HUE on the last date stamped below

UNj\/-'r " BRANCH

CALlFORNfA
L/BRARY

L-'S ANGELES. CAUF.

FREEHAND PERSPECTIVE
AND SKETCHING

FREEHAND
PERSPECTIVE
AND SKETCHING

PRINCIPLES AND METHODS OF
EXPRESSION IN THE PICTORIAL
REPRESENTATION OF COMMON
OBJECTS, INTERIORS. BUILDINGS
AND LANDSCAPES

DORA MIRIAM NORTON

INSTRUCTOR IN PERSPECTIVE, SKETCHING
AND COLOR, PRATT INSTITUTE, BROOKLYN

FOURTH EDITION

BROOKLYN
PUBLISHED BY THE AUTHOR

1916

S9 18

By Doua Miriam Norton

C139

THE UNIVERSITY PRESS, CAMBRIDGE, V. S. A.

TO THE

MEMORY OF WALTER SMITH

FIRST DIRECTOR OF THE MASSACHUSE'lTS NORMAL ART SCHOOL
INSPIRING CRITIC AND JUDICIOUS FRIEND

THIS BOOK IS DEDICATED

WITH THE WISH THAT IT MAY HELP OTHERS AS ITS
AUTHOR HAS BEEN HELPED

D. M. N.

iz/rs'

PREFACE

REVISED FOR THE SECOND EDITION

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 here set forth
have been taught by the author for some years in the above school,
and have been found practically effective in that direction. In
revising it the author has drawn upon its use as a text-book
for large classes, as well as in other directions.

As offered to the public this course is intended to form a text-
book for classes in high, normal, and technical schools and in
colleges ; also as a book of reference for supervisors and teachers
of drawing, for draughtsmen and artists whose training in per-
spective needs to be supplemented, and for the instruction of
students so situated that personal art teaching is beyond their
reach. Since manuals for the teaching of drawing to children
already exist, the methods here presented are primarily such as
have been found effective with maturer minds. Its relation
to the teaching of children is thus like that of a grammar to
the ** language lessons ' ' of the primary schools. From it the
teacher, whether of children or of adults, may select material for
courses according to age or aims in study.

In the case of older students, though perspective books excellent
in certain directions have been published, it has been found diffi-
cult to direct inquirers to anything at once applicable to immedi-
ate use and comprehensive enough to give a working knowledge of

PREFACE

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.

The author gladly acknowledges indebtedness to many sources
in the making of this volume. Although the naming of all would
be impossible in this brief space some are so preeminent that
mention cannot be forborne. The experiences of teaching the
subject under the care of Mr. Walter Smith, then Director of
the Massachusetts Normal Art School, and later with Mr. Walter
Scott Perry, Director of the School of Fine and Applied Arts of
Pratt Institute, himself a teacher of great originality and force in
the subject for some years, have been most fruitful in the accumu-
lation of subject matter for this course. The author's drawings
have been so largely and sympathetically supplemented by the
work of Mr. Ernest W. Watson, now teaching in the Institute,
as to merit appreciation beyond that due the ordinary illustrator.
In the bringing out of the book the practical advice of Mr.
C. Franklin Edminster, many years an instructor in the same
school, and the critical taste of Mr. Henry Lewis Johnson, editor
of Tlie Printing Art, to whose suggestion is owing the form in
which the book appears, have been of great value. Of these aids,
and of others not mentioned, it is a pleasure to here express a
grateful appreciation.

D. M. N.

Brooklyn, June 24, 1910.

CONTENTS

Page

Introduction xi

Chapter

I. General Directions 1

II. Pencil Measurement and the Picture Plane 4

III. The Ellipse 8

IV. A Cn-INDER 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

VIII. A Group of Cylindrical Objects 26

IX. Cylindrical Objects jGrouped with Fruit 29

X. A Group of OsjECTS^-wtOM 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. The Cylinder and Rectangular Block ^ A Problem for Original

Study 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. The Square Frame Leaning on the Rectangular Block — A Prob-
lem FOR Original Study 9j

CONTENTS

Chapter Page

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 110
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 — A Problem for Original

Study 131

XLI. The Study op 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

SOLUTIONS OF PROBLEMS , I6I

INDEX 171

INTRODUCTION

FREEHAND 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 tvhen placed at 7'ight 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

/y\ y\ when held as in B. All the phenomena of free-

' A 'I B hand perspective, however complicated and per-

^^^' ^ 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

FREEHAND PERSPECTIVE

instance, we hnoiv 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 looks
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 sufficiency of a description was expressed, and in
this case 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 dratving by
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

xii

INTRODUCTION

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.

xiu

Chapter I

GENERAL DIRECTIONS

MATERIALS. — Any paper having a fine and fairly soft
texture can be used. 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
made.

Fig. 3

Fig. 4

FREEHAND PERSPECTIVE

Fig.

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
shaking them. Unfamiliar or diffi-
cult exercises should be first carefully
sketched with a thin, light line. If
wrong, draw 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. Exact knowledge is to be acquired only
that artistic interpretations may be expressed with ease and
certainty.

Models for "Work. — 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

GENERAL DIRECTIONS

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 called 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. — All 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 AND THE
PICTURE PLANE

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. 12).

The Picture Plane. — Here we must learn to keep tlie pencil
parallel tvith the face in order that the pencil measurement
may be reliable. For this, go to the window, and stand facing

Fig. 7

PENCIL MEASUREMENT, ETC.

WINDOV(/ USED AS

PCCTt/BE

, Plane

•0

B. Plan of A

A. Showing use

OF WINDOW AS
PICTURE PUANE

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
house.

This leads us
toseethata//j!;fr-
spective drcming
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

other words, 07i a plane parallel ivith and in
front of the face. This imaginary transparent
plane is called the Picture Plane, and is a
most important factor in all freehand draw-
ing. Thus, by turning or revolving the
pencil on the glass in front of the face,
that is, 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

.•iflHH

Fig. 9

FREEHAND PERSPECTIVE

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. Any distance between the eye and
the object may be assumed for the picture plane. But for
accuracy this assumed distance must be kept the same while
comparing sizes. This is easily done by sitting erect and
measuring at arm's length, supporting the elbow with the other
hand if needful. The student should then mentally see the pic-
ture plane, recalling that it is vertical^ or parallel with the face
when looking at the middle of the objects to be drawn. That is,
it is at right angles to what we may call the Central Direction
of Seeing.

The Central Direction of Seeing. — This extends from the eye to
the center of the objects observed, 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

PENCIL MEASUREMENT, ETC.

it when 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 (Fig. 12).
Thus as the book is turned, the pencil,
if it remains on the picture plane, shows
^ the book to appear narrower or be fore-
shortened. 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 im-
portance to distinguish sharply between
actual facts of form and size and the per.
spective appearance of them as presented
to the eye. Fig. iz

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
XXXIV, XLI, and XLIII.

Fig. 11

Chapter III
THE ELLIPSE

AVING 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,

Fig. 12a

we naturally look for the same change in the Circle.

Making a Cylinder. — Fold

the long edges of a piece ^^
of stiff paper (A in Fig.
12a) and roll it into a cylin-
der, tucking one end of the
paper under the folds of
the other (B in Fig. 12a).

Seeing the Ellipse. — 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 fore-
shortened that its surface is entirely lost to sights
leaving only its edge visible. Now, keeping the
cylinder vertical, lower it till the eye sees into it
perhaps half an inch. Observe carefully the shape
formed
by the
top.
Turn it
so the
top appears as a cir-
cle (A in Fig. 14),
then, iiolding it ver-
tically again (as at
B), compare men-
tally the apparent shapes as the top is placed in the two different
positions. Now (keeping it always vertical) raise and lower the

Fig. 14

THE ELLIPSE

Fig. 15

cylinder slowly, and note how the form of the top changes,
appearing rounder as it is lowered.

Symmetry of the Ellipse. — This peculiar shape, varying in round-
ness between the straight line and the circle, represents the ap-
pearance of the circle seen ohliqueli/, 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 ellipse
is constantly changing in the degree b(
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 Diiime-
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.

Testing 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 rhythmically varying curves which compose ellipses as seen in
real objects, correcting and comparing till the eye is satisfied. If
this be faithfully done, the time will be short before
ellipses, often deemed a bugbear of freehand draw-
ing, become a pleasure instead of a penance.

Roundness of Ellipses According to Position. — 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

Fig. 16

FREEHAND PERSPECTIVE

side and compare the shape of this traced ellipse with that of the
top ellipse (Fig. 16). Also compare both with that part of the
cylinder bottom which can 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.

Practice of Ellipses. — The student should now practice drawing
ellipses, both vertical and horizontal, until they can be formed

Fig. 18

THE ELLIPSE

with ease and exactness. Mark the 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 for drawing ellipses as directed m
Chapter I for straight lines, using a position of the hand that
will brmg the pencil at right angles to the long diameter. If the
elhpse 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 cer-
tainty of movement.

Chapter IV
A CYLINDER AND A CYLINDRICAL OBJECT

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.

KVSSMSafi^-^VAC*.

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

CYLINDER AND CYLINDRICAL OBJECT

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-
j hind them indicates that they stand on the

same surface, and thus links them together.

The size of the space between them, being no

^°" ^^ more 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-

FREEHAND PERSPECTIVE

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) i*

Corr^c//<ys ^^ ^^^ ^^ SVLGII Si CaSC.

^^^' ^^ 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

Fig ^^

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

CYLINDER AND CYLINDRICAL OBJECT

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
perspective sixths. That is, they appear successively smaller as they
recede 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
draw the long diameter of each of these two ellipses where it
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

FREEHAND PERSPECTIVE

Fig. 2J.

any fall at the same height as one on the cylmder, 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. 2-i) 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 yl
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 the 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
16

CYLINDER AND CYLINDRICAL OBJECT

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 inarl^: 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

AN OBJECT ABOVE THE EYE AND THE
CONE PRINCIPLE

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
placed.

Dra-wing the Object.
I — Proceed as in the
\ previous exercise.
\ Observe that the slop-
\ ing side boundary
I lines of the shade join
I the ellipses in front
\ of their ends. The
\ flaring shade is like
I the lower half of the
rose jar reversed. Its
smaller part is far-
thest from the level
of the eye, as was the
base of the rose jar,
and we therefore do
not see half way
round it. To make
this clear, hold a cone^
in various positions

Fig. 27

^ One may be made from paper (Fig. 28).
18

OBJECT ABOVE THE EYE, ETC.

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
oblique 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 ivhen 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 ivay round
tvhen 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
A CREAM JUG

HE Model. — Provide an object similar to the cream pitcher

here shown (Fig
should be made.

30), from which the student's drawing
If inexperienced he will be helped by

„____ first making a copy

, from this example.
I 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

t —

Fig. 30

A CREAM JUG

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 may be 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 the ends
of these profile curves rest 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, of the greatest diameter.
Fig. 34 q^^^ ^j ij^^g^ profiU curvcs, shapcd 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

FREEHAND PEKSPECTIVE

Fig. 35

Fig. 36

according to position, from a profile vieiv 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 (0
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

A CREAM JUG

the other with the
A

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
apex down (B, Fig. 38). See Chapters
ly 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

ti\f !<• ! of profiles similar to those for
h'^^i placing the handle and spout.

Fig. 38

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.

\^'f

Fig. 39

Chapter VII
A TIME STUDY

HAVING- 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 the
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

A TIME STUDY

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.

2.5

Chapter VIII
A GROUP OF CYLINDRICAL OBJECTS

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

Fig. 41 effcct of a compositiou, 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
group look best. Little trial or
*' thumb-nail" sketches (Figs. 42
and 43) should also be made to
determine the best arrangement.

In Fig. 44, for example, we
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

A GROUP OF CYLINDRICAL OBJECTS

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.

FREEHAND PERSPECTIVE

Drawing the Group. — In drawing this exercise, observe that
though the bottom ellipses of the two objects are at the same
level the nearer appears slightly rounder (Solutions of Problems,
p. 162). 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

A ^--:5^"~~-v, ^® ^^ ^ ^^^^^ passing through the perspective (that is,

^V^^^S^w actual) center (o) of this ellipse, as were the handle

'y^^-- — ^^;\>1 and nose of the pitcher in Chapter VI. The

J3 ^,j'^2_^^ cover is arched (B in Fig. 45), so that it conceals
J— — i. the back of its elliptical edge. This arched shape

PROFILE VIEW .,.., • ^ o o •

6How.NGARCHOFcovtR jg distuictly socu 111 tiiG 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

CYLINDRICAL OBJECTS GROUPED
WITH FRUIT

A N example of grouping is here given in which part of the
/\ group is cut by the margin, while the apples illustrate
-A- ^ 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

FREEHAND PERSPECTIVE

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

PESWtCTIVE.

X-Y roRESHORTENED.

Fig. 47 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

A GROUP OF OBJECTS FROM MEMORY
OR INVENTION

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 YIII, and using the best of these in his

:-;;>:>^,---fC==CXAM\\

_ -..J

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 ivithout the object in view.

FREEHAND PERSPECTIVE

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.
XXVIII.)

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
?i 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

OBJECTS FROM MEMORY

Chapter IV. Note the foreshortening in its details, and how the
Hnes 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

THE CYLINDER CONE AND BALL GROUPED
—A PROBLEM FOR ORIGINAL STUDY

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 tvitliout 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-
' ncTUBc'^ Pu^^4E )' not be expected that any object should be drawn
' ' ' until opportunity has been given for its thorough

^^^- ^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 4'' in

CYLINDER CONE AND BALL

diameter. The group is to be drawn as if resting on a surface
which is twice 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

THE STUDY OF STRAIGHT-LINE
OBJECTS

A Book with Back Parallel 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

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

^ It may not at first be realized that the ends of the cover are horizontal, as M'ell 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
position.

Fig. 52

But as we drop it slowly

STRAIGHT-LINE OBJECTS

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 IV. If (as with the house in
Ch. II) , a pane of glass were standing in place
of the imaginary picture plane, a tracing of
the cover on that would be a true perspec-
How to draw on the paper such a
Stand the pen-

FiG. 53

V tive of it.
perspective is our problem.

Fig. 54

cils against the nearest corners 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 toivard 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 Book 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. 55

4^zm

FREEHAND PERSPECTIVE

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

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

STRAIGHT-LINE OBJECTS

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 Jceej) the book horizontal, the point toward ivhich 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 booh
covers (like the cylinder top in Ch. IV) appear
foreshortened according as they approacli the eye
level, whether above or below it (Figs. ^6
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

FREEHAND PERSPECTIVE

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 Fig. ei

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 until the fir^st two converging ones ivill
appear to converge ivith 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

STRAIGHT-LINE OBJECTS

ill 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
^^^- 6^ than the one before it,

and the cover ends all falling in the same con-
verging lines, until a point would be all that could
rei3resent 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

FREEHAND PERSPECTIVE

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
DRAWING THE BOOK IN TWO POSITIONS

HE 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,
be understood by holding the cylinder horizontally

Fig. 67

This will
(Fig. 69).

Fig. 68

FREEHAND PERSPECTIVE

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

of seeing. The farther these lines are from
coinciding with this direction (in this case
to right and left) the more apparent is their
curvature.

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 width. 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
THE BOOK WITH A CYLINDRICAL OBJECT

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
picture.

FREEHAND PERSPECTIVE

PUN OF AJ

9HOWINO

PLANE

TO BOOK

This brings us to reflect that whatever the number of objects
we include in our picture, it is always drawn tvith 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 equall}^ 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
book.

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

Fig. 71

PlANOrX

OSJECTSX
PLACED \

M
^

/the SAKIE
/with the bOOK
/at TKt RiOHT.

TIltOEMTIM
13 MOVED, \

/of the PICTUOC
/and TMt PICTvnC

TDTTIB '

/loNOSH PARALLtl.
/SOOK.

Fig.

Fig. 73

THE BOOK WITH A CYLINDRICAL OBJECT

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
observed.

Chapter XV

THE CYLINDER AND RECTANGULAR BLOCK
- A PROBLEM FOR ORIGINAL STUDY

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.

_-/6,»

<■

-• ^

i
si

,-.!

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.

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

Chapter XVI

THE FURTHER STUDY OF STRAIGHT-
LINE OBJECTS — A CUBE AT ANGLES
WITH THE PICTURE PLANE

THE Model. — For this study make a cube, four inches on a
side, from cardboard cut as in the ilhistration (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,

16;^-

the front, ^,

begins

to be turned
away, ceasing to be
parallel with the pic-
ture plane, and tend-
ing toward coinciding
with the direction of seeing

:^i

Fig. 76

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 frout (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

FREEHAND PERSPECTIVE

Fig. 78

Now turn the cube slowly back to the position of Fig. 78,
and with the stnngs 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. Wo 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, «/, 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. 80

£yc Cc^&l

// '

Fig. 81

STUDY OF STRAIGHT-LINE OBJECTS, ETC.

ArjAloT A CufiE.

A.B,AA<oC ARE.
TOO/TEEO Fob
Tut WIDin OF

X; AND D, E

ANO F ABE
AJOT yiEEP
ENOUOM FOB
THE roRE-

OF Y-

G-CoWECTIeH

OF A, 8Y
CHANGINO THE
WIDTH OFTHa

THEomecnoN

OF THE VANI/H-
^ ec f?ioriT.

B— Correction

OF A, BY CMANO-
IklG THE _fuANTOF
THE VANI/HINO

eoosf, whem the
wicth of the ■

51 D£.^ 1; FOl/NOTO
flS «I&MT.

Fig. 82

foreshortened. As x widens, and its horizontal edges (A and B)
seem less steep, the other side narrows, and its horizontal edges
(E and F) appear more steep. Steep-
ness of the horizontal edges, therefore,
goes ivith 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 Toward 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
point.

Fig. 83

FREEHAND PERSPECTIVE

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 Mm as a
line, hence it is always drawn as a line.

Chapter XVII
THE CUBE IN TWO DIFFERENT POSITIONS

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

r-- —

Fig. 8-1

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

5«i

FREEHAND PERSPECTIVE

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- vuv^ or a plah of b
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. IV). 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)

DlAGPAM
SHcM/lNG A
CONVENIENT
ORDER FOR
DRAWING THE
LINES OF THE
CUliC, AMD OF
RECTANGULAR
OBJECTS IN GENERAL

Fig. 87

THE CUBE IN TWO DIFFERENT POSITIONS

give the order iu 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
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). 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

Fig. 88

Fig. 89

the paper,
sh o win g

Fig. 90

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 VP2 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).

Fig. 91

FREEHAND PERSPECTIVE

(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

^ ? ' \ lui ^^^® ^^ directed, and proceed as

^L^^^,^^.' ill with the first cube. In this case

I "//

Fig. 92

yP2 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 oivn vanishing
point.

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-

5()

Fig. 93

THE CUBE IN TWO DIFFERENT POSITIONS

rieuce 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

A BOOK AT ANGLES TO THE PICTURE

PLANE

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.
XVII). In studying this position begin with the book directly in

'«aaya<ii>^DeaaiCfflagia»aas:!Wai/.j;

»rn*irmj«»n'Wt(i*(-'<^icnr2:3r:iHSH|

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

A BOOK AT ANGLES TO PICTURE PLANE

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
spots.

FREEHAND PERSPECTIVE

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 be draivn as they actually are, or vertical.

Chapter XIX

TWO BOOKS AT DIFFERENT ANGLES TO
THE PICTURE PLANE

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,

FREEHAND PERSPECTIVE

EYiUvU-

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

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 ^-^vf
VP4). Its length ap- ^
pears lessened, and its
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 when Imes cease to he parallel, their
vanishing points are different.

Fig. 97

•*• TjCf A J<,

Fig. 99

Chapter XX

THE ACTUAL CENTER OF THE CIRCLE AND
MEASUREMENT INTO THE PICTURE BY
PARALLEL LINES

PRELIMINARY Study. — Does the eye see half way round the
cylinder I 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
mw as the greatest dimension. As shown in Fig. 103 this appar-
ent greatest dimension (A B) forms the long diameter of the ellipse

FREEHAND PERSPECTIVE

Showing apparent middue from

FffONT To BACK. OR LONG DIAMETER

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
details.

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

R-ANOFABCVC,
5HO"0t/lN& THE

ACTyAu Place

OF THE LON&
DIAMETER, A B

ACTUAL CENTER OF CIRCLE, ETC.

Geometric diagram.

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-
tivelij 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

1 Students of geometry will recognize in this the problem
of dividing a line proportionately by means of parallel lines cross-
ing a triangle.

5 65

Perspective representation of above.
Fig. 103

Fig. 104

FREEHAND PERSPECTIVE

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 GJ 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
G-J 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: Whatever meas-
urements can he obtained geometrically by the use of actually parallel
lines, can be obtained in perspective by the use of perspectively parallel
lines.

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
BOOKS WITH A CYLINDRICAL OBJECT

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

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

FREEHAND PERSPECTIVE

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
THE STUDY AND DRAWING OF A HOUSE

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-

FREEHAND PERSPECTIVE

8!4-^

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

THE STUDY AND DRAWING OF A HOUSE

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
other.)

The sloping sides of the gable may then be drawn
to the house corners, and the ridgepole to VP2. 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.

c ^

E.ND View
Fig. 113

FREEHAND PERSPECTIVE

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 Gr) 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. Hoiv to find that point is the
question.

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 uptvard, or in a vertical line
passing through VPl. This continues until the cover becomes

Fig. 114

THE STUDY AND DRAWING OF A HOUSE

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 l)y the visible ends of the hooJcs 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 hook ends in this case will vanish in this line,

Fig. 115

FREEHAND PERSPECTIVE

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 (P 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 on 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

I E

Fig. 117

THE STUDY AND DRAWING OF A HOUSE

BAC»\ VIEW

End View

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
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 VP2 fornas 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 0, 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

FREEHAND PERSPECTIVE

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

-D-

DD D DD

fEONT VIEW

Fig. 120

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-

--4- IN

Fig. 121

THE STUDY AND DRAWING OF A HOUSE

iug 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

VPl. 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),

FREEHAND PERSTECTIVE

r^w^ra^s^

a!&

A- Plan or Cmimney.
g- "Profile smowino orna-
mental. BAND AT TOP.

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 YP2, 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, cutting 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

THE STUDY AND DRAWING OF A HOUSE

<»

.^B

/■^

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 O 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 VP2, 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

FREEHAND PERSPECTIVE

U 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
determined.

Chapter XXIII

A BUILDING FROM THE PHOTOGRAPH OR

A PRINT

HE 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

Fig. 128

next work, which should be the drawing of a building or part of
one from a print of his own selection.

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

6 81

FREEHAND PERSPECTIVE

Fig. 129

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.

In Fig. 130, on the other hand,
the long,
low mass
of farm
buildings
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 Jeml
of the eye must he deter'
mined and marked, and the vanishing points of the principal sets

'H'iTi

Fig. 130

^■■^■^■KT^^

[ ^'^% ffi JVMBKl

f ■

k "1

BHHr

^^^K|s£|^|u^^

Fig. 131

BUILDING FROM PHOTOGRAPH, ETC.

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 dearly 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

study.

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

FREEHAND PERSPECTIVE

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
lines.

Chapter XXIV

TYPE FORMS HELPFUL IN UNDERSTAND-
ING THE HOUSE ^ — THE SQUARE FRAME

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.

iaia»soKi^^Viz:xsSiS¥i^»^¥iSs:^->^S:^:S:S»ij,a:iVSi^Ui,

Fig. 134

FREEHAND PERSPECTIVE

A ,/

TRONT

Fig. 135

Fig. 13G

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 ne:!rt 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 VPl. 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 [z) 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 z^ D is drawn to YPl. The
lower inner edge (F) at the back is parallel to the outer thick-

Fig. 137

Fig. 138

TYPE FORMS HELPFUL, ETC

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 Gr 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.

Fig. 139

Chapter XXV

THE SQUARE PYRAMID AND SQUARE

PLINTH

HE 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 [xij
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
x-y-3 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 stands on the plinth,

SQUARE PYRAMID AND SQUARE PLINTH

T
i

A. Pattern

FOR MAKING
PYRAMID

«\rith 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, one

perspective sixth

must be marked

from each end

on line AB (Fig.

143). To get

these points (1 and 2) a diagonal of the side

ABCD can be used (as in Ch. XX). The vertical

edge, AC, being unforeshortened, is first divided

into six actually equal parts. Lines from the up-

B. Pattern for

MAKINO PLINTH

16 in.

Fig. 141

B. [Plan

Fig. 149

EVg LIVCl -

ii£7

per and lower division points

(8 and 9) to VPl transfer these

divisions proportionately 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 (O)

will divide f^^- i*^

AB into perspective halves at 3, when each
half can be divided into thirds by the eye.
One method can loe used to prove the other.
From points 1 and 2 draw lines to VP2. Where they cut a

diagonal, as AE, will be two corners (4 and 5) of the pyramid.

Geometric view or

SIDE piF PLINTH .
Fig. 144

FREEHAND PERSPECTIVE

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 convenient
to use, hence from A the vertical line is con-
tinued, and four times AC is measured on it,
giving AG 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 actually horizontal line
above AE, that is, a line parallel to it, conse-
quently vanishing to the same point. There-
fore continue the diagonal AE to the eye level,
giving VP3, and draw^ the parallel line from
G to VPS, which will mark on the vertical
from the desired perspective height at x.
Complete the pyramid by drawing its oblique
edges to the corners of its base.

Applications. — The difficulty of making

Fig. 145

a church spire or a tower (Fig.
145) " stand true " will be readily
recognized. The use of the di-
agonals (AB and CD) will aid
in placing its axis and apex.

Any upright rectangular ob-
ject with equally sloping sides is
easily constructed on the same
principle, as the grape basket in
Fig. 145a. If such an object is
leaning, the meeting of its cor-
ners continued can still be used
as there shown.

6PAPr-BA«K»T

Fig. 143a

Chapter XXVI

THE SQUARE FRAME LEANING ON THE

RECTANGULAR BLOCK — A PROBLEM

FOR ORIGINAL STUDY

THE Models. — These have ah^eady been described on pages
48 and 85, respectively.
Position — The block rests on one long face, with its
square ends making angles of sixty degrees with the picture plane.

Plak

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 XXFII

Fig. 1-17

CYLINDRICAL OBJECTS WHEN NOT
VERTICAL

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

^ 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. To study

them in other positions begin with the
cylinder model held horizontally, with its
middle on 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

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. They will continue

Fig. 148

CYLINDRICAL OBJECTS NOT VERTICAi.

If WJ I I-

UMmMliliiuif

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.

150)
will
help

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

.iiiMim

LUJlU

Fig. 149

Fig. 151

FREEHAND PERSPECTIVE

cTBKps.

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 tlie 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

Fig. 152

Fig. 1o4

cylindrical ^^^- ^^^

object are actually at right angles to its axis, they

ivill appear so whatever the position of the object.

Consequently, cylindrical objects alivays appear

symmetrical.

Tests. — A drawing of such an object may
be tested by turning it to bring the object in
question vertical (Fig. 156, Ch. XXVIII), 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
A GROUP OF FLOWER POTS

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. ise

the group.

Chapter XXIX

THE CIRCULAR FRAME IN A SQUARE

FRAME

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

^ experienced.

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

Fig. 1.57

CIRCULAR FRAME IN A SQUARE FRAME

1 -

-c

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

Front view.
Fig. 158

Fig. 159

axis

,.,.^

Top AS SEEN SY EYE,

lookino at a .
Fig. 160

the object (Fig. 160),
line ivill 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 ij 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

Fig. 161

FREEHAND PERSPECTIVE

right angles to x ?/, and through its apparent middle from front
to hack; 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
perfect.

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. Or four points for the
inner circle (e, e',/and/. Fig. 161a) may be found by the method
given for concentric circles in Chapter XX. Thus points E and
F are set off their proportionate height on the nearest vertical
edge xy, and lines vanished from them to VPl. Where these lines
cross AD are e and /, the two points for the inner circle corre-
sponding to A and D in the outer circle. For the points on its
horizontal diameter (e' and /', corresponding to C and B in
the outer circle) vertical lines are
drawn from the points (1 and 2)
where the vanishing lines cross the
diagonal.

The inner ellipse is then con-
structed as was the outer one. ^j^ Fig. leia
' 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
convergence measures the smaller length of the desired back el-
lipse, which lies with its ends in these lines as does the front one.

/>«

^ k

CIRCULAR FRAME IN A SQUARE FRAME

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.

^° J-

A- Showing application of methods.

B. WRONO. INNER BACK ELUPSE
MADt Too 5H0RT, AS IF ACTUALW
SMALLER, INSTEAD OF A DUPLICATE.

Fig. 162

Chapter XXX .
A ROUND WINDOW

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
k from a building or

photograph.

Having drawn
^'' ' the straight-line part

i i > of the exercise, the

round window is
'^-- " ( ^\ \\^ next to be consid-

^ ered. This is actu-

ally a cylindrical
<^P'' • ^ opening in the wall.

Being above the eye,
Affc.., .''^M^''" the circles of thewin-

dow appear as slant-
ing ellipses like the
cylinder ends in Fig.
.^i, 149 (Ch. XXVII).

^''^ ^ To make sure that

the slant of these
ellipses shall agree
with the straight-line
[ part of the building,

U the axis of this cylin-

,j X drical window is

'' ^'-h used. This axis is

S^Ji actually at right an-

^^^- ^^^ gles to the- wall, and

100

A ROUND WINDOW

is therefore parallel to the lines already vanishing in VPl.

Hence the apparent middle of the window ellipse (O 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,
and through it a vertical line (EF) and a
horizontal one (GH, 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.)

A. FRONT VIEW

Fig. 164

'-^^

PERSPecTive
Fig. 165

101

f*^

)

f
\

Chapter XXXI
THE CLOCK A PROBLEM

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 proJolems 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
chapters.

Fig. 166

102

Chapter XXXII
THE ARCH

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

the

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,

103

FREEHAND PERSPECTIVE

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
used.

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 Viemv
Fig. 169

104

Chapter XXXIII

INTERIORS

A ROOM PARALLEL TO THE
PICTURE PLANE

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

the picture plane, and the eye level a little less than two thirds
of the way up. If desired, the v^indows, 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

105

FREEHAND PERSPECTIVE

(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 w^ill
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
model.

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 oi- inventive drawings should be advised before study from
a room. But in this case we meet a subject of which all have something 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.

106

Fig. 171

ROOM PARALLEL TO PICTURE PLANE

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) can 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 OYPl 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
VPl.

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 is a line at 45° with its sides. By lines at 45°,

107

Fig. m

FREEHAND PERSPECTIA^E

distances into the picture can be measured equal to others paral-
lel to the picture plane (Fig. 103, p. 65). In Fig. 173 (and per-
spectively in Fig. 172) AC is made equal to AB by drawing BC at

c 45°, or parallel to the floor diagonal.^

I /"'^ I ^^>/^| 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-

~SSZSZZZZZZZZZZZZZ2ZZZZZA

iM—-\'

\^J^

,0^

lei to these edges.

Fio. 173

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.

108 , ^

ROOM PARALLEL TO PICTURE PLANE

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

* See Chapter XLII for fiu-ther coDsideratiou of a room parallel with the picture plane.

109

Chapter XXXIV

INTERIORS CONTINUED — A ROOM AT
ANGLES TO THE PICTURE PLANE

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
■j::=^e-'- v-'^" ^^-^ -s 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 very un-
equal angles with the
picture plane (B in
Fig. 175). Both
sides and the top
and bottom are now
foreshortened and
their horizontal lines
vanish respectively
in VPl and VP2
(Fig. 177).

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

110

Fig. 174

ROOM AT ANGLES TO PICTURE PLANE

relation of the subject-matter within that space to the picture
plane. Thus in the last study, 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 (A in Fig. 175).
Therefore we could
not, without ab-
surdity, vanish the
lines on this wall
in either direction
(A in Fig. 176),

^ Picture

Plane

\ z

\ °

\ \-

\ ^

■ui ,

\ ^

u /
•^ /

\ <

^ /

\ 1^

o /

\ t-

Fig. 175

still less in both

(B in Fig. 176).

The only way to

give a truthful

impression of the

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. 171) 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, xy, which was
before parallel to the pic-
ture plane, now recedes
from it. Hence the height
of the room at the corner,
being farther into the pic-

FiG. 176

111

FREEHAND PERSPECTIVE

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

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

112

ROOM AT ANGLES TO PICTURE PLANE

that sixty degrees of the horizon circle is the most that
be taken, and that usually thirty degrees is better. The
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, — Tlie picture plane is dif-
ferent for each neiv picture selection.

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

3. the pa[zt selected fob a picture .
Fig. 178

should
rule is

118

Chapter XXXV
FURTHER STUDIES OF INTERIORS

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-

i ~ ' r<,5\

114

Fig. 179

FURTHER STUDIES OF INTERIORS

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 pur-
posely quiet in detail to aid in con-
centration of interest.

Fig. 180

Fig. 181

(Figs. 180, 181,
artistic value of
principal sets of

and

the

The chair in the
extreme right of
the fore-ground
is subordinated
also.

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
selection. Avoid equal angles in the

should be drawn

vanishing lines. If both

at forty-five degrees, or nearly that, th^

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. 181.) A subject may

115

Fig. 183

FREEHAND PERSPECTIVE

Fig. 184

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 such a relation to the
margin or inclosing rectangle that they
do not conspicuously point to the picture
corners. The rectangle which bounds a
composition is an orderly conventional
shape, and
hence unno-
ticeable, leav-
ing the atten-
tion to be
concentrated
on the picture.
Lines to its corners direct the atten-
tion there, and defeat this end.

Vignetting. — It is not necessary p^^ jg^

that an inclosing margin should always be

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

116

Fig. 186

FURTHER STUDIES OF INTERIORS

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

117

Chapter XXXVI
A CHAIR

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

t 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

J having planned

its place on the

118

Fig. 188

A CHAIR

L I \

PLAN OF 5CAT

paper and located the eye level, the next step is to take the direc-
tion of the front and side of the seat (lines A and B, Fig. 190).
Leaving line A for the present we carry line B out, obtaining
VPl. The feet, being on the horizontal floor, fall in lines vanish-
ing to the eye level — on the right to VPl, and on the left in lines
E and F. But as the chair sides are not parallel, line E, being
turned away from the picture plane more than F, appears steeper,
and its vanishing point, VP2, is nearer ;

than VPS, to which line F vanishes. |

The near side of the seat, line A, being
in the same vertical plane with line E
(directly over it) must vanish where it
will cut the vanishing trace of this ver-
tical plane, which is a^ vertical line
through the vanishing point of line E.
(Compare the roof, p. 72.) Carrying out
line A, therefore, to this vertical trace
gives its vanishing point, OVPl, to
which we vanish line Gr, which is parallel with A.

In the same way, line F, vanishing to a point on the eye level
a little further away than VP2, gives the location of a vertical
trace for 0VP2, in which the other (invisible) side of the seat
must vanish. The crossing of these invisible seat edges (H and I)
and of the lines C and F aid in shaping the further leg.

raoNT vitw

Fig. 189

Fig. 190

As the seat is evenly slanted, 0VP2 is on a level with OVPl.

All edges parallel to the front and back converge to VP2.
The curving front of the seat and the curves of the back are
drawn so their ends rest on a line to VP2. A rocking chair is

119

FREEHAND PERSPECTIVE

simply a chair (A, Fig. 191) placed on rockers (B, Fig. 191). The
weight of its back generally tips it backward when at rest, hence

the usual appearance of a rocking chair is
that of an ordinary chair plus its rockers.
In B, Fig. 191, lines A and B, which above
are actually horizontal, here actually in-
cline downward at the back; and must
vanish to a point below the eye level (like
A, G and H in Fig. 190). The seat and
arms are more foreshortened, and the back
less erect. But the horizontal lines C, D,
E, F and their parallels, remain actually
horizontal, and vanish to the same point
as in A. The directions of the slanting
back, the legs and the arm supports should
be carefully taken, remembering that their
ends and ornamental details (as G, H and
D) fall in lines to VP2. In sketching
straight guide lines for the rockers, recall
that they are farther apart in front, to agree with the sides of the
seat. The lines where the cylindrical rungs enter the legs are
actually modified circles, hence they will be seen as shapes modi-
fied 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.

Fig. 191

120

Chapter XXXVII

THE HEXAGONAL PLINTH IN TWO
POSITIONS

HE 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.
The Model. ^

This can be

made from card-
board according

to the diagram

(Fig. 193). Tor , ' \.__.^.,_.,„__^^'

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

plane.

The Greomet- ,, , ^ ,

ric Hexagon. — ] " . | \/

This should be

constructed

first. Divide

the line AB

(Fig. 194) in \ --^^.^___^_._ ^ I

halves, drawing f,^ 192

121

FREEHAND PERSPECTIVE

■•^ 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 (AO 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 (AC and BD) and its middle (0) divide
this diagonal (EF) into four equal parts at x, 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

122

Fig. 194

HEXAGONAL PLINTH IN TWO POSITIONS

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
plinth.

VPl

VPI

fVf LEl/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 VPI and VP2 if the drawing
is correct.

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

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

123

FREEHAND PERSPECTIVE

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

CYE LEVEL

iPlOTVRtl PlONC
R- PI-AM or HEXAGON 8£EN IN A.

Fig. 196

A. Wrono. Lines I, 2, 3

AND 4 DO NOT CONVERGE
TO THE EYE LEVEL.

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

jestimated. 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
Oy^ because a little nearer. Hence E:c should be set off a little
larger than xO, and y¥ a little smaller than Oy. Similar cases,
as the cylinder top in Chapter IV, 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,

124

B. Wrong-, lines J, 2,3

AND 4 ARE TOO &TEE.P.

hexagon appears tiltep.
Fig. 197

HEXAGONAL PLINTH IN TWO POSITIONS

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
slight.

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

125

<mM»"g "'""' « " >'" '

ni rm

INTERIOR WITH A TILED FLOOR

HE plan of this floor is shown in Fig. 199. 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). As with
the previous hex-
agons the rectangle
ABCD is drawn
first. And since
much depends on
the correctness of
this first rectangle,
it is worth while to

■iW ■jatnuttf iirii<».j

Fig. 198

126

INTERIOR WITH A TILED FLOOR

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.
Thus these tiles are three and a half inches
on their edges, making the rectangle width
actually half the height of a seven-inch step. In
the drawing this width, being foreshortened as

much as the

steps, ap-
pears only
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

Fig. 199

Fig. 200

width; or (what is the same thing) one eighth less than the

height of the step. Being slightly turned e^pView of^tep

away, it will appear a very 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. 201

127

Chapter XXXIX
THE HEXAGONAL PRISM AND FRAME

HIS exercise may be drawn from the objects, if they are
at hand. If they cannot readily be had, the drawing
juay 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-
sions.

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-
ture plane. The

Fig. 202

128

THE HEXAGONAL PRISM AND FRAME

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 its 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 VP2. Vanish
the other long edges with it, and set off on the one first drawn
its apparent length (AGr). 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 Gr, 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 fi?ame
Fig. 204

XVI.

This method is given ia a slightly different form under Solutions of Problems, Chapter
9 129

FREEHAND PERSPECTIVE

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 C^ 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. If one fifth of CB (CO in
Fig. 205) be found, and three quarters of
this {Cy) be taken, it will serve the purpose.
Mark the same distance from B up (point
2). From these points {y and b) vanish the
horizontal lines of this inner hexagon with
their parallels to VPl. The corners of the
inner hexagon 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
OVPl.

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.

130

Chapter XL

THE TRIANGULAR PRISM AND FRAME —
PROBLEM FOR ORIGINAL STUDY

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.

A TEIANQULAR FACE
OF FPAME

B. 5IC)E VIEW
OF FPAME

Positions. — The objects are
placed four feet away, and one
below the eye, and are 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. — The 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.

C END OF PPISM

d 51 de view of pei3m
Fig. 206

131

Chapter XLI
THE STUDY OF PARALLEL PERSPECTIVE

Fig. 207

ROM 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. 208^ 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.

132

Fig. 208

THE TABLE I5WB0N0

Fig. 209

STUDY OF PARALLEL PERSPECTIVE

COCTSECTION

Fig. 210

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 reSviSn orp™E plane

TO TABLE. 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.
In Fig. 212 this is shown by the conver-
gence of the lines at right angles to these
surfaces ; their vanishing point being a little
out of the center of the picture. Yet if
all these vertical surfaces are drawn in per-
spective (A in Fig. 212) the result is mis-
leading or impossible, and the eye protests.
But the drawing is perfectly 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 of the upper drawer is cut out, the

133

'i^''^^/'^//////'/.

PLAN OF
INTERIOR
jHOWN IN

GuRE ais..

Fig. 211

FREEHAND PERSPECTIVE

A UntpuE drawing

OF VIEW FROM X, IN P(./\N

B CORRECT DRAWING
or TME SAME VIEW.

Fig. 212

remaining 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 to right or left of the house fronts.

In this view, although the beholder has
passed on to y, the conditions are still
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 the view
must be drawn as shown in Fig. 216.

It may therefore be concluded, that
171 any picture having a domhiant imrt
parallel with the picture plane and conse-
quently clraivn in its true shape^ all por-
tions of that picture which are parallel tvith the picture plane must
also he drawn in their true shape.

134

i^^^crr

■<p

.-4P-

# J ^^

Fig. 213

STUDY OF PARALLEL PERSPECTIVE

Also, even such dominant parts as are not quite parallel
with the picture plane must
be drawn in their true shape
in certain cases ivliere draiving
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
^een.

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
subject.^

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

Plan OF the

^TEEET 6HOWN
IN PEESPECTIVE

Fig. 214

-Ti

VIEW or JTEEET FBOM Y.

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

are therefore far from satisfactory.

135

FREEHAND PERSPECTIVE

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.

view of coenee fbom y
Fig. 216

136

Chapter XLII

A STREET FROM THE PHOTOGRAPH

THIS exercise (Fig. 217) may be drawn first if judged best,
noting carefully the changes made in rendering from
the photograph shown in Fig. 218. The student should
then select a print

street and

drawing

of a
make a

from it. All
sketches should
be thoroughly |
thought out, hav-
ing the level of
the eye carefully
placed, and all the
vanishing points
located, either
actually or men-
tally. It will
probably be neces-
sary to correct
some distortions
of the camera (see
Ch. XLIII).

This drawing
should be followed
by studies from the
street and from
memory. Notes
and sketches made

137

Fig. 217

A STREET FROM THE PHOTOGRAPH

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

Fig. 218
Entrance to the Ponte Vecchio, Florence

188

Chapter XLIII

EXCEPTIONS TO THE USE OF THE FLAT
PICTURE PLANE

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
picture.

139

Fig. 219
Cloisters of the Monastery of San Martino, Naples

FREEHAND PERSPECTIVE

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
to say that the aim of freehand

^^=-

—

. r

1 I

c1l

Jit*

Cylinders as found by mechanical perspective,
Side cvlindebs vntsue as REPREseivTATiONa.

Fig. 220

perspective is the draiving 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

A. Plan
Showing

Vi>E OF5PEC-
JAL PICTURE
PLANES- THE
EQUIVALENT OF
THE CylINDRi-
CAU PICTURE.
PLANE .

EYE

objects, however placed, should be

B. Appearancp of group at a, drawn by

\}i>e OF special -THATiS, CYHNDRlCflU, PLANED

FiQ. 221

140

EXCEPTIONS TO FLAT PICTURE PLANE

^ " Side viev/

' Showing the u4E or

PICTURE PLANES INCLINED
FROM THE VERTICAL-THftr 15,

THE Sphericau picti;re pw\n&.
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 onlj^ 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 ohjecfs 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

141

•''^•^iDI§J^

Fig. 223

FREEHAND PERSPECTIVE

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 ivithout noticeably
moving the eyehaUs (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.

142

Chapter XLIV
SHADOWS

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. 221

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,

143

FREEHAND PERSPECTIVE

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 Lave
traveled such an incon-
ceivable distance, that to
us they are parallel, as
are the paths of fall-
Fi«- 225 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

first:

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, line C, the shadow of

its vertical axis will be parallel with the shadows of the vertical

line AB on the box floor, and on the horizontal book cover.) The

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

144

Fig. 226

SHADOWS

shadow (F^^) of the vertical book corner (FD) will be parallel
with these lines. If we push the box back or forward
they all change direc-

EYE LEVEL

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 Fig. 227

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, on both the back and the floor of the
box. Also in Fig. 228 the shadow (b-e) falling on the hori-
zontal book cov-
er from the hori-
zontal box edge
(BE) is parallel
to the shadow
(d-g) falling on
the horizontal
box surface from
the horizontal
line Da.

Wemaythere-
FiG. 228 fore say that the

shadows' of actually parallel lines ivill lie actually parallel to each
other on the same surface or on surfaces parallel with each other.

But in perspective, parallel lines vanish; and if they are

horizontal lines they vanish to the level of the eye. We should

therefore expect these parallel shadows to also vanish thus, and we

find they do vanish, in the same manner as any lines or objects.

10 145

FREEHAND PERSPECTIVE

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 pm'allel to the
object or line casting the
shadow, the shadow will 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 (h) of point B.

The evident truth is that the direction of the light-ray from any
point in the ohject 7narhs the same point in its shadow. 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

146

Fig. 230

SHADOWS

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 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-^) 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 Bb and E-e) appear to converge to this vanishing point.

Hotv to find where the light ray mid the receiving surface
meet is the iiext consideration. In the
case of D-^ we had a vertical line, DC,
and its shadow, C-^, cut by the light-ray,
T>-d. From E we can imagine a vertical
line, similar to DC, 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 MJ 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 Q-d (vanishing in VPS) cuts the
light-ray in the desired point, e. This is essentially the way most
shadow points are found: — hy a vertical line from the point on the
object to the receiving surface^^an^ 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 maybe 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
several vertical planes as above mentioned.

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

147

FREEHAND PERSPECTIVE

ou the back of the box in a slantmg Hue 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 h, vanishes on the floor of
the box to yPl till it reaches the box edge in point i. The shadow
must start at H, hence H-i is the line in question. Many shadow
lines can be found thus, — 1)1) 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-

^.■5HOWiNOr A
Veeticau plane
at eioht anoue5 to

CENTER or^hAPOW

Perspective

or ABOVE.

FACE VIEW OF
VERTICAL PART
OF 5HAPOW.

Fig. 232 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, vieived from above it mahes right angles ivith the light
rays (as shown in A, Fig. 233), the shadoiv on the vertical surface

148

SHADOWS

A. JnowiNG-
VERTICAL Plane.

ATUNEQOAU ANGLES
TO 5MAD0\A/ CENTEE

B Pei?spe:ctive of above
— shadow distorted

Fig. 234

also ivill he actuaUy symmetrical; though it may appear fore-
shortened (B in Fig. 233), 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, y, 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 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 drawing the
horizontal hat-pin in Fig. 227 and the
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

149

FREEHAND PERSPECTIVE

eye, though it could be constructed by 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 shadow line on the book cover. This line
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 shadoiv is itself oblique. A vertical line from D to the

SHADOWS

A. Plan, J MOWING

T?At)IATINQ RAVJ.

(%P^ POINT O, THE
V""^ CrAS TLAMC

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
D.

Shadows 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, hid of the infi-
nite plane containing the roof. (It therefore can be drawn as long
as needed, being really infinite in length.) So we have only to
draw the trace from OVPl to VP2, and mark 0VP3 on it verti-
cally over YP3, 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

151

Perspective

OF ABOVE

Fig. 238

FREEHAND PERSPECTIVE

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
D gives d where
cut by a line on
the ground di-
rectly under it
^'^- ~^^ (from through

E). The shadow of Gh falls on the floor in the direction of
o-H till it reaches the wall. On the vertical wall, the shadow
of the vertical GrH is also vertical. It is ended by the light-ray
from O through Gr. The shadow of the edge GI 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

152

SHADOWS

extends in an almost opposite direction from that of the chair.
These shadows are produced by the diffused dayhght 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-l) and c-d vanish with AB and CD, while points a
and & 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.

153

Chapter XLV
OUT-OF-DOORS WORK

A LTHOUGH the same perspective principles apply to out-
/\ of-doors work the conditions of the study vary, and some
-A. ■■^ 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

51OE VIEW or i-iousr SMOwiMG 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 x) 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 al) and ccl, and on picture
plane 2 by a'l) and c'cT. 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

154

OUT-OF-DOORS WORK

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

.aJ^^^.,,

SHAPE OF HOUSE AS
5CEN FROM X

B SHAPE OF HOUSE AS

SEEN reoM Y,

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 large the 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 ivholJy 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

155

FREEHAND PERSPECTIVE

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
place.

TJie size of objects ac-
cording to their distance
iiito 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
heisfht 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 ^^^- -^^

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

156

OUT-OF-DOORS WORK

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

-7-x

A. Perspective-

SHOVl/lNG STAKE
FORESHORTENED

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.

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

157

FREEHAND PERSPECTIVE

It is therefore evident that in case of reflections on a horizon-
tal surface J 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
plane.

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

WRONG. Reflections not ver-
tically UNDER OBJECTS RE-
FLECTED. WATER APPEARS

sloping- not leueu.

Fig. 245

Ihls ctistonce (E f) further into the
picture thar\ tb>e bocJ'Wou3e.,

Fig. 216
158

OUT-OF-DOORS WORK

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, b, 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 07i
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

159

Fig. 247

FREEHAND PERSPECTIVE

path " of the moon in rippUng 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. YIII) 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
selection.

160

SOLUTIONS OF PROBLEMS

CHAPTER XI — PAGE 34

THE CYLINDER CONE AND BALL

HE 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, p. 34). In per-
spective 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 center a ver-

:^^^s^'r:^ *:rjtoL«fSs2t!i3»p;«;2s^«*sss*i:.n-ts:

►•«-^«Mt»ii^>Sti^jifo:&2iS»itjK«a<ii;S>;iC-,i?ia««iSi;^^

ZXSajJUSVl.flt.gX.is:^.,

Fig. 248

161

FREEHAND PERSPECTIVE

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. If right, this true diameter should measure a little greater
than the diameter of the cylinder. The base of the cone, though actually
of the same size as the cylinder, will ajyjiear both
a little larger and 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 long diameter (the
light line in front of AB) is next drawn, exactly
in the middle of CD and a very little longer than
AB. The ellipse is then sketched through the

four points A, B, C, and D, taking care to have it touch the base
of the cylinder, and to make the greatest length not on AB, but on
the long diameter. The cone is completed by drawing its sides from
the apex tangentially to this elliptical base.

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

^ r>eiiig actually of the same height, they lie in the same horizontal plane. This plane,
being below the level of the eye, appears to recede upward, 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.

162

Fig. 250

SOLUTIONS OF PROBLEMS

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. Being nearer the eye, its diameter is greater than AB.

CHAPTER XV

THE CYLINDER AND THE RECTANGULAR BLOCK

PAGE 48

The Block. — This should be drawn first
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 XH.
In setting off the
apparent width of
the top, we remem-
ber that it is actually
narrower in propor-
tion than the book
cover.

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

Since its front face is

Fig. 251

163

FREEHAND PERSPECTIVE

51DE VIEW or GCOUP.

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 s 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

CHAPTER XXVI — PAGE 91

THE SQUARE FRAME LEANING ON A REC-
TANGULAR BLOCK

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, where a diagonal (AB) of the horizontal top face

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

others.

164

SOLUTIONS OF PROBLEMS

of the first cube is carried to its vanishing point VPS, from which a
perspectively parallel line is drawn back to C, cutting the back edge

Fig. 254

in D, which becomes the farthest corner
of the second cube, and of the whole block. v${^,
A line through D, vanishing in VPl gives
the corner E.

The Frame. — For this the loiver face
only may be first considered. This face
rests on AE in points one eighth of AE
from either end, which can be easily found
by the method used on page 79 for the steps. (The vertical edge
AH is divided into eighths, and lines from points 1 and 2 vanished
to VP2. Where they cross the diagonal Al verticals are erected,
marking points F and G.)

Now by placing a card against one of the side thicknesses of the

165

PERSPECTIVE VIEW.
THE LI IVES ABAND CD

are actually parallel
am0van16hin vp3. _. /

Fig. 255

FREEHAND PERSPECTIVE

Fig. 256

frame.

frame (Fig. 255) we readily see that these thicknesses are vertical, and
that their vertical planes (one of which is indicated by the card)

are parallel to the block ends, and cut
the block side A E I H in vertical lines
(FJ and GK, already drawn), and also cut
the ground in horizontal lines through J
and K, parallel to the horizontal lines of
the block ends. In these horizontal ground
lines rest the lower corners of the leaning
We therefore vanish the lines through J and K to VPl,
continuing them forward indefinitely. The distance JL (Fig. 257)
is actually equal to half the block width ; that is, to J3, and can
be easily measured by a line through J parallel to H3 (vanishing
with it to VP4). From where this line meets the continuation of the
edge HN (in point 4) a line parallel to HI (vanishing in VP2) gives
the corners L and M, and the ground edge of the frame.

The leaning edges of the square are drawn from L and M through
F and G, and will be found to vanish in OVPl. They vanish a little less
(have a more distant vanishing point) than the horizontal edge LM, and
should therefore be made slightly longer, but shorter than if standing
erect from L. We can check our estimate by comparison with LO,
the height of the frame at that point (EH). To obtain this, the height
of the block at J is carried for-
ward to P on the vertical (by
lines vanisliing in VPl) and one
half of LP added at the top,
making LO.

The Thickness Edges. —
These edges must vanish sharp-
ly (or have a near vanishing
point) because the edges at
right angles to them (the ones
to OVPl) are foreshortened but
little. Hence 0VP2 is placed
but little below the group. To this point these short edges are drawn,

IGG

-X -TiVPi: — -^

T^^HSi-*

Fig. 257

Fig. 258

SOLUTIONS OF PROBLEMS

carrying them forward 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 a corner is marked accordingly

at Q. From this corner a horizontal edge van-
ishes to VP2, and a long oblique one to OVPl,

giving respectively R and S where each crosses

a thickness edge. From S the other horizontal

edge vanishes to VP2 and from R the other long oblique one to OVPl,

completing the square.

The Inner Square. — This is drawn as on page 86.
The frame width can be measured perspectively on
LT by continuing the edge TS to cut the con-
tinuation of LO in point 6, dividing L5 into six
actually equal parts and vanishing lines to 0VP2

THl^LiN^E'l'^HreHll°na as? froui the first and last of these division points, eriv-

VANISH IN bvP* '^RE HERE SEEN ■«• ^ <=>

To«.AeTUAuuy.«.RALLx.. -^^^ Q ^^^ ^^ ^j^^ ^^^^^^ nccded.

Fig. 259

CHAPTER XXXI
PAGE 102

THE CLOCK

The dial of the clock is expressed
by the methods used for the circu-
lar frame in the square frame (Ch.
XIX). After the rectangular por-
tion of the clock has been sketched,
an enclosing square for the dial (AB
CD) is drawn and the clock axis
vanished from the actual center of
this square. The axis is continued
forward indefinitely (upward to left)
to give the direction of the short di-
ameter (which always apjjears to lie in line with the axis,
actually at right angles to it — p. 97, Fig. 160) . For the ends (G

1G7

Fig. 260

though
andH)

FREEHAND PERSPECTIVE

of the short diameter, slightly curved lines are sketched up to the right
from E and in reverse direction from F, giving the front and back of the
ellipse. The long diameter is then sketched, of an indefinite length,
midway between G and H and at right angles to the axis. It is then cut
by curves from I to the right, and from J to the left, being careful to
keep K and L equally distant from the axis. The remainder of the el-
lipse may then be drawn, correcting the curves as needed ; and even
moving K and L if necessary for symmetry, but not changing points E,
F, I or J. The inner ellipse is drawn in the same way, as explained on
page 98.

CHAPTER XL -PAGE 131

THE TRIANGULAR PRISM AND FRAME

The Triangular Frame.

— This solid is easily con-
structed 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).
The Triangular Frame. — I

This is also readily drawn Fig. 261

168

SOLUTIONS OF PROBLEMS

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 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 perspectively, crosses this lower edge is a corner of the inner triangle.

Fig. 262

Fig. 263

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).

169

INDEX

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

to acquire artistic judgment, 80, 83,
120
Apparent size of objects, according to dis-
tance, xi, 156
in out-of-door work, 155
in relation to other parts of picture,

108
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
Baskets, 90
Benefit of perspective study, xi

acquiring of artistic judgment, 80, 83,
120

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 cylindrical 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,
140
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,

xii
see obliquely, 9
Circular frame within square frame, 96-99
application of its principles, 99; see
Cylindrical objects not vertical
Clock, 102, 167
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

171

INDEX

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, 50

making the drawing, 54

order of drawing edges, 54

proportions used in estimating other
objects, 51, 86, 129, 164-165,
167-168

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,
55

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 ellipses, 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

Desk, 179

Diagonals, use for measuring, concentric
circles, 65, 66
door in room, 108

square frame, 86
square plinth, 89
Drawing from a description, xii;
Problems

see

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, 13-15

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
Foundation truths of perspective, two, xi
Freehand sketching defined, xii
Freehand work entirely, 3
Fruit grouped with cylindrical objects,

29-30
Furniture, chair, 118

clock, 102, 167
( desk, 179

good examples, 120

rocking chair, 118

proportioning to other objects, 108

172

INDEX

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, application of study,
126-127

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-

113
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,

116
parallel to picture plane, 105-109,

132, 133
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,
107-108
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-
168
Mechanical perspective, value alluded to,
140
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

Hght 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

173

INDEX

greater distance of vanishing points,

154-155
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,
note
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 planes,

139-142
may be assumed at any distance, 6
method of using, 7
position relative to central direction

of seeing, 6
relation to group of objects, 46
relation to subject in drawing in-
terior, 110-112
study of, 4-7
Position, for drawing, 1

of hand for ellipses, 1 1 ; for lines, 1-2
of models, 2
Practical use of methods, xii, lOl
Practice, of ellipses, 11

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

conditions, general, 34; special, 102
cylinder, cone and ball, 34, 161-163
reasons for giving, xii
rectangular block and cylinder, 48,

163
square frame leaning on block, 91,

164
triangular prism and frame, 131, 167

Profile lines, 21, 22

QUATREFOIL, 101

Railroad track, illustrates vanishing

lines, 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
Rocking chair, 118
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 oblique edge, 150-151

cast by parallel rays (sun, moon),
144-151

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,
147

located by two points, 148

of a cube, 147

of curves, 149

of natural objects, 151-152

on an obHque surface, 151

on a house, 151

on curved surfaces, 149-150

vanishing of light rays, 146-147

vanishing of shadow-directions, 145-
146
Shoulders of cylindrical objects, 16; re-
lation to cone principle, 19
Solutions of problems, 161-168

cylinder and rectangular block, 163

cylinder, cone and ball, 161

square frame leaning on rectangular
block, 164

174

INDEX

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,
164

91,

Table line, explained, 3

high enough on paper, 43

position with 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,
57

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 difl^erent angles, 61-62

with cylindrical object, 67-68

Vanishing lines, " converging," 37-38

example of, railroad, 41-42

oblique, 73; see Oblique vanishing
lines

taking direction of, with pencil, 55

tests of, 56, 57
Vanishing of parallel planes, 52, 73-74,

162, note
Vanishing points, abbreviation of, 44

numbering, 50

oblique, 73; see Oblique vanishing
lines

use without marking, 56, 80, 124
Vanishing traces, 73, 151
Vertical lines drawn vertical, 60
Vignetting, 116

175

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