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Copyright, 1908 
By Dora Miriam Norton 







D. M. N. 



THIS book presents essentially the course of study in Free- 
hand Perspective and Sketching as developed during its 
teaching at Pratt Institute since the founding of the 
institute in 1887. It consists of a series of illustrated exercises 
with explanatory text, so covering the subject that students who 
follow the course as directed acquire the power to draw with ease 
and intelligence, not only from objects, but from memory and 
from descriptions. The principles and methods thus set forth 
have been taught by the author for some years in the above school, 
and have been found practically effective in that direction. 

As offered to the public this course is intended to form a text- 
book for high, normal, and technical schools and for colleges ; 
also as a book of reference for supervisors and teachers of draw- 
ing, and for draughtsmen and artists whose training in perspective 
needs to be supplemented. Where a less extended course is de- 
sired, selections can be made from it at the discretion of the 
teacher. Moreover there are students so situated that personal 
art teaching is beyond their reach, though books could be had. 
But while perspective books, excellent in certain directions, have 
been published, it has been found difficult to direct inquirers to 
anything at once directly applicable to immediate use and com- 
prehensive enough to give a working knowledge of the subject. 
For several years, therefore, the need which this book is intended 
to meet has been increasingly felt. In the hope that it may pass 
on to others the aid received in the past it is sent forth. 

In making these illustrations the author has been aided largely 
by Mr. Ernest W. Watson, a graduate of the Massachusetts Nor- 
mal Art School, and later a student and now an instructor in 
Pratt Institute. Of this efficient and valued assistance it is a 
pleasure to thus express a cordial appreciation. 

D. M. N. 

Brooklyn, July 14, 1908. 



Introduction xi 


I. General Directions 1 

11. Pencil Measurement and the Picture Plane 4 

III. The Ellipse 8 

rV. A Cylinder and a Cylindrical Object 12 

V. An Object above the Eye and the Cone Principle 18 

VI. A Cream Jug 20 

VII. A Time Study 24 

VIIL A Group of Cylindrical Objects 26 

IX. Cylindrical Objects Grouped with Fruit 29 

X. A Group of Objects from Memory or Invention 31 

XI. The Cylinder Cone and Ball Grouped — A Problem for Original 

Study 34 

XII. The Study of Straight Line Objects 36 

XIII. Drawing the Book in Two Positions 43 

XIV. The Book with a Cylindrical Object 45 

XV. A Problem for Original Study — The Cylinder and Rectangular 

Block 48 

XVI. The Further Study of Straight-Line Objects — A Cube at Angles 

WITH THE Picture Plane 49 

XVII. The Cube in Two Different Positions 53 

XVIII. A Book at Angles to the Picture Plane 58 

XIX. Two Books at Different Angles to the Picture Plane .... 61 
XX. The Actual Center of the Circle and Measurement into the 

Picture by Parallel Lines 63 

XXL Books with a Cylindrical Object 67 

XXII. The Study and Drawing of a House 69 

XXIII. A Building from the Photograph or a Print 81 

XXIV. Type Forms Helpful in Understanding the House — The Square 

Frame 85 

XXV. The Square Pyramid and Square Plinth 88 

XXVI. A Problem for Original Study 91 



Chapter Vxqb 

XXVII. Cylindrical Objects when not Vertical 92 

XXVIII. A Group of Flower Pots 95 

XXIX. The Circular Frame in a Square Frame 96 

XXX. A Round Window 100 

XXXI. The Clock a Problem 102 

XXXII. The Arch 103 

XXXIII. Interiors — A Room Parallel to the Picture Plane .... 105 

XXXIV. Interiors Continued — A Room at Angles to the Picture Plane HO 
XXXV. Further Studies of Interiors 114 

XXXVI. A Chair 118 

XXXVII. The Hexagonal Plinth in Two Positions 121 

XXXVIII. Interior with a Tiled Floor 126 

XXXIX. The Hexagonal Prism and Frame 128 

XL. The Triangular Prism and Frame — Problem for Original Study 131 

XLI. The Study of Parallel Perspective * . . . . 132 

XLII. A Street from the Photograph 137 

XLIII. Exceptions to the Use of the Flat Picture Plane 139 

XLIV. Shadows 143 

XLV. Out-of-doors Work 154 


INDEX 169 



REEHAND Perspective teaches those few principles 
or truths which govern the appearance of things to 
the eye, and the application of these principles to the 
varied conditions encountered in drawing. Strictly speaking, 
there are but two foundation truths in perspective, namely: 

First. Things appear smaller in proportion to their dis- 
tance from the eye. A house ten rods distant can be 
wholly seen through one pane of glass (Fig. 8, 
Ch. II). 

Second. The eye can see surfaces in their true 
shape only when placed at right angles to the direc- 
tion in which the eye looks, or, generally speaking, 
parallel to the face. When not so placed they ap- 
pear lessened in one dimension, that is, either nar- 
rowed or shortened, in proportion as they are 
turned away from the face or tend to coincide 
with the direction of seeing. This apparent change of shape is 
Foreshortening. The cylinder top held at right angles to the 
direction of seeing appears as a circle (A in Fig. 1). When 
turned away from this direction (as at B), it appears nar- 
n rowed, or foreshortened. So the pencil seen its 

.-^^^'^^-^ full length at A in Fig. 2 appears foreshortened 
^/| when held as in B. All the phenomena of free- 
A J * hand perspective, however complicated and per- 
FiG. 2 plexing, may be simplified by referring to one 

or both of these principles. 

One great obstacle to the ready mastery of these prin- 
ciples is our knowledge of the actual shapes of objects. For 



instance, we hnow the top of a cylinder (B, Fig. 1) to be 
in fact a circle, and therefore we tend to mentally see a circle, 
though it is just as truly a fact that the top can only appear 
to the eye as a circle when the cylinder is held so as to lose 
sight of all other parts of it, as at A. Consequently, the first 
aim and benefit in studying perspective is the learning to see; 
that is, to know what is the image really presented to the eye. 
Therefore no step should ever be passed without clearly see- 
ing the appearance under consideration. And in all drawings 
the final test must be the eye; for, unless the drawing loolis 
right, it is not right. All rules and tests are only means to 
this end. 

Furthermore, the right study of perspective, which is think- 
ing and drawing in perfect coordination, enables the student 
to draw objects singly or combined or in unfamiliar positions, 
without having them in sight. Also he should be able to 
draw an object which he has never seen if a description of it 
can be supplied. That this last is quite possible any prac- 
tical artist will agree. The writer recalls hearing a popular 
illustrator ask in a company of friends, " Does any one know 
what a cider press is_like?" adding that he must put one 
in an illustration with no chance to see the thing itself. No 
doubt of the suflS.ciency of a description was expressed. In 
fact it must suffice — a not uncommon situation. Hence the 
necessity of memory work and dictation problems, such as 
form part of this course of study. 

Finally, it is not intended that in later practical work drawings 
should be actually constructed by the explanatory methods here 
given. These exercises should be drawn as directed, since only 
by the actual experience of doing it can their principles be mas- 
tered,, but a rigid clinging to these methods in practice would 
result in very little art. Freehand Sketching means drawing hy 
the trained eye and judgment, only using constructive methods to 
test new or doubtful points. It is to make such sketching valu- 
able by a foundation of definite knowledge that these methods 



are given. The trained artist draws a vase in his flower study, or 
a round tower in a landscape with no distinct recalling of ellipse 
laws, feeling only joy in the living curves as they spring out 
under his hand. But he would labor long and wearily over their 
shaping had he not this foundation knowledge, which he uses 
almost unconsciously. 





Chapter I 


MATERIALS. — Any paper having a fine and fairly soft 
texture can be usedgfc It should produce an even 
grain in both vertical and horizontal pencil strokes. 
Pencil exercises such as those reproduced in this book are 
usually drawn on paper of quarter imperial size (11" x 15") , 
on which at least an inch and a half of margin is allowed. 
This is a good size for the student's drawings, whether copied 
from these exercises or drawn from objects. Have two 
pencils, one fairly soft (as No. 2 Faber, SM Dixon, or 
\2 B Koh-i-noor), and a harder one; also a good eraser. 
Line Practice. — Cut the pencil like the illustration 
(Fig. 3), and rub on practice paper ^ till a broad line, 

firm at the edges 
and transparent 
(that is, with the 
grain of the paper 
slightly showing 
through it) can be 
made. Sit erect, 
with the paper directly 
in front, and have the 
desk top inclined, or use 
a drawing board (Fig. 4), 
that the paper may be as 
nearly as possible parallel with the face. Hold the pencil almost 
flat, as in the illustration (Fig. 5), and as loosely as is consistent 

^ Save spoiled sheets for this. Practice paper should be like that on which drawings are 

Fig. 3 

Fig. 4 


Fig. 5 

with a steady control. For horizontal lines use position A, 
Fig. 5, moving the pencil from left to right; for vertical lines 
use position B, moving from the top downward. Practice 
vertical, horizontal, or oblique lines persistently; moving the 
hand freely from the shoulder, not resting it on the wrist or 

elbow. If the muscles acquire an 
unpleasant tension, relax by dropping 
the hands at the sides and loosely 
shq^ng them. Unfamiliar or diffi- 
cult exercises should be first carefully 
sketched with a thin, light line. If 
wrong, (iraw over without erasing 
until a satisfactory form is obtained. 
Erase the incorrect part, and ren- 
der expressively (Ch. IV). But after 
the composition of the. exercise is 
planned, such straight lines as mar- 
gins, cylinder sides, and many ellipses may be drawn in full at 
once. And as the student gains in skill, more and more of the 
work should at the first touch be put on the paper as it is 
intended to remain. The aim is to acquire exact knowledge, 
that artistic interpretations may be expressed with ease and 

Models for Workr — Objects in common use have been chosen 
for most of these exercises. Geometric solids are assigned only as 
needed for the clearer elucidation of perspective truths. Neces- 
sary models, as the cylinder, the cube^ and others, should be made 
by the student as directed. For forms (as the hexagonal frame) 
too complicated to be easily made, the well-known wooden 
models have been used. But after thorough mastery of the 
simpler forms, most of the later lessons can be understood with- 
out models. 

Placing of Models. — All objects. for study should be placed so 
as to present their vertical surfaces in nearly their true shape to 
the student. Thus if the model is to be near, as on the table 


at which the student sits, it is better to raise it a few inches 
(Fig. 4). This will not be necessary if it can be placed four or 
five feet distant. If the study is seen too much from the top, 
the perspective will be unpleasantly violent, as in a photograph 
where the camera has been pointed too much downward. 

The Table Line. — To indicate a supporting surface under the 
objects a horizontal line (A, B in Fig. 6) is used. It stands for 
the back edge of the table or other horizontal support- 
ing surface, and is caljpd the Table Line. It should be 
^- represented as further back than any portion of the 
study. As will be observed later, it need not be used 
if the supporting surface is otherwise suggested, as by 
a cast shadow (Fig. 34). 

All Work Freehand. — AH work is to be done freehand, that 
is, with no ruling, and no measuring other than by the eye 
and pencil. 

Fig. 6 

Chapter II 


PENCIL Measurement. — Before studying the exercises which 
follow, the beginner should become familiar with Pencil 
Measurement. Place a book upright directly in front of 
the eye. With one eye shut and the arm at full length (to ensure 
a uniform distance from the eye) measure on the pencil held hori- 
zontally the apparent width of the book. Then turning the pen- 
cil, compare this dis- 
tance with its height 
(Fig. 7). (It is bet- 
ter to take the smaller 
distance first, and to 
measure it into the 
larger.) Compare the 
proportions so found 
with those obtained 
by actual measure- 
ment of the book. 
But always get the 
pencil measurement 
first, for this compels the eye to do all that it can unaided 
before showing by actual measurement how much better it can 
learn to do. 

Now turn the book away a little, and compare this new ap- 
pearance of the width with the height (Fig. 11). 

At this point the student must learn to lieep the 'pencil 
parallel with the face in order that the pencil measurement 
may be reliable. For this, go to the window, and stand facing 

Fig. 7 






















B. Plan of A 

A. Showing osc 

or WIN0O>V AS 

the glass, so the face is parallel with it- Choose some object 
seen through the window, as another house, and resting the 
pencil against the glass measure its width and compare that 
with its height 
(Fig. 8). 

Observe that 
if the outline of 
the house could 
be traced by the 
pencil on the 
glass it would 
form correctly 
the apparent 
shape of that 

This leads us 
to see that all per- 
spective drawing 
may be regarded 
as placing on 
paper the equiva- 
lent of such a tracing on the glass. It will therefore be apparent 
at once that pencil measurement, to be correct, must be taken 
with the pencil held as if laid on such a pane of glass; or in 
^<>r- other words, on a plane parallel with and in 

^^^jc^\ front of the face. This imaginary transparent 
) plane is called the Picture Plane, and is a 

flu. <, ^ ' 

fS I Hil^ J most important factor in all freehand draw- 
V ^ fl PTT i^S- Thus, by turning or revolving the 
pencil on the glass in front of the face, 
thatis, by revolving the pencil in the picture 
plane, it can he made to cover the appearance of any possible line or 
direction. For example, the sloping gable edge of the outside 
house, though retreating from the eye and therefore foreshort- 
ened, can be covered by the revolving pencil (Fig. 9) , thus giving 


Fig. 8 



the appearance or picture of its direction. Its apparent or fore- 
shortened length can also be taken on the pencil and compared 
with any other dimension, as the height of the nearest corner. 
The essential requirement is that the pencil shall constantly lie flat 
on this pane of glass; that is, on the picture plane. 

We have therefore, in the use of pencil measurement on the 
picture plane, a ready and accurate means of ascertaining any 
direction or any proportionate dimension seen by the eye. It 
cannot give us actual sizes, as the length of the gable in feet ; but 
it will tell us how long the slanting line representing the gable 
must be drawn in proportion to other parts of the house. In this 
case, for instance, the sloping edge appears, three-fourths of the 
gable width. The difficulty in using this valuable aid with exact- 
ness lies in the beginner's trouble in keeping the pencil always in 
his invisible picture plane. To do this, he should always mentally 
see this plane, recalling that it is always parallel with the general 
position of the face — that is, vertical. And since the eyes look 
mostly straight out, or at right angles to the face, the picture 
plane is at right angles to what we may call the Central Direction 
of Seeing. 

The Central Direction of Seeing. — This central direction of see- 
ing must also now be imagined very definitely. It extends from 
the eye to the center of the objects observed. In the case of the 
house seen through the pane of glass, the central direction of see- 
ing extends from the eye to the center of the house ; while the 
face and the picture plane are parallel to each other and at right 
angles to it. The picture plane may then be thought of as a 
transparent vertical plane pierced in its middle by the direction 
of seeing. 

We have said the central direction of seeing is at right angles 
to the face. Since the face is generally vertical, the direction of 
seeing is generally horizontal (A in Fig. 14, Ch. III). The com- 
monest exception is that of being directed slightly downward (B 
in same Fig.). In this case it cannot be at right angles to the 
picture plane. It will, however, always appear at right angles to 



it ivhen looked at from above. That is, it is at right angles from side 
to side, and in a plan will always be shown at right angles, as in 
Fig. 8. 

Return now to the seat (Fig. 7), and try pencil measurement 
on the turned book. Imagine as clearly as possible the trans- 
parent picture plane at arm's length, 
on which the pencil may be revolved, 
but through which it must never be 
thrust. Starting with the pencil erect 
(Fig. 10) drop it directly over to the 
left (Fig. 11), watching carefully to 
keep it from leaning back or forward. 
Let another person help by turning the 
book away while you measure it and at 

the same time Fig. lo 

keep the pencil from following it back- 
ward as it is turned away. Thus as the 
book is turned, the pencil, if it remains on 
the picture plane, shows the book to ap- 
pear narrower or be foreshortened. What 
is now sought for 
is that which the 
eye really sees as 
the width, not what the mind knows it to 
be. It is of great importance to dis- 
tinguish sharply between actual facts of 
form and size and the perspective appear- 
ance of them as presented to the eye. 

An excellent object for practice is a 
door. Stand facing a closed door, and take its proportions by 
pencil measurement. Then let some one open it, and observe the 
apparent decrease in width. 

For further consideration of the picture plane see Chapters 

Fig. 11 

Fig. 12 

Chapter III 


HAVING learned that the book cover and door appear 
foreshortened in proportion as they are near to co- 
inciding with the direction in which they are seen, 
we naturally look for the same change in the circle. Roll a 
stiff piece of paper into a cylinder, the thickness of which shall 
be half its length. Fasten it with a pin or rubber band. Holding 
the cylinder vertically, as in A, Fig. 13, and with 
one eye closed, raise it slowly till on a level with 
the eye. The top now appears as a straight line 
(B, Fig. 13). It is so foreshortened that its sur- 
face is entirely lost to sight, leaving only its edge 
visible. Now, keeping the cylinder vertical, lower it 
till the eye sees into it perhaps half an inch. Ob- 
serve carefully the shape formed by the top. Turn 
it so the 
top ap- 
p e ar s 
as a cir- 
cle (A 
in Fig. 14), then, 
holding it vertically 
again (as at B), com- 
pare mentally the 
apparent shapes as - ^'°" ^^ 

the top is placed in the two different positions. 

Now (keeping it always vertical) raise and lower the cylinder 
slowly, and note how the form of the top changes, appearing 




Fig. 15 

rounder as it is lowered. This peculiar shape, varying in round- 
ness between the straight line and the circle, represents the ap- 
pearance of the circle seen obliquely, and is the Ellipse, one of the 
most beautiful, spirited, and subtle of curves. While the circle 
is formed by a curve bending equally 
in all parts, the outline of the ellipge 
is constantly changing in the degree e( 
of its curvature. From the middle 
of each side (A, A in Fig. 15) this 
curvature increases smoothly to the 
ends (B, B). Thus the ellipse may be divided by lines through 
the middle of its sides and ends into four duplicate curves or 
quarters. These lines are known as the Long and Short Diame- 
ters. On these two lines the ellipse must be symmetrical, what- 
ever the proportion of the diameters to each other; that is, 
whatever the roundness of the ellipse. 

A test useful to determine the correctness of a drawing of the 
ellipse is sighting with one eye along the long diameter. If the 
ellipse is perfect it will appear foreshortened to a circle having a 
diameter equal to the short diameter of the ellipse. But there is 
no test of the ellipse like the ellipse itself as seen in objects. The 
student should compare his drawing of ellipses with the rhythmi- 
cally varying curves which compose ellipses as seen in real objects, 
correcting and comparing till the eye is satisfied. If this be faith- 
fully done, the time will be short before ellipses, 
often deemed a bugbear of freehand drawing, be- 
come a pleasure instead of a penance. 

Since the top ellipse appears rounder as it is 
<^^~^ dropped below the eye level, it must be concluded 
that could the bottom be fully seen it would appear 
as a rounder ellipse than that of the top. Place 
the cylinder on the table and trace around the bottom with a 
pencil. Move the cylinder to one side and compare the shape of 
this traced ellipse with that of the top ellipse (Fig. 1^)^ Also 
compare both with that part of the cylinder bottom which can 


Fig. 16 



. OF . T' 


be seen. There is no difficulty in perceiving that the ellipses in 
a vertical cylinder below the eye are rounder as they are farther 
below the eye level. 

Now, keeping the cylinder vertical, raise it slowly. When the 
bottom ellipse reaches the level of the eye, it appears as a straight 
line (A in Fig. 17), like the top ellipse when at the 
same height. When the cylinder is moved on 
above the eye, the bottom becomes an ellipse (B), 
which as we raise it farther above the eye level 
appears rounder. We perceive that it appears 
rounder or less foreshortened in proportion as it is 
farther from coinciding with the direction in which 
the eye looks to see it, as was the case with the 
book cover in Chapter II. Furthermore, if the 
cylinder be turned horizontally and held at the level 
of the eye with its length parallel to the picture 
plane, and one end be brought in front of the eye, 
we shall again see this circular end as a straight line (B in Fig. 18), 
because it coincides with the direction of seeing. If the cylinder 
be moved horizontally to one side, still keeping its length parallel 
with the picture plane (A in Fig. 18), 
the ellipse appears to widen exactly 
as when the cylinder was held verti- 
cally and moved above or below the 
.eye level. The circular top appears 
as a circle only when its surface is 
at right angles to the direction of 
seeing (A, Fig. 14). When oblique to this direction, as at B, it 
appears as an ellipse, or foreshortened circle. The ellipse is 
plainly, therefore, an illustration of the second great principle, 
that of Foreshortening. 

The student should now practice drawing ellipses, both vertical 
and horizontal, until they can be formed with ease and exactness. 
Mark ijie extreme points (A, A, B, B, Fig. 15) first taking care to 
have B B exactly opposite the middle of A A. Hold the pencil 


Fig. 18 


for drawing ellipses as directed in Chapter I for straight lines, 
using a position of the hand that will bring the pencil at right 
angles to the long diameter. If the ellipse is horizontal, begin it 
a little to the left of the middle of the upper side, drawing to the 
right first. If vertical, begin below the middle of the left side, 
and draw up. Make the whole outline with one movement, first 
carrying the pencil evenly several times over the paper without 
touching it, to gain confidence and certainty of movement. 


Chapter IF 


THE student should draw this exercise, following carefully 
the directions given. After doing so he should draw a 
cylindrical object of his own choosing, putting in practice 
the principles taught in this chapter. 

Fig. 19 

Planning the Drawing. — The Design, or Composition, or Decora- 
tive Arrangement of the exercise, which is that kind of beauty 
secured by a harmonious and artistic relating of the work and its 
spaces, is to be considered first in all drawings, and should always 
be kept in mind. For this exercise (Fig. 19) we first consider 
how to place most effectively in a drawing these two separated 



objects, a cylinder whose height is twice its width and some 
simple cylindrical object (in this case a rose jar). To this end 
after drawing the margin the extreme points in the boundaries 
of the objects are lightly indicated on the paper (Fig. 20), taking 
care that the spaces between them and the margin are such as to 
give an agreeable and interesting division of the 
inclosed surface. The continuous table line be- 
hind them indicates that they stand on the 
same surface, and thus links them together. 
The size of the space between them, being no 
Fig. 20 morc than that between either and the side 

margin, also helps unite them ; and the position of the ornament 
on the jar, near the middle of the sheet, attracts the eye to the 
'center in comparison with the whole. 

Drawing the Cylinder. — For this the paper cylinder model used 
in Chapter III is placed as shown in the illustration (Fig. 19), 
that is, a little below the eye level, and at least six times its 
height from the eye. The apparent proportion of the top (that 
is, if the width of the ellipse appears to be one third, one fourth 
or some other part of its length) should be carefully judged by 
the eye and then tested by pencil measurement. Four points for 
this ellipse should then be lightly marked, and it should be drawn 
through these points as previously directed. The bottom ellipse 
is sketched directly under the upper and in the same way, re- 
membering that it must be of the same length, but rounder. The 
back or invisible part of each ellipse is left light. The straight 
lines for the sides must be tangential to the ends of the ellipses, - 
so they will join with perfect smoothness. If they do not thus 
join, the ellipse is the part most likely to be wrong. 

Now is the time to put the drawing back by the paper model 
and compare the two. Look longest at the model, glancing briefly 
at the drawing ; the aim being always to form in the mind a clear 
image of the model's true shape, and to correct the work by it. 
The student should ask himself if the cylinder in his drawing 
appears to press evenly on the ground like the model. The com- 



Fig. 21 

monest error is that of bending the outline of the partially visible 
ellipse too much at A in Fig. 21, and not increasing the curvature 
toward its ends (B, B), thus making the curve 
more circular than elliptical, and causing the cyl- 
inder in his drawing to look as if it would rock 
on its base, instead of resting firmly on every 
part of it. Sketching the ellipse entire (C, C) is 
corrccr/<vs ^^ ^^^ j^ such a case. 

Next a pencil line is drawn around the paper 
cylinder half way between the top and bottom. Points (E, E, 
Fig. 21) are then marked on the drawing, one half way between 
the fronts, and the other half way between the backs of the 
ellipses. These points should be tested after marking and made 
correct, but never measured till the eye has been made to do its 
utmost. Observe that these marks give a short diameter for the 
middle ellipse half way in size between those of the upper and 
lower ellipses. 

In the same way the two other lines around the cylinder may 
be made on the model and represented in the drawing. 

Now another paper cylinder, of the same length as the first 
but only two thirds its diameter, must be made, and placed within 
the first. Have the space between them even all the way around, 
so that the circular tops of the two cylinders form concentric cir- 
cles. They appear as ellipses. Observing carefully the space be- 
tween these ellipses, the student easily sees that it appears widest 
between the ends, and a little wider between the front sides than 
between the back sides. As we 
made the inner cylinder two thirds 
of the diameter of the outer, the 
horizontal space between the ends 
of the two ellipses will each be act- 
ually made one sixth of the length of the outer ellipse. They will 
also appear as sixths, because the ends of the ellipse are equally 
distant from the eye. The ends of the inner ellipse (C, C) are 
marked by light vertical lines. For its front and back we divide 


Fig. 22 


the width of the outer one also into sixths, but as these sixths 
are in perspective or at varying distances from 'the eye, they are 
^ persj^ective sixths. That is, they appear successively smaller as they 
7'ecede from the eye. This perspective division is here made wholly 
by the eye (though later another method is given). The per- 
spective middle (point G) is first marked on the short diameter, 
making the near half considerably larger than the far one. Each 
perspective half is then divided into perspective thirds, after 
which the six divisions are tested to see if they are successively 
smaller, as directed above. Draw the inner ellipse, making its 
ends tangential to the vertical lines (C, C), and exactly opposite 
the middle of its short diameter. It will now be found, if we 
have drawn the long diameters of these two ellipses where they 
must always be, in the apparent middle from front 
to hack, that the long diameter of the inner one 
falls higher on the paper than that of the outer 
one. From this we conclude that neither long 
diameter represents the actual diameter of the 
circle. Fig. 23 shows the plan of a circle with 
its true diameter, A A. The eye at x sees B B, a 
line connecting two tangentials, as the longest line 
in that circle. It therefore becomes the long diam- 
eter of the ellipse which the eye in that position 
sees. Meanwhile the actual diameter appears both 
shorter and farther back than B B, because far- 
ther away. That part of the circumference back 
of B B, though actually larger, is so foreshortened as to appear 
exactly like the part in front, producing the symmetry which is 
the wonderful and unfailing characteristic of the ellipse. 

Since the inner circle is smaller, the eye can see farther round 
it, as shown in Fig. 23. This furnishes another reason for its long 
diameter falling farther back, and agrees with the fact that the 
really even space between the two circles appears greatest in front. 

The Rose Jar. — For the second object proceed as with the 
cylinder, drawing lightly all of the ellipses entire first. Should 



Fig. 24 

any fall at the same height as one on the cylinder, it must be 
made of the same roundness, since the two objects are shown 
by the table line to be on the same surface, and are equally near 

the eye. Compare the size of the ellipses 
with the extreme width of the space occu- 
pied on the paper by the jar. Compare 
also the lengths of the top and bottom 
ellipses, and the length of each with the 
extreme width of the jar. Observe that 
the sides of its short cylindrical neck slope 
outward slightly toward the body of the jar. 
The Shoulders and Base of the Jar. — Before 
drawing the side outlines, hold the jar verti- 
cally at arm's length, and with the top on a 
level with the eye. Mark the point (A in 
Fig. 24) where the body and neck boundaries meet. Holding this 
point, lower the object till the top appears as a fairly round ellipse. 
It will be plain that we now see a portion of surface beyond the end 
of the ellipse, and more than half way over its shoulder. The 
boundary line which marks the limit of our seeing has moved back 
on the shoulder, so that it passes out of 
sight behind the neck. A little experi- 
menting shows that the surface visible be- 
yond the end of the ellipse is in exact h 
proportion to the roundness of the ellipse. 
Now place a sheet of paper on the 
table, and first holding the jar so its base 
is on a level with the eye (Fig. 25), mark 
the extreme point of the base, B. Lower 
the jar slowly, till the bottom rests on the 
paper. Mark point B on the paper and 
then tha points (C, C) where the side 
boundaries now appear to meet the bottom ellipse. Trace 
around the bottom and lift aside the object. It will be found 
that the projecting mass of the jar, being nearer the eye than 


/ -: 

Fig. 25 


the ellipse, had hidden from sight more than half of it. It is 
also evident that on the lower and receding part of the jar the 
boundary line advances, so that we see less than half of the sur- 
face, instead of more, as at the top. If we now trace on the jar 
its boundary line and turn it around, the tracing will be seen to 
cross the object obliquely (Fig. 26). With these facts 
in mind, complete the drawing of the rose jar. 

Tangential Joinings. — All meetings of boundary lines 
with ellipses must he tangential. That is, they must 
touch so that if smoothly continued they would not 
cut the ellipse. 
Artistic Rendering. — The jar should be first drawn with thin 
light lines, corrected to accuracy, and afterward rendered with sig- 
nificance. For though outlines are entirely conventional, never 
being seen in nature, yet they may not only be made to mark off 
beautiful and interesting shapes, but by their character to sug- 
gest other truths and qualities for the enhancement of charm. 
Thus in the rose jar the front edge is shown to be nearer than 
the back by the heavier line, and the rounded thickness of the 
top is indicated by the absence of nearly all of the inner ellipse 
at the back and of 'the outer one at the front. The sides are 
drawn with a little lighter lines at the top, and though firm 
enough to clearly present the shape of the jar, are lighter than 
the front part of the top, because representing a part of the jar 
further from the eye. 

The ornament may help to express the rounded form of the 
jar by its foreshortened shape as it nears the boundary, and by 
the greater clearness and emphasis of that portion of it most 
thrust forward. Its outlines, emphasized on one side, with the 
other side light or lost, and the detail shown in the side lines of 
the jar, indicate it to be in relief. The expression of color in 
places is used to strengthen the projection of the jar.. 

These remarks, however, must not be understood as rules. 
They are but suggestions for the incitement of the student to 
use his own artistic judgment. 

2 17 

Chapter V 




HE electric lamp shade here shown (Fig. 27) is vertical 
and above the eye, and its ellipses therefore increase in 
roundness toward its top. The student should draw this 

exercise, and make 
another study from 
some object similarly 

Dravring the Object. 
— Proceed as in the 
previous exercise. 
Observe that the slop- 
ing side boundary 
lines of the shade join 
the ellipses in front 
of their ends. The 
I flaring shade is like 
the lower half of the 
rose jar reversed. Its 
smaller part is far- 

Ithest from the level 
of the eye, as was the 
I base of the rose jar, 
I and we therefore do 
j not see half way 
I round it. To make 
I this clear, hold a cone^ 
_,...«,.»»^««.™»...:^.^-.l .^ various positions 

Fig. 27 
1 One may be made from paper (Fig. 28). 


Fig. 28 

(that is, above and below the eye), and with apex up and down 
(Fig. 29). 

The button through which the cord is drawn forms an 
obHque ellipse. But by turning Fig. 27 so as to bring this 
ellipse horizontal, the button will be found symmetrical 
on its axis (see Ch. XXVII), and as it is arched or 
thickened in its middle, the space in front of the holes 
for the cord appears a good deal wider 
than that back of them. Each side of 
the button is a very flat modified cone. 
Notice in the outline of the cylinder the 
slight depression marking where the key 
enters its side. 

It may be now noted that when the cone apex 
(that is, its decrease of diameter) is nearer the 
eye than its base, we see more than half way round 
it. Conversely, we see less than half way round 
when the apex is farther from the eye. This char- 
acteristic of curved or sloping surfaces in cylindrical objects 
may be termed the Cone Principle. Broadly speaking, it is the 
expression, hy outline merely, of Belief, or Solidity, or TJiird 
Dimension. The rose jar in Chapter IV, the cream jug in Chapter 
VI, and indeed all cylindrical objects with flaring or bulging 
sides, are examples. More advanced applications of this principle 
are found in the drawing of such natural objects as trees and 
mountains, also in drawing the human face and figure. 

Fig. 29 



Chapter VI 

HE Model. — Provide an object similar to the cream pitcher 
here shown (Fig. 30), from which the student's drawing 
should be made. If inexperienced he will be helped by 

first making a copy 
from this example. 

The Handle. — Place 
the jug a little below 
the eye (according to' 
the directions in Ch. 
I). Draw the cylindri- 
cal body entire first as 
if it had neither handle 
nor spout, and with 
light lines (Fig. 31). 
Then hold the model 
with its center at the 
level of the eye and 
with the handle in 
profile (A in Fig. 32). 
Observe that a center 
line for the joining of 
the handle with the 
model would fall in 
the curved boundary 
line or profile of the 
jug. Turn it to bring 
the handle directly in 
front, when this same center line (x in Fig. 32) will appear 
straight and vertical. Now, turn the jug slowly back, bringing 


Fig. 30 

Fig. 31 

Fig. 32 


the handle again into the boundary line. It is apparent that as 
the handle revolves, its center line of joining changes in appear- 
ance from a straight line in front through a succession 
of curves that increase in roundness till at last it coin- 
cides again with the profile of the jug (C in Fig. 32). 
These curves are lines such as would be produced on 
the surface by cutting vertically through the jug cen- 

^ ter, as an apple is 

halved; and maybe named 
Profile Lines or Profiles. 

Replace the model on the 
table and revolve the handle 
to the side again, when it 
will be seen that these pro- 
file curves now begin and end on the top and bottom ellipses 
of the jug (Fig. 33). And the side boundary of 
the jug does not now coincide with the profile 
at the side, as it did (A in Fig. 32) when the jug 
was held at. the eye level. This is because of 
the change in the position of the boundary. 
As the jug is placed below the eye the bound- 
ary advances from A to B, recedes from B to 
C, and advances again from C to D, in accord- 
ance with the cone principle (Ch. V). 
In Fig. 34 is shown by a dotted tracing of this 
boundary how it actually differs from 
the profile curves in Fig. 33. In 
sketching a profile curve, therefore, allowance must be 
made, as shown in Fig. 33. Note how x, x, x, the 
points of the smallest diameter, fall in an ellipse at that 
height ; also the points y, y, y, of the greatest diameter. 
One of these profile curves, shaped according to its 
nearness to the boundary of the object, should he sketched as a 
guide for the attachment of the handle. 

In the same way we observe the shape of the handle itself to vary 


Fig. 33 

Fig. 34 


Fig. 35 

Fig. 36 

according to position, from a profile view at the side (A in Fig. 32) 
to a view of its outer surface (B in Fig. 32). 

The Spout. — Looking directly into the jug from above (Fig. 
35), we note that the spout is directly opposite the handle, so that 
a horizontal line through the middle of both would 
pass through the center of the circular top. We there- 
fore mark the perspective middle of the top ellipse (O 
in Fig. 36) (that is, making the nearest half larger), 
draw a line through it from the center of the handle top, 
and mark the end of the spout on this line. For the 
width of the spout, set off perspective distances from this line 
either way on the top edge of the pitcher, remembering that the 
half nearest the end of the ellipse is much 
the more foreshortened; and that the dif- 
ference is greater the more the top is fore- 
shortened. From these points to the tip 
of the spout straight lines may be sketched 
as guides for drawing the edges, which 

may be straight but usually curve both upward 
and sidewise. The profile or center line for the 
spout is sketched like that for the handle (A in 
Fig. 36). 

The Foot. — The drawing of the foot also needs 
some explanation, though covered by the cone 
principle of Chapter V. In profile it would ap- 
pear as at A in Fig. 37, with the circles as straight 
lines; and the student should raise his model to 
the eye level and observe it thus. On lowering 
the model these seeming straight lines appear as ellipses (B, 
Fig. 37), and the lower part of the side boundary lines of both the 
pitcher and its foot move forward of the ends of these ellipses 
till tangential joinings are made at C and D. The upper part 
of the boundary of the foot moves back, joining the upper ellipse 
at E. In consequence, this side outline of the foot (E D) is a 
little lengthened, making its curve less round than in profile. 


Fig. 37 


The lower half of the jug, as we can now see, is a modified ver- 
tical cone with its apex down (A, Fig. 38). The foot is modified 
from two cones; one with the apex up, the other with the 
apex down (B, Fig. 38). See Chapters 
IV and V. 

The Ornament. — The principle fol- 
lowed in suggesting the perspective of 
the ornament will be readily seen from 
the illustration (Fig. 39). The 
curved guide lines are parts 
of profiles similar to those for 
placing the handle and spout. 

The student will now begin to understand that it is 
possible to recognize and suggest the solid rounding 
surface of the object by every line and touch upon it. 
To this end that part of the ornament nearest the 
eye is more emphasized in the final drawing. And 
looking carefully at the object, we see that besides 
its foreshortening, that part of the ornament near 
the boundary is less distinct, and is often lost in the reflec- 
tions from its surroundings. 

Fig. 38 

Fig. 39 


Chapter VII 

HAYING carefully studied the principles of cylindrical 
objects, it is now best to take a specified time, as fifteen 
minutes, for the more free drawing of such an object, 
choosing a simple one at first. Proceed as before, except that 
most of the measuring and testing must be omitted. This leaves 

■ Fig. 40. A Time Study. 

time to draw slowly and thoughtfully, making the unaided eye do 
all that is possible. Study the general shape, looking long at thB 
object, and moving the pencil several times, without marking 
over the paper where the lines are to be drawn to acquire confi- 
dence and certainty of touch. Require yourself to work with 



no erasing (except of such construction lines as may show when 
the drawing is done), and to stop when the time is up. It will 
be found a valuable exercise to draw in this way the same object 
several times. After one drawing is done, carefully examine 
and test it to find the errors, but do not correct them on that 
drawing. Instead, make those points right in your next attempt 
at the same object. 

Observe in Fig. 39 how the effect of glass is given by a few 
lines selected from those many graceful curves of delicate dark 
and light which appear in the object; also the sketching of its 
high lights, or window reflections, and the wavy distortion of 
lines seen through it. The. straight lines give a firmness to the 
\ composition which is needed, since the bowl consists wholly of 
curved lines. 



FiG. 41 

Chapter VIII 

FIG, 44 is an exercise in the grouping of cylindrical objects 
agreeably and appropriately together. The student is 
advised to first draw this example, using carefully the 
explanations given. After that, he should arrange and draw 
another group of two cylindrical objects. 

Making the Composition. — In composing this second group, 
experiments should be made with a number of objects, combin- 
ing them in different ways. A Finder, which is a 
card having a small rectangular opening cut in it 
(Fig. 41) , will greatly assist in judging the pictorial 
effect of a composition, especially in a rectangular 
margin. The student should look through it at his arrangement 
with one eye, letting its edge take the place of a margin, and 
moving it back and forth till the ^ 

place is found where it makes the ^db 

group look best. Little trial or ^^^ 
*' thumb-nail" sketches (Figs. 42 —y 
and 43) should also be made to 
determine the best arrangement. ^-=^ 

In Fig. 44, for example, we Fig-42 

observe that the objects are such as might naturally be placed 
together, and are placed in positions that are not unusual. Next 
their shapes make a pleasant relief or contrast to each other with- 
out harsh or awkward opposition ; one being tall and slender and 
the other lower and round. Yet the teapot is not so low nor 
wide but that it echoes in some degree the dominant height of 
the candlestick, thus aiding harmony. Its spout is allowed to 


Fig. 43 


project across the candlestick, thus contributing to the unity of 
the composition. The leaning bowl, by passing behind both, 
also strengthens 
unity, and by its 
lighter and more 
interrupted lines 
furnishes a tran- 
sition, or connec- 
tion, between the 
nearer objects 
and the white 
paper. These 
results might be 
secured by other 
groupings. But 
had the candle- 
stick been in 
front, for in- 
stance, its pro- 
jection above and 
below the teapot 
would have been 
so nearly equal as 
to seem uninter- 
esting (Fig. 42). 
Yet we could 
have remedied 
this somewhat by 
placing the can- 
dlestick a greater 

distance in ad- Fig. 44 

vance, or raising 

the teapot handle. Or the cover could have been placed on the 
ground in front (Fig. 43) — indeed, many possibilities will be 
suggested by a little study. 



Drawing the G-roup. — In drawing this exercise, observe tliat 
though the bottom ellipses of the two objects are at the same 
level the nearer appears slightly rounder (Solutions of Problems, 
Ch. XI) . Care must be taken in placing these ellipses to allow 
for the bulging of the teapot sides. Remember that in propor- 
tion as the bottoms are drawn foreshortened so must all spaces 
on the table be regarded as foreshortened. Note also that all 
ellipses in the candlestick which are nearer the eye level than the 
top of the teapot will be less round than those of the teapot. 

The Teapot Ears. — In placing these, an ellipse may be used as 
a guide (A in Fig. 45) . The middle points of the two ears should 

^^<r:E?r^\ be on a line passing through the perspective (that is, 
/^^^^^^y,^^ actual) center (o) of this ellipse, as were the handle 

"^ and nose of the pitcher in Chapter VI. The 

Q ^^^^i^^ cover is arched (B in Fig. 45), so that it conceals 

^X- -X^ the back of its elliptical edge. This arched shape 

sMowNOARCM orcovtR jg dlstluctly sccu 111 the form of its top boundary 

and is a very different shape from the ellipse. 

But this arched boundary does not fall in the actual middle of 

the cover (since we are looking down on it), but a little beyond 

that. The knob is in the actual middle. 

The rendering of two things is more complicated and inter- 
esting than that of one alone. As the candlestick is farther away 
than the teapot, its lines are made lighter, and in places are quite 
lost. The lines of the glass rim, or hoheche are thinner, more 
interrupted and more smoothly sweeping. The leaning bowl 
may be omitted at this time, if found difficult. The principles 
of its construction are given later (Ch. XXVII). If omitted it 
will be found necessary to make the farther lines of the candle- 
stick lighter yet, to serve in place of the bowl as a transition. 


Chapter IX 


A N example of grouping is here given in which part of the 
/\ group is cut by the margin, while the apples illustrate 
jL JL the combination of natural forms with cylindrical ob- 
jects. As in the preceding exercise, the student should compose 

Fig. 46 

a group corresponding to this exercise and draw it ; and if inex- 
perienced should draw this before making his original one. 

Study of the Group. — In locating the pitcher on the paper, see 
that its base is far enough from the dish for the two objects to clear 



each other. Observe the generally elliptical shape of the curves 
in the glass pitcher ; and how the edge of the plate is seen warped 
and interrupted through it. The plate is made subordinate, as 

forming part of the background for 
the other two objects. Its position, 
appearing in its actual shape as a 
simple circle, contributes to the de- 
sired effect of quietness or subor- 
dination, as does its being cut by 
the margin line, and its lighter and 



^^«- 4'' slightly interrupted lines. 

Since it is standing vertically, it must be supported by a 
vertical surface behind it. Consequently the table line (Ch. I) 
must be placed only far enough on the paper above the lowest 
point of the plate edge to express the foreshortened necessary 
distance of this point from the wall behind it (Fig. 47). 


Chapter X 


THIS example illustrates the drawing of objects from 
invention or memory. The student may sketch this 
exercise as directed; then should invent or draw from 
memory one of his own arrangement, making small trial 
sketches as in Chapter VIII, and using the best of these in his 

Fig. 48 

final composition. Should his memory not be clear enough 
for this, it may be refreshed as often as necessary by study 
of the objects he chooses to draw, the only condition being 
that the drawing he done without the object in view. 



Drawing the Above Study. — In this exercise the Japanese 
luncheon carrier is placed first. Its ellipses are sketched in 
full, whether entirely seen or not. The bowl-shaped top, being 
slightly inclined, is drawn on a leaning axis (A, B in Fig. 49). 
But it is perfectly symmetrical on this axis (Ch. XXVII) . This 
symmetry should be tested in the drawing by turning it to 
bring the axis vertical, when any error is easily detected. (Ch. 

The Flat Dish. — It was desired to draw the flat dish as it 
would appear if touching the luncheon carrier. Its height (x y) 
is therefore measured upon the front of that object from its lower 

edge, and an ellipse of 
the proper roundness 
drawn at that height. 
The top ellipse of the 
dish would touch the 
other object somewhere 
in this ellipse, and so 
was drawn tangential 
to it. To obtain the 
bottom ellipse of the 
dish, this same height, 
increased to allow for 
its slightly greater near- 
ness to the eye, was measured downward from the dish top. 
But as the sides of the dish are flaring, this measuring was done 
from the estimated true middle (0 in Fig. 49) of the top of the 
dish, giving O' for the true center of the lower ellipse. The foot 
is like a very short cylinder. The flaring sides of the dish are 
drawn tangentially from the rim (F, F) to the upper ellipse of 
the foot. 

The Ornament. — In drawing the ornament on the luncheon 
carrier the explanation in Chapter VI is recalled. On the cover 
the band of fret decoration appears narrowed at its front, and 
widest at the ends. It is a modification of the cylinder top in 


Fig. 49 


Chapter IV. Note the foreshortening in its details, and how the 
lines of the fret express the curving form of the cover. It will 
be seen that the stripes on the object and some lines of the fret 
follow the profile lines mentioned in Chapter VI. 

The Fan. — Like the plate in Chapter IX, the fan is purposely 
placed so that it is not foreshortened. Therefore the two points 
(G-, Gr) at which it rests on the table appear, as they actually are, 
in a horizontal line. It also appears in its true shape, symmetrical 
on an axis passing through its handle (H, H). It is more easily 
drawn entire first, erasing later the part not needed. 


Chapter XI 


GENERAL Conditions for Perspective Problems. — Problems 
are to the student both, a test of his comprehension of 
the subject thus far, and an exercise by which the subject 
becomes firmly fixed in his mind. To this end the drawings 
must be made without the models in sight, though they should be 
studied, and if necessary even sketched in the required positions 
before drawing. If the student is at loss to recall 
their appearance while engaged in work they may 
be studied as often as needed ; provided only that 
neither the models nor sketches of them are be- 
fore the student as the drawing is made. It can- 
' \"^«efpuANE )' not be expected that any object should be drawn 
' ' ' until opportunity has been given for its thorough 

Fi«- ^0 study ; but on the other hand it is not mastered 

until it can be correctly drawn from unaided knowledge and 
memory. The stated dimensions are important, giving training 
in the expression of proportion, though drawings need not be 
full size. 

Drawings should of course be made without assistance, and 
without referring to the explanations in the back of this 
book. When the student has under the required conditions 
made his drawing, he may then test his work by consulting 
the explanation. 

Conditions of this Problem. — In this problem the cylinder and 
cone are to be 4" in diameter by 8" high, and the ball ^' in 



diameter. The group is to be drawn as if three fourths of the 
cylinder height below the eye, and at least six times its height 
distant. The cylinder stands on one end and the cone on its base, 
touching the cylinder and a little in front of it at one side. The 
ball also touches the cylinder, and is a little more in front of it 
on the other side. The plan (Fig. 50) will make this clearer. 


Chapter XII 



A Book with Back Paeallel with the Face 

FOR this study provide a book, two long pencils, and three 
yards of fine twine, also paper for sketching. Choose 
a book of interesting appearance; a somewhat worn, 
leather-bound book is best. Place it well back on the table in 

front of you and below the 
eye with its back next to 
and parallel with the pic- 
ture plane, and its ends 
equally distant from you 
(Fig. 51). 

The Book Below the Eye. 

— Two surfaces are visible, 
the back and one cover. 
Count the edges seen (seven), 
then decide how many of 
these are actually horizontal.* 
If the 

Fig. 51 , i . 

book IS 

placed as directed, its back, being parallel with 
the picture plane, will be seen in its true shape 
if traced upon it. Lifting the cover till it 
is vertical (Fig. 52), we see that the cover 
also now appears in its actual form. But as we drop it slowly 

* It may not at first be realized that the ends of the cover are horizontal, as well as its 
sides. But as they are contained in a horizontal surface (in this case the cover), they also 
must be horizontal. Their perspective appearance must be distinguished from their actual 


Fig. 52 



back till horizontal, we observe that the further edge seems to 
grow shorter because moving from the eye, and that the whole 
cover becomes foreshortened or narrowed from front to back, 

like the circular ends of the cylinder in 
Chapter lY. If (as with the house in 
Ch. II), a pane of glass were standingin place 
of the imaginary picture plane, a tracing of 
the cover on that would be a true perspec- 
V tive of it. How to draw on the paper such a 
perspective is our problem. Stand the pen- 
cils against the nearest comers of the cover 
(Fig. 53) ; then closing one eye, and keeping the other exactly 
opposite the middle of the book, incline the 
pencils toward each other (being careful 
not to lean them back or forward) until 
they appear to lie just along the retreat- 
ing ends of the cover (Fig. 54). Let 
another person hold a ruler against the ^-^ 
pencils, moving it down until its edge seems 
to coincide with the further edge of the 

cover (Fig. 55). Now the pencils and the 
ruler together picture the apparent shape of 
the cover, and we plainly see how the ap- 
parent shortening of the back edge (caused by 
its greater distance from us) makes the ends 
appear to converge toward each other. The 
question now is : Can the law of that conver- 
gence be so determined that it may be applied in any drawing? 

The Converging B,ook Ends. — Substitute for the pencils the 
string slipped under the cover to the back, and using one eye as 
before, bring the ends together so that the strings will appear to 
exactly coincide with the ends of the cover as did the pencils. 
(Be sure to keep the string vertically over the front edge of the 
book, not letting it fall back or forward.) The pencil may now 


Fig. 54! 

Fig. 55 

,. ^f" THE 



be taken in the other hand, and slipped down on the string, to 
form again the shape of the foreshortened cover (A in Fig. 59). 

Still holding the string as before, raise the book and string 
a few inches, keeping the book level and the string taut (A in 

Fig. 56). The string does not now cover the book ends, and 
the joining must be brought lower (as in B) that it may do so. 
If the book is raised more, the joining is yet 
nearer to the book, as in C; until when the 
book cover is at the level of the eye (Fig. 57) 
the string and the book cover both disappear 
in, or coincide with, the upper edge of the book. 
Now, starting with the position last shown, 
(Fig. 57) hold the thumb and finger firmly at 
that place on the eye level (this can be done 
by noting a point behind it on the wall), and 
let the book drop slowly. Keep it exactly 
horizontal, and let the string slip through the 
stationary thumb and finger, so that their meet- 
ing point remains at the eye level. If this is carefully done, it 
will be seen that as the book descends, the string continues 
to cover the converging ends (as in C and B, Fig. 56). At the 
same time the cover appears to grow wider, and its ends more 
and more nearly vertical. 


Fig. 57 


These experiments should also be tried with the lower 
cover, holding the book above the eye (Fig. 58), and raising and 
lowering it. 

From the foregoing study it is easily perceived that, provided 
we Jceep the hook Jiorizontal, the point toward which its ends appear to 
converge remains always at the level of the eye. 
We have therefore only to sketch the eye level 
at its right height compared with some measure- 
ment on the object and mark the point of con- 
vergence in the right place on it, to be able to 
use it for drawing these converging lines. 

We have also found that the horizontal hook 
covers (like the cylinder top in Ch. IV) appear 
foreshortened according as they approach the eye 
level, whether above or below it (Figs. 56 
and 58). 

Sketching the Book. — The book may now be replaced as at first. 
Then, holding the strings, as before, take the pencil as in B, Fig. 

Fig. 58 

59, that the thumb nail may be used as a sliding gauge. With it 
measure the length of the back of the book on its upper near 
edge and compare its length with the vertical distance from this 
edge to where the strings join (C in Fig. 59). (In this case it 
takes one and one fourth of the book length to reach the joining 
of the strings.) Now the back of the book may be sketched in, 
the point of convergence (the joining of the string) marked on 



Fig. 60 

the paper one and a half book lengths above its middle, and lines 

drawn from the upper corners of the back to this point. On 
these lines the ends of the cover are to be marked 
off. The perspective or apparent width of the 
cover may be found by measuring it with a pen- 
cil held vertically as in Fig. 60, and comparing 
this dimension with the length of the book. In 
this case the apparent width is one fourth of the 
book length. 

The Level of the Eye. — This will be found of 
the greatest importance in all drawings. It should 

be carefully marked in the drawing as soon as the position of the 

objects on the paper give a basis for locating it. At first, another 

person may assist (Fig. 61), but a 

little practice will enable the student 

to find it for himself. The top of 

a pencil, held vertically over the 

objects, will appear as a straight 

line when at the height of the eye 

(Fig. 62). Or if any part of the 

study is as high as the eye, the eye 

level will be where any horizontal 

surface or any receding horizontal lines appear as straight lines. 

See Fig. 63. 

Parallel Lines. — By holding one string down on 

the near end of a margin line on the book this line 
will be seen to converge to the same point with 
the, two ends (Fig. 64). By placing a second book 
on and parallel to the first, we can show that all 
lines parallel with the first two converging ones will 
appear to converge tvith them to the same point. An 
Fig. 62 important deduction from this is that parallel lines 

appear to converge to the same point. 

It is also evident that since the whole book cover is fore- 
shortened from front to back, the margins will be foreshortened 


Fig. 61 


— tr «. _ ,«--- 

in the same direction. And we find that the side margins are 
foreshortened in length, but not in width; while the front and 
back margins are foreshortened in width, and 
the back one more than the front. This fol- 
lows the principle of the top of the hollow 
cylinder in Chapter IV. 

The Vanishing Point. — We see that the 
book ends seem to converge in proportion ^^^- ^^ 

as the back edge of the cover appears shorter. If a second 
book like this were placed back of, and touching it, its front 

edge would appear of 
the same length as the 
back of this, and its back 
edge shorter, while its 
ends would converge in 
a line with those of the 
first book (Fig. 65). This 
can be imagined as re- 
peated infinitely, each 
book appearing smaller 
than the one before it, 

Fig. 64 

and the cover ends all falling in the same con- 
verging lines, until a point would be all that could 
represent the last book. The 
row of books might be said to 
vanish in this point, which is 
therefore called the Vanishing 
Point of such lines as converge 
toward it, as do the ends of 
the book cover. Vanishing 
Points, like the Level of the 
Eye, play a most important part in the study 
of perspective. 

A familiar example of vanishing lines, as those which appear 
to vanish, or converge perspectively, are called, is found in a 



receding railroad track (Fig. 66). The ties appear shorter 
as they are successively farther from the eye; and the rails 
appear and converge, till the whole track, if it could be seen 
for a long enough distance, might seem to disappear, or vanish 
in a point. 


Chapter XIII 


THE student may copy this example, but in any case 
should place a book successively in these positions and 
draw from that; having it high enough or far enough 
from the eye, to see it 
in a normal position as 
explained on page 2. 
He should also make 
drawings from mem- 
ory of a book in both 
positions, expressing 
them as artistically 
as possible. 

In the first position 
on this sheet, the 
eye level falls off the 
paper; and may be 
marked for use on a 
piece of paper fastened 
to the drawing (Fig. 
68) . See that the table 
line is high enough on 
the paper to clear the 
lower back corners of 
the book. 

It will be observed 
that the back of the 
book is not quite flat 
but slightly curved — a modification of the cylinder. This will 
be understood by holding the cylinder horizontally (Fig. 69). 


Fig. 67 

Fig. 68 


The curve opposite the eye is seen as a straight line, since it 
coincides with (or lies in a plane passing through) the direction 

j _ -vft^ 1 of seeing. The farther these lines are from 

^o» / \ gI^^^-^^^^I coinciding with this direction (in this case 
to right aj|d left) the more apparent is their 

For the second position in this exercise 
the book is opened and turned around so that 
its ends are parallel with the picture plane. 
They may therefore be drawn in their true 
shape like the back of the book. The sides 
and all lines parallel with them now vanish 
to VP ^ (on the eye level directly in front of 
the student). Note that points A, B, and C, 
where the book rests on the horizontal table (Fig. 68), are in a 
straight line that is parallel to the picture 
plane, and therefore drawn in its true 
direction, which is horizontal. Observe 
the projection of the covers beyond the 
leaves, and that it extends backward at 
D and E. The thickness of the covers 
must be recognized, though the wearing off of the edges and 
corners may obliterate their sharpness. Since the right and Left 
corners of the book are equally distant from the eye care must 
be taken that the covers are drawn of equal wddth. The clasps 
must be long enough to allow of their being fastened when the 
book is closed. Their ends are in a line converging to VP. 

The table line, being a subordinate element, should be so 
placed that most of its length is covered. Avoid anything which 
would tend to emphasize it, as making it coincide with the back 
corners of the book. 

1 Used as an abbreviation for the vanishing point. 

Fig. 69 


Chapter XIV 

THIS exercise (Fig. 70) combines a book in one of the 
two positions previously studied with a cylindrical ob- 
ject. The student may draw this example or not, 
according to his proficiency; but should compose and sketch 
a similar group, arranging and making trial sketches of several 

Fig. 70 

compositions. Observe that the extreme points of the book 
must be equidistant from the eye, as in Chapters XII and 
XIII. But as soon as we place another object with the book, 
the two must be considered together as forming one group or 







This brings us to reflect that whatever the number of objects 
we include in our picture, it is always drawn with the eye directly 
opposite the picture as a whole, so that the center of seeing is in 
the middle of the group from side to side. We also 
recall that the picture plane is always at right 
angles (viewed from above) to the direction of see- 
ing. So, if the cylindrical object is placed on, or 
in front of the book (as in Figs. 70 and 71), the 
central direction of seeing the picture is not 
changed ; and the picture plane continues parallel 
to one set of lines in the book as in the preceding 
exercise with the book alone. (See plan, Fig. 71.) 
If, on the other hand, the cylindrical object is 
placed at the side and the picture thus enlarged 
in one direction only (Fig. 72), the direction of 
seeing is immediately thereby 
moved to correspond, and the 
picture plane moves with it. 
The book will cease to be equi- 
distant at its ends from the 
picture plane and cannot be drawn as previ- 
ously studied. This subject is considered 
more fully in Chapters XXXIV and XLI. 
It would of course be possible to add objects 
to the book equally at both 
sides (as in Fig. 73), but dan- 
ger of stiffness in such an 
arrangement must then be 
remedied by some such device as the string of 
beads, making a more complicated study than 
is desirable at present. 
For this exercise, therefore, place the cylindrical object some- 
where within the extreme points from side to side of the 

It will be observed that a cylindrical object is always placed so 


Fig. 71 

PLAN or 


W^ANE 16 MO 







Fig. 72 

Fig. 73 


that a part of its base is seen, if only a very small part. For 
this reason, it is not put behind the book unless the foot 
can be left partly visible. The reason for this precaution is 
the uncertain effect produced by a study in which it is not 


Chapter XV 


FOR general directions see Chapter XI. 
The Models. — The rectangular block is 4" square by 8" 
long. It may be made of cardboard, cut as in the diagram 
(Fig. 74), and glued/ like the cube in Chapter XVI. Or two 
cubes, made as there directed, may be used in its place. The 
cylinder is 4" by 8", and has a circle about its middle. 

' » 



• ^ 

Fig. 74 

Fig. 75 

Positions. — The block lies on one long face, its long edges par- 
allel with the picture plane. The cylinder stands on one base in 
front of the block, touching it at its middle (Fig. 75). The 
models rest on a surface three times the height of the block below 
the eye, and are four feet distant. 

^ The light lines indicate where it is scored and bent for the edges of the block, 
quarter-inch projections are laps for fastening. 



Chapter XVI 


THE Model. — For this study make a cube, f oui* inches on a 
side, from cardboard cut as in the illustration (A, Fig. 76). 
Pass a string under one edge and out of adjacent corners 
(B, Fig. 76) before glueing together. 

Study of the Subject. — Turn the cube so the string comes from 
the upper front corners, and place it 
as the book was placed in Chapter 
XII (Fig. 77). Now, holding a front 
corner of the cube firmly, revolve 
the cube on that corner, bringing 
the side x into sight (Fig. 78). The 
moment the cube begins to revolve, 
the front, y^ begins to be turned 
away, ceasing to be 
parallel with the pic- 
ture plane, and tend- 

ing toward coinciding 

Fig. 76 

with the direction of seeing. In proportion as it 
is turned away, its right edge (H) becomes shorter, 
so that its upper and lower edges (E and F) appear 
to converge. The cube may be revolved until 
these edges (and their parallel, D) in their turn 
converge directly in front (Fig. 79), as A and B did 
at first. Then the side x, becoming parallel with the picture 
plane, will in turn be seen in its true shape, while its top and 
bottom edges appear horizontal. 

4 49 

Fig. 77 


Fig. 78 




Now turn the cube slowly back to the position of Fig. 78, 
and with the strings find the converging point of A and B. 

Figure 80 shows the cube in this position, 

and the vanishing of A, B, and C by the use 

of three strings. The lines at right angles 

to them, which in Fig. 79 vanished directly 

in front, here (in Fig. 80) vanish so far to the 

right that the strings cannot reach their 

vanishing point. 

If the cube is now revolved in the opposite direction, this 

vanishing point (which we may call VP2 ^) again moves 

inward, as seen in Fig. 81. 

It will be now readily seen that though any set of 
parallel horizontal lines (as A, B, and C) are directed 
more to the right or left, according as the cube is turned, 

they are never actually raised 
or lowered. Hence their van- 
ishing point does not move 
up or down, but is always 
found on the eye level. We may 
therefore conclude that receding hori- 
zontal lines always vanish in the eye 
_ level. 

We also confirm what was observed 
in Chapter XII, that 'parallel lines vanish 
to the same point. 

Let us now study the effect on the 
shape of its faces of revolving the cube. 
In Figure 77 the front face, y^ appears 
in its true shape, while lines at right 
angles to this face (as A and B) vanish 
directly in front, and the sides x and y 
are invisible. As the cube is revolved 
(Fig. 78) so that x comes into sight, so y is turned away, or 

1 In distinction to that already found. Vanishing points are numbered in order of finding. 


Fig. 79 

■£.yf Lcii'^l- 


Fig. 81 


Ar/^OT A CUI5E. 
A,B,a«dC ASt 

THE wimn or 
X; ANO D, E 

THE ro(?e- 
OF y. 


ofA 8y vi/ioth of the ' 



Tb ee leoHT. 

Fig. 82 

foreshortened. As a; widens, and its horizontal edges (A and B) 
grow less steep, the other side narrows, and its horizontal edges 
(E and F) become more steep. Steep- 
ness of the horizontal edges, therefore, 
goes with foreshortened surfaces. Good 
judgment on this point is very impor- 
tant, as the cube is the basis for later 
estimates of foreshortened surfaces. 
For this reason much space has 
been given to its study. It should 
be drawn with great care till thor- 
oughly mastered. 

The Recession of Horizontal Surfaces Tovrard the Eye Level. — It 
will be interesting here to place several cubes in a receding row, 
and see how the vanishing lines, being all included in one or 
the other of two sets, will vanish accordingly to one or the other 

of two vanishing points. Taking out 
every second cube (Fig. 83), we find the 
vanishing of those left to be unaltered. 
We also perceive that the table on 
which all rest seems to rise as it re- 
cedes, apparently tending to vanish or 
merge itself in the line marking the 
eye level. 

Looking at the tops of the cubes, all 
situated in one horizontal plane, and recalling the horizontal 
surfaces in previous drawings (as the book covers and the 
cylinder ends) we conclude that all horizontal surfaces appear 
to approach the level of the eye as they recede. This is seen 
to be true whether they are below the eye or above it. The 
vertical distance between receding horizontal planes must ap- 
pear less as it is farther from the eye, till at an infinite dis- 
tance it would be entirely lost, and the parallel planes would 
vanish in a line (the eye level) as parallel lines vanish in a 



"~To ^ZpI- 

FiG. 83 


The Eye Level. — The eye level, or level of the eye, is not 
actually a line ; but a height, or invisible horizontal plane, which 
may be said to extend indefinitely. Thus if the student's eye is 
^ five feet above the ground, his eye level passes through and 
includes every point at that height. But as each one's eye level 
is " edge to" him, it would (if visible) always appear to him as a 
line, hence it is always drawn as a line. 


Chapter XVII "^ '^^ 

THIS exercise should first be drawn from the objects, and 
then from memory, according to the general directions 
for memory work (Ch. XI). Two drawings on one 
sheet, showing the cube in different positions, are to be made. 
They should be represented as of the same size, which may be 

i iii i i iJMj w iffy yi w i k '^awyMMtapii 

wta% »:* Wiii>M»t'^< I I Mi Milw tf 

Fig. 84 

done by making their nearest vertical edges of the same length 
and at the same height on the paper, and using the same eye 
level for both. 

Position of Models. — For the first drawing, place the card- 
board cube so that its front faces are equally turned away, or 
make angles of forty-five degrees with, the picture plane (A in 



Fig. 85, also plan). Notice that its upper back corner will then 
appear exactly behind the upper front one, the vertical sides 

Fig. 85 

of equal width and the side corners opposite each other and 
equidistant from the center. In the second position the cube is 
turned so its right face makes an angle of 
sixty degrees with the picture plane (B in 
Figs. 85 and 86). 

Making the Drawing. — Fasten the paper 
in its place on the desk or have its posi- vlaa or a plah of q 
tion so marked that it can be accurately ^^^- ^^ 

returned to the same place. Draw the margin lines and lightly 
mark the extreme points for the two cubes (Ch. lY). Note that 

in the first position (A, Fig. 85) the cube 
occupies slightly more space, both horizon- 
tally and vertically. Since the cube is a 
type solid the lines in its final rendering 
are simple and firm, only varying slightly 
in thickness to suggest distance. Begin 
the first cube with the easiest part, which is its nearest vertical 
edge. This is parallel with the picture plane, and so is drawn 
in its true position. As soon as this line is placed mark the 
eye level (in this case it falls off the paper) finding its height 
as directed for the book. The numbers on the diagram (Fig. 87) 



shovinq a 
order for . 
drawing the 
lines of the 
cui3e, and of 
objects in general, 

Fig. 87 


Fig. 88 

give the order in which not only cubes, but rectangular ob- 
jects generally, should be drawn. Get the direction of lines 
2, 2 by pencil measurement (Ch. II) with espe- 
cial care, as their meeting with the eye level 
determines the vanishing points. Hold the pen- 
cil vertically in front of, or even touching the j<^"^:::jin 
nearest end of line 2 (Fig. 88). Then keeping 
it parallel with the picture plane 
(that is, not receding as the line 
does, but resting in an imaginary 
vertical plane) revolve it down- 
ward to the right until it seems to cover line 2 
Fig. 89 (Fig. 89). Holding it thus, with the other hand 

slip the paper (on which the 
drawing has been started) up 
vertically behind it till the pen- 
cil touches the upper end of 
the vertical line already drawn, 

and lies on 

C^\ the paper, 

sh o win g 

the direction line 2 should take (Fig. 90). 
(This puts the paper in the position of 
the picture plane.) Draw this first line 2, 
and mark its vanishing point on the eye 
level (VPl). The direction of the other 
line 2 could be found in the same way 
but as in this case they make equal angles 
with the picture plane, their vanishing 
points will be equidistant from the center, 
and YP2 can therefore be so located, and the second line 2 drawn 
to it. Lines 3, 3 are then drawn (recalling that parallel lines 
converge to the same vanishing point). 

For lines 4, 4 compare the apparent width of a near vertical 
face (A in Fig. 91) with the front vertical line (B in Fig. 91). 

55 • 

Fig. 90 

Fig. 91 


Fig. 92 

(This front line, being seen in its actual position and unfore- 
shortened, is the best for use as a unit of measurement.) Mark to 
right and left from line 1 in the drawing the proportionate dis- 
tance so found (in this case two thirds of line 1) and draw lines 

4, 4. From their upper extremities 
draw lines 5, 5 to their respective 
vanishing points. 

For the second drawing place the 
cube as directed, and proceed as 
with the first cube. In this case 
VP2 falls so far away that it can- 
not be shown in the illustration 
(B, Fig. 85). But we know that 
it must fall somewhere in the eye 
level. (It will be so found in the 
illustration, if tested.) At this stage a string pinned to VP4 will 
aid in detecting errors of vanishing, and will also make real the 
fact that these lines must vanish precisely to their own vanishing 

A Valuable Testing Method. — After this the following far more 
speedy and convenient method of testing should be acquired: 
With one eye closed hold the drawing close to the eye level, and 
turn it so that 
one set of van- 
ishing lines are 
directed to the 
open eye (Fig. 
92). Push the 
drawing back 
or forward as 
needed till the 

eye occupies the place of the vanishing point for the lines in 
question. Now sight back over this set of converging lines, 
when it will be found that any failing to properly vanish are 
quickly seen and easily noted for correction. A little expe- 


Fig. 93 


rience is needed to do this successfully, but it is well worth 
the trouble. 

Testing Before a Class. — An impressive method of demonstra- 
ting the vanishing of lines when teaching a class is the following. 
Draw a long horizontal line on the blackboard and mark it " Eye 
Level." Tack each pupil's drawing in turn on the blackboard so 
that the blackboard eye level coincides with the eye level of the 
pupil's drawing. With a long ruler follow out one of the vanish- 
ing lines (Fig. 93), and find its vanishing point on the blackboard 
eye level. Holding the ruler at this vanishing point as a pivot, 
swing it over the other lines of the set that should vanish to that 
point. The test is convincing, even to children; and helps 
greatly to form a standard of accuracy. 

It should always be remembered, however, that such measur- 
ing is only for testing^ never for drawing the lines. 


Chapter XVIII 



THE student may copy this example but must in any case 
draw from a book similarly placed ; and finally make a 
correct and spirited drawing of the same from memory. 
The position of this book is like that of the last cube (Ch. 
XYII). In studying this position begin with the book directly in 



VrftMiittw rirjfl\*^ihi^iniiii-'hiiai\f»mman<o 

■ o iij^ tfW * '' w %S Bwt> ^ '^gg»Jr»»'vwgw 

Fig. 94 

front as in Chapter XII. Note the convergence of its ends ; then 
turning it slowly into the required position for drawing (Figs. 94 
and 95), observe how the ends change in their convergence and 
how their vanishing point moves to the right on the eye level 
as the book is turned. Look also at the long edges of the 
book and see how, at the first movement of revolving it, they 



cease to appear horizontal, and vanish toward a point which, 
though at first infinitely distant, must nevertheless fall on the 
eye level. 

Drawing the Book. — Sketch the margin lines, and plan a good 
position of the book in relation to the inclosed space. Mark the 
height of the eye level as soon as a dimension (as xy. Fig. 95) by 

Fig. 95 

which it can be estimated is decided on. Find the direction of the 
book edges (corresponding to lines 2, 2 in the cube in Ch. XVI) 
with especial care. Sketch in the book with delicate lines, pro- 
ceeding in the order observed when drawing the cube, and 
correcting where necessary. 

Artistic Expression. — Finally, the subject should be rendered 
artistically. To accomplish this, the line is adapted to the qual- 
ity of that portion on which it is used. Certain features may be 
selected for use to augment interest ; as the curving ridges, the 
ornament, and the title space on the back, or even the worn 
corners. But having expressed in these details the point intended 
(as a worn corner by the shape of its boundary line) take care to 
do no more. It is wearisome, for instance, to see lines on these 
corners to represent the separation into layers caused by wear. 
Lines also produce a dark color, while worn corners are generally 
light; and are also undesirable places for the use of dark 



As the vertical edges of the cube are drawn vertical because 
parallel with the picture plane, so the corners of the book must be 
made vertical in the drawing, as they are in reality. For in- 
stance, points C and D being in a vertical line, must be so placed 
in the drawing. The same is true of the curves on the back of 
the book. 

At this point the student readily sees that all vertical lines 
(since the picture plane is vertical) will he parallel to the picture 
plane, and must invariably he drawn as they actually are, or vertical. 


Chapter XIX 



EGIN the study of this subject by placing the books as 
in Fig. 97. Observe that in this position there is but 
one vanishing point for the two objects, the ends of 

Fig. 96 

the books being all parallel, and their other horizontal edges 
parallel with the picture plane. Now turn the whole group, 
as in Fig. 98, and see that we have two vanishing points, 




one for the ends and the other for the long edges of the 

Now revolve the upper book a little more (Fig. 99), so 
that its horizontal edges cease to be parallel 
to those of the other, 
and it will have its 
own points of con- 
vergence (VPS and -t^i 
VP4). Its length ap- 
pears lessened, and its — i ^^— _ 
ends longer, for this Fig. 98 

change. The shortened edges vanish more steeply, and those 
which have become longer appear less steep. We find, as 
would be expected, that ivJien lines cease to be parallel^ their 
vanishing points are different. 

Fig. 97 

-£y£ LiVEL 

■Tb'iff . 

Fig. 99 


Chapter XX 


PRELIMINARY Study. — Does the eye see half way round the 
cylinder! The question is best answered by experiment. 
Holding the cylinder vertically and rather near (to 
more easily see the facts), mark on it the points where the side 
boundaries appear to meet the top (A and B in Fig. 101). It will 

Fig. 100 

be found that they are actually less than half way from the front 
to the back (Fig. 102). Yet the pencil has marked what the eye 
saw as the greatest dimension. As shown in Fig. 103 this appar- 
ent greatest dimension (A B) forms the long diameter of the ellipse 



Showing apparent middle from 


Fig. 101 

in the perspective view. It is evident, therefore, that the eye does 
not see half way round the cylinder, and (as seen in Ch. IV) that 

the long diameter of the ellipse is 
not an actual diameter of the circle, 
while that portion of the circumfer- 
ence beyond the long diameter (A B) 
is actually more than half of the circle, 
the part in front of A B appearing equal 
to it only because nearer to the eye. 

The actual position of the apparent 
greatest dimension (the long diameter) 
changes with the position of the observer. 
The plan (Fig. 102) shows that C D would appear as the greatest 
dimension if the eye should be at 2. This also may be seen by 
experiment (as in Fig. 101). 

Planning the Exercise. — In 'placing this exercise observe that 
the perspective of the concentric square and circles is made 
much larger than the geometric diagram, 
to show more clearly the perspective 

Drawing the Circles. — When the square 
has been drawn in perspective (like the 
top of the cube in Fig. 77) its actual 
center (o in Fig. 103) is found at the 
crossing of its diagonals, as in the geo- 
metric diagram above. In the diagram 
the ends of its diameters mark the points 
(C, D, E, and F) where the circle touches 
the square, and they will do the same in 
the perspective. The diameters pass through the true center and 
one is parallel to the picture plane. It can therefore be drawn 
in the perspective in its actual direction, giving two points 
(c and d). The other diameter, being parallel with the receding 
sides of the square, vanishes with them in the eye level directly 
in front (at VPl) giving points e and /. Now, though the actual 


showing' the 
actual place 
of the long 
diameter, a b 

Fig. 102 


THE \ 



diameter of the circle touches the square in c and d, the ellipse 

appears longest at a part nearer than c and d, which seems to 

be exactly half way between e and /. Through this half-way 

point {x) the long diameter can 

be drawn; making it longer than 

c-d, and yet not. quite touching 

the square. The ellipse is then 

easily sketched through these six 

points (a, 6, c, d, e, and/), making 

it symmetrical on a-b and e-f. 

For the other ellipses the points 
where they cross the actual diam- 
eter of the circle {c-d) are marked 
by lines from 1, 2, 3, and 4 which 
vanish in VPl, giving four points 
(9, 10, 11 and 12), two for each 
of the smaller ellipses. 

Measuring Distances into the 
Picture. — For the front and back 
points of these ellipses, line ef 
must be divided into six perspec- 
tively equal parts, as EF in the 
diagram is divided into six actu- 
ally equal parts. This can be done, and in practice usually is 
done, by the eye (as for the cylinder in'Ch. IV), noting that 
the true center (o), already known, is one point of division. But 
the use of the diagonal for such distances is simple and often 
a convenience. Thus it is easy to see that in the diagram the 
vertical lines from 1, 2, 3, and 4 cut the diagonals proportionately 
to the divisions on GH* in this case into six equal parts. These 
divisions can in turn be transferred to EF by horizontal lines 
from the points on the diagonal HI, giving 5, 6, 7, and 8, the four 

^ students of geonojetry will recognize in this the problem 
of dividing a line proportionately by means of parallel lines cross- 
ing a triangle. 

5 64 

Perspective representation or above. 
Fig. 103 

Fig. 104 


points needed. In the perspective the method is the same, using 
lines perspectively parallel to e-f — that is, the lines already 
drawn from 1, 2, 3, and 4 to VPl. This use of the diagonal 
occurs further on, as for the steps in Chapter XXII. 

A Second Method. — The vanishing point of the diagonal can 
also be used to obtain these points. Thus the diagram shows 
that lines from 1, 2, 9, and 10 parallel with the diagonal GrJ will 
mark on EF the same divisions. In the perspective these lines 
will appear perspectively parallel to the diagonal — that is, drawn 
to the same vanishing point. Since they are horizontal, that 
vanishing point will be on the eye level. Therefore the diagonal 
GrJ can be carried out to the eye level to find its vanishing point 
(VP2) to which the parallel lines are drawn. 

The principle to be remembered for use is: WJmtever meas- 
urements can he ohtained geometrically hi/ the use of actually parallel 
lines, can he ohtained in perspective hy the use of perspectively parallel 

It must, however, be noted that these are only relative meas- 
urements. A first distance into the picture — the foreshortened 
width of the square in this case — is determined freehand by 
past experience (as with the cube). Mechanical perspective 
gives methods of obtaining this first distance, the position of the 
eye and the picture plane being given. It can also be obtained 
from a side view, by using the same data. Both these methods 
are too complicated for common use in freehand work. Such 
proportions are so easily estimated by recalling the cube that it 
is better to rely on a trained judgment for them. 



Chapter XXI 


HE student should take this exercise as previous ones, 
copying first if he needs to do so, then composing and 
sketching a similar study, and finally making a drawing 










zl^ "' 




8K ■^■^^;i^^"^^>^-| 


ISr- ' ^£^& \r<v\'lr- .£l.' ' 







i--v '€^: 


m?- — 


Fig. 105 

from memory. For both of the latter several different arrange- 
ments of objects, with trial sketches, should be made; and the 
best chosen to use in the final drawing. 

The Finder (Ch. VIII) should be used to compare the 
effect of different compositions, also the effect of cutting out 



compositions from larger ones by different margins. Note 
that in Fig. 106, with a tall object, the books are turned so 

that their horizontal dimensions are 
not great enough to neutralize the 
dominant vertical effect. In Fig. 
107, on the other hand, the long 
horizontal dimensions of the books 
and the low flat dish harmonize 
very well, and this arrangement 

Fig. 106 

Fig. 107 

necessitates a marginal rectangle longer from side to side. 
The books and dish alone make a good simple arrangement, 
but the tray may be added if desired. 

Fig. 108 


Chapter XXII 


ODEL for the Study. — Make an equilateral triangular 
prism from cardboard cut as in the diagram (Fig. 110), 
and place it on the top of two cubes. Put a box or 

Fig. 109 

books on the table under this model, raising it so that the level 
of the eye will fall one-fourth way up on the cubes (Fig. Ill) . 
Place the model about sixteen inches from the eye, and turn it 
so its long edges will make angles of thirty degrees with the pic- 
ture plane (Fig. 119). It may now be regarded as the type form 
of a house, seen (in proportion to its size) from an ordinary posi- 



Fig. 110 

tion for viewing a house. By aid of the imagination, it may 
be regarded as a house of two stories, with a front door in the 

middle of a side, the box top taking 
the place of the ground. 

This exercise should be first drawn 
in thin, light lines, studying the dia- 
grams, and following the directions. 
The construction lines should then be 
erased, and the drawing rendered as 
shown in Fig. 109. 

Drawing the Exercise. — Begin with 
the nearest vertical edge of the house. 
The model was placed so that the 
level of the eye should be one-fourth way up the height of its 
rectangular part because the eyes of a person standing might 
be about five feet above the 
ground, and the height of a 
two-story house at its eaves 
about twenty feet. Mark the 
eye level on the paper there- 
fore, one fourth of the height 
of the nearest edge from its 
bottom, and take the direction 
of lines A and B (Fig. 112) 
to determine the two vanishing points, exactly as was done 
with lines 2, 2 in the cube (Ch. XVII) although, being above 
the eye, they appear to tend downward. Draw the two lower 
horizontal lines and the side vertical lines as those of the cube 
were drawn. 

To construct the roof recall that the end of our model is 
an equilateral triangle (Fig. 113) with its apex over the center 
of the house end. Draw the diagonals of this square house end 
and carry up a vertical line of indefinite length from its center, 
on which the apex of the gable is to be marked. The actual 
roof height of our small model may be found by this diagram ; 


Fig. Ill 


but as that makes a roof steeper than is usual, we will set it 
off less in the drawing, that is, making EC (Fig. 112) but a 


Fig. 112 

little more than CD. (Since these distances are in the same 
vertical line, and so at the same distance from the picture plane 
they are seen and drawn in their true proportions to each 

The sloping sides of the gable may then be^^ drawn 
to the house corners, and the ridgepole to VE2. The 
gable apex on the other end may be found by drawing 
a vertical line from the center (x) of the invisible end 
to cut the ridgepole. Its slanting sides are drawn to 
complete the blocking-in lines thus far of the house. 


Fig. 113 





Oblique Vanishing Lines. — We have said little about the slop- 
ing end lines of the roof. But now, looking again at our 
model (Fig. Ill), we see that the ridgepole, R, because it is 
farther away than the eaves, appears shorter, so that the 
slanting ends (F and G) of the front surface of the roof appear 
to converge upward. Turning to the drawing, we find in' con- 
firmation of this that (if the drawing has been carefully made) 
these lines do thus converge. Now let us search for the general 
truth governing that convergence. These slanting ends are not 
horizontal, so that we should not expect them to tend toward the 
eye level ; and we observe that they do not. But they are actu- 
ally parallel to each other and therefore must vanish or appear 
to converge to the same point. How to find that point is the 

Put some books in the place of the house model, and arrange 
them so that their edges vanish like the house edges. Now raise 

the upper cover (A, 
'^ '' ~ - Fig. 114), and observe 

that its ends 1 and 2, 
though still parallel 
with each other like 
the ends of the roof, 
have ceased to be 
parallel with the 
other book ends. 
They therefore no 
longer vanish toward 
VPl, but to a higher 
point. We have not, 
however, turned these 
edges to right or left, but have simply lifted their farther ends 
or revolved them in parallel vertical planes. Therefore their 
vanishing point cannot move to right or left; but as they are 
revolved, must appear to move directly upward, or in a vertical line 
passing through VPl. This continues until the cover becomes 


Fig. 114 


^. vertical ; when its ends appear in their true position and cease 
to vanish, like all vertical lines. 

Place strings under the cover, as in Chapter XII. Holding 
the strings with the left hand as in the illustration (B in Fig. 
114) raise the cover with the right (keeping the strings parallel 
to the picture plane). By this experiment their convergence 
toward a point in the vertical line from VPl is more plainly 
shown. Since these slanting book ends are neither horizontal 
nor vertical but oblique to both directions, their vanishing point 
or that of any set of oblique lines, may be distinguished as an 
Oblique Vanishing Point, or OVP. 

By revolving the upper cover farther, or opening the lower 
cover and using the string (A in Fig. 115), oblique vanishing 
points below the 
eye level may be 
determined. And 
the apparent di- 
rection of oblique 
lines can be found 
with the pencil ex- 
actly as that of 
any line (B in 
Fig. 115). 

Vanishing Traces. 
— By turning one 
of these illustra- 
tions around, to bring the eye level vertical, as in Fig. 116, it 
will be seen that the line containing OVPl and 0VP2 serves 
a purpose similar to that of the eye level. We note that 
the surface formed hy the visible ends of the hooks appears to 
recede, or vanishes, toward this line. If a larger book be placed 
against the other ends, the surface of the larger book, being 
parallel to ,the visible ends of the other books, will be found to 
vanish toward the same line. It may be concluded that all sur- 
faces parallel to the book ends in this case will vanish in this line, 


Fig. 115 


/ exactly as all horizontal surfaces appear to vanish toward the 
eye level. We may call this line a Vanishing Trace. The eye 
level is such a vanishing trace for all ^horizontal surfaces. See 

note, Chapter XI, in Solu- 
tions of Problems. ' 

First replacing the house 
model as in Fig. Ill, we now 
turn to the drawing and test 
these oblique lines (F and Gr 
in Fig. 112). If correctly 
drawn they will be found 
to converge toward a point 
(OVPl) directly above VPl. 
At once use is made of this 
point for drawing the ends 
of the roof projection (sug- 
gested in the model by pin- 
ning cardboard as in Fig. 
117). These edges are par- 
allel with the corresponding roof edges, like the book margins; 
so their width can be set off oil the upper line of the house (B, 
Figs. 112 and 117) to right and left, 
(points X and y) remembering that 
the nearer distance appears a little 
greater. Through these points draw 
lines vanishing to OVPl. A similar 
projection is measured downward on a 
continuation of the oblique gable edge 
F beyond its lower end (z) ; and through 

this point a line parallel to B (that is, vanishing with it in VP2) 
forms the eaves. (The estimation of these last measurements by 
the eye forms an important part of the student's training and 
should be carefully thought out. Thus the eaves projection from 
line B forward is more foreshortened than the gable projection 
from F to the left; and distances should be set off accordingly.) 


Fig. 116 


Fig. 117 


dA.C\\ VI tW 

Fig. 118 

The lower oblique vanishing point (0VP2) is used for the 
projections on the back slope of the roof. Continue the back 
oblique roof edge (line H) to meet the vanishing trace through 
VPl, giving 0VP2. Draw line K from the near end (I) of the 
ridgepole to 0VP2, and cut it by a line from the nearest eaves 
i corner to VPl. Where this line cuts the oblique edge K will be 
the eaves corner (M) for the far side of the roof ; and a line from 
it to yP2 forms the eaves on that side. 

There is another way of getting the projections on the 
further slope of the roof, which is useful in case 0VP2 falls too 
far away to be conveniently used. 
Turning the model we see that the 
invisible line (L in back view, Fig. 
118), if carried to the edge, ends 
in O, horizontall}^ opposite x (end 
view). A line from x through O 
would therefore vanish in VPl. 
Hence, to obtain O, line L is carried forward indefinitely, and 
cut by a* line from x to VPl. The desired edge is then drawn 
from point I through O indefinitely, and cut by a line from J to 
VPl, giving the corner, M. 

The "L" Part of the House. — The plan (Fig. 119) shows its 
position. Its width is marked off on the farther (invisible) 

end of the house, and it is drawn 

as was the main part of the 

house. Note the less steep slope 

of the porch roof (Fig. 112), so 

that its oblique lines are nOt 

parallel with those of the other 

roofs but have another vanishing 

point, 0VP3, lower in the same 

vertical line. 

Windows and Doors. — The windows and doors may be marked 

on the model (Fig. 120). It will be easily seen that their top 

and bottom edges are all parallel to the horizontal lines of the 


Fig. 119 



n 1 

'd d 

D n 

DD □ DD 


Fig. 120 

side on which they are located and therefore vanish to the same 
point. Mark their heights on the nearest vertical edge of the 
house, and draw lines (as P, in Fig. 112), thence to the vanishing 

points. On these lines their perspec- 
tive widths are to be set off. Begin 
with the door. Find the middle of 
the house front by its diagonals, and 
make the near half of the door a 
little wider than the far one. Check 
this by seeing that the remaining 
distances (from the door to the front corners of the house) are 
also perspectively equal, that is, the near one larger. Mark the 
sides of the windows in the same way. Remember that since 
the space between the near window and the near corner is con- 
siderably nearer to us than that between the far window and the 
far corner, more difference should be made in their size than 
between the halves of the door. 

The width of the windows on the end of the house should be 
to that of the front ones as the right face of the second cube in 
Chapter XVII (Fig. 84) is to the right one. The height of the 
windows in the "L" is made the 
same perspectively by carrying their 
measurements from the right front 
corner of the main house on lines 
vanishing to VPl. These lines lie 
on the invisible end of the main 
house, and from where they reach the 
" L " are continued along its front by 
lines running to VP2. 

The Chimney. — To better visualize this part of the house, cut 
and fold cardboard as in Fig. 121. Get the slope of lines 1, 2, 3, 
and 4 by laying the cardboard against the apex of the gable 
and marking around it. Stand this model on the roof in its 
middle, and after marking on the roof around it, cut out the 
space so marked and push the chimney down through the open- 


Fig. 121 



ing until it projects the proper distance above the roof. Lay a 
pencil on the roof against the chimney (Fig. 122) , and move it to 
the left without changing its direction till it coincides with the 
gable edge. This shows the gable 
edge and the oblique line where the 
chimney passes through the roof to 
be actually parallel. This oblique 
line, therefore, has the same vanish- 
ing point as that of the gable line, 
which is OVPl. The top of the 
chimney front and the line below 
it (AB, Fig. 123) are parallel to the 
eaves and ridgepole. Turn the model (Fig. 120, end view) and 
see that the top edge of the chimney is parallel with the hori- 
zontal lines on the house end, which we have already drawn to 

Fig. 122 


Fig. 123 

YPl. A pencil held horizontally and moved slowly up in front 
of the model will help to see this as will marking the chimney in 
the model off into bricks (Fig. 124). 

Drawing the Chimney. — Continue on the roof the center line 
used for the door (that is, vanish a line from its top to OVPl), 



A.- 'Plan or Chimney. 

Q- "Profile snowwe obna- 


Fig. 124 

and mark down from the ridgepole on this line half the thickness 
of the chimney (judged by the eye). Draw a line through this 

point toward VP2, and on it set off to right 
and left perspectively equal distances for 
the breadth of the chimney (AB). Draw 
line C to OVPl. Where it crosses the ridge- 
pole (D) is the middle of the chimney from 
front to back. Make the far half of the 
chimney proportionately as much smaller 
than the near half as the far half of the 
house end is smaller than its near half. 
The projecting band at the top of the chimney is shown in 
plan and profile in Fig. 124. Its perspective is drawn as are 
projecting book covers" Be careful to represent the backward 
projection on the farther side. 

The Steps. — For these the detail drawing (Fig. 125) is first 
made. The height under the threshold of the door is a little 
less than two feet, or about 
one third of the height of 
the eye — enough for three 
steps. Divide the vertical 
line under the near edge of 
the door therefore into three 
equal parts, and draw lines 
of indefinite length to VPl 
through the four points of 
division. On the lower line, 
B, mark off the proper dis- 
tance (as four feet), which 

may be estimated by comparison with the windows on the end 
of the house (their width being parallel with these lines, and 
usually about three feet). Divide this distance into perspective 
halves. A vertical line from the near end of line B, cuttins: line 
E in Point 2, completes the rectangle, 1-2-3-4, the middle of 
which can be found by its diagonals, giving the perspective 


1 Mu^JIJU 

Fig. 125 















7 8 

C J ^\ 



B ^x 

Fig. 126 

halves required. (See A, Fig. 126.) The further half is for the 
wide top step. The near half of the rectangle can be divided 
again in the same way for the two lower steps. Where the 
vertical line from the near end of B cuts 
line C is the upper near corner of the lower 
step. A vertical line through will mark its 
width on C, and continued to cut D forms 
the nearest front corner of the second step. 

Another method of sketching the steps 
is shown in Fig. 126. When the first step 
has been drawn its diagonal is continued 
through 6, cutting line D in 7, and forming 
the diagonal for the second step, which is completed by con- 
tinuing line D to cut a vertical from 6 in 8. This can be con- 
tinued for as many steps as needed. The diagonal can also be 
used as a test for steps drawn by the first method. 

The long edges of the steps vanish in YP2, and are cut alter- 
nately by lines vanishing in VPl and vertical lines. 

The Dormer Window. — This is constructed in principle like 
the gable of the roof. The detail drawing (Fig. 127) should be 

carefully studied, and drawn sepa- 
rately if desired, before sketching 
the window on the house. 

On the center vertical line of the 
house front continued upward, mark 
the height of the dormer from line 
B. (In this case it is not so high 
as the main house.) Through this 
point (S) the dormer ridgepole is 
drawn to VPl, and cut by the 
oblique middle line on the roof (in point T). The width (1-2) 
of the dormer is then marked perspectively to right and left on 
line B. Through these points (1 and 2) the " valleys," or meeting 
lines of the dormer with the main roof, are drawn from the roof 
end (T) of the dormer ridgepole to the edge of the eaves (points 


Fig. 127 


TJ and V) . From these points the edges of the dormer roof pro- 
jection run parallel respectively to A and C. 

Oblique Lines in the Dormer. — These two lines A and C, 
though oblique to line B, are in the same vertical plane (as the 
gable lines F and H in the main house are in the same plane 
with line A in Figs. Ill and 112), Therefore draw a second 
vanishing trace for oblique lines vertically through VP2, and 
continue A upward until it cuts this trace in 0VP4, to which 
draw the edge D. The other oblique edge (E) vanishes in the 
same vertical below VP2. 

When experience has been acquired, such oblique lines can 
be satisfactorily drawn without actually finding their vanishing 
points. Such convergences are generally estimated in practical 
work. But estimates are much more valuable when made 
with a knowledge of methods by which they can be definitely 


Chapter XXIII 



THE example given in Fig. 128 is from the old church of 
San' Apollinare in Classe, near Ravenna. 
The beginner may draw this as a preparation for his 

next work, which should be pilding or part of 

one from a print of his own seieciiou. 

Making a Selection. — This choice should be carefully made, 
care being taken to secure unity, or an appearance of one whole 
6 81 


thing having a center of interest and parts which are subor- 
dinated, or catch the eye less quickly. It should be well placed 

in its rectangle (Chs. VIII and XXI). 
For instance, the tall tower in Fig. 129 
needs a margin that is longest vertically, 
and quite narrow, to produce a harmony 
of lines. The smaller buildings with it 
give variety, and by a contrast which is 
not too great emphasize its height, be- 
ing subordinated that the tower may 
remain prominent in the composition. 

130, on the other hand, 

In Fig. 
the long, 
low mass 
of farm 
set well 
back into 
the pic- 
ture re- 
quires a rectangle that is longer horizontally. 

Some of the different selections that may be made from 

one print (Fig. 131) are 
shown in Fig. 132. The 
beginner can by such 
means obtain an example 
simple enough to be with- 
in his powers and often 
a better composition. 

Drawing from the Print. 
— As soon as the place 
of the building on the 
paper is fixed, the level 
of the eye must he deter- 
mined and marked, and the vanishing points of the principal sets 

Fig. 129 

Fig. 130 

Fig. 131 



of horizontal lines must he found on that. It is of course easier 

for the beginner to use such vanishing points as are near enough 

to be marked. But the 

student must fully un- 
derstand that a point too 

far away to be marked 

can be mentally located, 

and the lines drawn 

toward it with closely 

approximated accuracy. 

The essential thing is to 

have the position of such a 

point clearly thought out, 

— even, for instance, as 

specifically as that it is 

" the width of the board," 

or " three times " that, 

distant. The power to 

do this accurately grows 

rapidly, and can be 

attained by students 

of moderate ability. It 

is one object of this 


Rendering from the Print. — As more complex sketches are 

made, certain parts may be expressed in color (that is, covered 

with a tone of pencil lines) , as was 
done with the title space of the 
books in Chapter XIX. The door- 
way, windows, and shaded sides of 
the buildings in this exercise (Fig. 
131), are examples of this. Such 

use of color is intended sometimes to attract the eye to the most 

important or interesting parts, or to bring out the beauty of such 

details as the majestic forms of the trees in A, Fig. 132. 


Fig. 132 


'jp ^-x c: 

Fig. 133 


The Comparative Simplicity of Perspective. — By experience in 
mentally grouping each new vanishing line with the set to which 
it belongs, the perspective of apparently difficult studies becomes 
simple. In Fig. 133 a seemingly complex group of buildings is 
shown to need but four vanishing points for nearly all of its 


Chapter XXIF 



HE Model. — The model for the square frame is six inches 
on a side, and one inch square in section. Looked at 
from the front, 

it appears as two con- 
centric squares one 
inch apart (Fig. 135). 
It is placed with one 
set of long edges ver- 
tical, and the other 
horizontal and mak- 
ing angles of sixty 
degrees to the left 
with the picture plane 
(Fig. 136). 

In considering its 
shape it may be first 
regarded as a Plinth, 
or one-inch rectangu- 
lar slice from a six- 
inch cube, and there- 
fore one sixth of the 
cube in thickness 
(Fig. 137). 

^ Some of the geometric 
solids here and later given may 
be omitted at the discretion of 
the teacher. Those selected 
for study should be such as to 
supply any deficiencies in the 
student's mastery of the subject. 

»* K <» Ka i.B i ■ mift »ii l1w i W" ' lu !WWBPMMIiWi<!P<mi>.' 

-— ..Tri. l-.n-. »,.•■.- — ,W„^^p>^-.yg;. , ..^.^y|.- g Y».-^— '.^-S 

Fig. 134 




Fig. 135 

Fig. 136 

Drawing the Model. — The lines of this solid may be drawn 
and its proportions established in the same manner as those of 
the cube (p. 54). Remember to place the eye 
level at once after drawing the first vertical edge. 
Like the cube, these type forms should be lightly 
sketched first and later rendered with the firm, 
simple lines appropriate to them. Being more 
complicated than the cube, their visible edges 
may, if necessary, be strengthened (that is, be drawn as they 
would finally appear) as soon as determined, to avoid confusion. 

The inner square of the frame may next be marked 
out on the plinth (Fig. 137). To do this place points 
one sixth of the front vertical edge from each end, 
and from them vanish lines to YPl. In these lines 
the edges A and B of the inner square must lie. By 
the front view (Fig. 135), we perceive that the corners 
of the inner square lie in the diagonals of the outer 

one. One diagonal, C, will mark two corners 
{x and y) of the inner square. Its other two 
corners are found by drawing the vertical edges 
of the inner square, from corner x down to line 
B and from y up to A. 

If this inner square were cut out, leaving a 
frame, parts of the inner thickness of the frame could then 
be seen. Of this inner thickness, line D (Fig. 138) lies in the 
back surface of the frame, parallel 
to B, and at actually the " same 
height. Hence it can be started 
at a point (s) on the right hand 
vertical edge of the frame, obtained 
by drawing line E from the near 
end of B to VP2. This may be 
called carrying line B " around 
the corner." From this point 0, D is drawn to VPl. The 
lower inner edge (F) at the back is parallel to the outer thick- 

FiG. 137 

Fig. 138 


ness edges, and therefore can be drawn to VP2, cutting off line 
D. A vertical line from where D and F meet completes the 
inner thickness. 

Tests. — The correctness of this drawing can be tested by 
adding the invisible portions, shown 
by dotted lines in Fig. 138. Thus 
if line A be carried around the cor- 
ner, giving the invisible edge H, the 
inner invisible edge I should cut it in 
line G continued. 

The Application of Type Form 
Principles. — The application of the 
foregoing work to the drawing of 
such parts of the house as windows and doors may be seen 
in Fig. 139, where the inner edges of door and window frames 
converge with the set of horizontal lines at right angles to the 
door. For instance, lines A and B converge with the dormer 
eaves and other lines tending to VP2, and line C vanishes with 
the set to VPl. 


Chapter XXV 



THE Models. — The plinth is two inches high and six inches 
square : the pyramid four" inches square at base and eight 
inches high. The models can be made (Fig. 141). If 

made, note that in 
order to secure the 
required height in 
the completed pyra- 
mid, the length (a;^ 
in Fig. 141) of each 
triangular side piece 
of the pyramid pat- 
tern is measured 
from xy in Fig. 142, 
where the true 
length of a side face 
is shown. The face 
ocr-y-z in Fig. 142 
leans back, making 
x-o foreshortened. 

Position. — The 
plinth rests on one 
square face, with its 
sides at angles of 
thirty and sixty 
degrees with the 
picture plane (Fig. 
142). The pyramid 
Fig. 140 stauds ou the plinth, 



A. Pattern 


with its base parallel to and equidistant from the edges of the 

plinth top. 

Drawing the Models. — Proceed 

with the plinth as with the cube, 

remembering that its height is 
but one third as 
much as the cube 
in proportion to 
its breadth. 

For the pyra- 
mid base, mark 
on line AB (Fig. 
143) one perspec- 
tive sixth from 
each end. For 
these points (1 

and 2) a diagonal of the side ABCD can be 
used (as in Ch. XX). The vertical edge, AC, 
being unforeshortened, can be divided into six 
equal parts. Lines from the upper and lower 


B. Pattern for 



Fig. 141 

B. ! Plan 
Fig. 14;^ 



division points to VPl trans- 
fer these divisions propor- 
tionately to the diagonal, AD. 
From the diagonal they are 
transferred by vertical lines 
to the edge AB. (See also 
Fig. 144.) 

Or a diameter through the 
center (0) 

will divide ^'«- ^^^ 

AB into perspective halves, when each half 
can be divided into thirds by the eye. One 
method can be .used to prove the other. 
From these two points (2 and 3) draw lines to VP2. Where 
they cut the diagonal, AE, will be two corners (4 and 5) of the 


Geometric view of 
side of plinth. 

Fig. 144 


pyramid. Lines to VPl through these points will give the two 
other corners (6 and 7), and complete the base of the pyramid. 

The apex of the pyramid will be vertically over the center of 
its base, point O. On a vertical line from O must be measured 

the perspective height of the pyramid. Its 
actual height is four times that of the plinth. 
The nearest corner of the plinth is conven- 
ient to use, hence from A the vertical line 
is continued, and four times AC is measured 
on it, giving AGr for the pyramid height as it 
would appear at that point. If now this 
height, AGr, could ' be moved back on the 
diagonal AE to O it would appear to shorten 
as moved. Its top would describe an actu- 
ally horizontal line above AE, that is, a line 
parallel to it, consequently vanishing to the 
same point. Therefore continue the diagonal 
AE to the eye level, giving VPS, and draw 
the parallel line from Gr to VP3, which will 
mark on the vertical from O the desired 
perspective height at x. Complete the py- 
ramid by drawing its oblique edges to the 
corners of its base. 
AppUcations. — The difficulty of making a church spire or 
a tower (Fig. 145) " stand true " will be readily recognized. The 
use of the diagonals (AB and CD) will aid in placing its axis 
and apex. 

Fig. 145 


Chapter XXFI 

THE Square Frame Leaning on the Rectangular Block. — 
These models have already been described in Chapter XV 
and Chapter XXIV, respectively. 
Position. — The block rests on one long face, with its square 
ends making angles of sixty degrees with the picture frame. 


Fig. 146 

The frame leans against the block, equidistant from its ends, 
and with a distance equal to half the width of the block between 
its lower edge and the block. 


Chapter XXVII 

Fig. 147 


A LTHdUGH in Chapter III the cylinder held horizontally 
/\ was mentioned, we have only studied cylindrical objects 

jL JL. when vertical. In this position they have been found 

symmetrical, the ellipses and the axis (which is the middle 

from end to end) being at right angles to each other. I'o study 

___^_ them in other positions begin with the 

cylinder model held horizontally, with its 
middle oh the eye level, and its ends 
equally distant (Fig. 147), It will be read- 
ily seen that the ends now appear at right 
angles to the axis, as when the object was 
vertical. Turning it a little so the right 

end can be seen (still keeping it horizontal and at the eye level), 

it will be observed that the apparent directions of the axis and 

ellipses are unchanged. (The axis being 

a horizontal line and at the eye level 

remains apparently horizontal, and the 

ellipses still appear vertical. The further 

ellipse has become a little shorter and 

rounder, and the side boundaries, like all 

parallel receding horizontal lines, appear p^^ ^^^ 

to converge to the eye level.) 

Now lower the model, keeping it turned away, till it rests a 

foot below the eye on some horizontal support , (as the box in 

Fig. 149). 

The side boundaries and the axis, being below the eye, vanish 

upward to a point on the level of the eye. ^^^hey will continue 






1 1 




^ 111 













to vanish to the eye level, whether below or above the eye (B 

in Fig. 149), as long as the cylinder is kept horizontal. The 

ellipses should now be exam- 
ined to see if in this position 

they appear, as formerly, to be 

at right angles to the axis. Do 

this first with the head erect 

as usual, looking with care, 

and deciding mentally. Then 

try inclining the head (in this 

case of A, Fig. 149, to the right 

and downward) to bring the 

face in relation to the model 

as it would be if both were 

vertical. Two pencils held in 

the shape of a letter T, held in 

front of the cylinder (as in Fig. 


make sure that the axis of the cylin- 
der and the long diameters of its 
ellipses unmistakably appear at right 
angles to each other. 

To understand how this can be 
the case, 
^''■'"' hold the 

cylinder again at the eye level, as 

in Fig. 151, and with a pencil mark 

on it the points (A and B) where 

the side boundaries meet the ends 

of the ellipse. Lower the cylinder 

again (Fig. 152) when it will be seen that these points are not 

now at the ends of the ellipse, and that the line AB is not now 

its long diameter. It has now a new long diameter, CD, at right 


IlllillnW illiiii i 'iliilllill \tiiiiiiiii' 
Fig. 149 

Fig. 151 


angles to the axis in its new apparent direction; and also new 
side boundary lines from C and D toward the vanishing point. 

As the long diameter is 
always at right angles to the 
axis, it must change its position 
when the object is moved so that 
its axis appears changed in 
direction. We see therefore 
that the long diameter is movable. 
These experiments may be 
tried with other cylindrical ob- 
jects, as a tumbler, or the flower pots in the next chapter. The 
leaning dish in Chapter 
YIII, and the tilted cover 
in Chapter X are examples. 
It will invariably be found 
that, provided the circular de- 
tails of a 

I'liMiillilliin \'A ' 'A Mm~ 
Fig, 152 

Fig. 153 

^^^C^^y oZy'ec^ are actually at right angles to its axis, they 

Fig. 154 

will appear so whatever the position of the object. 
Consequently, cylindrical objects always appear 

Tests. — A drawing of such an object may 
be tested by turning it to bHng the object in 
question vertical (Fig. 156, Ch. XXYIII), when 
errors in symmetry will be more apparent. 

A wheeled vehicle (Fig. 153) is a common 
illustration of this principle; also a clock (Ch. 
XXXI) and the round arches in Chapter XXXII. 
Others will readily occur to the student. 
Flowers (Fig. 154) are striking examples, and 
many awkward drawings of flowers are so be- 
cause drawn in ignorance of this beautiful and 
simple principle of the symmetry of the cylinder. 



Chapter XXVIII 

S with previous examples, drawing this exercise is optional 
with the student, according to his proficiency. But he 
should compose a similar group, that is, having in it at 

Fig. 155 

least one cylindrical object not vertical. 
And he should take especial pains to se- 
cure the symme- 
try of such non- 
vertical objects. 

The illustra- 
tion (Fig. 156) 
shows how this 
symmetry may be 
tested by turning Fig. uq 

the group. 




»NO A 


lb BIhMCV 




Chapter XXIX 





HIS is an example of rectangular and cylindrical forms 
in the same object. The explanation should be carefully 
studied, and the exercise drawn unless the student is 


The Circular Frame. — 
After the square frame 
(Ch. XXIV) is drawn, 
look at the model from 
the front (Fig. 158), and 
note that the outer sur- 
face of the ring touches 
the square frame at four 
points only — where the 
diameters of the square 
cross it (A, B, C, and D). 
These diameters, being 
parallel respectively to 
the sides of the square, 
are represented in the 
perspective by a vertical 
line, and a line vanish- 
ing in VPl, both pass- 
ing through the true 
center 0, at crossing of 
the diagonals. Through 
these four points the 
outer edge of the cir- 
cular frame must pass. 


Fig. 157 










Disregarding at first the opening in it, the circular frame may 
be temporarily thought of as a slice from the cylinder, one inch 
thick, and therefore a very short cylinder, with 
an axis only one inch long. As placed within 
the square frame (Figs. 157 and 159) its axis and 
side boundaries are parallel with the short edges 
of the square, and its actual centre and that of 

the latter coincide. 
Hence we may draw its 
axis from this center (0) 
to VP2. Its circular outer edge will 
be seen as an ellipse at right angles to 
this axis, and passing through the four 
points A, B, C, and D, previously found. 
As the short diameters of such ellipses 
always appear to lie in a line with the 
axis of . 

the axis ^-^-^f^ 

Front view. 
Fig. 158 

Fig. 159 


Looking at A . 
Fig. 160 

the object (Fig. 160), 
line will give the apparent direc- 
tion of the short diameter of the 
ellipse. (Here the beginner is 
advised to turn the paper round, 
bringing the axis as a vertical 
line, the better to secure the 
symmetry of the cylindrical part 

of the model. See Fig. 161.) 

To obtain the length of this short 
diameter, slightly curved lines, perpen- 
dicular to the short diameter, are 
sketched from B to the right and from 
C to the left, giving x and y for the 
extreme front and back of the ellipse. 
Now (since an ellipse is always symmet- 
rical) the longest dimension or long diameter of this ellipse is not 
on the vertical line through A and D (Fig. 159), but is on a line at 

7 97 

Fig. 161 


right angles to x y, and through its apparent middle from front 
to lack; making one end fall in front of D, and the other back 
of A. It will also be a little in front of O, the true center of 
the circle, and a little longer than A D, but not touching the 
frame. Mark lightly and accurately this apparent middle and 
sketch the long diameter through it. Mark the ellipse ends 
by sketching rounded curves, from A back and from D for- 
ward, making them symmetrical on the long diameter, and 
equidistant from the axis. Complete the ellipse by connecting 
these ends and sides, correcting if necessary, till the ellipse is 

For the inner ellipse proceed as with the top of the cylinder 
in Chapter IV, remembering to make its proportions as shown 
in Fig. 158, also that the perspective halves of the short diameter 
are already found by the true center, O. The side boundary 
lines of the circular frame may now be drawn to VP2, when 
there will remain only its inner thickness to draw. The edge 
of this is an inner circle on the back of the frame, actually like 
the inner one in front. Draw lines (E and F) from the ends 
of the front inner ellipse to VP2. These may be called side 
boundary lines of the cylindrical opening; and their conver- 
gence measures the smaller length of the desired back ellipse, 
which lies with its ends in these lines as does the front one. 
To find these ends it is necessary to obtain the apparent thick- 
ness of the frame at that distance into the picture. From where 
E begins (on the front inner ellipse) carry a vertical line to the 
upper near edge of the square frame. From this point G, vanish 
a line in VP2, and drop a vertical from where it crosses the back 
edge of the frame to line E again. This measures on E the 
perspective thickness desired, and gives a point for the new 
ellipse, which corresponds to the end of the front inner ellipse. 
Set off the same distance on F for the other end of the new 
ellipse. Sketch the ellipse, remembering that the thickness at 
the back (1-2) is a little less, and at the front (3-4) is a little 
more than at the ends. 



The round arch (Fig. 162, also Ch. XXXII) is an interesting 
application of this principle. Errors in drawing these and 
kindred forms (B in Fig. 162) so common with beginners are 
easily avoided when these principles are understood. 

A. Showino application of methods. 

B. WRONG. Inner 8ack eu.\pse' 


Fig. 162 


Chapter XXX 



HIS exercise (Fig. 163) and the methods of sketching it 
should be carefully studied, and if necessary it should 
be drawn. After this the student should sketch a 

similar example 
from a building or 

Having drawn 
the straight-line part 
of the exercise, the 
/ . round window is 
next to be consid- 
ered. This is actu- 
ally a cylindrical 
opening in the wall. 
■ Being above the eye, 

the circles of the win- 
dow appear as slant- 
ing ellipses like the 
cylinder ends in Fig. 
149 (Ch. XXVII). 
To make sure that 
the slant of these 
ellipses shall agree 
with the straight-line 
part of the building, 
the axis of this cylin- 
drical window is 
used. This axis is 
actually at right an- 
gles to the wall, and 


Fig. 163 


is therefore parallel to the lines already vanishing in VPl. 

Hence the apparent middle of the window ellipse (0 in Fig. 

164) is first located, and the axis is i 

drawn through it to VPl, extend- 
ing forward indefinitely. The long 

diameter (AB) is sketched at right 

angles to the axis, and the short 

diameter (CD) is set off on the 

axis line. The ellipse is then drawn 

through these four points. The 

inner ellipses of the window are 

shorter as 
well as far- 
ther back 
than the 
outer one. 

For the 
partial ellip- 
ses of the 

quatrefoil, the actual center of the inner 
ellipse (1 in Fig. 165) must be marked. 


Fig. 164 

and through it a vertical line (EF) and a 
horizontal one (GrH, vanishing in VP2) 
drawn. The real centers of the quatrefoil 
circles fall each on one or the other of 
these lines. Their long diameters (a little 
in front of these real centers) are parallel 
with the other long diameters of the window. 

This quatrefoil is especially an example of objects which 
should in practice be sketched freehand first, and afterward tested 
by the constructive methods here given. (See Introduction.) 

B, PeRSPecTive 
Fig. 165 



Chapter XXXI 

HIS example (Fig. 166) is given as an aid in rendering, 
though it may be drawn first if desired. 

The Model. — Any clock containing rectangular forms 

and the usual cir- 
cular face will serve 
as the model. It 
should be placed 
above the eye. 

Conditions. — Un- 
like problems in 
general, this draw- 
ing may be made 
from the object, 
but must be done 
without assistance. 
The aim is to test 
the student's abil- 
ity to apply the 
principles taught 
in the last few 

Fig. 166 



Chapter XXXII 


HESE arches from the cloister of St. Paul's Without the 
Gates, at Rome, also illustrate the symmetry of the 
cylinder, and can be drawn by the same method as 

Fig. 167 

the round window in the last chapter. (See Ch. XXX.) They 
are semi-cylinders and their openings are semicircles (Fig. 169). 
The semicircles are sketched on a horizontal line (A) which, 



being above the head in this instance, vanishes downward (Fig. 
168) . The true centers of the semicircles are on this line, and from 
these centers the axes are 
drawn, vanishing with 
other lines to VPl. The 
joints of the stones form- 
ing each arch, being lines 
really tending to meet in 
its true center, are so 
drawn in perspective. 

Pointed arches, and 
other modified forms can 
be readily drawn on the 
same principles here 

The student should 
draw this exercise unless experienced, when he should instead 
select a print of artistic interest, illustrating the same principle, 
and make from it a careful and ^expressive sketch. In either 
case, he should follow his first drawing with another involving 
the use of this principle, from a building. 

Fig. 168 

Front View 
Fig. 169 


Chapter XXXIII 


THE Cube as a Model. — With a penknife loosen one face of 
the cardboard" cube; and turn it back or take it off. Place 
it within a foot of the eye, with the opening parallel to 

Fig. 170 

the picture plane, and the eye level a little less than two thirds 
of the way up. If- desired, the windows, doors, rug, and pictures 
may be marked with a pencil on the inside of the cube. It now 
serves to illustrate the room shown in Figs. 170 and 172, as the 
cubes and prism illustrated a house in Chapter XXII, In this 
room the floor, ceiling, side walls, and all details on their surfaces 



(as the window, the side door, and the rug) are foreshortened ; 
while the side edges converge to VPl directly in front. The back 
wall and all surfaces parallel to it (as the end of the table and 
one side of the stool), being parallel to the picture plane, appear 
in their true shape. 

Directions. — This example (Fig. 170) should be drawn by the 
student if a beginner. After this the end of a room, also with its 
farther wall parallel to the picture plane, should be drawn from 
memory or invention.^ A hall, a 
kitchen, a street car, or a piazza will 
be recognized as especially adapted 
to such views. 

The Apparent Width of the Sides. 
— To aid in estimating this, recall the 
appearance of the most foreshortened 
side of the cube in Chapter XVI. 
This estimate may then be tested 
by pencil measurement of the cube 

The Pictures. — Although the pic- 
ture on the right wall is inclined 
slightly forward, its sides are still parallel to the picture plane ; 
and therefore appear in their true direction and shape. With 
the picture on the back it is not so. Its top is slightly nearer to 
us than its lower edge ; and must therefore appear longer, mak- 
ing the sides appear to converge ' downward (though almost 
imperceptibly). To determine the direction of this convergence, 
revolve it forward on its lower edge in imagination until hori- 
zontal (Fig. 171). It will at once be seen that the sides (A and 
B) in this position are parallel with the lines already vanishing in 

^ It may be asked why memory or inventive drawings should be advised before study from 
a room. But in this case we meet a subject of which all have somethings in memory ; and 
drawing from memory when possible (when one has something remembered) is not only far 
pleasanter, but much less laborious. The object or place itself presents to a beginner a confus- 
ing mass of detail, much of it not needed" for the drawing. The student must make later many 
drawings from the place to accumulate knowledge ; but will always do his most free and 
individual work from this knowledge, not directly from the object. 



VPl. If now the picture should be slowly revolved upward, the 
converging point for the sides would descend as the picture rises. 
As the picture is not moved sidewise at all, this point of con- 
vergence (OVPl) call only move downward in a vertical line 
from VPl like the roof ends in Chapter XXII. When the 
picture is returned to its original position, it varies but slightly 
from the vertical ; consequently OVPl is too far away to locate 
and the vanishing of its 
sides must be estimated. 
Make sure it is slight 
enough, and toward a 
point vertically under 

An Open Door. — So 
far we have found but 
one vanishing point on 
the eye level. If we 
begin to open the far- 
ther door, its horizontal 
lines will instantly ac- 
quire a vanishing point, 
but at an infinite dis- 
tance. In proportion as the door swings toward being parallel 
with the sides of the room (that is, with the direction of seeing), 
this vanishing point will move inward. When the door becomes 
quite parallel with the walls of the room, this point will coincide 
with VPl. If the door is swung still farther back, this point 
moves on toward the left. The apparent width of the door 
thus opened may be measured by an ellipse on the floor 
(Fig. 172) representing the path of its near corner as it swings 
in a circle. The short diameter of this ellipse is proportioned 
to the foreshortening of the floor in which it lies. It can 
also be found by the use of lines parallel to the floor diagonal 
(Fig. 173). From this diagram it is seen that since the floor is a 
square, its diagonal measures equal distances in its sides. Any 


Fig. m 


• ii>///^/////M//MM///A 


Fig. 173 

distances in these sides or in lines parallel to them can therefore 
be measured by lines parallel to the floor diagonal. In the per- 
spective drawing (Fig. 172) these lines can be used there perspec- 

tively in precisely the same way.^ 

It will also be noted that the 
thickness edges of the door, being 
horizontal and at right angles to 
its top and bottom, have their 
own vanishing point upon the eye 
level (VP4). Also the door knob 
is cylindrical, and its axis is paral- 
lel to these edges. 

The Stool. — The proportion of 
the stool is found in the same 
manner as the height of the pyra- 
mid in Chapter XXV. The room 
may be considered as nine feet high, and the stool as approxi- 
mately eighteen inches, or one sixth the height of the room. 
Mark any point (as H in Fig. 172) where it is desired to place it 
on the floor of the room. Its height cannot be compared directly 
with that of the room here, for we cannot determine where a 
vertical line from H will touch the ceiling. Therefore imagine 
the stool moved from point H in a straight line to any place on 
the front edge of the room, as I, where its height (IJ) can be 
measured by that of the room. If now it were moved back on this 
same line (IH) , its top would move in a horizontal line directly 
over IH, that is, actually parallel to it, or vanishing in the same 
point on the eye level. Both lines (HI and one from J) may 
therefore be carried to this point VP5. The height of the stool, 
when placed at any point on the line from 1 to VP5 will be 
the vertical distance (as at H) between these lines. 

The near side of this stool is now drawn in its true shape, and 
the parts at right angles to this side found by vanishing lines 
to VPl. 

* The ellipse is made horizontal, as explained in Chapter XLIII. 



Finally, to improve the composition some of the ceiling and 
a little of the floor are cut off, as shown by the dotted lines in 
Fig. 172. This gives a more generally favorable shape (Fig. 170) 
to the inclosure, and keeps the oblique lines from running to its 
corners, which should always be avoided. (See p. 116.) ^ 

^ See Chapter XLII for further consideration of a room parallel with the picture plane. 


Chapter XXXIV 


THE Model. — The cube model may be prepared for illus- 
trating this study by removing a side adjacent to the 
opening made for the previous study. 

Position. — The 
eye level and the dis- 
tance from the eye 
are the same as in 
the last chapter. 
Place the model so 
that its receding 
faces make angles 
of thirty and sixty 
degrees with the pic- 
ture plane. Both 
sides and the top 
and bottom are now 
foreshortened and 
their horizontal lines 
vanish respectively 
in VPl and VP2 
(Fig. 174). 

Selection of Sub- 
ject, and Use of the 
Picture Plane.— It will 
be seen that the room 
in Fig. 174 is the same as in the previous chapter. The differ- 
ence is in the selection of subject-space, and in the consequent 


Fig. 174 






Plane j 

\ Z 


\ ° 


\ ^■ 

o / 

\ '^ 

Z f 

\ " 

Zi / 

\ ^ 

ui / 

\ < 

^ / 

\ ^ 

O / 

\ w 




relation of the subject-matter within that space to the picture 

plane. Thus in A, the extreme points {x and y) of the back 

wall of the room ^^ 

are equally distant ^ ^^ ■ *^^ ^ 

from the picture 

plane as well as 

from the eye. 

Therefore we can- 
not, without ab- 
surdity, vanish the 

lines on this wall 

in either direction 

(A in Fig. 176), 

still less in both 

(B in Fig. 176). 

The only way to 

give a truthful 

impression of the ^ Pj^, 175 

back wall is to 

draw it in its true shape (C in Fig. 176), as was done in the 

last chapter. 

But in the present ex- 
ercise (Fig. 174) part of 
the room is left out. The 
central direction of seeing 
is therefore moved to the 
right, and with it the pic- 
ture plane is turned (B, 
Fig. 175). Consequently 
the wall (0) which was 
before parallel to the pic- 
ture plane now recedes 
from it. Hence the height 
of the room at the comer, 
Fie 176 being farther into the pic- 



ture and from the picture plane ^ appears less than at the right 
and left; and all horizontal lines on both walls appear to 

In this drawing the rectangle incloses the parts of most 
interest and cuts off the awkward outer lines (Fig. 177). An 

Fig. 177 

alternate selection is indicated by dotted lines. This brings us 
to the consideration of a new point, namely: — 

What to Include in a Picture. — This cannot be all that it is 
possible to see from any point, for the head can be turned to 
see all parts of the horizon circle. Such a view when painted 
forms a panorama, which is a continuous cylindrical picture 
surrounding the spectator. 

It is evident that the legitimate picture must include, or cut 
out from what it is possible to see, only such a space as can be 
perceived by the eye in a single effort of seeing. It should 
leave out whatever cannot be seen without turning the head, 
or even noticeably moving the eyes. It is generally understood 



that sixty degrees of the horizon circle is the most that should 
be taken, and that usually thirty degrees is better. The rule is 
that the greatest dimension of the 
selected space (whether height or 
width) shall not exceed the artist's 
distance from that dimension. 

It will be seen by the diagram 
(Fig. 178) that this is equivalent 
to not exceeding sixty degrees in 
the picture. It is also apparent 
that it does not prevent the in- 
clusion of objects nearer than that 
greatest width if they come into 
the picture space, as the table and 
part of the rug in this illustration. 

In conclusion it should be 
noted : 

First. — The picture plane is dif- 
ferent for each neic picture selection. 

Second. — The picture angle 
should not he over sixty degrees^ and 
is better less. 

b. the f^pt selected fop a picture. 
Fig. 178 


Chapter XXXV 

THE student should copy carefully Fig. 179, always deter- 
mining the level of the eye, and locating the vanishing 
points either actually or mentally. Note that the em- 
phasis of contrast and interest is concentrated on the old- 


Fig. 179 


fashioned desk while the details on the wall beyond it are very- 
quiet. The settle comes forward more (that is, has more con- 
trast and emphasis) than the wall, but 
less than the desk. The floor is pu]> 
posely quiet in detail to aid in con- 
centration of interest. The chair in the 

extreme right of 
the fore-ground 
is subordinated 

Original Work. — Following the 
above copy several interiors should be 
drawn from the place. The finder 
should be used, and thumb-nail sketches 
182) should be made to test the 
Avoid equal angles in the 
If both should be drawn 

Fig. 180 

Fig. 181 

(Figs. 180, 181, and 

artistic value of the selection. 

principal sets of vanishing lines 

at forty-five degrees, or nearly that, the 

composition would have a stiff effect. 

The contrast between surfaces turned 

away much and others turned away 

little is generally pleasing. Should it 
be impossible to avoid 
equal angles of van- 
ishing, relieve the stiff- 
ness by using details 
(as the sofa and win- 
dow in Fig. 183) that 

are widely different in effect. The rug, 
being parallel with the sofa, assists further in 
overcoming the monotony. It is undesir- 
able to show the floor and ceiling as occupy- 
ing equal space in the picture ; and it is 

usually better not to show both floor and ceiling. (Compare 

Fig. 182 with the same subject in Fig. 184.) A subject may 


Fig. 182 

Fig. 183 


often be improved by cutting it with a different margin 
(Fig. 185). 

Care should be taken to place the principal oblique lines 
of the study in good relation to the 
margin or inclosing rectangle that they 
do not conspicuously point to the picture 
comers. The rectangle which bounds a 
composition is an orderly conventional 
shape, and 
hence unno- 
ticeable, leav- 
ing the atten- 
tion to be 
on the picture. 
Lines to its corners direct the atten- 
tion there, and defeat this end. 

Vignetting. — It is not necessary 

Fig. 184 

that an inclosing margin should always be 

Fig. 185 

used. The drawing 
may be vignetted, or 
blended off into the 
white paper (Fig. 
186). This is less 
easy to do as the 
tendency of the ir- 
regular outer edge 
is to contrast sharply 
with the white paper, 
detracting from the 
effect of the more 
important central 

parts. The edge details therefore must be carefully " quieted," 

or rendered inconspicuous. 

Interesting effects are often produced with a partial margin 



Fig. 186 


line (Fig. 187). This is advisable if the center of interest is 
too near the edge of the paper to blend off well. 

So many principles of perspective and of artistic rendering 
are included in the drawing of interiors that they form a most 
important division of the subject. In figure compositions they 
are constantly used, especially by the illustrator. 

Fig. 187 



Chapter XXXVI 


HE chair in Fig. 188, like most chairs is different from 
the stool in our first interior study (Ch. XXXIII) in 
that the sides are not parallel to each other (see plan, 

^.^..^ ,__--_. — , ^. Fig. 189). Also 

the seat is a 
trifle lower 
in the back, 
so that lines 
B, B (Fig. 190) 
slope down- 
ward toward 
the back. But 
the chair is 
symmetrical on 
its center line 
(front view, 
Fig. 189), there- 
fore the hori- 
zontal lines in 
its front and 
back are par- 
allel, having a 
common van- 
ishing point on 
the eye level. 

Drawing the 
Chair. — After 

! having planned 

Fig. 188 its place ou the 



paper and located the eye level, the next step is to converge the 
horizontal lines of the feet and side rung (C and D) to their 
vanishing point (VPl). Then sketch the line for 
the front of the seat, one for the top of the back, 
and line E on the ground to VP2. 

The near side of the seat (lines B, B) is 
drawn next. Although not horizontal, the side 
is in the same plane with (directly over) the 
horizontal lines C and D, so that its point of 
convergence must be directly under theirs. The 
direction of the upper line B is therefore taken, 
and OVPl is located where this direction crosses 

a vertical line from VPl. 
On this line and the lower 
line B (vanishing in the 
same point) the curves of 
the side of the seat are 
drawn, after which the legs 
and back on that side should be sketched. 

The same method is pursued with the 
other side of the seat, but as it is a little 
more facing the beholder, the vanishing point for its horizontal 
lines is VPS, farther away on the eye level. As the seat is 


Fig. 189 

Fig. 190 

Fig. 191 

evenly slanted, the other oblique vanishing point (0VP2) is on 
a level with OVPl. 

All edges parallel to the front and back converge to VP2. 
The curving front of the seat, the front rung and the curves of 



the bars across the back are drawn so their ends rest on a line 
to VP2. Since the back is hollowed from side to side, more 
space will be visible between its vertical center and the far side 
than between that and the near side. Mark its principal points 
on lines to VP2, and draw the curve between them. The lines 
where the cylindrical rungs enter the legs are actually modified 
circles, hence they will be seen as shapes modified from the 
elliptical, and should be carefully considered and drawn. 

Following this, the student should choose and draw at least 
one other piece of furniture. Selection should be made of 
something interesting in itself; that is, well designed and con- 
structed, and agreeable in association. In chairs the old-fash- 
ioned rush or splint bottom ones, the wooden rockers of our 
grandmothers, the beautiful examples by Chippendale, Sheraton 
and others of the colonial period, and good examples of modern 
mission shapes may be mentioned as among those satisfactory 
for study. If one is fortunate enough to get a really fine old 
cradle, it is a most instructive subject, as is an antique desk or 
a tall clock. 

The student who has faithfully done the work and study 
prescribed thus far will find himself possessed of a rapidly 
increasing power of judgment in applying the principles of per- 
spective, and will be able to express that judgment with ease 
and certainty. The artistic quality of this expression should 
also have gained ; though for this so much coordinate study and 
observation are needed, that its degree cannot so certainly be 


Chapter XXXVII 



THE student should draw this example from the objects, 
following the directions here given; and should then 
sketch them from memory, as by the general directions 

in Chapter XI. r- _ ., ,^ 

The Model. — 
This can be 
made from card- 
board according 
to the diagram 
(Fig. 193). For 
the first posi- 
tion it should 
be placed about 
three feet from 
the eye and 
nine inches be- 
low it, and with 
two vertical 
faces parallel 
with the picture 

The Geomet- 
ric Hexagon. — 
This should be 
first. Divide 
the line AB 
(Fig. 194) in 

halves, drawing Fig. 192 



.<^ h 

Fig. 193 

a perpendicular at the point of division. Measure the distance 
AB, taken on the pencil, from B against the perpendicular. 

Where it falls (at O) will be one corner 
of an equilateral triangle (AOB) which 
will form one sixth of the desired hexa- 
gon. Sketch vertical lines of indefinite 
length from A and B, and cut them by 
continuing the sides of the equilateral 
triangle (AG and 
BO) to D and C, 
then draw DC. The 
constructive rec- 
tangle ABCD, which 
we have now com- 
pleted, will be al- 
ways essential in 
drawing the perspective of the hexagon. 
Its diagonals (A and B) will form two di- 
agonals of the hexagon, and its center (0) 
will be the center of the hexagon. The 
other diagonal, EF, is drawn through O parallel to the rectangle 
ends, half on each side of the center. Its length should be tested 
by that of the diagonals already found. The other four sides 
complete the geometric hexagon ABCDEF. 

Drawing the Hexagonal Top. — The rectangle ABCD (Fig. 195) 
is drawn first. Although it is actually nearly twice as long as 
its width it will be found to appear less than half as long. The 
diagonals of the rectangle will in perspective, as in the geo- 
metric view, form two diagonals of the hexagon. The other 
diagonal, EF, is set off on a line of indefinite length through 
their crossing. Looking at the diagram, it is observed that the 
sides of the rectangle (AD and BC) and its middle (O) divide 
this diagonal (EF) into four equal parts at x, O and y. In the 
perspective drawing (Fig. 195) the two middle fourths (xO and 
Oy) are seen to be already measured. Since these fourths are 


Fig. 194 


all equally distant from the picture plane, they appear equal, and 
are so set off from x and y. The hexagonal top is completed by 
drawing its last four sides. 

The Thickness of the Plinth. — The front face of this thick- 
ness, being parallel to the picture plane, is drawn in its true 
shape. The lower edges of the other two faces are parallel to 
the receding horizontal edges (AE and BE) above them, and 
will therefore vanish with these edges respectively to VP2 and 
VPS. Vertical lines downward from E and F will complete the 

fVf LE\/EL 

Fig. 195 

A Test for Vanishing Lines. — The other two diagonals and 
the receding back edges of the hexagon are in reality each 
parallel to one or the other of the two sets of vanishing lines 
just drawn. Therefore when carried out to the eye level, 
they should meet respectively in VPl and VP2 if the drawing 
is correct. 

The Hexagonal Plinth Slightly Turned. — For this drawing turn 
the model so its front edge will make an angle of thirty degrees 
with the, picture plane. Draw the constructive rectangle, ABCD, 
as before; taking with the pencil the direction of the front edge 
and of the imaginary left side (AC in Fig. 196)* of the rectangle. 
Take especial pains to have these lines correct in direction and 

1 Corresponding to the receding edges (2, 2) first drawn in sketching the cube. 



length, as an error here causes a particularly unpleasant repre- 
sentation. In vanishing the back edge with the front one be 
careful to keep it also tending upward. 


A. Wrong. Lines 1,2,0 


Fig. 196 

The vanishing point (VP2) of these front and back edges 
is so far away that their convergence will naturally be only 

estimated. Hence the necessity of locating 
definitely in mind what cannot he seen — 
that is, the vanishing point, not only of 
these two lines, but of their parallels, 
the diagonal EF and the lower edge of 
the front face. 

In this position the last diagonal, EF, 
is in perspective, and therefore its four 
actually equal divisions will appear de- 
creasing in size, or perspectively equal. 
Consequently, if the work so far done is 
correct it will be found that the space 
xO is (almost imperceptibly) greater than 
O?/, because a little nearer. Hence Eic should be set off a little 
larger than icO, and 2/F a little smaller than Oy. Similar cases, 
as the cylinder top in Chapter lY, and the concentric circles in 
Chapter XX are readily recalled. 

Testing the Drawing. — The test used for the other drawing 
of the plinth is equally effective here. It is more needed here, 


B. Wrong, lines J, 2,3 


hexagon appears tii-ted. 
Fig. 197 


since the vanishing point for one set of lines has not been 
actually found. But if these lines have been carefully thought 
out as to direction, the errors will be found encouragingly 

The test of placing the eye at the vanishing point (Ch. XYH), 
to sight back along the lines which should converge to them, is 
especially applicable here. 



Chapter XXXVIII 

HE plan of this floor is shown in Fig. 197. As drawn in 
the example, the foreshortening of the tiles is proportioned 
to that of parallel surfaces, such as the receding " treads " 

or tops of the steps. 
From the vanish- 
ing of their edges, 
we may judge 
these treads to be 
foreshortened about 
one half. Conse- 
quently lines paral- 
lel to the staircase 
edges (vanishing in 
VPl) must be fore- 
shortened as much. 
The tile adjacent 
to the lowest step 
is the best to begin 
with (since it is the 
same distance into 
the picture) , and as 
with the previous 
hexagons the rectan- 
gle ABCD is drawn 
first. And since 
much depends on 
the correctness of 
this first rectangle, 
it is worth while to 

Fig. 198 



take especial pains with it. Observe that as the tiles are here 

placed, it is the width (AD) of this rectangle which is 

parallel with the receding staircase edges. 

Now if it is desired to represent tiles of 

a certain size, a step is one of the best 

objects to use for comparison, as steps 

do not vary much from seven inches 

in height 

(Ch. XLV). 

Thus these 

tiles are 

three and 

a half inches on a side, making 

the rectangle width actually half 

the height of a seven-inch step. 

In the drawing this width, be- 
ing foreshortened as much as the steps, appears one fourth 
as wide as the height of the step. The 
actual length of the rectangle is seen by the 
diagram to be one eighth less than twice its 
width; or (what is the same thing) one 
eighth less than the height of the step. 
Being slightly turned away, it will appear 
a little shorter in comparison than that. In 
this case it was made one sixth less than the height of 
the step. 

It will be readily seen how if the rectangle proportions are 
right, the lines of this first hexagon, when carried out forward 
and back, will give points for the other tiles, making them fall 
harmoniously into their proper perspective. 

Fig. 200 

EnpView or step 

Fig. 201 


Chapter XXXIX 


HIS exercise may be drawn from the objects, if they are 
at hand. If they cannot readily be had, the drawing 
may be made from these directions, using the cardboard 

plinth made for 
Chapter XXXVII. 
The objects should 
then be drawn 
from memory. 

The Prism. — Be- 
ing the simpler to 
draw, this object 
should be taken 
first, though it 
should be placed 
at the bottom of 
the sheet, on ac- 
count of its greater 
horizontal dimen- 

This model is 
eight inches long, 
and the diameter 
of its hexagonal 
bases is four inches. 
It is placed so that 
the bases make 
angles of sixty de- 
grees with the pic- 
FiQ. 202 ture plane. The 



Fig. 203 

dotted lines in Fig. 203 show how the cardboard plinth may be 
placed in the same position as a help in study. 

The Nearest Vertical Hexagonal Base. — Sketch the construc- 
tive rectangle previously used, noting that i|;s width is ^^fore- 
shortened as much as the most foreshort- 
ened side of the cube in Chapter XVII. 
Set off the third diagonal of the hexagon 
(EF) perspectively on a line vanishing to 
VPl through the rectangle center (as on 
page 124). 

The Long Edges of the Prism. — Take 
the direction of the nearest upper edge, giving YP2. Vanish 
the other long edges with it, and set off on the one first drawn 
its apparent length (AG). This can be easily estimated by re- 
calling the cube. That is, its actual length as given is twice that 
of the diameter of its base (the near vertical line AB). AGr will 
therefore be as long as two cubes placed side to side.^ 

The Further Base. — For the horizontal top edge of the further 
base draw a line parallel to the same line in the near one (that 
is, vanishing in VPl), which gives GH. For the corner corre- 
sponding to B in the near base drop a vertical 
from G, giving J. The upper oblique edge 
(GI) is parallel to AF in the near base, and 
their vanishing point is OVPl, vertically 
above VPl. Another oblique line, from the 
nearest point, I, to the lowest, J, completes 
the prism. 

The Hexagonal Frame. — This model is three 
inches on a side, and is one inch square in sec- 
tion (Fig. 204). It stands on one rectangular 
face with its hexagonal faces at an angle of 
thirty degrees with the picture plane (Fig. 205) . For the outer 
hexagon and the outer thickness proceed as in the prism. 

Face view or fpame 

Section of frame 
Fig. 204 


^ This method is given in a slightly different form under Solutions of Problems, Chapter 



For the inner hexagon we may first study the actual shape in 
Fig. 204, where it is seen that the vertical frame thickness can 
be conveniently carried across to measure it at Qy on the 
nearest vertical, BC. We know this thickness to be one inch, 
which is more than one sixth and less than one fifth of the 

vertical CB. H one fifth of CB (CO in 
Fig. 205) be found, and three quarters of 
this {Qy) be taken, it will serve the purpose. 
Mark the same distance from B up (point 
z). From these points (y and z) vanish the 
horizontal lines of this inner hexagon with 
their parallels to VPl. The corners of the 
Fig. 205 iuucr hexagou are on the diagonals of the 

outer one, so the crossings of the diagonals 
by these two vanishing lines give four corners of the inner 
hexagon (1, 2, 3, and 4). From 2 an oblique line vanishes down- 
ward with, its parallels to 0VP2, marking point 5 on the hori- 
zontal diagonal EF. Another from 3, also vanishing in 0VP2, 
is drawn from 3 upward to cut EF in point 6. Lines from 6 to 
1 and from 2 to 5, complete the inner hexagon; and should, if 
the drawing is correct, vanish with those parallel to them in 

The inner edges- of the thickness which are visible vanish 
from 4 and 5 to VP2, being parallel to the outer thickness edges. 
For the visible part of the further inner hexagon, a line of the 
near inner hexagon, as 4-3, may be carried " around the corner " 
and back, as in the square frame (Ch. XXIV). From the point 
where this vanishing line (7-8) cuts the line from 4 to VP2, an 
edge (8-9) vanishes with its parallels (CF and others) to OVPl. 
This point is even further away than 0VP2, so that the con- 
vergence of its vanishing lines must be slighter. Where line 8-9 
crosses the one from 5 to VP2 it meets the last visible edge of 
this back inner hexagon, a line vanishing in 0VP2. 


Chapter XL 


THE MODELS. — The prism is eight inches long, and its 
triangular ends are four inches on a side. The frame is 
six inches on a side, and one inch square in section. 
See diagrams. Fig. 206. 


B. 5ide: view 


Positions. — The objects are 
placed at the usual distance and 
height, and are drawn separated, 
as were the models in the last 
chapter. The prism rests on 
one long face, with its long edges 
making angles of thirty degrees 
with the picture plane. The 
frame rests on a rectangular face 
with its triangular faces at thirty 
degrees with the picture plane. 

Arrangement on the Sheet. — Two drawings are to be placed on 
one sheet. The position of the paper (whether with its long 
edges horizontal or vertical) and the placing of the drawings 
on the sheet must 'be such as to produce the most agreeable and 
satisfactory effect. 



Fig. 206 


Chapter XLI 

FROM Chapters XXXIV and XXXV it is seen that inte- 
riors follow the law of the cube. This, however, leads to 
what may seem an inconsistency. Why, it may be asked, 

does the table in Fig. 207 differ from 
the cube in Fig. 2081 In the cube 
B was made shorter than A because 
farther into the picture. But in 
the table B was not drawn shorter 
than A. 

The answer to this is that the 
table was not studied 
alone, as was the cube. 
It was part of a pic- 
ture in which the dominant part (the back of the 
room) was parallel to the picture plane. Having 
drawn the side of the table parallel to the sides 

of the room, it is absurd (Fig. 209) 
to draw its end otherwise than 
parallel with the back of the room. 
Fig. 207 satisfies the eye and gives 
a true impression of the room and 
its contents. Could the room be 
erased, leaving the table alone, it 
would present the error shown in 
Fig. 210. Here it forms the whole 
picture and its picture plane makes 
an angle with its ends (see plan in Fig. 210), hence it must be 
drawn as below. 


Fig. 207 


Fig. 209 

f Uf THE 






It is undeniable that in Fig. 207 B is farther from the eye 
than A. But since drawing that corner smaller produces the 
false impression seen in Fig. 
209, we are guided by the dis- 
tance of these points not from 
the eye, but from the picture 
plane. The picture plane simply 
forms the best means of attain- 
ing our fundamental object — a 
truthful representation. Hence 
the necessity of determining 
the limits of the picture (Fig. 
211) and of clearly fixing in mind re i^'tiono^pI^I^pe plane 

.- , 1 T ,. /. . ROTABLE. WHEN TABLE 

the central direction of seeing '* ^'-°''^ 

and the picture plane. 

Under some conditions, sur- 
faces may even be drawn in 
their true shape when not quite 

parallel to the picture plane. In Fig. 211 
none of the vertical surfaces, as A, B, and C, 
are exactly parallel with the picture plane. 
This is shown to the beholder at x by the 
convergence of the lines at right angles to 
these surfaces. They vanish a little out 
of the center of the picture as seen in 
Fig. 212. Yet if all these vertical surfaces 
are drawn in perspective (A in Fig. 212) 
the result is misleading or impossible, and 
the eye protests. But the drawing is per- 
fectly satisfactory in B, Fig. 212. 

A convincing illustration of dominant 
surfaces parallel to the picture plane is the 
familiar form of a bureau. With an un- 
broken top (A in Fig. 213) it is easily drawn like the book 
and cube. If now the middle drawer is cut out, the remaining 


Fig. 210 






Fig. 211 


A Untpue drawing 



Fig. 212 

small ones are seen to occupy positions similar to that of the 
table in Fig. 1. 

The Street. — The street is another example of these con- 
ditions. Viewed from the middle of a crosswalk {x in plan, 

Fig. 214) the fronts 

of the houses pre- 
sent to the beholder 
a perspective like 
that of the interior 
in Fig. 207. They 
vanish to the center 
of the picture, and 
surfaces at right 
angles to them are 
drawn in their true 
shape. This is done 
even if the con- 
vergence is not 

toward the exact middle of the picture (Fig. 215), provided it 

does not fall outside of the house fronts. 

The beholder has passed on to 2/, and the 

conditions are then like those of the 

interior in Fig. 211. 

But if instead of using both sides 

of the street for our picture, we choose 

one of the corners, the picture plane for 

this forms a different angle with the 

principal surfaces (Fig 214), and must be 

drawn as shown in Fig. 216. 

It may therefore be concluded, that 

in any picture having a dominant part ^J^^= 

parallel with the picture plane and conse- ^ 

quently drawn in its true shape, all por- 
tions of that picture which are parallel with the picture plane must 

also he drawn in their true shape. 



Fig. 213 



Also, even such dominant parts as 
with the picture plane must 
be drawn in their true shape 
in certain cases where drawing 
them in perspective produces 
false or misleading results. 
Finally, it is of great import- 
ance to include in the pic- 
ture only what can easily he 

Parallel perspective, as 
work under such conditions 
is called, involves no depart- 
ure in principle from free- 
hand perspective in general. 
It is merely an adaptation 
of perspective methods to 
certain conditions in the 

Space has been given here 
to a somewhat extended con- 
sideration of the subject. 

are not quite parallel 

. // Plan OF THE 


Fig. 214 


Fig. 215 
1 The terms " parallel " and " angular " perspective, though used for lack of better ones, 

are therefore far from satisfactory. 



because the confusion concerning it that frequently exists is 
deemed unnecessary. It has been found that students may be 
easily led to distinguish when such conditions are present, after 
which there is no difficulty in dealing with them. 

vitw or coENca tbom y 
Fig. 216 


ef XLII 


THIS . exercise (Fig. 217) may be drawn first if judged best, 
noting carefully the ch-anges made in rendering from 
the photograph shown in Fig. 218. The student should 
then select a print 
of a street and 
make a drawing 
f'rom it. All 
.sketches should 
be thoroughly 
thought out, hav- 
ing the level of 
the eye carefully 
placed, and all the 
vanishing points 
ocated, either 
ictually or hj^ml- 
talb It wilj^- 
probably be noeesi 
K^ . « distortion* 
of tli camera (see 
Cb. N. LIII). 

Tils drawing 

■ le 


; ^ es 

etches made 

-,FiG. 217 



by the student at the place chosen may be used to help this 
memory work. 

Chapter XLIII 


IT will be observed that in photographs the circular tops of 
columns near the edges of the picture often appear as slant- 
ing ellipses (Fig. 219). And all who have an acquaintance 
with mechanical 
perspective will 
recall that in cer- 
tain problems the 
ellipses of cylin- 
ders do not work 
out at right an- 
gles to the axis 
(Fig. 220). While 
the eye sees ob- 
jects pictured on 
the inside of the 
spherical eyeball, 
the camera forms 
its pictures, and 
mechanical per- 
spective projects its problems on a flat surface. Therefore the 
camera cannot wholly reproduce objects as seen by the eye, 
and certain results obtained by mechanical perspective are 
untrue representations. ^ 

* The photographic error has been recognized, and a camera is now made in which a 
clockwork attachment brings each part of the plate in turn directly facing the part of the 
subject it is to receive, and gives horizontal ellipses to columns wherever placed in the 


Fig. 219 


As for mechanical perspective, though useful in many cases, 
it has sometimes obscured the real aim of representative drawing. 

It has even been taught that the 
flat picture plane should be used 
for all representative work as in 
mechanical perspective, logically 
to the end, regardless of any pro- 
test of the eye as to its results. 
To this error it is sufficient reply 

Cylinders as found by MtECMANicAL per&pective, , ^^ i it • u n i i 

Side cyuinoebs vntwc as reprcscntations. to Say that thc aim 01 treehand 

^^^- ^^^ perspective is the drawing of objects 

as they appear; and that the eye never sees a column as in Fig. 

219, nor a cylinder as the outer 

ones in Fig. 220. When, there- 
fore, the use of the flat picture 
plane produces an untrue draw- 
ing, it is evident that an excep- 
tion must be made in that case. 

The Cylindrical Picture Plane. 
— Looking at Fig. 220, we find 
that the middle cylinder, which 
does appear right to the eye, 
extends equally each side of the 
central direction of seeing, so 
that the picture plane is parallel 
to the apparent breadth of the 
cylinder. By drawing the other 
cylinders as if each had such a 
central direction of seeing and 
such a picture plane of its 
own (A, Fig. 221) a result is 
obtained that appears true to 
the eye (B, Fig. 221). 

In other words, cylindrical 
objects, however placed, should he 


A. Plan. 
Showing \ 

K5E or SPEC- . 


THE Cylindri- 
cal picture: 



B. Appearance of group at A,o(^awn by 


Fig. 221 




^ Side view 

' Showing the viz of 




Fig. 222 

drawn as if for those objects alone, the picture plane was bent or rolled 
into a cylindrical picture plane. But this does not apply to the 
straight-line portions of the 
picture (as the block in Figs. 
220 and 221), nor to the plac- 
ing of the cylindrical parts, nor 
to their height. These must 
be determined in the ordinary 
way, by using the flat picture 
plane. We only abandon the 
flat picture plane where we 
cannot otherwise produce a 
representation which the eye 
will accept as true. 

The Spherical Picture Plane. — Another exception occurs in a 
vertical direction. Thus, the only outline that will truthfully 

represent a ball to the eye is a 
circle. To obtain that, we must 
regard its special central direction 
of seeing as directed to its middle, 
not only from side to side (as in 
case of the cylinder), but from 
top to bottom also. If the ball is 
above or below the eye therefore, 
its special picture plane is slanted 
accordingly (Fig. 222). In this 
case the picture plane (again /or 
such objects alone) may be called 
a Spherical picture plane. 

An example of its application 
is the case of a model posed 
higher than the student who is 
drawing (Fig. 223). The head is 
foreshortened vertically, and the forehead appears smaller in pro- 
portion than the lower and nearer features. At the same time the 


Fig. 223 


window beyond that model is drawn on the usual flat picture 
plane; that is, with its vertical lines vertical, as always. 

These distinctions will be found not only necessary, but 
natural and easy to make;^ especially if care is taken to include 
in the picture space only what the eye can see without noticeably 
moving the eyeballs (Ch. XLI). The picture plane should be 
regarded as limited to what will cover this selected space, and we 
have no concern with what lies outside of that. 

When working from a photograph therefore, as must often be 
done, such camera distortions as the columns in Fig. 219 should 
be corrected to agree with what is pictured by the eye. And in 
freehand work only such truths of mechanical perspective should 
be used as produce results which the eye confirms as true repre- 
sentations. Where the eye and a train of reasoning are in 
conflict, the reasoning should be scanned for errors. Unless a 
drawing looks right, it may safely be pronounced not right. It 
may look right, and still be wrong ; but if the eye refuses to be 
satisfied, it is certainly wrong. 

1 So natural and easy, in fact, that space for this explanation is hardly needed, except to 
guard against false reasoning in the subject. 


Chapter XLIV 

WHILE it is unnecessary for the mastery of freehand 
sketching to study this subject exhaustively, there 
are a few simple facts which have been found funda- 
mentally useful in practice, and which may be easily understood. 

Fig. 224 

To that end the student should follow these explanations 
carefully, making experiments and sketches as needed. He 
should then compose and draw a group similar to Fig. 224, 
also should make other studies involving the use of the truths 
here developed. 

Light may be regarded as composed of an infinite number 
of rays. From a lamp they extend outward in all directions, 


Fig. 225 


forming what may be called a sphere of light. The shadow of 

the apple on the right of the lamp in Fig. 225 extends toward the 

right; that of the book 
on the left in an almost 
opposite direction. The 
sun, on the contrary, is 
so much larger than the 
earth, and its rays have 
traveled such an incon- 
ceivable distance, that to 
us they are parallel, as 
are the paths of fall- 
ing raindrops. Fig. 226 

illustrates this familiar truth. (This, of course, is also true of 

moonlight.) There are therefore two classes of shadows: those 

cast by the sun, and those 

produced by near light, as a 

lamp. Under those formed 

by the sun may be studied 


Small Objects in a Room. — 

If a shadow box be placed 

near and a little back of 

the window,^ as shown in 

Fig. 227, the shadow edge 

(A-6) cast by the vertical 

box edge AB will lie on the 

floor of the box in a line actually parallel to that of the vertical 

hat-pin. The shadow of a vertical vase (Fig. 228) also casts a 

shadow in the same direction. (That is, the center line C of 

its shadow will be parallel with the shadow of AB, both where 

it falls on the box floor, and on the horizontal book cover.) The 

* In this case the window is larger than the box; so that as far as the box is concerned the* 
rays of light are parallel. As will be seen later in this chapter, the diffused light from a win- 
dow causes radiating shadows in the room itself. 


Fig. 226 



shadow (F^) of the vertical book corner (FD) will be parallel 

we push the box back or forward 

EYE • ueveu 

Fig. 227 

with these lines. If 
they all change direc- 
tion, becoming more 
nearly parallel with 
the picture plane 
as the box moves 
forward, and vanish- 
ing more steeply if 
we put it further 
back of the window. 
But they always 
remain actually parallel to each other. 

In the same way we see in Fig. 227 that the shadow of the 
horizontal edge BH and of a horizontal hat-pin (EF) are actually 
parallel to each other, both on the back and the floor of the 
box. Also in Fig. 228 the shadow (b-e) falling on the hori- 
zontal book 
cover from the 
horizontal box 
edge (BE) is 
parallel to the 
shadow (d-g) 
falling on the 
horizontal box 
surface from 
the horizontal 
line DG. 
F^°-228 " Wemaythere- 

fore say that actually parallel lines or objects cast actually parallel 
shadows on the same or parallel surfaces. 

But in perspective, parallel lines vanish; and if they are 
hoHzontal lines they vanish to the level of the eye. We should 
therefore expefet these parallel shadows to also vanish thus, and we 
fln<] they do vanish, in the same manner as any lines or objects. 

10 145 


Fig. 229 

We next observe (in Fig. 228) that the vertical vase casts 
a vertical shadow on the vertical back of the box. Then we 
recall that the horizontal edges BE and DGr cast on horizontal 

surfaces horizontal 
shadow edges h-e and 
d-g. As these edges 
vanish, their shadows 
vanish with them to 
the same point, as any 
parallel lines would. 

It thus appears that 
when the receiving sur- 
face is parallel to the 
ohject or line casting the 
shadow, the shadow tvill also he parallel to the object or line. 

This brings us to consider how to find the extent of shadows. 
If the shadow box is lowered from its usual place on the table 
to the floor, the shadows will be found shorter (Fig. 229). The 
light, falling more steeply, cuts off the shadows nearer the 
objects. When the box is lifted back to the table (Fig. 228) the 
shadows will be seen to lengthen. 
With a light ruler or " straight- 
edge " (Fig. 230), take the actual 
direction from the hat-pin top (D) 
to its shadow {d) on the floor of 
the box. Keeping the ruler in the 
same actual direction, move it to 
the left till it grazes the top corner (B) of the box. It will be 
found also to mark the shadow (6) of point B. 

The evident truth is that the direction of the light-ray from any 
point in the object marks the same point in its shadoiv. Therefore 
to find the shadow of any point, as of the other hat-pin head 
(E, Fig. 227) we have only to draw the light-ray from E to where 
it strikes the receiving surface. 

But since in this case the window is nearer than the box, the 


Fig. 230 


light-rays are receding slightly, hence must appear to converge 
a little, like any parallel receding lines. Therefore to draw them, 
we first find their vanishing point. Imagine one of these rays 
(as T>-d, Fig. 227) dropped vertically to the floor of the box (as 
the gable edge in Ch. XXII was dropped). It would then lie 
in the horizontal shadow line (C-^) directly under it, and would 
vanish in VPS. When lifted again to its former oblique position 
(D-d) its vanishing point would have moved down in a vertical line 
from VPS, and become OVPl. All other light-rays in this illustra- 
tion (as B6 and E-e) appear to converge to this vanishing point. 

Hoiv to find where the light ray and the receiving surface 
meet is the next consideration. In the 
case of D-d we had a vertical line, CD, 
and from it a shadow, cut by the light- 
ray. From E we can imagine a vertical 
line, similar to CD, dropped to the floor 
of the box. (We can find where this 
vertical line will touch the floor by a vertical from F to the box 
edge (IJ) at point K, and a vanishing line from VPl through 
that point (K) to cut the vertical from E (in L). From L a line 
actually parallel to the shadow C-^ (vanishing in VPS) cuts the 
light-ray in the desired point, e. This is essentially the way most 
shadow points are found : — by a vertical line from the point on the 
object to the receiving surface, and from that a shadow line on the 
receiving surface to cut the light- ray. In other words we pass an 
imaginary vertical plane through the point and the light-ray. 
Fig. 2S1 shows a simple application of these principles. When 
the cube has been drawn, the direction and length of the shadow 
edge Ba may be assumed (or taken if drawing from the object). 
This gives the direction of the light-ray A-a. The shadow a-c 
vanishes with AC till cut by the light-ray from C, and c-d 
vanishes with CD. These points can be tested by vanishing the 
light-rays from the other corners to OVPl, thus completing the 
vertical planes above mentioned. 

The shadow of the left horizontal box edge (BH) falls partly 



on the back of the box in a slanting line which may be thus deter- 
mined. Take the book out (Fig. 227), when it will be seen that 
the near part of the shadow, beginning at &, vanishes on the floor of 
the box to VPl till it reaches the box edge in point i. From this 
point to the box corner lies H-^, the line in question. Many shadow 
lines can be found thus, — hy locating any two points in the line. 

Looking again at the shadow 
of the vase on the back of the 
box, we observe that the shadow 
of its horizontal circular top f all- 

Vetzticau plane 
at riomt anoue5 to 


A. Pjlan-3mowino 


Fig. 232 


or ABOVE . 


Fig. 233 

ing on the back of the box is not a horizontal curve. To 
understand this, we will begin with the shadow of any vertical 
cylindrical object on a horizontal surface, as in Fig. 232. It will 
be found actually symmetrical. In perspective it will be fore- 
shortened (B in Fig. 232) ; and lines marking its horizontal details 
(as AB, CD, and EF) will vanish, as would any parallel horizontal 
lines. If now we move this object toward a vertical surface, 
placed so that, viewed from above it makes right angles tvith the light 
rays (as shown in A, Fig. 233), the shadow on the vertical surface 



A. 3nowiNG- 


B Pet?spe:ctive or above 
— Shadow distot?ted 

Fig. 234 

also will he actually symmetrical] though it may appear fore- 
shortened, as in this case. 
Now if the receiving surface be 

turned so it is not at right angles 

(viewed from above) with the 

light (Fig. 234), we get what we 

observed in Fig. 228, — an actu- 
ally one-sided shadow. The reason 

for this distortion is made clear 

from the plan in Fig. 234. The 

descending light-ray from y has 

farther to travel before striking 

the receiving surface, hence its 

shadow, yj is lower than the 

shadow from z. Such variations 

of original shapes are of the same 

nature in producing beauty (and 

consequent enjoyment) as theme 

variations in music. Thus the bottle with shoulders (Fig. 235) 

acquires a charm from the proximity of its interestingly altered 

shadow-self which it cannot have alone. 

For the shadows of curves (as for 
that of the horizontal hat-pin in Fig. 
227) vertical planes through several 
points (as x^ y, and z) are taken and 
the curve then sketched freehand. 
The use of this method is also shown 
in the vase shadow in Fig. 228. But 
as soon as the underlying truths are 
clearly understood, the actual taking 

of points is seldom needed. 

So far the shadows have fallen on flat surfaces, but the 

shadow of the vertical box edge in Fig. 228 falls partly on the 

curved book back ; and on this it forms a vertical curve — that is, 

with its ends in a vertical line. The curve is sketched by the 


Fig. 235 


eye, though it could be constructed /fey points. In Fig. 229 the 
shadow on the book back is cast by a horizontal edge and is 

therefore an oblique curve. In this 
case its upper point (M) is found by 

imagining the book cover continued 
until it cuts the back of the box in a 
line from N vanishing with the long 
box edges. Where the shadow line 
HI cuts this line (point o) will be 

the farther end of the line on the book cover. This shadow will 

vanish in VPl, and where it cuts the upper edge of the book back 

will be M, the upper end of the curve. 

The book in Fig. 236 shows the use of points when the edge 

casting the shadow is itself oblique. A vertical line from D to the 














Fig. 237 







edge 1-2, and a line from that point (E) to C, constructs one 
vertical plane. The light-ray from point A cuts the shadow- 
direction of AB at a. The shadow of CD travels from C through 
a to the edge 1-2, 
and from there to 

-p. :S:^^^^^^ ^- Plan, SHOWING 

-L'- ^^^^^^^^>i>^ T?At)lATINa RAVJ. 

Shado'ws on a 
House. — The truths 
thus developed ap- 
ply to out-of-door 
work, as shown in 
the house (Fig. 237). 
Here one new con- 
dition is met, — the 
shadow of the ver- 
tical dormer edge 
falls on the oblique 
surface of the roof, 
and hence has an 
oblique vanishing 
point. This vanish- 
ing point is easily 
found, as we already 
have two points in 
the vanishing trace 
of theroof, — OVPl 
and VP2. The line 
containing these 
points is the vanishing trace, not only of the roof, hut of the infi- 
nite plane containing the roof. Hence the line can be drawn as 
long as needed — it is really infinite in length. So we have only 
to draw the trace from OVPl to VP2, and mark 0VP3 on it, 
vertically over VPS, exactly as we marked OVPl over VPl. 

The shadow of the bush is an instance of the ease with which 
shadow laws are applied to natural objects. The shadow is sketched 



Fig. 238 


freehand ; but with much greater certainty for knowing that its 
center must fall on the ground in the direction of VPS, and that it 
can extend no farther than its meeting with the ray of light, HI. 
Shadows from a Lamp. — The radiating rays from an artificial 
light can all be contained in an infinite number of radiating ver- 
tical planes through the light itself. Some of these radiating 
planes are seen in the plan (OA, OB and others in Fig. 238). 

These radiating 
planes are used 
instead of the 
parallel vertical 
planes previously 
explained. Other- 
wise the methods 
are the same as 
with light from 
the sun. Thus in 
the shadow of the 
stool the light-ray 
from O through 
T> gives d where 
cut by a line on 
the ground di- 
rectly under it 
^^°- ^^^ (from through 

E). The shadow of Gr^ falls on the floor in the direction of 
o-H till it reaches the wall. On the vertical wall, the shadow 
of the vertical GH is also vertical. It is ended by the light-ray 
from O through Gr. The shadow of the edge GrI will be parallel 
to it, and like it will appear as a vanishing line to VPl. The 
near part {d-j) of the shadow of DI will be parallel to DI, and 
will vanish to VPl till it reaches the wall at J. A line from J 
to I completes the shadow of DI. 

Shadows in an Interior. — These are partly like the lamp 
shadows. For instance the shadow on the couch in Fig. 239 



extends in an almost opposite direction from that of the chair. 
These shadows are produced by the diffused daylight radiating 
from the window. On the other hand a patch of sunlight falling 
through the window would follow the laws of sunlight generally. 
The edges a-b and c-d vanish with AB and CD, while points a 
and h are marked by the meeting of light-rays from A and B 
with shadows of verticals from A and B. In this case the light 
comes from beyond the window, hence the light-rays recede up, 
and appear to converge or vanish in that direction. 


Chapter XLV 

A LTHOUGH the same perspective principles apply to out- 
/% of-doors work the conditions of the study vary, and some 
-^ -^ cases need explanation. 

Vanishing Points. — In drawing the house (Ch. XXII), we 
placed ourselves proportionately in relation to the small cube 
as we should naturally be in relation to the real house. Thus 
the sixteen inches of distance from the eye, or four times the 

51DE view or nousc showing two positions op eye 

Fig. 240 

height of the cube, was equivalent to only four times the height 
of a twenty-foot house, or eighty feet — less than five rods. At 
this short distance the vanishing of the lines is very decided ; but 
at a half mile from the same house, those lines appear nearly 
horizontal. The reason for it is seen in Fig. 240. When the eye 
is near the house (at ic) the apparent difference in length between 
the edges AB and CD is greater than when the eye is at y. 
This is shown on picture plane 1 by ab and cd^ and on picture 
plane 2 by do' and dd!. The horizontal edges of the house 
as seen by the eye from x would therefore vanish more steeply, 
causing the vanishing point to fall nearer, as shown in A, Fig. 
241. As seen from y^ the horizontal edges are less steep ; there- 
fore in B the vanishing points fall much farther away. 

It follows, therefore, that the greater the distance of the eye 



from an object, the farther to right and left will the vanishing 
points fall. When the house is a half mile away they fall so far 
to left and right that its horizontal lines appear almost level. 
Hence the beginner in landscape work, accustomed only to near 
objects, is sometimes puzzled, because distant houses seem to 
have no perspective. And the landscape artist who has *' no 


[! PI n 



Fig. 241 

trouble with houses " in the distance may shrink from attempting 
them in the foreground. 

Size of Objects Seen. — The image formed on the retina of the 
eye is always exceedingly small, and with distant objects becomes 
microscopic. All mental picturing of the size of objects pro- 
ceeds from our mental knowledge of their actual dimensions. 
Size judged from seeing alone can be but a matter of comparison. 
This is easily proved by asking two persons how largejthe moon 
appears to them. Here we have an object whose real size and 
distance are so great as to be no guide in comparison with other 
objects and it will probably appear of a different size to each 
person. There is consequently no such thing as the drawing 
of objects ''the size they appear." Size in draiving is merely 
relative; and the scale on which a drawing is made is wholly a 
matter of choice. We may choose to make a drawing what is 
termed " actual size," but this means that we regulate its size by 
a mental knowledge obtained either from measuring in the ordi- 
nary way, or by putting our sketch back by the object to compare 
them by the eye. 

The absolute size of objects varies so much, also, that unless 
the picture contains something the size of which is well known 



and but little variable, we cannot be sure of the sizes repre- 
sented. The human fig- 
ure serves best for such 
a standard, but some 
objects always adjusted 
to the human figure in 
size, as steps, and often 
doors, will answer in its 

The size of objects ac- 
cording to their distance 
into the picture is impor- 
tant in out-of-doors work 
also. Here the indis- 
pensable picture plane 
becomes again useful. 
In Fig. 242, for instance, 
the gondolier must not 
be too large for the 
buildings. Lines drawn 
from his head and a 
Fig. 242 poiut ou the watcr di- 

rectly under it to the eye level will contain between them his 
height above the water 
all the way to their van- 
ishing point. If we 
wish to know, for in- 
stance, whether the door 
on the left is large 
enough, we have only 
to draw horizontal lines 
from its top and from a 
point directly under it on 
the plane of the water f^^- 243 

continued. Where this water line would cut the water line from 



the figure to the vanishing point a vertical line is erected, on 
which the two heights can be compared. 

Reflections. — If a mirror be 
laid on a table, and a cup placed 
on it (Fig. 243), the reflection 
will appear precisely like the 
cup reversed, with its bottom 
resting against the bottom of 
the real cup. The reflection 
will not present to the eye the 
same shape as the real cup, 
for besides being reversed it is 
farther below the eye, making 
its inside invisible while its 
base is covered by that of the 
real cup. We can also see 
farther around on its flaring 

A. Perspective- 


B. Side view of stake -showing how 
its foreshortening occurs. 

Fig. 244 

surface, because its decrease of diameter is toward the eye level, 
while in the actual object it is away from the eye level. 

Now since it is like the cup reversed we see that any point 
(as A) in the cup, must be reflected directly under itself. So a 
stake, thrust into a pool of still water (Fig. 244), will produce 
a reflection like itself reversed ; ^ and each point in the reflec- 
tion will be directly under the same point in the real stake. 

* In this case appearing longer than the real stake, as explained a few pages later. 



It is therefore evident that in case of reflections on a horizon- 
tal surface, the image formed must he vertically under the reality. 

Consequently, as long as the 
reflecting surface remains 
horizontal, reflections on it 
cannot be thrown to one 
side, but must be shown di- 
rectly under the real objects, 
even if the reflecting surface 
be broken (as in Fig. 242). 
And if in drawing reflections 
we represent them out of 
the vertical (Fig. 245) the 
reflecting surface (in this 
case the water) appears to be 
sloping, like rapids in a river. 
We may therefore take as 
our rule that reflections are 
invariably like the reflected 
object reversed on the reflecting 

When the Object is Sep- 
arated from the Reflecting 
Surface. — In Fig. 246 the bungalow is separated from the reflecting 

WRONG. Reflections not ver- 

Fig. 245 

.The bonqalow/ is estimated lb b« 
this ctistance (E r) forthev into the 
picture thar\ the boo^House.. 

Fig. 246 


surface (the water) by a high bank. But by using a water line 
of the boathouse (which stands parallel to it and directly on the 
water), the points (A, B, and C) where the bungalow edges 
continued would strike the plane of the water can be closely 
approximated. From there the points (a, 6, c, and d) for the 
reflection of the bungalow are measured vertically. 

Even if we had not the boathouse to give parallel vanishing lines 
on the water, the necessary points (A, B, and C) could be esti- 
mated with sufficient accuracy after a little experience. The main 
thing to remember is that it is on the reflecting surface or on 
its plane continued that the object is reversed in its reflection. 

Reflections on Vertical Surfaces. — With reflections on vertical 
surfaces the problem is very simple. In Fig. 247 the box appears 
reversed as far back 
of the mirror surface 
(the thickness of its 
frame) as it actually 
stands in front of it. 

Length of the Re- 
flection. — The verti- 
cal length of the re- 
flection, while the 
reflecting surface is 
unbroken (as in Fig. 
243) is actually the 
same as that of the real subject. This does not mean that the 
reflection will always appear of the same vertical length as the 
object, as that depends on its position and on the location of 
the point from which it is viewed. In Fig. 244 the stake is 
seen from a higher point and leans toward the beholder. It is 
consequently seen foreshortened, as the roundness of its top 
indicates. The reflection, being reversed, appears practically in 
its true length. A point (x) on the surface of the water directly 
under the top, appears lower than where the stake enters the 
water, because nearer the eye. Cases like the familiar " silver 


Fig. 247 


path " of the moon in rippling water, or like Fig. 242, where 
the reflections of upright objects appear lengthened vertically 
as well as broken, are caused by the many curved surfaces of the 
waves on which successive bits of the reflection fall. 

Use of the Finder. — Nowhere will the finder (Ch. VITI) be 
of more use than in out-of-doors work. The difference in distance 
between the near and far objects in a landscape is so great, 
that the beginner finds it hard to realize how much difference 
he must make in size. The finder serves as a measuring unit 
for these differences, besides being invaluable as an aid in 






THE Cone. — After the cylinder is drawn, the base of the cone is 
next placed. This is actually a circle, and of the same size as 
the cylinder base. Its position will be clear from the plan (Fig. 
249). In perspec- 
tive it is best placed 
by its true center. 
If the cone were 
moved on the 
ground around and 
touching the cylin- 
der, this center 
would describe a 
circle, twice the 
diameter of the 
cylinder base, and 
equally distant at 
every point from 
the cylinder. This 
circle is sketched in 
perspective (Fig. 
250) as an ellipse 
(see Chs. IV and 
XX). The true 
center for the base 
of the cone is placed 
on this ellipse (at 
0). From this 
true centpr a ver- ''^''''^"'''''!''^^'^"*^^i iti ^'<f Ti -'~'<~[ t'^ -r > "^v^-ir-r\~n - rt—r — 

Fig. 248 
11 161 

■ '11 m Di—itxowiiilgMi— 



Fig. 249 

tical line of indefinite length may be erected, on which to set off the 
axis of the cone. Its height, being actually the same as that of 
the cylinder, will appear slightly greater because nearer the eye. At the 
same time its apex (F), being nearer, cannot appear 
quite so high on the paper as even the nearest edge 
of the cylinder top.^ Through its lower end (0) the 
real diameter of its circular base passes. Being at 
the same distance into the picture as the axis, and 
like it parallel with the picture plane, it appears 
in its true proportion to the axis (one half). It is 
therefore so set off, equally on each side of 0, 
giving AB. The base of the cone, though actually of the same 
size as the cylinder, will appear a little larger (and also a little 
rounder), because nearer. The short diameter 
of this base (CD) is therefore set off greater 
than that of the cylinder base, remembering 
that as is the real center, DO must be 
larger than CO. The ellipse is then sketched 
through the four points A, B, C, and D, tak- 
ing care to have it touch the base of the 
cylinder, and to make the greatest length not 
on AB, but at a point a little in front, 07i 
the apparent middle from front to hack — that is, on the long 
diameter (the light line in front of AB). The cone is completed 
by drawing its sides from the apex tangentially to this elliptical 

The Ball. — If the ball be rolled about and touching the cylinder it 
will follow the same path as the center of the cone base, so that its 
resting point will always be somewhere in the ellipse representing that 
path (its center being always vertically above the resting point). We 
should therefore mark some point in the large ellipse (in this case x) 
for the resting point of the ball. If we stoop to bring the eye nearly 

^ Being actually of the same height, they lie in the same horizontal plane. This plane, 
being below the level of the eye, appears to recede upwai'd, as the table does. This will be 
better understood if a sheet of paper is laid on the tops of the two objects, when it can be seen 
that it appears to recede upward. The following chapter will further illustrate this truth. 


Fig. 250 


to the table level, and look at the ball, we shall see it resting on this 
spot. But if we return to the point from which the group was to be 
viewed, we shall find this point hidden by the projecting mass of the 
ball. The circle which represents the boundary of the ball is therefore 
drawn with its lower edge a little below x, and its center vertically over 
that point. 



The Block. — This 
parallel with the pic- 
ture plane, it will 
appear in its true 
shape, and the block 
ends will vanish in 
VPl like the book 
ends in Chapter XII. 
In setting off the 
apparent width of 
the top, we remem- 
ber that it is actually 
narrower in propor- 
tion than the book 

The Cylinder. — 
The cylinder rests 
against this block 
(side view. Fig. 252), 
so we can measure 
the height of its back 
(AB, Fig. 253) actu- 
ally, making it twice 
the height of the block 

should be drawn first. Since its front face is 

Fig. 251 




Fig. 252 

front. The lower base is actually the same in width as the block, but be- 
ing nearer the eye it will appear larger. Just how much can be easily 
determined. Draw the invisible lines of the block (the dotted lines in 
Fig. 253), and carry the di- 
agonal of one half {x) for- 
ward in a line of indefinite 
length. Cut this by line y 
of the invisible edge of the 
block. From the point (C) 
so found draw line 3 to the 
right, to cut another invisi- 
ble edge continued. This constructs another 
rectangle the actual size of the side of the block, 
but nearer, hence appearing larger.^ The middle 
of the front of this rectangle (point D) will be 
the front of the base of the cylinder. Its back 
will be A, and its long diameter, EF, can be set off on a line sketched 
half way between this front and back, marking E and F half way 
between AD and the ends of the construction rectangle. Through these 
four points the bottom ellipse is drawn. 

For the top ellipse a similar rectangle can be constructed directly 
above it. This will give a much more foreshortened ellipse, as would be 
expected. The back and front of the middle ellipse are drawn as 
directed in Chapter IV. 

Fig. 253 



The Rectangular Block. — This solid is drawn in the same manner 
as the book similarly placed. Recall that it is equal to two cubes, as 
shown in Fig. 255. Here a diagonal of the first cube measures the 

^ This use of the diagonal for measuring will be found in Chapters XXII, XXV, and 



width of the second, as in the steps in Chapter XXIL (Thus BD and 
DE, being the same distance from the picture plane, are made actually 

Fio. 254 

equal. A line from E is then vanished with 
those from A and C to VP2, and the diagonal 
continued to cut it in point F. From F a 
vertical line gives GH, the edge desired.) 

The Frame. — For this the loioer face only 
may be considered first. Place a six-inch 
square of cardboard in the required position 
(Fig. 256). Mark in the drawing its touch- 
ing points (A and B), 
one eighth of the block 
length from each end. 
Now push up the card- 
board till it rests verti- 

III ' 

X 1 \ \ ' 



Fig. 256 


Fio. 'ioo 

\ -t^^^- 


cally against the block (Fig. 257) and observe that its lower corners 

move in lines parallel to the block ends and rest directly under points 

A and B. Sketch vertical lines downward from 
these points, giving C and D. Draw lines 
through C and D, vanishing in VPl, and ex- 
tending forward indefinitely. On these lines 
the distance of the frame from the block can be 
measured. This distance is actually one half the 
width of the block width, hence is made perspec- 

tively that, or apparently a little larger than the near half (Ca^, Fig. 258) of 

the block width at that point. It ^^^ -r- eve leweu- 

is measured on the line through ^^- 

C, giving E. From E the lower 

edge of the square is vanished 

to VP2. Where it crosses the 

line through D will be F, the 

other lower corner of the square. 

The leaning edges of the square 

are drawn from E and F through 

A and B, and will be found to 

vanish in OVPl. They vanish 

a little less (have a more distant vanishing point) than the horizontal 

edge EF. They should therefore be made shghtly longer, but shorter 

than if standing erect from E. We can check our 
estimate by comparison with the vertical height of 
the frame at that point (EH). To obtain this, 
the height of the frame standing vertically at C is 
measured (one and a half times the block height), 
and a line through its top vanished in VPl. This 

gives EH, the apparent height of the square if erect at that point. When 

leaning it wdll appear slightly less (Fig. 259). 

The Thickness Edges. — These edges must vanish sharply, or have a 

near vanishing point, because the edges at right angles to them (the ones to 

OVPl) are foreshortened but little. Hence VP2 is placed but little below 

the group. To this point these short edges are drawn, carrying them for- 


Fig. 259 


ward of the corners indefinitely for a short distance. The foreshortening 
of these edges (actually one sixth of the long edges of the square) will be 
much greater than that of the long edges ; and one may be set off accord- 
ingly, as at I. From this corner a horizontal edge vanishes to VP2, and 
a long oblique one to OVPl, giving points J and K where they cross the 
thickness edges. From K the other horizontal edge vanishes again to VP2 
and from J the other long oblique one to OVPl, completing the square. 

The Inner Square. — On this we may draw the inner square and its 
thickness as in Chapter XXIV, remembering that the sixths to be meas- 
ured on IK are perspective sixths ; and that since IK does not vanish 
much, the difference in their apparent size is very slight. 


The Triangular 
Prism. — This solid is 
easily constructed by the 
use of the cube (see dotted 
lines). The length of AC 
is found by the diagonal 
(ED) of one side of this 
imaginary cube, DF being 
made equal to DG (p. 
165), The steps in Chap- 
ter XXII illustrate this 
method. The end view 
shows how the triangle 
is related in shape to the 
square face of the cube. 
Its vertical center line is 
located by the diagonals 
of the square, and the 
height of its apex is meas- 
ured to X on the near verti- 
cal edge of the cube (AE). 

Fig. 260 



The Triangular Frame. — This is also readily drawn by the help 
of one face of the cube. After the triangular outline is sketched, the 
height of one lower bar (one sixth of the height of the cube) is marked 


Fig. 261 

Fig. 262 

upward from C, giving point E, and the lower edge of the inner triangle 
is vanished through E. Where BF, drawn so as to divide AD per- 
spectively, crosses this lower edge is a corner of the inner triangle. 
Through this corner (G) another edge of the inner triangle vanishes to 
OVPl. The other edge is drawn toward 0VP2. The thicknesses are 
found as in the square frame (Ch. XXIV). 



Aims of perspective, drawing objects as 
they appear, 140 
learning to see, xii 

to acquire artistic judgment, 80, 83, 
Apparent size of objects, according to dis- 
tance, xi, 156 
in out-of-door work, 155 
in relation to other parts of picture, 

relative only, 6, 155 
Arch, errors in drawing, 99 

pointed and other forms, 104 
round, 99, 103-104 
Artistic judgment, 80, 83, 120 
Artistic rendering, book, 59 
buildings, 83 
glass, 25, 28, 30 
rose jar, 17 

Background, subordination of objects in, 
fan, 33 

leaning bowl, 26 

plate, 30 
Bases of cylindrical objects, see Foot 

always partly visible, 47 

location on horizontal surfaces, 29, 30, 
32, 161-162 
Benefit of perspective study, 

acquiring of artistic judgment, 80, 83, 

learning to see correctly, xii 
Book, artistic rendering of, 59 

at angles to picture plane, 58-60 

back of, 44 

clasps of, 44 

cover thickness, 44 

in two positions, 43, 44 

margins, 40, 41 

projection of covers, 44 

use of pencils to show convergence of 
lines, 37 

use of strings, 38, 39 

vertical edges, 60 

with back parallel to face, 38-42 

with cyUndrical object, 46-47 

Books, two, at different angles to the pic- 
ture plane, 61, 62 
•with, a cylindrical object, 67, 68 

Boundary, movable, 16, 17, 21 
tangential to ellipses, 13, 17 

Buildings, camera distortions in, 139, 142 
few vanishing points for, 84 
from photograph or print, 81 
house, the, 69-80; see House 
round arch, the, 99, 103 
round window, the, 100, 101 
spire or tower, 90 
type forms useful in, 85, 88 

Camera distortions, 139, 142 
Carrying lines "around a corner," 86, 130 
Central direction of seeing, alluded to, 10, 

explained, 6 

moves with changed picture center, 46 
Chair, the study of, 118-120 
Circle, actual center of, 63 ; see Ellipse 

concentric circles, 14, 63-66 

location of its center in ellipse, 15, 65 

only position in which seen as circle, 

seen obliquely, 9 
Circular frame within square frame, 96-99 

application of its principles, 99; see 
CyUndrical objects not vertical 
Clock, 102 
Color, in buildings, 83 ' 

of book, 59 

on rose jar, 17 
Composition, cylinder and cylindrical ob- 
ject, 12 

cylindrical objects grouped, 26; with 
books, 68 

in selecting from interior, 112, 115; 
from photograph of building, 81 
Concentric circles, 14 

with square, 63-66 
Cone model, 18 
Cone principle, 19 
Cover of teapot, 28 



Cream jug, foot, 22 

handle, 20 

ornament, 23 

spout, 22 

study of, 20-23 
Cube, at 45° with picture plane, 53, 56 

at 30° and 60° with picture plane, 56 

making the drawing, 54 

order of drawing edges, 54 

proportions used in estimating other 
objects, 51, 86, 129, 164-165, 

recession of horizontal surfaces to eye 
level, 51 

relation of foreshortening to vanish- 
ing of edges, 50, 51 

study of, 48-52 

taking direction of edges with pencil, 

tests of vanishing lines by string and 
by eye, 56 ; on blackboard, 57 
Cylinder, errors in ellipses of, 14 

hollow, the, 14, 15 

inner cylinder, 14 

models for, 8, 14 

perspectively equal divisions, 15 

position of model, 13 

roundness of ellipses, 14-16 

sides tangential to ellipses, 13 

study of, 12-15 

symmetry of elUpses, 15 

true diameter of circle, 15 
Cylindrical objects grouped, 26-28 
Cylindrical objects not vertical, 92-94 

application of principle, 94 

other examples, button on cord, 19; 
circular frame, 96-99; clock, 
102; flower pots, 95; leaning 
bowl, 27; luncheon carrier, 32; 
round arches, 99, 103-104; round 
window, 100, 101 

symmetry of appearance, 93-94 

test for drawing of, 94, 95 
Cylindrical objects with fruit, 29, 30 
Cylindrical picture plane, 140-141 

Diagonals, use for measuring, concentric 
circles, 65, 66 
door in room, 108 
square frame, 86 ' 
square plinth, 89 
Drawing from a description, xii ; see Prob- 

Ears of teapot, 28 

Ellipse, at right angles to axis in cylindrical 
objects, 93-94 

common errors in, 14 

diameters of, 9, 15 

drawn entire first, 14, 15 

from concentric circles, 14, 63-66 

measurement on its diameters, 14, 15 

position of hand in drawing, 11 

practicing, 10, 11 

roundness according to position, 9, 10, 

study of, 8-11 

symmetry, 9, 15 

tangential to boundary lines, 13, 15, 17 

test of shape, 9 

true diameter of circle, 15 

varying curvature of boundary line, 9 
Exceptions to the use of the flat picture 
plane, 139-142 

cylindrical picture plane, 140-141 

spherical picture plane, 141-142 
Eye level, explained, 52 

finding, 39, 40 

importance, 40 

way of using 39 

Fan, 33 

Finder, 26, 67, 160 

Flower pots, 95 

Foot of cylindrical objects, at least partly 

visible, 47 
of cream jug, 22, 23 
of rose jar, 16, 17 
Foreshortening, xi, 10 
Freehand sketching defined, xii 
Freehand work entirely, 3 
Fruit grouped with cylindrical objects, 29- 

Foundation truths of perspective, two, xi 

Geometric solids, omitted at the teacher's 
discretion, 85, note 

Geometric measurements, obtained per- 
spectively, 66, 108 

Glass, bowl, 25 

pitcher, 30 . ' 

Handle, cream jug, 20-22 
Hexagonal plinth, appUcation of study, 

test for, 124-125 

two positions, 121-125 



Hexagonal prism and frame, 128-130 

estimating length of prism, 129 
Horizontal surfaces foreshortened, 37, 40, 41 
Horizontal surfaces recede to eye level, 51 
Horizontal vanishing edges, 37-39, 50 
House, 69-80 

chimney, 76 

dormer window, 79 

eaves projections, 74 

"L" part, 75 

model, 69 

porch, 75 

roof, 70-73 

steps, 78 

windows and doors, 75-76 
How much to include in the picture, 112 

Interiors, at angles to picture plane, 110- 

ceiling, little or none shown, 115 
door in an interior, 107 
from memory, 106 and note 
further studies of, 114-116 
lines must not point to corners, 109, 

parallel to picture plane, 105-109, 132, 

picture on wall, 106 
relation of subject-space to picture 

plane. 111 
selection of subject-space, 110 
stool, 108 
with tiled floor, 126-127 

Knob of teapot cover, 28 

Lamp shade, 18, 19 

Line, directions for drawing, 1, 2 

expressive, 17 

texture of, 1 
Lines of the picture must not make equal 
angles, 115 

nor run to corners of margin, 109, 116 

Margin of picture, 1, 13 

cutting the group. 27, illus., 28, 30 

moving, to improve picture, 116 

partial, 116 
Margins, of the book, 40, 41 
Materials, pencil and paper, 1 

models, 2 ; see Models 
Measuring, by the diagonal, 65, 86, 89, 

distance into the picture, 65-66 
height within the picture, 90, 108 
only relative, 6, 66 
Measurements obtained geometrically can 
be so obtained perspectively, 66, 
107-108, 122-123, 164-165, 167- 
Mechanical perspective, value alluded to, 
errors of, 139 
correction of, 140 
limitations of, 142 
Memory work, conditions of, 31 
from interiors, 106 
group of objects from, 31-32 
less laborious, note, 106 
necessity for, xii 
specially advised, 43, 53, 58, 67 
Methods, their subsequent use in practical 

work, xii, 101 
Models, in general, 2 

making, cone, 18, 19; cylinder, 8; 
cube, 49; rectangular block, 48; 
hexagonal plinth, 121 ; triangular 
prism, 131 
position for drawing, 2 

Oblique vanishing lines, chair, 119 

dormer, 80 

hexagonal frame, 129, 130 

light rays, 147 

roof of house, 72 

shadow on roof, 151 
Obstacle to mastery of perspective, xii 
Ornament, constructive principles of, 23 

on Japanese luncheon carrier, 32, 33 

rendering of, 17 

use in composition, 13 
Out-of-doors work, 154-160 

greater distance of vanishing points, 

reflections, 156-160 ; see Reflections 

size of objects, 155 

Parallel retreating lines, convergence of, 

40, 50 
Parallel retreating horizontal lines, meet- 
ing at eye level, 50, 93 
Parallel perspective, bureau, 134 
interiors, 105-109, 132-133 
street, 134, 136-137 
term "parallel" unsatisfactory, 135, 



Pencil measurement, difficulties of, 6 

essential requirements for, 6 

gives relative size only, 6 

of a door, 7 

of the book, 39, 40 

of the cube, 55, 56 

of the ellipse, 13 

on a window-pane, 5 

on the picture plane, 7 

study of, 4-7 
Picture plane, different for each picture, 
46, 113 

exceptions to use of flat picture plane, 

method of using, 7 

position relative to central direction of 
seeing, 6 

relation to group of objects, 46 

relation to subject in drawing inte- 
rior, 110-112 

study of, 4-7 
Position, for drawing, 1 

of hand for ellipses, 11 ; for lines, 1-2 

of models, 2 
Practical use of methods, xii, 101. 
Practice, of ellipses, 11 

of lines, 1, 2 
Principles of perspective, two founda- 
tion, xi 
Problems, clock, 102 

conditions, general, 34; special, 102 

cyUnder, cone and ball, 34, 161-163 

reasons for giving, xii 

rectangular block and cylinder, 48, 

square frame leaning on block, 91, 

triangular prism and frame, 131, 167 
Profile lines, 21, 22 


Railroad track, illustrates vanishing fines, 

' 41, 42 
Reflections, lengthened by waves, 160 

length of vertical, 159, 160 

on a horizontal surface, 156-158 

on a vertical surface, 159 

when separated from reflecting sur- 
face, 158 
Rose jar, artistic rendering, 17 

foot, 16, 17 

ornament on, 17 

shoulders, 16 

study of, 15-16 

tangential joinings, 17 
Round arch, 99, 103 
Round window, 100-101 

San' Apollinare, Church of, 81 
Selection for picture, from photograph, 81 ; 
see Composition 

from interior, 112, 115 
Shadows, 143-153 

cast by an obfique edge, 150-151 

cast by parallel rays (sun, moon), 144- 

cast by rays from lamp, 143, 152 

distorted, 148-149 

in an interior, 152-153 

in a shadow-box, 144-150 

located by imaginary vertical planes, 

located by two points, 148 

of a cube, 147 

of curves, 149 

of natural objects, 151-152 

on an oblique surface, 151 

on a house, 151 

on curved surfaces, 149-150 

vanishing of light rays, 146-147 

vanishing of shadow-directions, 145- 
Shoulders of cylindrical objects, 16; re- 
lation to cone principle, 19 
Solutions of problems, 161-168 

cylinder and rectangular block, 163 

cyfinder, cone and ball, 161 

square frame leaning on rectangular 
block, 164 

triangular prism and frame, 167 
Spherical picture plane, 141-142 
Spout, of cream jug, 22 
Square frame, 85-87 

test of, 87 

application of study, 87 

leaning on rectangular block, 91, 

Table fine, explained, 3 

high enough on paper, 43 
position vdth plate on edge, 30 
significance in composition, 13 
subordination of, 44 

Taking direction of vanishing edges with 
pencil, 55 



Teapot, cover, 28; ears, 28 

knob, 28 

study, 26-28 
Tests, by blackboard for vanishing lines, 

by eye, for vanishing lines, 56 

cylindrical objects not vertical, 95 

hexagonal plinth, 124 

square frame, 87 

the ellipse, 9 

the eye a final test, xii, 142 
Thumb-nail sketches, interiors, 115 

still-life objects, 26, 68 
Tiled floor, 109, 126 
Time study, glass bowl, 24, 25 
Triangular prism and frame, 131, 167 
Two books, at different angles, 61-62 

with cylindrical object, 67-68 

Vanishing lines, "converging," 37-38 

example of, railroad, 41-42 

oblique, 73; see ObUque vanishing 

taking direction of, with pencil, 55 

tests of, 56, 57 
Vanishing of parallel planes, 51, 73-74, 

162, note 
Vanishing points, abbreviation of, 44 

numbering, 50 

oblique, 73; see Oblique vanishing 

use without marking, 56, 80, 124 
Vanishing traces, 73, 151 
Veri;ical fines drawn vertical, 60 
Vignetting, 116. 








14 DAY USE ^e/reuJ 








FORM NO. 00. «,., ,Z''''ll^l^l°Z%'''''''' 





General LiB>_, 

University of California