in a direction opposite to the other face C ; 
 i.e., downwards. Now take the two faces C and B, with the dividing 
 line a d, and with the side-lines bf and eg. They will converge in a 
 direction opposite to the other face A; i.e., to the right. 
 
THE HEXAGON AND THE HEXAGONAL PRISM. 
 
 39 
 
 In the same way the side-lines of the two faces A and C, i.e., a c, 
 b Cy and dg, will converge in a direction opposite to the other face B ; 
 i.e., to the left. Hence the rule : In drawing any rectangular solidj 
 three faces being visible^ the side-lines of any two faces will seem to con- 
 verge in a directio7t opposite to the third visible face. It will be seen 
 that the third visible face always indicates the ends of the lines nearest 
 the eye. 
 
 THE METHOD OF DRAWING THE HEXAGON AND THE HEXAGONAL 
 
 PRISM. 
 
 mg. 23 
 
 In drawing the hexagon and the hexagonal prism and the pyramid, 
 we have first to consider the elements of the hexagon as a geometrical 
 quantity. Describe a circle, and, with the radius from each end of 
 the horizontal diameter as a center, cut the circumference in points 
 above and below. By this means the circumference is divided into 
 six equal arcs : drawing the chords of these arcs, we complete the 
 figure of the hexagon (Fig. 23). Draw radial lines from the outer 
 angles to the center, thus dividing the hex- 
 agon into six equal equilateral triangles, all 
 having their inner angles at the center of 
 the hexagon (Fig. 24). If we draw the alti- 
 tudes of the two triangles having the com- 
 mon base a 0, we shall have the line b f 
 dividing the base a into two equal parts ; 
 for it is evident that the altitude of an 
 
 equilateral triangle will always bisect the base. Again, if the altitudes 
 of the two triangles having the common base d are drawn, we shall 
 have the line c e, dividing the base d into two equal parts. Since 
 
40 
 
 MODEL AND OBJECT DRAWING, 
 
 Fig. 24 b 
 
 a and o d are equal, it is plain that the diameter is divided into four 
 equal parts, which we will number i, 2, 3, 4, beginning at the left. 
 
 Let us now turn the hexagon into a posi- 
 tion oblique to the eye, so that the point a will 
 be nearer to the eye than the point d : it will 
 be seen that the points c and e will appear 
 nearer to each other than b and /, because the 
 line bf is nearer to the eye than c e (Fig. 25). 
 Hence the two lines be and f e^ which are 
 parallel to a d, will appear to converge : also, 
 the four geometrically equal parts of the diameter, 
 being at unequal distances from the eye, will appear 
 unequal ; the nearest part, i, will appear to be the 
 longest ; and 2, the next in length ; 3, the next ; 
 and 4 the shortest of all. Again, let us suppose we 
 have the hexagonal prism before us, with one end 
 visible in an oblique position. We first read from 
 the model the central rectangle b c ef; that is, we 
 observe these four lines, and draw them in their 
 
 relative positions and relations. Thus, as 3 ^ and 
 f e converge upwards, supposing the eye to be a 
 little above the model, we have the central rect- 
 angle beef drawn in its true position (Fig. 26). 
 Draw the diagonals b e and ef : they will cross 
 each other in O, the true center of the rectangle. 
 Now draw the diameter through O, parallel to the 
 two lines b e and f e ; that is, so that it will con- 
 verge at the same point with them. We find that we have the two 
 
 mg. 26 
 
THE HEXAGON AND THE HEXAGONAL PRISM. 41 
 
 central divisions of the diameter, 2 and 3, represented in their pro- 
 portional lengths ; and 2 will appear to be longer than 3. Comparing 
 these two divisions, we have the ratio between the several divisions of 
 the diameter; for, by as much as 2 appears to be longer than 3, by 
 exactly the same proportion will i appear to be longer than 2, and 3 
 than 4 : so that we can point off the first and the last divisions of the 
 diameter by observing the ratio of the two middle divisions. Having 
 thus placed the points a d on the diameter, we have only to draw the 
 adjacent sides to complete the apparent form of the hexagon in this 
 position. 
 
 It will be seen, that, to draw the hexagon from the model, it is 
 only necessary to read and draw the central rectangle ; and all the 
 rest follows necessarily, without any further examination of the 
 model : and, provided these four lines of this rectangle are correctly 
 located, the whole hexagon is easily represented in its true propor- 
 tions. 
 
 Any two opposite sides may be taken for the ends of the rectangle, 
 but it is usually best to choose the upper and the lower (when there 
 is an upper and a lower). The four lines must be drawn with great 
 care, allowing no error of observation or of execution to occur ; since 
 the rest of the hexagon depends upon them. 
 
 This analysis covers every conceivable -Frg. 27 
 
 position of the hexagon. Let us suppose / \ 
 
 that one of the possible positions of the cen- ^,1 A^ 
 
 tral rectangle is represented by the figure 
 
 d c ef, <5/and c e being the longer lines (Fig. 27). Draw the diagonals 
 cutting each other at 0, the center of the rectangle. Through this 
 point draw the diameter as before, parallel to the ends be and f e. 
 
42 
 
 MODEL AND OBJECT DRAWING. 
 
 Tig. 28 
 
 Fig. 29 
 
 We shall then have the two central divisions 2 and 3, giving the ratio 
 
 (Fig. 28). Laying off the points a and d, on the diameter, so as to 
 give the four divisions of the diameter in their 
 diminishing ratio from i to 4, draw the other 
 four lines a b, c d, af, and d e^ and the hexagon 
 is completed. 
 
 In the same way, if we have the central 
 rectangle in the position b c e f, draw the 
 
 diagonals to ascertain the central point (Fig. 29) ; and through o draw 
 
 the diameter as before, parallel to the 
 
 lines fe and b Cy which, in this case, 
 
 have but slight convergence : next, lay 
 
 off the points a and d, as before, and 
 
 then complete the hexagon ab c d ef. 
 
 These several positions may be regarded as typical, all others 
 beins: referable to the same. 
 
 Fig. 30 ^ 
 
 ^ Let us now suppose we have before us the 
 
 ""---^I n^ X ""^--^^^ ^ hexagonal prism standing on one of its bases, 
 
 la P 
 
 the upper base being visible : we should draw 
 the nearest line of that visible base a b (Fig. 
 30). Next, by observation, determine the posi- 
 tion of a c, the nearest side of the central rect- 
 angle, and compare its length with a b (in this 
 case, it is two-thirds of a b). Determine ^ ^ in 
 the same way, and draw c d and the diagonals : 
 through the center draw the diameter parallel 
 to a b and c d. Now, since ^^ is longer than 7i, make eg- longer 
 than ^^ by the same ratio, and nf shorter than n by the same ratio. 
 
THE HEXAGON AND THE HEXAGONAL PRISM. 
 
 43 
 
 Fig. 31 
 
 and complete the hexagon. Draw the vertical lines of the prism. 
 
 Make / h parallel to a b ; hi conver- 
 ging with bf, c e, and a d, j k with a e 
 
 and f d. The amount of convergence 
 
 is to be determined by observation. 
 Let us next suppose the hexagonal 
 
 prism placed in a position oblique to 
 
 the eye, and inclined ; a b representing 
 
 the nearest line of the central rect- 
 angle of the visible base (Fig. 31) : 
 
 observe and draw the two side-lines of 
 
 the same rectangle, ac and b d, and 
 
 join c d ; drawing the diagonals, we 
 
 find the center, through which, as before, draw the diameter paral- 
 lel to the lines a b and c d. 
 We fix the points e and / in 
 due proportion from the two 
 central divisions of the same 
 line, and complete the hexagon. 
 Observing the inclination of 
 . the side-lines of the prism, draw 
 them in the correct position 
 I with the proper convergence. 
 Next, draw the visible lines of 
 . the invisible base, converging 
 with their respective parallels 
 of the visible base, g h with a by 
 
 g i with a e. It will be seen that there will be four systems of con- 
 
44 
 
 MODEL AND OBJECT DRAWING, 
 
 verging lines, and that u b may be taken for the initial line of the first 
 system, ^^ of the second, the diagonal b c oi the third, and the diago- 
 nal ad oi the fourth. A fifth system would be indicated by a c and 
 b dy but it is not essential. Following the method here indicated, the 
 hexagon is an easy subject to draw in all possible positions (Fig. 32). 
 
 THE CIRCLE. 
 
 A circle seen in various positions, in whole or in part, appears to 
 the eye as a circle, a straight line, an ellipse, a parabola, or as a hyper- 
 bola ; that is, mathematically speaking, as one of the conic sections. 
 
 First, A circle is seen as a true circle when the central ray of light 
 from the plane of the circle is perpendicular to that plane ; that is, 
 
 when it forms a right angle above 
 and below, and to the right and left, 
 with the plkne. Thus, let a b repre- 
 sent the side view of a circle (Fig. 
 33), and the central ray of light from 
 n form a right angle on all sides with 
 the plane of the circle. Then the circle will appear as a true circle ; 
 for if we cut the rays of light which come in the form of a cone, 
 from the circle to the eye, by a plane at //, perpendicular to the 
 central ray, we shall have a section of the cone of rays parallel to 
 the base of the cone, consequently a sub-section, and therefore similar 
 to the base, that is a circle. 
 
 Second, A circle is seen as a straight line when the rays of light 
 proceeding from the circle to the eye mov^e in the direction of the plane 
 of the circle. Let a b and a' b' , Fig. 34 and Fig. 35, represent the side 
 
 
 Tig. S3 P 
 
 __^ — 
 
 J 
 
 L— --^ — ""^^ 
 
 
 ^ 
 
 
 _^_ 
 
 
 p 
 
 ^^"^^ 
 
 -^b 
 
THE CIRCLE. 
 
 45 
 
 Fig. 34 
 
 view of a circle, with the eye placed in the direction of the plane of the 
 circle. Then the circle would appear as a straight line. In the upper 
 figure the circle is in a vertical, and in 
 the lower figure in a horizontal, posi- 
 tion. The rays from the circle to the 
 eye will be in a single plane : no part of 
 the upper or under surface, or right or 
 left surface, gives rays to the eye. 
 Hence, a section of the plane of rays at 
 PP and P'P' would be a straight line ; 
 that is, the circle, thus seen, would 
 have the appearance of a straight line. 
 
 Third, A circle is seen as an ellipse when the ray of light pro- 
 ceeds obliquely from the plane to the eye. Thus, let a b represent a 
 side view of a circle with the eye at E, and the central ray oblique to 
 the plane of the circle (Fig. 36). Then the figure of the circle, on the 
 plane of section, P P, will appear as a true ellipse. The proof of this 
 theorem will be better illustrated farther on ; but a real ocular dem- 
 onstration may be had by constructing of wood an oblique cone on 
 
 a circular base, making a cross-section 
 
 corresponding to P P. We may note 
 
 here, however, that, as the obliquity 
 
 of the central ray with the plane 
 
 of the circle increases, the diameter, 
 
 more oblique to this ray, becomes 
 
 the more foreshortened, and that the one which remains at right 
 
 angles to this ray will not be foreshortened at all. Hence, since 
 
 these two diameters are at right angles to each other, it will be evi- 
 
 IFig.Se 
 
46 
 
 MODEL AND OBJECT DRAWING. 
 
 Tig. 37 
 
 a;, - 
 
 l\ 
 
 dent, that apparently the circle becomes flattened in the direction of 
 one of its diameters, as in Fig. 37 A A, the diameter in the vertical 
 plane of the eye, a ^, the same when revolved into its oblique position 
 
 to the central ray E c. The inter- 
 section of the rays from each, on 
 the plane P P, shows their relative 
 apparent lengths A' A' without 
 foreshortening, and a^ a\ in its 
 '^^--^J oblique position, foreshortened. 
 
 We may learn still further, by the 
 examination of a cone, from which a section of the ellipse is made, 
 that the perimeter of the ellipse, in its oblique position to the central 
 ray, may be made to appear to cover and coincide exactly with the 
 circumference of the circle at right angles to the central ray. 
 
 Fig. 38. Let "Eab be a cone, and m n the section. It is a true 
 ellipse. Place the eye at E : the contour of the ellipse will appear 
 to fall against the circumference of the circle, because the rays of 
 light from the latter will pass di- 
 rectly through the former ; hence 
 they will appear to coincide. Now, 
 if an ellipse, in an oblique position, 
 may be made to coincide with a cir- 
 cle in a perpendicular position, it 
 is reasonable to suppose that a cir- 
 cle in a position oblique to the central ray may be made to coincide with 
 the outline of an ellipse at right angles to the central ray. In Fig. 38 
 the ellipse seen from the direction of P appears as a perfect ellipse. 
 While seen from the apex of the cone e, it appears as a perfect circle. 
 
 Fig. 38 P 
 
THE CIRCLE. 47 
 
 That the figure of the circle in a position oblique to the central 
 ray will appear to be a true and symmetrical ellipse is, moreover, 
 evident from the following diagrams. 
 
 In order fully to appreciate the nice conditions and relations of 
 the apparent ellipse to the parent circle, and several points of great 
 interest in all subsequent practice shown by these and following dia- 
 grams, careful study and attention to every particular is demanded 
 of the student. 
 
 Figs. 39 and 40 represent the circle in nearly the same position 
 with reference to the eye. 
 
 The first. Fig. 39, is the plan of the circle, with the eye in the same 
 horizontal plane as the circle. It shows the place of the apparent 
 diameter at a' U to the left, and nearer to the eye than the real 
 diameter : the place of the apparent diameter a b' is found by 
 locating the tangential rays a E, b' Y.\ these are drawn by bisecting 
 the line E ^, ^ being the center of the circle : taking the central point 
 thus found, as a center, with the half-length of the line as a radius, 
 describe the arc a' c U ; it will give the points a b' on the circumfer- 
 ence, from which tangential rays may be drawn to the eye. The line 
 a' b' must be the apparent diameter, because it subtends a larger visual 
 angle than any other line that can be drawn in the circle. 
 
 The second, Fig. 40, is a vertical projection of the same circle, 
 revolved a little so as to come into a position slightly oblique to the 
 eye, as indicated by the diameter m' ;/ .• m' has been moved downward 
 from m^ its position in Fig. 39, and n' upward from n. 
 
 In this position the upper face of the circle sends its rays to the 
 eye, and its image is formed on the retina. Now we wish to ascertain 
 whether that portion of the circle to the left of a! U appears to be just 
 
48 
 
 MODEL AND OBJECT DRAWING, 
 
 as wide as the larger part, to the right of that line, or whether that 
 part of the diameter m' U subtends as large an angle as the part b' n' : 
 this may be easily determined by bisecting in the usual way the 
 angle in' E ;/, the whole visual angle subtended by the diameter, and 
 through the bisecting point 3 draw a line from the eye to the diameter 
 
 Fisr. SO 
 
 m' n\ cutting it in the point b\ the place of the apparent diameter in 
 Fig. 39. It seenis, therefore, that the smaller part of the circle to the 
 left of the line a! b' appears to be just as large as the larger part of 
 the circle to the right of a! b' , and that the apparent form of the 
 circle in this position is an ellipse, with a! b' for its longer diameter, 
 dividing the ellipse into two equal parts. We may say here, that there 
 
THE CIRCLE. 
 
 49 
 
 Tig. 41 
 
 ^~-- — i— -11^ h" 
 
 is in the illustration a slight error, which will be noticed farther on, 
 but which does not, in this case, vitiate the conclusions very much. 
 
 Taking the two lines a'^ b'^ and i^' m^' on the P P, the plane of sec- 
 tion, as the longer and the shorter diameter of an ellipse, we shall 
 obtain very nearly the apparent form of the 
 circle, as seen in Fig. 41. Thus we have the 
 true figure of the apparent form of the circle 
 in this position, and by the same means we 
 
 can obtain its apparent form in all positions intermediate between 
 that in Fig. 39 and a position at right angles to the central ray. 
 
 A true picture of the circle, when seen obliquely, according to the 
 definition of a true picture given on p. 32, can only be obtained by cut- 
 ting the cone of rays from the circle to the eye by a plane perpendicular 
 to the central ray or axis of the cone of rays. Although other sections 
 than this may give ellipses, yet they will not possess the proportions 
 of the true picture (Fig. 42). Let A B be the vertical projection of 
 the circle at an angle of 45° to a line drawn from the apparent center 
 
 of the circle to 
 E, the position 
 of the eye : then 
 the oblique cone 
 of rays will be 
 formed upon the 
 circular base A 
 B. Now, all sec- 
 tions, as I, 2, 3, 4, 5, perpendicular to the axis, will present true pic- 
 tures of the circle : but, if we take an oblique section of the cone of 
 rays m n perpendicular to the plane of the circle A B, it is quite evi- 
 
 Fis. 42 
 
50 
 
 MODEL AND OBJECT DRAWING. 
 
 dent that the section can not present a true picture of the circle 
 A B ; because the section itself will be a circle. 
 
 Drawing the section rn n (Fig. 43) at right angles to the base A B, 
 and then revolving the part of the cone E in n about the axis E x, 
 through 180°, it is plain that the point m will be revolved into the 
 
 Tig. 43 
 
 position m\ and the point ;/ into the position ;/, and the line m n will 
 be found in the position ;;/ ;/, parallel to the base A B. Thus, the 
 section m' v! will be a section parallel to the base A B, and geometri- 
 cally similar : therefore it will be a circle. 
 
 The revolution of the plane of section, which is at right angles 
 to the base, through 180°, brings it into a position parallel to the 
 
THE CIRCLE, 51 
 
 same base, and shows at once that it must be a circle ; as all sections 
 of a cone parallel to the base must be similar to the base, and conse- 
 quently circles. 
 
 It will not alter the conditions, nor invalidate the conclusions at all, 
 to revolve the whole diagram about the point E, through an angle of 
 45°, so that the base, A B, will be brought into a horizontal position, 
 and the plane of section m n into a vertical position : the section m n 
 in the vertical position will still be a true circle ; and it follows, of 
 course, that it can not be a picture of the circle A B. It may, there- 
 fore, be asserted that a true picture of the circle in this oblique posi- 
 tion will be found by a section at right angles to the central ray of the 
 cone of rays, and that all other sections, not at right angles to that 
 ray, will differ, more or less, from the true picture, according to their 
 obliquity to this central ray. 
 
 The slight error in the illustration on p. 48, which results from the 
 change of position in the place of the apparent diameter in Fig. 40, 
 the circle being slightly turned into an oblique position, can now be 
 corrected if desired. This change of position of the apparent diame- 
 ter, and the method by which we may ascertain the true position of 
 the apparent diameter of the circle, when it is at any particular angle 
 of obliquity to the central ray, may be understood by reference to 
 Fig. 44. 
 
 In Fig. 44 we have the vertical projection, in ;/, of the circle in a 
 horizontal position ; the eye being at E' : in the lower figure we have 
 the plan of the circle m n, the eye being at E. With the eye in this 
 position with reference to the circle, we have already seen that the 
 apparent diameter will be at a! b'. Now, if the eye is revolved through 
 an arc of 90° to the position E'' immediately over the center of the cir- 
 
52 
 
 MODEL AND OBJECT DRAWING. 
 
 cle, it will be evident, that, in this position, the apparent diameter can 
 no longer be at a' b\ but that the apparent and the real diameters 
 will occupy one and the same place, and will be identical. If we move 
 the eye from E'^ back along the arc of 90° towards its former position. 
 
 Fig. 44 
 
 it is evident that the position of the apparent diameter will recede 
 from the position of the real diameter until it reaches the position 
 of b\ when the eye has returned to the position E', thus passing over 
 the entire space between a b and a' b'. Let us now see if we can 
 
THE CIRCLE. 53 
 
 determine the position of the apparent diameter when the eye is at 
 any particular point on this arc. 
 
 In the passage of the eye over the arc from E' to E'', it moves 
 vertically over every point of the radial line YJ d ; and, when it has 
 passed vertically over every point of the radial line E' c\ the apparent 
 diameter of the circle has receded from its extreme position c^ b' 
 to the position of the real diameter a b. Hence it follows, that, when 
 the eye has passed vertically over any particular portion of the line 
 E' c\ the apparent diameter will have passed over the same proportion 
 of the line x c, the difference between the extreme position of the 
 apparent diameter and the real diameter in the plan.* We may, there- 
 fore, find the position of the apparent diameter with the eye at any 
 given point E'^' in the arc E' E^' by drawing to E^ c^ a vertical line 
 from E^'^, the assumed position of the eye on the arc E' E'^, to D. This 
 line will divide the line E' c' into two parts, E' D and D /. Then, by 
 dividing the line x c into similar proportional parts, we can determine 
 the position of the apparent diameter with the eye at the given point. 
 
 To divide xc into proportionals similar to the divisions of E'r', 
 draw a line from E' to YJ\ produce b' a! so as to cut E' YJ' in x\ and 
 make x' c" equal and parallel to x c. From D, the point of division on 
 E' c, draw a line to YJ' cutting x^ c" proportionally to E' c' in the point 
 o". (See "Robinson's Geometry," Bk. 2, Theo. 17, et seq.) 
 
 By drawing a parallel to E'' c' from 0'' cutting x c m 0, we shall have 
 the point in ;r ^ through which we can draw a'^ b'\ the apparent diame- 
 ter of the circle in' n' or m 71, with the eye at the point E^''. 
 
 This method is true for all other positions of the eye on the arc 
 
 * This method of determining the apparent diameter is given, without entering upon 
 trigonometric principles. 
 
54 MODEL AND OBJECT DRAWING. 
 
 E' E'', since 1^" is any point in it. Hence, by combining this method 
 with one on p. 48, all error, however slight, may be eliminated from 
 that problem. 
 
 The correctness of the foregoing solution may be tested also in 
 another way. It is evident, when the eye is at E E', that if two 
 planes are drawn through the eye, and tangent to the circle on opposite 
 sides, and perpendicular to the plane of the circle, the planes will cut 
 each other in a vertical line passing through the eye, and will be tangent 
 to the circle at the extremities of the apparent diameter a! b' ; as the 
 tangent lines E b' and E a' would be the traces of these planes, and a 
 vertical line drawn through E would be the line of their intersection. 
 It is also evident, that, when the eye is at YJ\ these two planes would 
 be tangent to the circle at the extremities of the real diameter a b, and 
 that their intersection would be in a horizontal line passing through 
 E'^. Now, at any intermediate points along the arc E' E'', the intersec- 
 tion of these two planes, if extended, would cut the plane of the circle 
 extended. Thus, draw through the point E'^' a line tangent to the arc 
 E' E'^, and extend the line until it cuts the line / E' extended in J': 
 this will be the trace of the line of intersection of the two planes 
 passing through the eye at E''', and tangent to the circle at the 
 extremities of the apparent diameter a'^ b" . For if we project the point 
 y on to the horizontal plane at J, and then draw from J tangents to 
 the circle, by bisecting the line J c, and drawing an arc from its central 
 point with the half-length of the line as a radius, cutting the circum- 
 ference, the arc will pass through the two points a'^ b'\ the extremities 
 of the apparent diameter, thus showing that the two planes drawn 
 tangent to the circle, and intersecting at the eye in a line tangent 
 to the arc E' E'', at the point E''', the place of the eye, will also be 
 
THE CIRCLE, 
 
 55 
 
 tangent to the circle at the extremities of the apparent diameter 
 a f? . 
 
 Fourthy A. When a part only of a circle from a point somewhere 
 
 :m0,48 
 
 in a straight line drawn perpendicular to the plane of the circle at its 
 center is seen through a plane parallel to this line (Fig. 45). Let m n 
 be a vertical projection of the circle with the eye at E, over the cen- 
 ter c\ Then we have a cone of rays on a circle as a base, with E as 
 
56 MODEL AND OBJECT DRAWING, 
 
 the apex. Cut this cone by a plane, ]' Y parallel to the axis of the cone. 
 The rays from the circumference of the circle on this plane will trace 
 its true curve, as seen from the point E. The section will be a true 
 hyperbola^ since from geometry we learn that all sections of a cone 
 parallel to the axis will be hyperbolas. The true form of the curve 
 is projected on the horizontal plane between the points J P on the 
 straight line J P as a base. The elements of the curve are obtained 
 by laying off on J P from the points p p p and the vertical distances 
 P' \" , Y 2'\ etc., each on its respective radial lines C i, C 2, C 3, etc. 
 
 B. When a part of a circle is seen through a plane S' V parallel 
 to E 7t, m n being the circle as before, then that part of the circle traced 
 upon the plane S' V, as seen from E, will present the form of the 
 parabola^ because it is a section of the cone of rays parallel to one 
 side of the cone. From geometry we know that all such sections are 
 parabolas. The development of the curve in its true form is seen in 
 the full line V S V ; while the dotted curved line between V and V just 
 to the left of V S V is the horizontal projection of the curve of section, 
 and not its true form. The curve V S V is obtained by throwing down 
 from S' V all the elements or normals of the curve from V as a center 
 upon the horizontal nt n, and then projecting them upon the horizontal 
 plane, each upon its own normal drawn from V V. 
 
 All possible forms of the circle as seen in various positions are 
 referable to some one of the conic sections, all consequently taking 
 their places among the absolute mathematical figures. Let the student 
 thoroughly master these forms, and trust to no methods not referable 
 to fixed geometric formulas. 
 
METHOD OF DRAWING CIRCULAR OBJECTS. 57 
 
 METHOD OF DRAWING CIRCULAR OBJECTS. 
 
 The application of the principles already developed relating to the 
 circle will be found necessary whenever the student attempts to draw 
 any circular object, or objects having circular bases, such as the cylin- 
 der, cone, frustum of a cone, vases, cups, saucers, wheels, and a great 
 multitude of objects. But, in order to deal successfully with many of 
 them, it will be necessary to consider several other facts and princi- 
 ples, as applied to combinations of circles. Let us take first the 
 cylinder, as one of the simplest volumes, having two circular bases. 
 
 There are eight rules applicable to the dimensions and positions of 
 the cylinder. As the same rules apply with some slight modifications 
 to all objects having two opposite circular bases, as vases, goblets, etc., 
 they are in an eminent degree generic, and consequently important. 
 We will now consider several facts relating to the cylinder, and see 
 what deductions we can draw from them. 
 
 Firsts When the two bases of a cylijider are equally distant from the 
 eye^ both are invisible (Fig. 46). -p.^ ^g 
 
 An apparent exception to this rule would be found 
 by taking a cylinder of the dimensions of a silver 
 dollar. Placing it so as to be seen by both eyes, both 
 bases would be visible, the one to one eye, and the opposite to the 
 other ; but the rule requires that we should look with one eye only, in 
 which case the exception vanishes. 
 
 Second, The visible base of a cylinder is always nearer to the eye 
 than the invisible base. 
 
 Thirdy The visible base is always apparently longer than the invisi- 
 ble base. 
 
58 MODEL AND OBJECT DRAWING, 
 
 Fourth, The invisible base is always wider in proportion to its length 
 than the visible base. 
 
 The last two rules may be stated thus : The visible base is always 
 longer and narrower, and the invisible base is always shorter and pro- 
 portionately wider. 
 
 Fifth, The longer diameters of the ellipses, which represent the 
 bases of a cylinder, are always perpendicular to the axis of the cylin- 
 der. 
 
 Sixth, The shorter diameters of the ellipses are always coincident 
 with the axis of the cylinder. 
 
 Seventh, The side-lines of a cylinder always appear to converge in 
 the direction of the invisible base. 
 
 Eighth, When a cylinder is in a vertical positio7i, the plane of 
 delineation is supposed to be vertical also ; and the side-lines are drawn 
 vertical and parallel, and of course without convergence, in accordance 
 with the general practice in all architectural subjects. 
 
 In illustration of this last statement, reference may be made to 
 Geometrical Perspective, where all regular polygons which are parallel 
 to the picture plane are represented, in the picture, by regular polygons. 
 In all architectural subjects, the plane of delineation is always sup- 
 posed to be vertical. 
 
 To illustrate the third rule, that the ellipse representing the visible 
 base of a cylinder is always longer than the ellipse representing the 
 invisible base, we have only to consider that the diameter of the cylin- 
 der is a constant quantity, and therefore the same at either end. If it 
 is the same constant quantity at unequal distances from the eye, the 
 nearer end must appear the longer (see illustration on p. 33), as in 
 Fig. 47. Let a c represent the axis of a cylinder, b d the nearer, and 
 
METHOD OF DRAWING CIRCULAR OBJECTS, 
 
 59 
 
 Tig. 47 
 
 ef the farther diameter : then b d will be longer than ef^ because a 
 nearer line appears longer than an equal line more distant. 
 
 The rule that the invisible base is always 
 wider in proportion to its length than the 
 visible base, will be readily understood by 
 observing the following diagram, Fig. 48, 
 where E represents the position of the eye, 
 and a e^ bf, eg, and d h, four equal and par- 
 allel circles ; the lines a e, b f, etc., showing 
 the actual width from front to back. By 
 drawing rays from each of these circles to 
 the eye, it is evident that the circle a e will have no apparent width, 
 because it is in the same plane as the eye ; and consequently it 
 appears as a straight line : the circle bf will have some apparent 
 width ; while eg will 
 appear still wider, ^^s-As Y"^^^^ 
 and dh widest of 
 all. For, if we in- 
 terpose the trans- 
 parent plane T P, 
 the relative appar- 
 ent width of the 
 several circles will 
 be expressed by the 
 distances U f\ e' g\ 
 and d' Ji!y of which 
 
 d' Ji! is nearly twice as long as b' f ; and dh, of which d' h' is the appar- 
 ent width, is the most distant from the eye : hence the rule. 
 
 /"■-'«. 
 
 / 
 
 
 \1 
 
 1 
 
 c' 
 
6o 
 
 MODEL AND OBJECT DRAWING, 
 
 --/J^ 
 
 ms'^9 
 
 It will be seen from the diagram that the rays of light come more 
 directly from the surface of the circle dh than from either of the 
 others. The same holds good for every invisible base of a cylinder as 
 compared with the visible base in any possible position. The rule, 
 that the longer diameters of the ellipses are always perpendicular to 
 the axis, may be made clear to the pupil by walking around a circle 
 
 and observing its 
 greatest apparent 
 length. Let C be 
 the center of the 
 circle, and E E^ E'', 
 Fig. 49, represent 
 the successive 
 /' ^^-'^ positions of the 
 
 / / ^^^^ eye : the lon2:er 
 
 / y ^^,-^' diameters or major 
 
 ' / ^^'''' axes of the ellipses 
 
 ^' will join the tan- 
 
 gential rays ; i.e., 
 from E the longer 
 diameter will be at a b, from E' the longer diameter will be at c d, from E'', 
 at ef: and in each case the apparent axis of the cylinder of which this 
 circle is the base will appear to be perpendicular to these diameters at 
 their central points, because it is perpendicular to the plane in which they 
 are. This is a matter that should be determined by observation, by walk- 
 ing round the circle and noticing how the apparent diameter seems to 
 follow, keeping its position perpendicular to the central ray. And so 
 it will be for all possible positions in which the cylinder may be placed. 
 
 Ef> 
 
METHOD OF DRAWING CIRCULAR OBJECTS. 6i 
 
 The sixth rule, that the shorter diameters are coincident with the 
 apparent axis of the cylinder, may be readily understood from the fact, 
 that, if the longer diameter is perpendicular to the axis, the shorter 
 must be coincident with the axis, because it is perpendicular to the 
 longer diameter. The truth of the proposition should be confirmed 
 by observing the cylinder in various positions. 
 
 That the side-lines of a cylinder will always appear to converge in the 
 direction of the invisible base, is evident from the fact, that, the apparent 
 diameters of the cylinder being geometrically equal, the more distant 
 will appear the shorter : hence, as we have seen, the invisible base is 
 apparently shorter ; and the lines connecting the extremities of the 
 two bases will appear to converge in the direction of the invisible base. 
 
 The eighth rule, in regard to harmonizing the fundamental princi- 
 ples of model-drawing with architectural methods, where all vertical 
 lines are drawn vertical in the picture : we call attention to the fact, 
 that the principles already laid down have no reference to merely 
 vertical or horizontal positions, but simply relate to absolute relations 
 of the object to the eye in all possible positions, with the plane of 
 section of the rays, that is, the plane of perspective, perpendicular 
 to the central ray of light. 
 
 In architectural methods the plane of section is supposed to be 
 parallel to the vertical lines in the object; and, of course, the central 
 ray would be supposed to be horizontal. This would not always be 
 really the case. This point presents no difficulty to the student who 
 makes himself thoroughly acquainted with the principles here deduced. 
 
 It should be observed here, that the differences in the lengths or 
 breadths of the two bases of a cylinder are inversely proportional to 
 the distance of the eye : thus, if the eye is at an infinite distance, there 
 
62 MODEL AND OBJECT DRAWING. 
 
 would be no apparent difference in the length and breadth of the 
 ellipses representing the bases, because the rays of light would be 
 practically parallel. So, when the distance is great, the difference is 
 little ; and, when the distance is little, the difference is great. The 
 same principle, for the same general reasons, will be observed in 
 regard to the convergence of lines. 
 
 It follows, from the above statement of fact, that every drawing of 
 models, and every picture, can be best seen from one particular point, 
 and will appear accurate from no other point of view. Hence it 
 follows, as a matter of necessity, that the spectator, at the proper 
 distance from the drawing, should place his eye at the point from 
 which all the lines can be seen in their true proportion. 
 
 Having deduced our principles and rules, let us now place the 
 cylinder in a vertical position, with the upper base visible. First draw 
 the apparent axis, or, as we may call it, ^' the axis; " 
 ^ — 1^ it always being understood that we do not mean the 
 real axis of the cylinder. In this case the axis A B, 
 Fig. 50, is drawn in a vertical position of any length 
 desired. Compare the length of the upper base 
 with the height of the cylinder. Let us suppose 
 it to be one-half of the axis. Divide the axis A B 
 B into two equal parts, and the upper half similarly, 
 
 and on either side of A measure off horizontally a 
 quarter-length thus obtained, on a line perpendicular to A B. Next 
 compare the apparent width of the ellipse with its length : suppose, in 
 this case, it is found to be four times as long as it is wide. Divide, 
 therefore, one-half oi c d into two equal parts by a dot at i, and the 
 quarter oi c d into halves by a dot at 2. Place this eighth above and 
 
 Tig. SO 
 
ME2HOD OF DRAWING CIRCULAR OBJECTS. 
 
 63 
 
 ■e 
 
 Fig. 51 
 
 ^< 
 
 Fig. 5U 
 
 below the middle point of the line c d, the two eighths making the 
 
 quarter required (Fig. 51). Thus we have the length of the shorter 
 
 diameter of the ellipse, as well as its position. It only 
 
 remains to draw the curve. 
 
 The lower ellipse should be found in the same way ; 
 
 observing, however, that it must be wider in proportion 
 
 to. its length (which, in this case, is the same as that 
 
 of the upper) than the upper ellipse (see Rule 8). 
 
 Thus, if the width of the upper ellipse is one-fourth 
 
 of its length, the width of the lower ellipse must be 
 
 more than one-fourth of its length. The whole of 
 the lower ellipse should be indicated, the farther 
 half by a dotted or shadowed line only (Fig. 52). 
 Finally the side-lines may be drawn as tangents 
 to the two ellipses, thus completing the drawing 
 of the model. 
 
 If, now, the cylinder is placed on its side, so 
 that it appears in an oblique position, we must 
 first observe the apparent position of the axis, 
 
 comparing its direction with the horizontal, 
 
 with which we will suppose it to make an 
 
 angle of 20° (Fig. 53). Draw, as the axis 
 
 in this position, the line a b oi any length : 
 
 observe how long the nearer ellipse is in 
 
 comparison with the axis. To do this, hold 
 
 the pencil at arm's length at right angles to 
 
 a line drawn from the eye to the center of the 
 
 cylinder, perpendicular to its axis, so that it corresponds to the longer 
 
 -H-+- 
 
 Fig. S3 
 
64 MODEL AND OBJECT DRAWING. 
 
 diameter of the ellipse, and determine its length by moving the thumb 
 along on the side of the pencil towards the end. Having thus 
 obtained the apparent length of the nearest ellipse, turn the hand, 
 keeping the pencil at right angles to the central ray till it coincides 
 with the axis of the cylinder, with which compare the length of the 
 ellipse. We will suppose it to be two-thirds of the axis. Place the 
 point c to mark this : then ac \^ the length of the longer diameter 
 of the ellipse. Divide a c into two equal parts, and, drawing a 
 line perpendicular to ab, 2X a mark off the points d and n respec- 
 tively above and below a, each at a distance equal to the half of a c. 
 Proceed to find, by means of the pencil as before, the shorter diameter. 
 Suppose it to be one-third of the longer diameter. Divide, therefore, 
 dn into three equal parts by points i and 2. Since the shorter diame- 
 ter coincides with the axis of the cylinder, produce the axis, upon 
 which mark the points and w, each half a third from a. This gives 
 the position and the length of the shorter diameter. Then draw the 
 curve of the ellipse through the four points dm n 0. 
 
 Next ascertain the length of the invisible ellipse ; it must be less 
 than that of the visible : measurement with the pencil as before will 
 so determine it. Do not guess at it. Put aside guess-work until 
 thorough knowledge is obtained. Make it as much shorter as it 
 seems to be, and then proceed to estimate the shorter diameter of 
 the same by observing the half-ellipse which is visible. When these 
 points have been determined, complete the ellipse, drawing the whole 
 curve, the invisible half with a dotted line. Lastly complete the fig- 
 ure of the cylinder in its oblique position by drawing the sides tangent 
 to the ellipses. 
 
 The foregoing explanations and principles will enable the student 
 
METHOD OF DRAWING CIRCULAR OBJECTS. 65 
 
 attentive to them to draw the cylinder in any possible position it may 
 be placed. Let no accident of position or relation trick you out of 
 your knowledge of principles and facts. 
 
 There are many necessary modifications of the above principles 
 when we come to draw vases. The same general laws prevail, but 
 they are modified in their application. 
 
 For instance, the bases may not have the same actual diameters as 
 in the case of the cylinder. The same law, however, as to position 
 and masrnitude exists. 
 
 ^ 'Fig. S4 
 
 Thus, if the bases, or the circles at the top and bottom 
 of a vase, are unequal, the lower being the larger, still 
 the rule applies ; and the invisible will appear propoi-tion- 
 ately shorter and proportionately wider than the visible / ! 
 base (Fig. 54). The same principle will also hold good for 
 all the minor bands of ornament, if such there are. Thus, 
 as you move in the direction of the invisible base, all ellipses must 
 appear proportionately shorter and wider ; and this is true whether 
 they are actually larger or smaller ellipses than the visible one. 
 
 Other applications of this law are found in 
 Fig.ss /K drawing the cone, and some bands on vases. 
 
 / i_ \ Take the case of two parallel circles, sec- 
 
 A;^^£~V tions of a cone. It is evident that the ellipse 
 
 ^/l I _ '\ 43, representing the upper circle in Fig. 55, will 
 
 \-^_J___^-^ be proportionately longer and narrower than the 
 
 6 
 
 lower ellipse, i 2, according to the rule ; because, 
 if the top of the cone were removed, it would be the visible one. 
 Now, from the nature of the cone, we may be able to see more than 
 half of the curve of the ellipse if the eye is considerably above its 
 
66 
 
 MODEL AND OBJECT DRAWING, 
 
 plane; as in Fig. 56 we see all of the surface of the cone in front of 
 the line i 2, which joins the points where the lines to the apex are 
 tangent to the ellipse, and also much more than half of the curve of 
 the ellipse. So when we have two parallel ellipses, as in Fig. 57, we 
 may find that we see more than half the ellipses. It is possible that 
 the width of the band may appear to be greater at the sides than 
 in front, on account of the obliquity of the surface of the band at the 
 
 Fig. 57 
 
 Fig. 66 
 
 Fig. 58 
 
 front or middle point tending to foreshorten its width at that point ; 
 while the width at the sides will not appear to be foreshortened at all. 
 
 Take, again, the rim of a bowl, as Fig. 58. The width of the rim 
 may appear greatest at the sides, nothing at the back, and interme- 
 diate at the front, or as wide or wider at the front according to the 
 angle of obliquity, if it happens to be a portion of the surface of a 
 cone with its apex at a. 
 
 Quite an opposite modification would occur in the case of a surface- 
 band on the sides of a vase or bowl seen below the eye, as in Figs. 59 
 and 60. 
 
 In this case the band a b would seem to be widest at the front, 
 gradually tapering towards the sides, as shown in the figure. This is 
 because the band is practically on a section of a cone, the slant height 
 
METHOD OF DRAWING CIRCULAR OBJECTS. 
 
 67 
 
 Fig. 59 
 
 Fig' GO 
 
 Flg.Gl 
 
 of which is very obUque to the central ray, the opposite of the condi- 
 tion in the rim of the bowl. 
 
 Another very important application of the ap- 
 parent forms of circles is found 
 in the drawing of rims and hoops, 
 or raised bands. As to rims, we 
 may have a vessel, as in Fig. 61. 
 The rim would in this case present 
 a varying quantity from front to sides, and from sides to back. 
 Thus, at the sides its thickness would not appear to be foreshortened 
 in the least, as the line expressing its thickness would be at right 
 angles to the rays of light to the eye : but, 
 at the front and back, the reverse of this 
 would be true ; and the lines expressing the 
 thickness would be proportionately fore- 
 shortened, provided the inner and the outer 
 ellipses were in the same plane ; but the 
 front thickness, being nearer to the eye, 
 would appear greater than the thickness at 
 the back. 
 
 The principle will be at once seen if we 
 consider the rim to be one-quarter of the 
 
 diameter across the top of the vessel (Fig. 
 62). Then we shall have to take a quarter 
 from the ends of each diameter of the circle 
 represented by the larger ellipse, and 
 through these points draw the curve, a i 
 and b 2, on the diameter a b, will be real quarters of the line ; but on 
 
 Fig. €9 
 
68 
 
 MODEL AND OBJECT DRAWING. 
 
 the diameter c d, the real quarters being at unequal distances from the 
 eye, the farthest quarter will appear much smaller than the nearest 
 one. The quarter-points on the long diameter may be placed without 
 trouble, but those on the shorter diameter are more hable to error. 
 
 The precise difficulty in this division will be hereafter considered. 
 It will be readily understood, since the shorter diameter of the circle, 
 c d, was divided into four equal parts, that there will be presented to 
 the eye a series of diminishing quantities, the first or nearest of which 
 will appear to be the largest, and the farthest will appear to be the 
 smallest ; so that we should have a i=^ 2, while c 3 would be greater 
 than d 4. Hence the thickness of all rims having the faces at right 
 angles to the axis appears greatest at the sides or at the ends of the 
 major axis of the ellipse, and the rims appear thicker in front than 
 on the back. Thus we have the rule for rims. The appare7it thick- 
 ness of a rim at the ends of the short diameter bears the same p7'oportion 
 to the thichtess at the ends of the long diameter as exists betzveen the 
 
 lo7ig and the short diameters them- 
 selves. 
 
 The application of the fore- 
 going analysis is required for a 
 large class of objects (Fig. 63). 
 Take a hoop, for instance : by 
 the rule given, its upper rim is 
 readily drawn ; but the apparent 
 varying depth of the hoop from top to bottom requires a new applica- 
 tion of the same analysis. All difficulty will disappear if we draw 
 the five vertical lines, i, 2, 3, 4, 5, and note, that, by reason of their 
 increasing remoteness, i is the longest, being nearest ; while 2 is 
 
 Fig. 63 
 
METHOD OF DRAWING CIRCULAR OBJECTS. 69 
 
 shorter than i, and longer than either of the other three ; 3 is shorter 
 than 2, longer than 4, and intermediate between i and 5 ; and 4 is 
 shorter than 3 and longer than 5 ; 5 is the shortest of the series, 
 because it is at the greatest distance. 
 
 Thus the five lines representing the same constant quantity appear 
 unequal on account of their unequal distances from the eye. 
 
 A thoughtless pupil will always fail in these particulars, hence the 
 necessity of thorough work on these points. 
 
 The rim is an element which will require some further explanation 
 for its complete comprehension. Let c and c' be the center of two 
 concentric circles. The intermediate space between the two circum- 
 ferences is what we wish to draw. Placing the eye at E, let the circles 
 be tipped obliquely, as in Fig. 64. Drawing the outer or tangential 
 rays from the eye to the larger circle, we find the points of tangency 
 to be a! and b': joining these two points by a straight line, we shall 
 have the position of the major axis of the larger ellipse ; it will appear 
 on the perspective plane at a" b" . Now, if we join the points of 
 tangency of the outer rays of the inner circle d! e\ we shall have the 
 position of the major axis of the inner circle, seen in Fig. 64 on T P, 
 
 From Fig. 64 it will be seen that the foreshortened diameter of the 
 circle nm and all its points and quantities, viz., no, Cy c r, rm, will 
 be obtained in their true proportions on the intersecting plane T P. 
 
 Now construct Fig. 65 by making jf 0'" r" m'" and a'" d!" e'" b'" 
 the same as the corresponding quantities in Fig. 64. 
 
 Draw the two ellipses in their respective positions, as indicated by 
 these lines and points, and the true apparent form of the rim will be 
 obtained, as seen through the transparent plane T P, from E. The 
 
70 
 
 MODEL AND OBJECT DRAWING. 
 
 principle here developed holds good in the apparent forms of all rings 
 and rims whose surfaces reside in a single plane, and the application 
 of the principle becomes very frequent in the drawing of models. 
 
 It will be apparent from the foregoing analysis, that, to draw a 
 
 wheel in an oblique position, the hub can not be placed in the middle 
 of the ellipse which represents the full size of the wheel, but must be 
 pushed back of the apparent center of the wheel : the outer ellipse of 
 the hub will be somewhat off the center, because it projects. If the 
 
THE DRAWING OF ELLIPSES. 71 
 
 hub is long, there would be another modification of the form ; but, 
 when the object is placed before the draughtsman, there is no trouble 
 in reading the form by means of the explanation already given. 
 
 THE DRAWING OF ELLIPSES. 
 
 Ellipses seen in various positions appear under several modifica- 
 tions, some of which it is important to notice. First, an elliptical 
 form, as for instance an elliptical dish, seen 
 obliquely from a point in a plane which contains 
 
 '""V- 
 '^X-; 
 
 m 
 
 the shorter diameter of the ellipse (Fig. 66) ; 
 
 that is, the eye and the shorter diameter of the 
 
 ellipse being in the same vertical plane perpen- m 
 
 dicular to the longer diameter. The ellipse will ^t'ls- ^Q 
 
 appear to diminish in width, according to the \J 
 
 a 
 
 degree of obliquity. 
 
 Thus, let a 5 be an ellipse, n m being in the same plane a^ the 
 ellipse ; let E be the eye as far above that plane as m E. Then the 
 diameter a b will not appear to be foreshortened, but will appear of its 
 
 full length, while the 
 ^'■-■^<-^-.^_ shorter diameter cd 
 
 ^^S'^"^ will appear to be 
 
 """■^---..^ foreshortened; and, 
 
 the nearer the eye 
 is brought to m, the 
 shorter will the line 
 cd appear; the higher above m the eye is placed, the less foreshort- 
 ened the line c d becomes. Again (Fig. 6f}, let ;;^ be a point in the 
 
 -■■>. fe 
 
 3' 
 
72 MODEL AND OBJECT DRAWING, 
 
 extended plane of the ellipse abed, and E the position of the eye 
 above the point m, at a distance equal to the line E ;;2. Then d e, 
 the shorter diameter, will not appear to be foreshortened ; its true 
 length being perpendicular to the central ray of light from itself to 
 the eye. But a c, the longer diameter of the ellipse, one end being 
 nearer to the eye than the other, becomes foreshortened ; the amount 
 of foreshortening depending upon the nearness of the eye to the point 
 m. It will be observed, that to foreshorten the longer diameter of the 
 ellipse, the shorter diameter remaining the same, will have the effect 
 of bringing the ellipse more nearly to the form of a circle. 
 
 It therefore follows, that if the longer diameter of an 
 ellipse appears to be foreshortened, so as to make it seem 
 just equal to the shorter diameter, the ellipse will ap- 
 pear to be a perfect circle, and must be so drawn. 
 
 It will be seen, that the apparent form of an ellip- 
 tical dish might be represented as having a perfect circle 
 for the outline of the upper ellipse. (See Fig. 6S.) 
 
 DRAWING THE TRIANGLE AND TRIANGULAR FRAMES. 
 
 In drawing a triangle in an oblique position, it is only necessary to 
 find by observation the apparent inclinations and lengths of the three 
 lines, and to place them in their true positions, according to the read- 
 ing of the same-. 
 
 But, in relation to the triangular frame, there are a few points 
 requiring notice, in order to secure ready execution and accurate 
 work. 
 
 Let abh^ the position of the lower or base front-line of a triangu- 
 
DRAWING THE TRIANGLE AND TRIANGULAR FRAMES. 73 
 
 Fis.69 
 
 lar frame standing upon a horizontal plane (Fig. 69). First find the 
 apparent position of the central point of the line a b, by holding 
 the pencil vertically against the apex of the triangle c, noticing where 
 the point n falls on a b: compare the length 
 of c n with a b, and thus determine the point 
 c. Draw c n, then c a and c b, completing the 
 face of the triangular frame. Having deter- 
 mined their inclinations, draw c e and a dy 
 observing that they are convergent lines : 
 determine the amount of convergence, observe 
 the length of c e, and draw d e convergent with 
 a c in the direction of c. Now find the cen- 
 ters of the two sides a c and c by 2X 20i\^ p, and 
 draw from each of these two points dotted lines to the opposite angles. 
 
 Determine the width of the frame as compared with the line a b. 
 Let us suppose it to be one-sixth of that line : divide a n, half of the 
 line a b, into three parts, so that each part will represent the apparent 
 length of an equal third of the line a n, placing the points of these 
 divisions at i, 2. Draw from i the line i h, convergent with a c, in the 
 direction of c ; it will cut the vertical line en in h: draw kf, con- 
 vergent with c b, in the direction of b ; draw the line ap bisecting the 
 angle cab ; it will cut the line i h in g: draw ^2, convergent with 
 a b, in the direction of b. These lines will complete the right face of 
 the frame. 
 
 Extend f h to the point 3, and draw 3 4, convergent with c e and 
 a dy fixing the point 4. From 4 draw a dotted line convergent with 
 c b and hf, and fix the point 5. From 5 draw the line 5 6 convergent 
 with a b and ^2, completing the inner visible surface of the frame. 
 
74 
 
 MODEL AND OBJECT DRAWING. 
 
 The method here given for drawing the triangular frame in this 
 position will sufficiently indicate the method to be pursued in all other 
 possible positions. It is always important that the student should 
 determine and keep in mind the different sets of convergent lines, 
 always being sure to determine the direction of their convergence. 
 
 ■Fie. "70 
 
 THE FRAME-CUBE. 
 
 Construct the outline as in the case of a solid cube, a b being the 
 nearest vertical line (Fig. 70). In the first place determine how much 
 of this line represents the apparent width of the vertical piece of the 
 frame on the left side. If it is one-sixth, divide the line a b into as 
 many equal parts, placing the points i 2 ; now draw lines, both to the 
 right and to the left, from each of these two points, convergent with 
 
 a d and b c, and with a e and bf respec- 
 tively : draw the diagonals af, ac, be, 
 b d. These diagonals, cutting the lines 
 drawn from i and 2 to the right and 
 left, will determine the points h, k, /,/, 
 /, m, n, Oy from which complete the 
 inner squares of the right and left faces 
 of the frame. It will be observed, that, 
 to secure all the varying dimensions of 
 the framework, only one measurement 
 need be determined ; viz., in Xy the 
 apparent width of the nearest upright standard. From the determi- 
 nation of this one quantity, all the other remaining dimensions follow 
 as a matter of course, by means of the diagonals, and of the converging 
 
DRAWING THE SINGLE CROSS. 
 
 75 
 
 sets of lines. Extend m I to 3, and ^ >^ to 4, and draw lines from 3 
 and from 4 convergent respectively with a d and a e. Then draw the 
 diagonals of the upper face of the cube. Where these diagonals cut 
 the lines from 3 and 4, fix the angles of the inner square, as in the 
 case of the two side faces, and complete the square. 
 
 Now draw from n^ m, and s lines convergent with b Cy and from 
 k,jy and t lines convergent with bf: draw vertical lines from rand/, 
 and also from ^, u, and v, as far as visible. If other inner lines of the 
 frame are visible, as, for instance, lines from z and j/, they may be 
 represented with their proper convergence. The drawing of the 
 frame-cube will not be found difficult if the method here indicated is 
 diligently followed. 
 
 With this model the danger is, that a pupil will undertake to guess 
 at some things without strictly observing them, and following the 
 order and method here laid down. Such efforts will generally lead, 
 with a great loss of time, to an entire failure. 
 
 DRAWING THE SINGLE CROSS. 
 
 JFig-. 7i 
 
 Let it be in either a vertical, horizontal, or an inclined position ; 
 first draw the squares, 
 a b c dy ci b' c' d\ and 
 d'b" c" d" (Figs. 71, 72, 
 and 73), to inclose the 
 cross in the several posi- 
 tions. Next draw the di- 
 agonals to these squares, 
 and take the apparent middle division of one side of a square equal 
 
 a 
 
76 
 
 MODEL AND OBJECT DRAWING. 
 
 to the thickness of the arms. From these points draw lines through 
 the squares, parallel to the adjacent sides, cutting the diagonals in 
 points I, 2, 3, 4 : through these points draw two lines parallel to, or 
 converging with, the other two lines, as the case may be ; this will 
 
 Tig. 72 
 
 Xiff. 73 
 
 /;f 
 
 '~~~/ 
 
 /-V7 
 
 / 
 
 / \ 
 
 / 
 
 
 / 
 
 / \ 
 
 / 
 
 / / 
 
 / 
 
 / \ 
 
 /, 
 
 V / 
 
 / \ ' 
 
 -^ 7 
 
 r* — A 
 
 r^~~~~/~~ 
 
 ^io// 
 
 ~y~ 
 
 
 
 / v^ 
 
 /\""' 
 
 
 - / 
 
 //V 
 
 tp 
 
 y 
 
 / 
 \ / 
 \ / 
 
 '■s^-'<-.L 
 
 // J 
 
 / 
 
 \ / 
 
 0" 52 
 
 '^. 
 
 •■«,,^ 
 
 J:m 
 
 
 
 '^^ 
 
 -~'-'K<^ 
 
 -r-W 
 
 -</ / 
 
 complete the face of the cross. Next, in a similar manner, draw 
 lines from the points 5, 6, 7, 8, 9, 10, parallel or converging, to indi- 
 cate the thickness of the cross, and then complete the drawing by 
 the lines which make up the back face of the cross. Care must be 
 taken to draw each line with its own system of parallels, and with its 
 proper convergence, where there is any. 
 
 DRAWING THE DOUBLE CROSS. 
 
 The double cross naturally comes after the frame-cube, and it 
 should be drawn at least in two positions. First place it with one 
 shaft upright and the other two horizontal : draw, first, a b, the nearest 
 
DRAWING THE DOUBLE CROSS, 
 
 77 
 
 vertical of the upright shaft (Fig. 74). Compare the apparent thickness 
 of either of the horizontal arms of the shaft with the line a b. Let us 
 suppose it to be one-seventh : then divide the vertical line, a b, into 
 seven equal parts ; and from points a and b marking the middle division, 
 draw the lines 2 and 3, indicating the positions of the two horizontal 
 arms, observing the true proportion, taking care to make the nearer 
 longer than the farther arms. 
 Having drawn the lines 2 and 
 3, the inclination of every 
 other line in the drawing will 
 have been determined when 
 the amount of convergence of 
 the other lines has been fixed. 
 Draw the two remaining ver- 
 tical lines, 4 and 5, and the 
 apparent form of the square 
 on the upper end of the ver- 
 tical shaft : then construct the 
 two visible sides of the square 
 at the lower end. In the same 
 manner draw the visible ends of the horizontal shafts, and the two 
 visible lines of the invisible squares of the same shafts. It will be 
 seen that there will be, in this position of the double cross, two sets 
 of convergent lines, nine lines in each set : one set, i, 2, 3, 4, etc., 
 will vanish to the left ; and the other set, i\ 2', 3', 4^ etc., to the 
 right. To draw this model in this and in the following position 
 requires strict attention and close observation on the part of the 
 student. 
 
^S MODEL AND OBJECT DRAWING. 
 
 The next position in which we will suppose this model to be placed 
 will be that in which it rests upon three of its arms, one being directly 
 in front. In this case we should draw first the line a by indicating the 
 position of the arm, one end of which is nearest to us. The line, we 
 will suppose, appears to be nearly vertical, leaning a little to the right. 
 Divide ^ /^ so as to get the central division, as in the last case, observ- 
 ing that the seven equal divisions of the line a b will present to the 
 eye a series of diminishing quantities from b to a, because the line is 
 receding. Take the middle seventh for the thickness of the lateral 
 piece : draw, with their true inclinations, the lines c and d. Observe 
 carefully, and make these arms in their true relative proportion 
 (Fig. 75). 
 
 Complete the figure by drawing all the subordinate lines, each con- 
 verging with its own system. In this case there will be three sys- 
 tems of convergence, with nine lines in each system. It will be seen 
 that the lines a b, c, and d are lines belonging to the three different 
 systems of convergence ; and each may be considered the leading line 
 in its own system. The first set of lines, converging upward, is i, 2, 
 3, 4, etc. ; the second set, converging downward to the left, is i\ 2', 3', 
 4', etc. The third set, converging downward to the right, is i", 2'', 
 3'', 4'', etc. If the drawing has been accurately made, the curve of 
 an ellipse can be drawn through the eight points at the ends of the 
 four arms. The curve may be lightly sketched, as an aid in the con- 
 struction of the drawing. 
 
 Having studied the main principles of model-drawing from an 
 analysis of the geometrical conditions under which various forms 
 appear, these principles must be put into practice by the use of the 
 models. For this purpose each model should be drawn carefully in 
 
 i 
 
DRAWING THE DOUBLE CROSS. 
 
 79 
 
 several positions. Practice should be kept up until the student is 
 able to draw at sight any model in any possible position. No 
 reliance whatever should be placed upon the practice of making 
 
 copies of drawings of models. It is only so much time thrown away. 
 One might as well copy a poem in order to learn how to compose one. 
 And, although copying pictures of models is at the present time much 
 
8o 
 
 MODEL AND OBJECT DRAWING. 
 
 \ 
 
 ""^ 
 
 Ht;;^ 
 
 ^^ 
 
 [^ 
 
 \\ 
 
 \ 
 
 
 / 
 
 3 
 
 10 
 
 
 
 \ 
 
 / 
 
 /s 
 
 / 
 
 
 
 
 / 
 
 
 
 / 
 
 
 / 
 
 \ 
 
 
 
 --!2 
 
 
 ^A^ 
 
 
 
 
 
 9 
 \ 
 
 7... 
 
 
 "^ 
 
 
 
 ^ 
 
 practiced in many of our public schools, I am satisfied that the pupils 
 
 learn less and less of model-drawing as the 
 practice continues. 
 
 We have already seen the use of the 
 diagonals when drawing the frame-cube (Fig. 
 "jS). Take now the frame-square, and draw it 
 in several positions, drawing the lines in the 
 order of the numerals. Having placed the 
 first six lines, fix the points 7 and 8 by com- 
 paring the width of the pieces of the frame 
 with the line i, and draw lines from them con- 
 verging with 2 and 4. They will cut the 
 diagonals in points 9, 10, 11, and 12: these 
 points determine the inner lines of the frame 
 (Fig. 'jf). Place the points 13 and 14 by drawing lines from 7 and 
 8 converging with the end- 
 lines, which give the thick- 
 ness : draw also the visible 
 inner lines from 11 and 12 
 with the same convergence, 
 if there is any in this set of 
 lines. From 13 in the up- 
 right, and from 14 in the in- 
 clined figure, draw the visi- 
 ble line on the back side : 
 this will cut the line from 11 
 or 12, and give the thickness 
 on the inside. The remaining lines will follow in their appropriate 
 
DRAWING THE DOUBLE CROSS, 
 
 8i 
 
 Tig. 78 
 
 places without any difficulty. The success of the drawing will depend 
 upon following attentively the order here given. There is another 
 application of the use of diagonals of a rectangle in sketching build- 
 ings, which we may notice here (Fig. yZ). 
 
 Let us suppose we have drawn the vertical rectangle i, 2, 3, 4, 
 representing the end of a house, and that the gable, or point of the 
 roof, is vertical with the real center of the rectangular end : by draw- 
 ing the diagonals 5 and 6, we find the real center of the rectangle, 
 and from it draw a vertical line ; the angle of the roof will be some- 
 where on this line. Find the alti- 
 tude of the roof by comparison 
 with line i, and draw to the upper 
 angles of the end-lines 8 and 9. 
 
 As the roof projects over the 
 ends, the line of the ridge can be 
 drawn, and the projection made as 
 
 indicated in the drawing. The center of the ground-plan, or of the front 
 of the house, can be obtained in the same way, for the purpose of pla- 
 cing the front door, or any central feature ; and these rectangles can be 
 divided into halves, quarters, and eighths, etc., by means of the diagonals, 
 for the purpose of placing windovv^s, or other features of the building. 
 
 Observe always that the intersection of the diagonals of a rectangle 
 in any position, perspectively represented, gives the real center of the 
 rectajzgle, and not the apparent center. The place of the chimney, if 
 it is in the middle of the roof from one end to the other, can be 
 placed by drawing the diagonals on the roof and through their inter- 
 section, drawing the line upward convergent with the ends of the 
 roof 8 and 10 : this line will cut the ridge in the center. 
 
S2 
 
 MODEL AND OBJECT DRAWING. 
 
 Place the cube in two or three positions, and draw it at sight. 
 First, in a vertical position, on a horizontal plane, a little below the 
 
 Tig. 80 
 
 eye. There will be two sets of converging lines, with three lines in 
 each set, i, 2, 3, to the left, and i', 2', 3', to the right (Figs. 79 and 80). 
 
 Second, in an oblique position, drawing the lines in the order of the 
 numbers, observing the three sets of converging lines, i, 2, 3, and i\ 2', 
 3', as in the figure above, and i^\ 2" , 3'', converging downward. 
 
 Third, place the model in a vertical position, showing the right or 
 
 left side narrow (Fig. 81) : draw the wide 
 face first in its true proportion, taking 
 especial care to make the narrow side no 
 wider than it really appears. To do this, 
 remember to compare the horizontal width, 
 by means of the pencil held in the usual 
 manner, with the length of the first vertical 
 line drawn. Having drawn the narrow face b, and found the inclina- 
 tion of the lines i, 2, 3, the face c is easily represented by drawing 
 each of its further boundary-lines converging in their respective sets. 
 
 riS' SI 
 
DRAWING THE DOUBLE CROSS. 
 
 83 
 
 The four-sided prism should be drawn in several different positions, 
 taking care to note the sev- 
 eral systems of converging 
 lines and their directions 
 (Fig. %2). The amount of 
 convergence in all cases 
 should be determined by 
 close inspection of the mod- 
 els themselves, the degree 
 of convergence depending 
 on the distance of the eye. 
 The triangular prism may 
 be placed in a variety of 
 positions (Fig. 83). The 
 altitude c d must be drawn after the base-line a b, remembering that the 
 
 T^lj.83 
 
 nearer half of the base will appear the longer when the base ^ ^ is a 
 
84 
 
 MODEL AND OBJECT DRAWING. 
 
 retreating line, and that consequently the altitude c d must appear to 
 be beyond the apparent center of a b. 
 
 Tis. 85 
 
 CFig. 87 
 
 "Fig. 8B 
 
 /CEi— |— :f^ 
 
 
 / h \ 
 
 
 N^-aj---,^^ 
 
 
 Place the hexagonal prism on the triangular prism, taking especial 
 care to read according to the method we have indicated, observing all 
 the sets of converging lines (Fig. 84). 
 
DRAWING THE DOUBLE CROSS. 
 
 85 
 
 Vases should be drawn in various positions (Fig. 85). Beginning 
 
 with the axis, find the proportionate lengths 
 
 and widths of the ellipses of the bases, and 
 
 determine the greatest and least diameter, 
 
 and the position of each on the axis, as 
 
 points I and 2, thus fixing the height of 
 
 each. 
 
 Observe the use of section lines at right 
 
 angles to the axis, in drawing symmetrical 
 
 figures (Figs. ^6 and 87). As many may 
 
 be drawn as desired at equal or unequal 
 
 distances from each other, provided they are 
 
 always at right angles to the axis : they will 
 
 be bisected by the axis. In other words, the 
 
 two parts of these section lines will be equal. 
 
 See <2, b^ c, d, e,f,g, etc., in the illustration. 
 
 This model should be 
 drawn in two or three posi- 
 tions. First upright, and 
 then resting upon one side 
 (Fig. ^Z). The latter posi- 
 tion will try the skill of the 
 pupil in reading correctly 
 the apparent form (Fig. 89). 
 Care should be taken to fore- 
 shorten the length in the 
 proper proportion. For this 
 
 purpose, compare the apparent length of the axis with the greatest 
 
86 
 
 MODEL AND OBJECT DRAWING. 
 
 diameter ; as one end of the axis is invisible, the pupil must imagine 
 where on the surface of the model he could 
 place a point that would cover the invisible end 
 of the axis. Having drawn the axis in its ap- 
 parent proportion with the greatest diameter, 
 proceed to place the several apparent diame- 
 ters, as in the preceding examples. 
 
 This model is one of the more difficult 
 ones of the series (Fig. 90). Its form, how- 
 ever, is based on the oval, which may be 
 slightly modified in some parts of the outline. 
 In any of these forms of vases, after the 
 first sketch is made, turn the drawing upside 
 dow7i to see if it is correctly drawn : any 
 
 want of symmetry will thus be seen at once. 
 
 LIGHT, SHADE, REFLECTED LIGHT, CAST SHADOW, AND REFLECTIONS. 
 
 The department of light and shade, with all its interesting modifi- 
 cations, approaches more nearly to what one might call real art, than 
 the subjects hitherto discussed, which pertain to construction. But it 
 will be found here, as well as everywhere else in art, that we are 
 dealing with absolute law, and that there is no room for guess-work. 
 There will be in this department ample chance for the exercise of 
 close observation, quick apprehension of principles, and of great care 
 and taste in execution. 
 
 Light, as we treat it, in respect to objects is of two kinds. First, 
 direct light is that from the sun or from some other luminous body ; 
 
LIGHT, SHADE, REFLECTED LIGHT, ETC, Sy 
 
 second, diffused light is that which jDcrvades, in the daytime, an 
 ordinary room. 
 
 In the first form of light, the illuminated surfaces, parts in shade, 
 and cast shadows will all possess mathematical proportions and 
 definite limitations, and will have to be dealt with in reference to 
 geometrical formulas. Light and shade in this kind of light is 
 regarded as a part of Descriptive Geometry, and forms by itself a 
 separate subject. 
 
 But the treatment of objects in the diffused light of a room is 
 the subject of our present inquiry. In this light the appearance of 
 objects will be quite different from that of the same objects in sun- 
 light, and yet there is law pervading both classes of phenomena. 
 
 For the purpose of studying these effects, one should sit with the 
 left shoulder to a window, the only source of light. 
 
 Take, first, the common white cube, and place it in front, at a 
 convenient distance, say six feet or more, from the eye. If it be the 
 first effort of the student to read light and shade, he will most likely 
 not be able to see the nicer differences : he will have to learn /low to 
 see. For this purpose let the student close one eye, and then with the 
 other half closed study attentively the light and shade upon the 
 object : in order to see correctly, ample time must be given to this 
 process ; the pupil sitting in this manner one, two, or five minutes, 
 repeating the effort often, until all that is to be seen is fully appre- 
 hended. 
 
 First, it will be observed that a part of the surface of the cube is 
 in light, and a part in shade. Let the student make with a pencil a 
 very light sketch of the outline of the object. Shade with vertical 
 lines the right face, which is in shadow, and then darken it with 
 
88 
 
 MODEL AND OBJECT DRAWING. 
 
 oblique cross-lines in one or two directions, and fill in the open checks 
 with dots or dashes, to destroy or modify the netted appearance. 
 Now it will be important to note the modifications of the shade on 
 this side. It should be observed that it is not of a uniform depth over 
 the whole face, but is darkest near the front edge, and at the upper 
 part of the surface in front, at point A ; while it is lightest at the back 
 
 and lower part of the 
 *^ ^-s^^^s^-,.^ face near C (Fig. 91). 
 
 This last modification 
 results in part from 
 light reflected from 
 the plane on which 
 it rests, and in part 
 from its contrast with 
 the darker cast shad- 
 ow. The near part of 
 the same face is dark- 
 ened by contrast with 
 the high light on the 
 opposite side of the line A B. The illuminated left face will be lightest 
 along the line A B, lighter at A than at B, and darker along the line 
 F G, and darker at G than at F. 
 
 The top may be in a lower light than the right side, according to 
 the position in reference to the light : it will be darkest along the line 
 A F, and next to the illuminated face ; and lightest next to the dark 
 side, along the line A D. It will be seen that the three faces of the 
 cube, in this position, present a series of contrasts of light and shade, 
 along the three lines running to the nearest solid angle, A. It is 
 
LIGHT, SHADE, REFLECTED LIGHT, ETC. 
 
 89 
 
 exactly in the order of these contrasts that the drawing is made to 
 express relief, as will be seen hereafter. 
 
 It will be observed, that in Fig. 92 the proportion of light and 
 shade on the three faces is exactly reversed ; and, instead of appear- 
 ing as a solid cube, the arrangement of 
 light and shade shows that it represents a 
 half of a hollow cube. 
 
 In reference now to the cube (Fig. 91), 
 we find that the following facts have been 
 observed : — 
 
 First, On an illuminated plane, the 
 highest light is on the nearest part of the 
 plane. 
 
 Second, On a semi-illuminated plane, the deepest shade is adjacent 
 to the illuminated plane. 
 
 Third, When a plane is in shadow, the deepest shade is on that 
 part of the plane nearest the eye ; and reflected lights would most 
 likely appear on the more distant and lower part of the plane. 
 
 Fourth, The cast shadow is darker than the adjacent shaded surface 
 of the object which cast it, and the darkest part of the shadow will 
 always be nearest to the object casting it. 
 
 The principles developed above will be found applicable to all 
 rectangular solids, and, with some modifications, to all objects on 
 which light and shade may be conveniently studied. For however 
 small the object, if no more than the thousandth of an inch in 
 diameter, there would be the same facts of light and shade, high 
 light, half-light, shaded surface, affected more or less by reflected light 
 and cast shadow : so that an attentive study of the cube in light and 
 
90 
 
 MODEL AND OBJECT DRAWING, 
 
 shade will develop the principles of the whole system of distribution. 
 Let the student study the cube in light and shade, and draw it until 
 there is nothing more to learn from it. 
 
 For the next example in light and shade, let us take the cylinder, 
 giving us a curved surface (Fig. 93). 
 
 Place the cylinder so that the light will fall upon it over the left 
 
 shoulder, and observe the posi- 
 
 ■mj.9. 
 
 and darkest at the back and left. 
 
 tion of the lights, shades, re- 
 flected lights, and shadow. The 
 deepest shade comes on the 
 right side, a little way in from 
 the outline : it is the darkest at 
 the top, and, by reason of re- 
 flected light, less dark at the 
 bottom. The highest light ap- 
 pears on the left side, a little 
 way in from the outline: it is 
 the lightest at the top. The 
 upper base of the cylinder is in 
 half-light, lightest next to the 
 deepest shade on the right side, 
 In this position neither the highest 
 
 light nor the deepest shadow occurs at the outline of the model, as 
 in the cube, where the highest light and the deepest shadow are con- 
 tiguous. In natural scenery these contrasts frequently occur in juxta- 
 position. 
 
 In the study of the cube and cylinder we have become acquainted 
 with many of the first principles of light and shade. We may now 
 
LIGHT, SHADE, REFLECTED LIGHT, ETC, 
 
 91 
 
 take up the sphere ; and, since its apparent form is a circle from all 
 points of observation, we may compare it with other objects whose 
 apparent forms are represented by a circle such as the plane circle 
 and the hollow hemisphere, the cone with the apex toward the eye, a 
 hollow cone with the apex away from the eye, etc. For this purpose 
 set up, if obtainable, these five objects, so that each will be repre- 
 sented in outline by a circle, and then, 
 drawing five circles on the paper, proceed 
 to study and represent the several forms, 
 with all their nice modifications and distri- 
 butions of light and shade. Nothing can be 
 more useful than the faithful study of these 
 objects. 
 
 A represents the flat shaded surface of a 
 circle in nearly uniform tint (Fig. 94). B, 
 the study of the light and shade of the 
 
 sphere, as it appears to the pupil, with the high light on the upper 
 
 left-hand side, but a little in from the outline 
 of the circle (Fig. 95). The deepest shadow 
 is seen on the lower right side, but not dark- 
 est near the outline : reflected light is seen 
 on the shaded side, caught up from the plane 
 on which it rests. The cast shadow on the 
 plane would be, as far as visible, in the form 
 of an ellipse. 
 
 C represents the hollow hemisphere (Fig. 
 96). Its shaded surface, in a proper light, is a cast shadow thrown 
 upon the inner surface by the rim : it has, therefore, the disposi- 
 
 , it 
 
 '« ilii 
 
 m 
 M 
 
 IP 
 
 lEi^. 95 
 
92 
 
 MODEL AND OBJECT DRAWING. 
 
 \ 
 
 / 
 
 tion of a cast shadow, with reflected light which belongs to a shaded 
 ^ _^., surface. 
 
 D represents the cone, with the apex 
 towards the eye (Fig. 97). The deepest 
 shade will appear under the apex on the 
 shaded side, the highest light being on the 
 opposite side of the apex. It will be ob- 
 served that near the base, on the light side, 
 it must be slightly shaded ; and that near 
 the base, on the shaded side, it must be less 
 darkly shaded than under the apex : so that, 
 as we approach the base from the apex, 
 there is less difference between the light 
 and the shaded side than there is near the 
 apex. Fix clearly the fact that both the light 
 and the shade are focussed 7iear the apex^ 
 on exactly opposite sides. 
 
 E represents the hollow cone (Fig. 98). 
 Notice how all the conditions of the distribu- 
 tion of light and shade are reversed from 
 those in the cone. 
 
 The means to be employed in represent- 
 ing light, shade, and shadow are various : the 
 selection may be made according to the prefer- 
 ence of the teacher or pupil. We may use the 
 pen and ink, lead-pencil, crayon-point, char- 
 coal-point, and stump with charcoal or crayon, or the brush with India 
 ink, or with any monochrome. 
 
 ■i 
 
LIGHT, SHADE, REFLECTED LIGHT, ETC, 
 
 93 
 
 When lines are used, they should be laid as evenly as possible, 
 and with nice gradation in passages of varying depth. Flat tint 
 should be laid with one set of lines running in the same direction : 
 where only one set is used, it is called half-tint (Fig. 99). If more 
 depth is required, two or more sets of lines may be used : the differ- 
 ent sets should cross each other at an acute angle, as in the illustra- 
 
 l!l!il 
 
 111 iiii li! 
 
 ill 
 
 ill 
 
 100 
 
 .±0± 
 
 tion, Fig. 100. This process is called hatching. The lines should not 
 be drawn far at one stroke of the pencil : it is better to lift the pencil 
 as often as is convenient. A rather broad line is much better than a 
 fine, wiry line ; and the spaces between lines should be uniform, and 
 not wider than the lines themselves. Vertical lines are appropriate 
 for vertical plane surfaces, and horizontal lines for horizontal surfaces. 
 Straight lines should be used for plane, and curved lines for curved, 
 surfaces ; or both may be used on the surface of the cylinder and cone, 
 
94 
 
 MODEL AND OBJECT DRAWING, 
 
 where the surfaces are both straight and curved, but in different direc- 
 tions. 
 
 Stippling^ with dots between the lines in the open checks, may be 
 resorted to, in order to produce a uniform effect, and to cause the lines 
 to blend (Fig. loi). 
 
 The study of shading should be pursued by drawing and shading 
 as many vases^ and other objects as possible (Figs. 102 and 103). 
 There is no danger that the student will draw too many, or become 
 too familiar with these objects. After each study of a model, a rapid 
 drawing entirely from memory should be made. When completed, it 
 may be compared with the original drawing to test its accuracy. This 
 
REFLECTIONS, 95 
 
 practice is of the greatest value in fixing in the mind whatever knowl- 
 edge has been acquired in the study of objects. 
 
 REFLECTIONS. 
 
 Reflections are an important element in pictorial effect, and, in 
 connection with model-drawing, should receive a passing notice. 
 
 First, they are produced by a polished surface taking up the light 
 of an object, and conveying it to the eye. If an object, such as a 
 cube, is placed upon a polished table, there will be present all the 
 various modifications of light, shade, and shadow ; and, in addition 
 to these, they will all be reproduced in the reflection of the object, 
 with some exceptions and modifications. The cast shadow will, how- 
 ever, never be fully reproduced in the reflection ; because, in this 
 position of the cube, the shadow rests upon the plane of reflection : 
 hence the cast shadow will be modified in proportion to the perfection 
 of the reflecting surface. 
 
 A vertical line reflected by a horizontal plane surface will always 
 give a vertical reflection. See, in Fig. 
 104, a, with a! as the reflection on the 
 reflecting plane D E ; but o! will be shorter 
 than ay according as the eye is more or less 
 above the plane of reflection, if measured 
 in the usual way by holding up the pencil. 
 A line inclined to the right, but not to the 
 front or back, as by will have its reflection 
 more inclined, as the eye is at a greater distance above the plane 
 of reflection, as at b' : e and / will have their reflections / and /' 
 
 jFis. 104 
 
96 MODEL AND OBJECT DRAWING. 
 
 respectively. These statements may be easily verified by holding a 
 pencil in various positions against the face of a looking-glass, noticing 
 the position of the reflection in each case. In the Fig. 104 the line 
 m n is perpendicular to the central ray of light from the object to the 
 eye. 
 
 These statements will guide the student in his observations of the 
 facts in sketching, as he can amplify them in many ways. When 
 once the general principle is apprehended, there will be no further 
 difficulty in its application to all the modifications under which reflec- 
 tions may occur. 
 
 There is another class of reflections with w^hich the student will 
 have to become familiar; and that is, where the objects themselves are 
 more or less polished, giving reflecting surfaces, which catch up lights 
 and colors from any illuminated objects near them, producing numer- 
 ous modifications of all the lights and shades hitherto noticed. The 
 only law which governs this class of reflections, as, indeed, all others, 
 is^ that the angle of reflection is ahvays eqnal to the angle of incide7ice ; 
 and the position of a reflected light on a polished object, as on 
 a polished silver or glazed earthen vase, will be determined by the 
 position of the object which is the source of the light reflected by 
 the object. Thus, let A, Fig. 105, be a plan of a polished cylinder, 
 with the eye at E : let B, at the same distance from the object as E, 
 be the source of the light reflected from the surface ; E B being in a 
 plane perpendicular to the surface of the cylinder. To find the point 
 of illumination, draw lines from E and B to the point a' , making equal 
 angles with the circumference at that point : a' will be the point illu- 
 minated, because lines drawn from the point ci to E and B will make 
 equal angles with the arc at that point. These lines would also make 
 
REFLECTIONS. 
 
 97 
 
 equal angles with the tangent of the arc at a\ This is easily done 
 where the point of light and the eye are equally distant from the cen- 
 ter of the cylinder ; but, where they are unequally distant, it becomes 
 a difficult problem to find the 
 loais of reflection. (See Appen- 
 dix A.) 
 
 As these reflections are sub- 
 jects of observation rather than 
 of construction, it will be suffi- 
 cient in this connection merely 
 to indicate the law which gov- 
 erns them. 
 
 There is no class of phenom- 
 ena more interesting or captivat- 
 ing to the painter than reflec- 
 tions and reflected light. To the 
 landscape artist they are the 
 
 source of some of his most pleasing effects. So much is he depend- 
 ent upon reflections in water for entertaining the observer of his 
 works, that a landscape picture without water is often devoid of inter- 
 est ; while a very simple view with water, with its multitude of glan- 
 cing lights and fragmentary shadows, becomes at once pleasing and 
 delightful. 
 
 Reflections multiply the quantities which make up the rhythmical 
 and harmonic series from which the mind derives its pleasure, and 
 seem to suggest the idea of life and activity. 
 
APPENDIX. 
 
 THE FOLLOWING IS THE SOLUTION OF THE PROBLEM FOR FINDING THE POINT 
 OF REFLECTED LIGHT ON A POLISHED CYLINDER. 
 
 t 
 
 The following problem depends for its solution upon finding, upon the 
 surface of a polished cylinder, a point where the angle of incidence from the 
 hght will be equal to the angle of reflection from the point to the eye. 
 
 To find the place of the point of illumination on a polished cylinder, when 
 the place of the light and the place of the eye are given : — 
 
 The problem assumes four diiferent forms. 
 
 I^t'rsfj When the point of light and the eye are in a plane not at right angles 
 to the axis of the cylinder, and are equally distant from the cylinder. 
 
 Second, When the plane, in which the eye and light are located, is not at 
 right angles to the axis of the cylinder, and the points of the eye and light are 
 unequally distant from the cylinder. 
 
 Third, When the eye and light are in a plane perpendicular to the axis of 
 the cyhnder, and equally distant from the cylinder. 
 
 Fourth, When the plane, in which the eye and light are located, is perpen- 
 dicular to the axis of the cylinder, and the point of the eye and light are 
 unequally distant from the cylinder. 
 
 The first and second forms of the problem are not of easy solution. The 
 third form is solved by drawing tangents from the points of the eye and light 
 E and L, to the circumference of the cylinder on the near side, and by bisect- 
 ing the angle formed by the two tangents : the bisecting hne cutting the center of 
 the cylinder will also cut the circumference at the point of illumination. 
 
 99 
 
100 
 
 MODEL AND OBJECT DRAWING. 
 
 The fourth form of the problem is not so easy of solution, and it seems that 
 it can only be solved in the following manner : — 
 
 Let E and L be the points of the eye and light at unequal distances from 
 the cylinder A : draw several concentric circles B, D, G, etc. Draw tangents to 
 
 ^ppendSx 
 
 the circumference of the cylinder and to each of the concentric circles ; each 
 pair of tangents intersecting each other in a, b, d, e,f. Now, by constructing 
 a curve passing through these several points of the intersecting pairs of tangents, 
 the curve will cut the circumferences in the points of illumination, and will pass 
 through the center of the cylinder. This curve will cut the surface of the 
 cylinder in the point of illumination : for we shall find that the angles formed by 
 
 \ 
 
APPENDIX, lOl 
 
 lines drawn from these points of the curve, intersecting the concentric circles, 
 will form equal angles with the circumferences of the circles at the points of 
 intersection of the curve ; thus showing that the angle of incidence is equal to 
 the angle of reflection. Or, in other words, the lines drawn from L and E 
 to these several points on the circumferences, found by the intersecting curve, 
 will make equal angles with tangents drawn through the same points; thus 
 proving that the angles of incidence and reflection are equal, and showing that 
 the point found by the construction of the curve is the point of illumination, 
 the locus of reflection. 
 
 Note i. — The curve becomes a curve of the fourth degree by virtue of the 
 arrangement of the points through which it passes. 
 
 Note 2. — If tangents are drawn to the same concentric circles on the far 
 side of the cylinder, and the curve extended through the intersections of the 
 tangents, the curve will give the points of illumination on the inner surfaces of 
 cylinders arranged in place of the circles. There would seem to be no other 
 simple solution of this problem that could be worked out visible to the eye.