The science and art of model and object drawing; a text book for schools and for self-instruction of teachers and art-students in the theory and practice of drawing from objects

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i!w^?J!Sltl

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A

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LIBRARY OF CONGRESS.

.^u UNITED STATES OF AMERICA.

\

WHITE'S INDUSTRIAL DRAWING

THE SCIENCE AND ART

MODEL AND OBJECT DRAWING AND FOR SELF-INSTRUCTION OF TEACHERS AND ART-STUDENTS IN THE THEORY AND PRACTICE OF DRAWING FROM OBJECTS

^^4

BY

LUCAS BAKER

tf'

n

\

•art^fHastfr

FORMERLY SUPERVISOR OF DRAWING

IN

THE PUBLIC SCHOOLS OF THE

CITY OF BOSTON

ILLUSTRATED

^

^,^

,

,333

Copyright, 1883, by

IVISON,

BLAKEMAN, TAYLOR, AND COMPANY PUBLISHERS

NEW YORK AND CHICAGO

CONTENTS. PAGE

NTRODUCTION

.

.

.

.

.

.

.5

.

Terms and Definitions

11

Of

12

Limits

Of Extension

12

Quantities of the First Degree.

— Lines

— Surfaces Quantities of the Third Degree. — Volumes Quantities of the Fourth Degree. — Inclination

Quantities of the Second Degree.

Words Denoting

.

.

.

.12

.

.

13

.

16 .

.

17

.

Position and Relation

Orthographic Projections

.

How

TO Read x\pparent Forms

The

Diascope

17 18

.

.26 29

.30

Analysis of Apparent Forms

The Drawing of the Rectangle or of the Square The Apparent Forms of Angles The Drawing of the Cube Method of Drawing the Hexagon and the Hexagonal The Circle .

.

32

.

34 36 Prism

.

39

44 3

CONTENTS.

4

Position of the Apparent Diameter.

Apparent Form of Circle seen Obliquely

The Recession of the Apparent Diameter. Apparent Forms of Parts of

PAGE

— Illustration

Circles.

.

.

.

.

.

— Illustration

— Illustration

.

49, 50

.52

.

.

.

.48

.

55

.

Method of Drawing Circular Objects

57

Rules for Drawing the Cylinder

57? 58

Apparent Widths of the Bases

The Position of the Major Axis of the Ellipse

59

...

Bands and Rims

60

66-68

The Law of Rims Demonstrated

The Drawing of

70

Ellipses

71

Drawing the Triangle and Triangular Frames

.

,

.

72

.

The Frame-Cube

.

Drawing the Single Cross

Drawing the Double Cross Drawing the Frame-Square

74 75

76 .

.

.

,

.

.

.

80

.

The Use of Diagonals The Cube

81

82

Groups of Rectangular Solids and Triangular Prisms Groups with Hexagonal Prism.

.

.

^

— Vases

Light, Shade, Refleci^ed Light, Cast Shadow,

Light and Shade on the Cube

Light and Shade on the Cylinder Shading the Sphere, Cone, Etc

'^Z

84

and Reflections

.

86

%^ 90 91, 92

Methods of Shading

93

Reflections

95

APPENDIX

99

INTRODUCTION.

HE

tendency of the American people to study art

marks an era of

^

multiply rapidly

art

filled,

:

life.

art-schools

and private teachers are

mand. tion,

in our intellectual

in

Students are well

great de-

All branches of art are receiving atten-

and especially the industrial department.

There are two sources teacher,

and nature.

of

There are

art-instruction,

also

— the

two methods

— working

nature.

Multitudes of private pupils do nothing but

from copies, and working from

copy the work of others, and consequently they never

^Qx

acquire the power to

produce original work themselves.

The two

methods may be combined, but nature must always be regarded the great instructor.

We

to the

as

can do no greater service to our pupils

than to prepare them to learn from nature, to open

minds

of

practice,

harmonies and melodies which she has

their eyes in

and

ample store

for them.

There

is

no department

the development

of

of

the powers

public of

instruction better adapted to

observation than drawing from

objects. 5

MODEL AND OBJECT DRAWING.

6

art-student, in progressing through the various branches of

The

his study,

soon confronted with the necessity of making for himself

is

original drawings

this

guidance

for

he would read a book

At

He

from objects.

depend upon them

;

stage he

:

and he must give

presumed

is

his

own rendering

of them.

in

gation, not unlike the explorer of a

in

possession of some knowledge

Thus prepared he enters upon a tour

Geometry.

hand

to have acquired a ready

drawing from the copy, and to be of Plane

can not long follow copies, and

he must read forms independently, as

new

of investi-

He must

country.

the facts presented to his observation, and deduce

all

note

all

the laws dis-

coverable by his understanding.

To

the student

must be opened

to

His method

him.

emphatically a field of discovery.

is

it

new

facts,

of

seeing

which have been hitherto unnoticed by

changed from the casual and

to be

is

accidental to the accurate and discriminating

and comprehends the subtleties of light, shade,

knows see

shadow, reflections, and

The

to see.

method which penetrates forms of objects, and

of the apparent

that the principal part of

and how

His eyes

his

Every teacher

color.

work

pupil begins with

is

of art

teaching his pupils

little

knowledge

of

to

the

apparent forms of objects, and with no habit of observing them.

This knowledge must be acquired, and the habit of seeing must be formed.

This

respect, to

The There

is

draw

is

the only foundation for true progress.

is to

knotv ;

no guess-work

every question

and

not to know,

is

not to be able to draw.

Object-Drawing has a basis

subject of

can be

;

In this

of

fact

throughout.

mathematical precision pervades the whole settled

by reference

to

fundamental prin-

ciples.

Model-drawing

is

the best possible preparation for sketching from

INTRODUCTION. The

nature.

equipped

to

student graduating from the study of models goes

preparation the results of

this

be uncertain, and accurate only by accident. tific

basis for free sketching ;

its principles,

The forms,

first

no

part of

model-drawing,

the fixed laws of

of

viz.,

falls

light.

The

and conditions

also its fixed limitations

subject

it,

efforts

would

the scien-

and an understanding of

that relating

Geometry

shadow, and reflection,

part, viz., light, shade,

ince

and without

his

It furnishes

can count himself secure in his work.

artist

closely related to Descriptive

is

fully-

scenery or of architectural

the delineation of natural

Without

objects.

7

;

falls

within the

third division, :

apparent

to

while the second

color,

viz.,

hence the whole

prov-

field

has

of our

within the domain of science, and only partially within

that of taste.

The models used

in this

department are geometrical forms, and

based on these, as the sphere, cylinder, cone, cube, prism,

objects

pyramid,

vases,

plinths,

rings,

etc.,

supplemented

whose forms bear

objects of utility and

beauty,

to geometrical types.

To become thoroughly

ciples of the

numerous

by

close

relationship

familiar with the prin-

whole subject should be the aim of every student of

torial or industrial art; for

thus only will the

way become

pic-

clear for

any future advancement. Model-drawing also possesses an educational value that ought to

commend

it

to

every true teacher.

course of instruction in

the public

The

general tendency of the

schools, aside

from drawing,

is

toward the development of the world of ideas, and not toward the

development

of the

the case, that the ble, to

power

mind

is

of observation.

Indeed, so strongly

drawn away from the

real, visible,

the contemplation of the unseen and ideal.

is

this

and tangi-

Thus our

pupils

MODEL AND OBJECT DRAWING.

8

to belong to the class, that, ''having eyes, see

come

Emerson of the

says,

intellect

;

"The

and that

is

possession of his faculties.

full

study of art

is

of

the highest of

''

unfolds and necessitates attention,

It

all skills

and virtues."

Attention makes the scholar, the want of It is said that

and

high value to the growth

and Goethe called drawing " That most moral of

"

accomplishments," saying,

all

the artist

knows what

it

the dunce.

to look for,

and what he sees

;

almost equally true, that the untrained in model and object

it is

drawing do not know what to look

for,

or

what they

see.

It is for

these reaso7is that our subject has a high educational utility over

above

say,

from the practice of this subject

then, that the discipline derived

tends to put the pupil in

We

noty

all

considerations of

its

industrial or commercial value.

and

Model-

drawing in particular, and drawing in general, should be well taught

in

our public schools, in order to secure a more complete development of the mental powers.

Moreover, this subject opens to the pupil

ment

;

as

it

unfolds

vision, while

it

he derives from

new

powers,,

new

increases the value of his labor in it

enters into

all

sources of enjoy-

and extends the area

skills

and

labors,

of his

life.

mental

The power

and adds another

segment to the arc of his being.

The student has presented

to his

mind, for his comprehension, a

multitudinous series of facts relating to form, light and shade, shadow

and

reflection.

The whole

and made a part if

of

the

series

must be appropriated and digested,

student

:

he must assimilate the whole

he would attain to a complete mastery of the subject.

method

The

best

for the teacher to follow, is to place before his pupils a single

model, and then,



first,

to lead

them

carefully to recognize the several

INTRODUCTION. facts, relations,

and principles involved

in its

9 apparent form

and

to note the distribution of light, shade, shadow,

same tion

;

secondly,

on the

reflection

and, thirdly, to deduce the general principles which the observa-

;

and comparison of these appearances are found to

It is

establish.

not enough merely to set the pupil to work on the models.

At

His powers of observation are undeveloped, and need directing. the

rules should be deduced

same time, the

furnished ready-made by the

teacher.

The

by the

pupil,

and not

should be taken

pupil

into partnership with the teacher in the analysis of the subject,

taught to write down his own conclusions.

and assimilate the full

facts for his

own

He

use, so that

thus appropriate

will

he

and

will feel

he

is

in

possession of them.

The to lead

practice in

all

and direct the

than to

branches of our school instruction should be

pupil's

minds

in all their investigations, rather

impose upon them a burden

of arbitrary

dogmatism without

regard to their power of assimilation.

we

In the practice of model or object drawing

before us in suitable positions, and proceed to draw brush, or crayon, in choose.

The method

line, light, is

and shade, or

in

place the objects

them with color, as

pencil,

we may we

wholly a freehand process throughout

use no instruments but the pencil, brush, stump, and^ rubber

we proceed upon

certain general

are to be noticed hereafter, to

surfaces

we may

Drawing, then,

is

make

;

and

and fundamental principles which

make

the representation upon whatever

have chosen for that purpose.

Model and Object

a study for the artist as well as for the mechanic.

In Perspective Drawing, which

Geometry applied

:

is

really a

branch of Descriptive

to the representation of objects as they appear,

we

a drawing of an object or objects wholly or mainly with instru-

MODEL AND OBJECT DRAWING,

lO

ments

for

measurement and execution, following certain

determined laws

assumed

we may have It is

of

intersection of

and planes, from

lines

or fixed data or measurements,

it

is

certain

upon whatever plane surface

selected for that purpose.

a mechanical

and not a freehand process

:

hence

the ordinary method followed by the artist in securing his

but

and

fixed

generally the

method employed by the

is

it *'

architect to

not

views,"

render

apparent the results of his inventions and combinations. It will

be seen, therefore, that, in practice. Object Drawing and

Perspective Drawing are essentially different. ent the practice in these two departments

fundamental principles

common

to both

harmony, the one with the other. tions,

If

they are apparent only, and not

of understanding of the subjects

;

may

But, however differbe, there are certain

and they are

in

complete

there seem to be contradicreal,

and are owing

under consideration.

to a

want

Model and Object Drawing. TERMS AND DEFINITIONS. HE

terms used in drawing, so far as they relate

to

mathematical quantities, should be identical

with those used in Geometry

;

and they should

be given the same value. It

may be

useful, therefore, to insert

here

a partial analysis of geometrical quantities, with their definitions, for the use of those

who

are not other-

wise familiar with the same.

A and

class of beginners should

to define geometrical quantities as a preparation for

model or perspective drawing. of geometrical quantities,

class

:

this

is

be taught to distinguish

the

first

Let them begin with the four kinds

and learn

to refer

any quantity

to its

own

step in getting at the correct definition.

In Geometry there are four different kinds of quantities, some-

times called quantities of different degrees. First,

Quantities of Length

:

all

Second, Quantities of Surface Third, Quantities of

Volume

:

:

lines belong to this degree.

all

Fourth, Quantities of Inclination

surfaces belong to this degree. solids

all :

all

belong to this degree.

angles belong to this degree.

MODEL AND OBJECT DRAWING.

12

The first

degree, or kind, to which any quantity belongs determines the

word or words

of the definition of that quantity.

of the definition refers to the

manner

OF

The

last part

of limitation or boundary.

LIMITS.

Points limit lines, lines limit surfaces, surfaces limit volumes to

by

lines,

and

Or again points

the

we should have

reverse the statement,

following order

these limitations in the

by

quantities of the

points.

first

degree, or kind, are limited by

quantities of the second degree are limited

;

first

degree

or,

volumes are limited by surfaces, surfaces are limited

:

lines are limited :

;

and quantities

;

of

by quantities of

the third degree are limited by

quantities of the second degree.

Quantities of the fourth degree are limited by lines or planes.

OF EXTENSION. Extension

ultimately the occupation of space.

is

three dimensions,

— length

(lines),

Extension has

breadth (surface), thickness (limited

space or volume).

A

POINT

is

the zero of extension, as

three elements of extension

:

hence

it is

it

possesses neither of the

position only.

QUANTITIES OF THE FIRST DEGREE.-LINES. There are

A

straight, curved, broken,

STRAIGHT LINE

straight line

is

is

and mixed

the direct distance

lines.

between two points.

one without change of direction.

A

SURFACES.

A

CURVED LINE

The change

ging.

one

is

in

which the direction

of direction

may

lie

is

constantly chan-

constant, or constantly increasing

is

or diminishing by a certain law of ratio

curved line

13

or

;

it

may be

A

irregular.

wholly in a plane, or in a regularly curved surface,

or in an irregularly curved surface.

QUANTITIES OF THE SECOND DEGREE. -SURFACES. Surfaces are of several kinds, such as regularly curved surfaces,

those of the sphere, cylinder, cones, faces

broken and warped surfaces

;

;

etc.

;

and wrinkled

rolling

A

if

not limited

plane takes a plane

is

its

any straight surface.

is

Planes are considered

and hues Hmit planes, as stated above.

;

name from

the

manner

of its limitation.

which

CIRCLE, then, is a is

is

A

Thus, when

limited by a curved line, every point of which

distant from a point within the plane, the plane

A

all

these last are called Planes.

:

PLANE, therefore,

infinite

sur-

and surfaces which are neither

warped, broken, nor curved in any direction, but are straight in directions



is

equally

called a Circle.

plane limited by a curved

line,

every point of

equally distant from a certain point within the plane called

the center.

(It will

be observed here, that the distinction between the

plane of the circle and

its

limiting line

is

kept clearly in view.)

Again, a plane limited by three straight lines therefore,

a

TRIANGLE

is

a plane

straight lines.

Triangles are of five kinds.

angle triangles

(Fig.

A),

is

called a Triangle:

limited by three Right-

having one right angle

Right-angle Isosceles triangles, having a right angle

and two equal sides

(Fig. B)

;

Equilateral triangles, having the three

MODEL AND OBJECT DRAWING,

14

sides equal (Fig. C)

;

Isosceles, having two sides equal (Fig.

D)

;

and

Scalene, having the three sides and angles unequal (Fig. E).

From

the same analogy

we should have

the following definitions

of planes.

A

SQUARE

make

is

a plane hmited by four equal

A

RECTANGLE

is

lines,

which

A

RHOMBUS its

A

is

a plane limited by four straight lines, the opposite

and forming four right angles.

lines being equal,

only

straight

four right angles one with another.

a plane limited by four equal straight lines, having

opposite angles equal.

RHOMBOID

is

a

plane limited by four straight

lines,

only the

opposite lines being equal, and forming equal opposite angles.

A

REGULAR PENTAGON

forming

A

lines

A

a plane limited by five equal straight lines

is

equal angles.

REGULAR HEXAGON

forming

A

five

a plane limited

is

by

six equal

straight lines

six equal angles.

REGULAR HEPTAGON

is

a plane limited by seven equal

straight

forming seven equal angles. REGULAR OCTAGON

is

a plane limited

by eight equal straight

lines

forming eight equal angles.

A

REGULAR NONAGON

is

a plane limited by nine equal straight lines

forming nine equal angles.

SURFACES,

A

REGULAR DECAGON

15

a plane limited by ten equal straight lines

is

forming ten equal angles.

An is

ELLIPSE

a plane limited

is

sum

equal in the

they are

line,

every point of which

from two points within the plane

ellipse is said to

have two axes, or diameters

angles to each other; and they are called the major

at right

and minor

of its distances

An

called the foci.

by a curved

common

axis, or, in

language, the longer and the shorter

diameters.

Returning to the

circle

the definition of each part

which

it

belongs.

and the CIRCLE

its different

cumference of a

parts and their limitations,

dependent upon the kind

Thus, the CIRCUMFERENCE the plane limited.

is

A

figure of the circle.

(Fig.

and is

part of the

circle is called

is

of quantity to

the line of limitation

The circumference becomes

the

cir-

an Arc

I).

The

SEMICIRCLE

is

the half-plane of the

by the semi-circumference

circle limited

and the subtending diameter.

A

SECTOR

circle limited

cluded

A

a part of the plane of a

is

by two

and the

radii

in-

arc.

SEGMENT

circle limited

is

a part of the plane of a

by an arc and

its

chord.

It will

be observed,

that, in the

foregoing definitions of the several limited planes, the word ''figure'' is

not used.

in

some

It

seems that

cases, the

this

mind from

word tends

to confusion; preventing,

seizing at once the idea.

that every limited plane has a figure, but the figure

the circle has a figure

;

yet the figure of a circle

is

is

We may

say

not the plane

:

not the circle, but

MODEL AND OBJECT DRAWING.

1

We

the perimeter, or circumference, of the circle. area of a figure

because the figure

;

is

can never find the

only outline, and not area at

All figures, as such, belong to quantities of the

first

all.

degree.

QUANTITIES OF THE THIRD DEGREE.-VOLUMES. Extending on

all

of the universe in

Whenever any ited in fore,



A A

A A

VOLUME

is

any limited portion

is

the method of

SPHERE

and below,

is

the infinite space

worlds and beings have their existence.

portion of this infinite unlimited space becomes limof space

becomes a volume

of space,

;

there-

and the volume takes

its

its limitation.

a volume limited by a curved surface, every point of

is

equally distant from the center of the sphere.

CUBE

is

a volume limited by six equal squares.

PYRAMID

is

a

volume limited by a polygon and as many equal

isosceles triangles as the

A

all

any manner, such portion

name from which

sides of us, above

which

CONE

is

polygon has

sides.

a volume limited, both by a circle as a base, and a curved

surface which

is

straight in

the directions

of

all

lines

drawn from

the circumference of the base to a point in a line perpendicular to the

center of the circle, called the Apex; or a cone would be limited as described by the revolution of a right-angle triangle about one of

its

sides adjacent to the right angle.

A

CYLINDER

circles,

of the circles,

A

is

a volume limited by two opposite equal and parallel

and by a surface curved in the direction of the circumferences

PRISM

is

and straight at right angles to this direction. a volume limited by two equal, opposite, and parallel

POSITION many

polygons, and as

AND RELATION.

IJ

equal rectangles as either of the polygons has

sides.

QUANTITIES OF THE FOURTH DEGREE. - INCLINATION.

When two tion

Acute Angles

line

meets another

of the line met,

two

tion of

line,

same

point. is

An

more

The

When

Acute Angle

same plane having is

inclination

their vertices

An

than a right angle.

less

The

greater than a right angle.

The

point of intersec-

called the Vertex of the angle.

is

in the

inclination of

and intersection

two

of three

planes, at one point, form a Solid Angle.

WORDS DENOTING POSITION AND

Two

Right Angles

(Fig. H).

forming two equal angles on the same side

forming an angle

planes also forms an angle. or

three kinds,

both angles are Right Angles.

lines

Obtuse Angle

of

and Obtuse Angles

(Fig. G),

There may be four right angles in the

incline to each other, the inclina-

Angles are

called an Angle.

is

(Fig. F),

one

same plane

lines in the

RELATION.

other classes of definitions are important to the student

viz.,

;

those of words which denote position, and those of words which denote relation.

First,

Words denoting

flat, inclined.

position

All these terms

any other object save the earth

;

namely, vertical, horizontal,

level,

signify position, without relation to itself.

That

is

to say, a line in

any

MODEL AND OBJECT DRAWING.

1

of these positions

so of itself alone, without the aid of any other

is

line.

Second,

Words denoting

A

tangent, secant, etc.

some other

definite relation to

A

line.

PARALLEL LINE

from another line

from

its

A

;

relation

line in

any

;

and changes position with such

line,

one which

is

namely, parallel, perpendicular,

of these positions bears a certain

everywhere equally distant

is

while a vertical line

vertical alone,

is

and

of itself,

position only.

VERTICAL LINE

is

one

in

an upright position, pointing to the center

of the earth.

A A

HORIZONTAL LINE

horizontal

vertical line

line

one,

is

all

drawn through any point

drawn through the same

INCLINED LINE

A line is with

is

one,

point,

same

level.

perpendicular to a

is

and the

vertical is perpen-

*

dicular to the horizontal line.

An

points of which are on the

all

points of which are at different elevations.

perpendicular to another line

when

it

makes a

right angle

it.

A point,

line is tangent to another line

and would not cut

it

if

when

it

touches

it

at

a single

both were produced.

ORTHOGRAPHIC PROJECTIONS. In order to understand clearly some of descriptions which follow in this book,

the

we think

illustrations it

and

advisable to ask

the attention of the student to a brief preliminary statement of the leading principles

and methods

object of these projections

combinations of objects

;

is,

to

of

Orthographic Projection.

show the

The

real forms of objects, or

so that any one understanding these methods

ORTHOGRAPHIC PROJECTIONS, of representation can construct

from such drawings the things repre-

These methods are generally used by

sented.

and inventors,

ship-builders,

sions, combinations,

invent or design.

architects, machinists,

to represent in detail the forms,

and methods

They

19

of action, of

dimen-

whatever they

may

many

geo-

are also useful in demonstrating

metrical principles, with reference to perspective, forms of shadows, intersections of solids, etc.

Two

planes of projection at right angles

One

ployed.

of

these

is

named

the

to

each other are

em

Vertical plane of Projection,

the projection itself on this plane being generally called the Elevation

the other plane

:

the projection on

it is

is

named

the Hoidzontal plane of Projection, and

called the Plan.

The

plan and elevation of a

building or machine, drawn to dimensions, gives an idea of size,

and method

methods forms

of

will

Two

of construction.

may be drawn where tional details. By these means can be made apparent. The use we shall make

projections

or

objects,

form,

vertical or horizontal

they are required to determine addithe most

of

complicated combinations

these

be to show the apparent

some

more

its

X'iS.2

and to demon-

strate certain mathematical principles.

Let us suppose that we have, as Fig.

2,

two planes represented by sheets

paper at right angles to each other, in a vertical,

and the other

in

of

— one

in a horizontal,

G L.

These

will say

they are

position, intersecting or touching each other in the line

planes are represented in a perspective view, and

each one foot square.

we

Let us suppose, further, that the sun

is

in the

MODEL AND OBJECT DRAWING,

20

Place the vertical plane so the sun's rays will strike the plane

west.

they pass parallel to the horizontal

at right angles to its surface, while

plane.

Now,

if

we hold

a four-inch square plane, or piece of paper, parallel

to the vertical plane, at a little distance vertical, will

from

it,

with two of

its

sides

the paper will throw upon the vertical plane a shadow which

have the precise form and dimensions of the four-inch square.

We may

call this

shadow the

With the square

in

vertical projection of the square.

the same

position, suppose the

sun directly

over-head: the horizontal projection of the square will be cast

down

upon the horizontal plane. This projection

is

a straight line, four inches long.

the vertical projection of the square horizontal projection in the

It is

mg.s

the square A' B'

is

same position not

is

In the figure

C D',

the straight line

and

its

A B.

customary to represent, as above,

these planes of projection in a perspective view,

but simply to draw a horizontal line on the paper,

representing the intersection of the vertical and the horizontal planes, and to regard that part of the paper above the line as the vertical plane, and that part below the line as the horizontal plane.

This line

is

called

marked with the

the

letters

ground-line,

and

Rays

lines,

the line

G

L, cause the

of light

is

G L.

Let us analyze the case above described 3).

it

moving

in horizontal

(Fig.

parallel

perpendicular to the vertical plane above

shadow

of the

square to

fall

upon that plane

and rays of light moving vertically downward, in parallel

lines,

;

cause

ORTHOGRAPHIC PROJECTIONS.

21

the shadow of the square to be cast on the horizontal plane. first

shadow

a square, and the second

is

see the two elements projected,

is

we know

projections: since the horizontal projection

the object which

because

what form they are the

of

only a

is

possesses no appreciable thickness

it

Thus we

form of a square. object from

its

line,

we

see that

we know

;

and, since

we have

that the plane

are able to understand the form of an

further the two projections,

still

we

should also see

what position the object occupies with reference to both planes.

Since

we know that the square is parallel when we see that A' B' is parallel to the

to the ground-line,

to the vertical plane

ground-line,

a

in the

is

projections.

observing

A B is parallel

The

Hence, when we

line.

the origin of projection must be merely a plane,

is

square for the vertical projection,

By

a

;

we know

and,

that the lower and upper edges are parallel to

the horizontal plane.

In Figs. 4 and

the horizontal and vertical projections of several

shown.

solids are

First,

5

we have

tical projections,

a

the sphere at circle.

It

is,

A

;

having, for

of course, the

its

horizontal and ver-

same

in both

:

but

it

should be observed that two circles at right angles to each other, and intersecting at the horizontal diameter of each, would give the

projections as in Figs.

;

but,

CE

if

same

these planes were revolved into different positions,

H and K,

the projections would show that they were

planes, and not a sphere.

At B we have

the projections of a cube.

Two

angles to each other would give the same projections.

squares at right

At C we have

the cube revolved on the horizontal plane, so as to bring one diagonal of the

upper and lower sides perpendicular to the vertical plane.

MODEL AND OBJECT DRAWING.

22

In this position, two square planes would not give the horizontal and %

In this figure

vertical projections of the solid, as at C.

the horizontal

that

gives

projection

the

we observe

true form and dimensions

of a side of the cube, and that the vertical projection does neither.

D we

At

have the horizontal and vertical projections of a cone,

the horizontal being a circle equal to the base of the cone

;

and the

vertical projection, a triangle equal to a vertical section through the

axis of the cone.

At

E we Tig.

have the same tipped up, with

This projection

point

b'

as a

center, a!

plane, to the position d'

The

b' ;

in its

the right, from

D

ticals

from

a!' b' to'

eter,

a

will

made by

revolving, on the

on the

horizontal position

same

vertical

made by carrying forward

is

to

to E, the diameter c

d;

determine a

evident that this iatter diam-

b.

be foreshortened.

horizontal projection of the circle

The dotted

is

on this as a base constructing the triangle.

horizontal projection of the

b,

base and axis oblique

A

to the horizontal plane. b'

its

vertical, let fall

It is

Upon

letting fall the dotted ver-

these two

must be drawn

from the apex

:

it

diameters,

will

be an

the

ellipse.

e\ will give the place of the

ORTHOGRAPHIC PROJECTIONS, From

vertex e in the horizontal projection.

and the figure

to the ellipse,

At F we have projection

shows

the vertical

jfig.

5

projec-

four

:

and

isosceles

triangles,

oblique

:

of the base.

tion of the base

the

be complete.

the projections of a four-sided pyramid

horizontal

the

draw tangents

a triangle, equal to a vertical section through the axis

is

and diameter

The

will

this point

23

their

in

positions,

forming the sides of

the

pyramid,

projected

bee,

are

a be,

at

de a

c e d,

each having the position at

H

constructing

;

common

point

The

e.

projections

of

its

oblique

are obtained similarly to those of the cone, after first

its

projections at G, where

it

has been revolved on the

horizontal plane through a quarter circumference.

At

I, J,

K

(Fig. 6),

we

have, in succession, the projections of a four-

sided prism in several positions.

At

I

the sides of the prism are per-

pendicular, and parallel to the vertical plane

;

at J the

prism has been

revolved so as to bring the sides at an angle of 45° to the vertical

plane

;

and, at K,

it

is

tipped up so that the bases and sides

angles with the horizontal plane.

The method

jections will be readily understood

by what has preceded.

The

of

make

drawing these pro-

reader will further observe, that the projection of any particu-

lar line or

plane

may be

studied from

these projections of solids.

MODEL AND OBJECT DRAWING.

24

For instance,

at I the

edge of the prism, represented by the line

the vertical projection, has

e'

a! in

horizontal projection in the point a; and

same way the remaining edges, represented by the other

in the

tical lines in

in their

found

its

corresponding points.

We

conclude, therefore, from what was

in the case of the four-inch square,

gation, that the vertical projection of

Fig,

ver-

the vertical projection, have their horizontal projections

and

in the present investi-

a vertical line

is

a vertical

line

6

of the same length,

and

that

the horizontal projection of a vertical

line is a point. If

we take

the lines, ad, be, at

I,

in the

horizontal projection,

which are the projections of the two opposite sides of both bases the prism, the bases being perpendicular to the vertical plane, that their vertical projections are found in the points a'

Therefore, line is

we conclude

,

we

e' ,

b',

of

see

f.

that the horizontal projectiofi of a horizontal

a straight line of the same length ; and, if the line

dicular to the vertical plane,

its

is

perpen-

vertical projection will be a point.

By

ORTHOGRAPHIC PROJECTIONS. examination of the several planes bounding this

25

we

solid,

see that

the horizontal bases are projected on the horizontal plane in squares

same

of the to

these projections, the rays are assumed

size, since, in all

be parallel

;

and that

two side planes, which are

also the

same magni-

to the vertical plane, are projected in rectangles of the

We may

tude.

of projection,

say, then, that, zvhen

its

a plane

parallel to either plane

is

on that pla7ie will be equal

projection

parallel

to the

plane

itself.

If

we

we examine

see that

it

the front face of the prism, as projected in

has

f

e',

projection in the line ab, and the

horizontal

its

a' b'

other three sides of the prism have their horizontal projections in the lines b

c,

c d,

abed, and

da;

the two bases have their horizontal projection in

their vertical projection in the lines a!

whenever a plane

is pe7pe7tdicular to either plane

tion on that plane will be a straight

In J

we have the

positions projected on the vertical plane

same

size as the

plane

itself

:

and

e'

f:

ofprojection^

its

hence, projec-

line.

vertical planes of

with the vertical projections in

b'

I,

;

the

prism in their oblique

and we

see,

comparing them

that neither projection

but, in the horizontal

is

of

projection,

the

we

have the two bases of the prism projected in their true form and dimensions.

Compa.re with a b

By comparing

K

with

limiting the solid are of this subject

I

shown

c

and

d J,

'\x\.

\.

we

see that none of the planes

in their true dimensions.

The

analysis

might be carried on to any extent, and deductions

made, and processes developed, for showing various

and forms, intersections of construction, etc.

;

but

combinations

of solids, projections of shadows, principles

we have given enough

of the principles of

Orthographic Projections to enable the attentive student to under-

MODEL AND OBJECT DRAWING.

26

stand the illustrations given in the body of the book. is

This

is

that

all

necessary for our present purpose.

HOW TO READ APPARENT

FORMS.

one had the faculty, when looking at a house, for example, of

If

making

it

appear like a

flat

spot of a certain shape, disregarding the

fact that certain surfaces are retreating, thus

reducing the whole to one vertical plane, he

would have the most complete qualification for rapid sketching (Fig.

just

what the do

ble, to difficult to

ment

artist

the planes, and of their

Herein our knowledge

way it

It is

not

group of buildings, has

But our knowledge

many

secure the apparent form

of the retreating of

combinations, makes

of the of real

very hard to

it

whole group. forms and directions seems to stand

of our appreciation of other facts relating to

appearances

;

always happens that the beginner draws the forms as he

knows them

to exist, instead of representing

to his eye.

To draw what y 07 l

yonr knowledge leads yon admonition of the teacher.

and greatly to be prized

;

to

see, to

Works

his eyes.

them only

paint what yon

imagine you

see,

as they appear

see,

and

not

what

must be the constant

of imagination

may be

excellent,

but, at this stage, neither the imagination

nor the knowledge of the pupil

upon

endeavors, as far as possi-

in order to read forms.

of planes, constituting the house, or the

so that

is

read off rapidly the outline after the whole complex arrange-

been reduced to one plane.

in the

Indeed, this

7).

is

of

any

avail.

He must

depend only

Seeing with the eyes, and knowing from data in the

BOW

TO READ APPARENT FORMS.

mind, are very different acts

;

and the province

of

27

each

separate

is

from that of the other.

Taking the cube with three faces block appear like a

when seen

if

we can make

we can then draw

horizontally,

the whole

on a vertical plane

spot

flat

visible,

Fiff.8

the

various lines with accuracy by referring each to an

imaginary horizontal or vertical, passing through

one end (Fig.

8).

and by noting the angle

the same,

of

The

determined by

To sum up

inclination

reference

to

of

all

the

these suggestions,

may be

lines

or

to

say that

all

vertical

we

the

horizontal.

attempts at com-

parison of lengths and positions of lines

must

be made on a plane perpendicular to

the

of

in

sight,

or,

axis

other

words, perpendicular to the central ray from the

A

object to be drawn.

common way

is,

to hold

out the pencil at arm'slength, in such a position

that

as near to

one

end

is

the eye as

the other, and then to

compare two

lines

as

to

their apparent

lengths, or their positions

with regard to each other, or to a horizontal or a vertical

Thus,

relative

apparent

lengths,

and

relative

apparent

line.

positions,

MODEL AND OBJECT DRAWING.

28

may be

See cut

determined.

hands showing the positions of the

of

pencil.

A very of a

satisfactory

and conclusive method

drawing of a simple object, after

it

is

of testing the accuracy

made,

is

to cut out the

drawing with a pen-knife, running the point around the outside, or the outer lines of the whole figure, and folding back the different planes

Thus, in the case of the cube. Fig.

on certain

lines.

along the

full lines,

and

lines,

/"~

/

fill

will

be seen

\

\

\

""^^---./

at once.

appear to

will

Any error

in the

In the same

work

way

the

of

any separate plane may be tested

by putting

in place all the other planes, leav-

Care must be taken to hold the paper

in a

perpendicular to the

ray

position

central

from the object to the eye.

V'-''''''

A very simple method parent position of a

such a

ing the one to be determined folded back.

\

/

made

w^as

the opening.

just

drawing

/

at

distance from the eye that the model from

Tig. 9

\

run the knife

and then hold the paper

which the drawing

/

9,

back the several squares on the dotted

fold

line,

when

of finding the ap-

neither horizontal nor vertical,

is

to hold

out the pencil as above directed, so as to coincide with the line to be

determined, and, with the other hand holding up the paper, bring the pencil against

in a position

it

direction of the

The

pupil

may

line

corresponding to that of the

on the paper

also put

up

will

The

thus be readily determined.

in front of the eye a plate of glass, and,

holding the head fixed in one position, of the object.

line.

may

trace

upon

it

the outline

THE niASCOFE.

29

THE DIASCOPE.

The

DIASCOPE

is

a simple contrivance for testing apparent forms.

This instrument

simply'a frame, across which are drawn fine

is

wires or threads, at equal distances, in two opposite directions, divid-

ing the space inclosed into a

number

inches square, inside measure,

is

be made of some thin material, 'Fig.

of equal squares.

A frame

four

The frame should and provided with a handle. The

a convenient

size.

10

\ \

1 1

1

may then be divided into half-inch spaces, and small holes should be made near the inner edges at the points of division. Small wires or threads may be drawn through these

inner lines of the frame

holes from opposite sides, dividing the whole space, for instance, into sixty-four equal squares.

When tion,

completed, the Diascope

between the object

to

may be

held up in a vertical posi-

be drawn and the eye, so that the central

ray of light from the object will pass through the Diascope at right angles to

its

plane.

With

it

in

this

position, the observer will

be

30

MODEL AND OBJECT DRAWING.

enabled to read

off

many

without difficulty

of the apparent inclinations

and magnitudes.

The wire,

side of a cigar-box,

and two or three yards

of fine iron or copper

the material required in the construction of this instrument

is all

(Fig. 10).

ANALYSIS OF APPARENT FORMS.

Every light

visible object transmits to the eye of the observer rays of

from every part

in straight lines

The

of its visible surface.

rays of light

and converge as they approach the eye

whole bundle of rays from an object

is

body

of the eye,

received on the inner side of the posterior-wall, called the retina.

On

it

the image of the object

ent form of the object

itself,

is

formed, exactly similar to the appar-

only greatly reduced in size and reversed

In order to understand the explanations which follow,

in position. is

move

so that the

able to enter the eye through

the small opening called the pupil, and, traversing the is

;

it

important to consider attentively this bundle of converging rays

which the eye receives from every object upon which

Every object seems light,

which

the eye

is

is

to

of light as

It suits

moving

is

turned.

be charged with the luminous quality we

profusely diffused abroad in

directed to any object,

ous vibrations.

it

it

all directions.

call

Whenever

receives a shower of these lumin-

our present purpose to regard these vibrations

in straight lines

an object converging to the eye.

;

that

is,

The form

a bundle of lines from of

the bundle of rays

depends upon the form of the object. Thus,

if

a square be placed directly in front, so that the eye

equally distant from each of the four corners, of light

from

this square,

it is

converging to the eye,

is

plain that the rays

will

form a true right

ANALYSIS OF APPARENT FORMS, pyramid, having four sides, with the square for

31

base, and its apex

its

in the eye as in Fig. 11.

In this case the sides of the pyramid of rays would be bounded

by four equal isosceles triangles

and the central ray

;

of light

the square, would be the axis of the pyramid of rays.

pyramid

and

parallel

from

now, this

by a plane

of rays is cut

perpendicular to the axis or central ray,

If,

Cy

f^s-

"

the

to the base,

section will be geometrically simito the base, that

lar

The

a square.

is

section will, therefore, be

a

true picture of the square, and will correspond in form to the

little

spot in the eye formed by the square. If

the square

light are

thrown

is

turned obliquely to the eye, so that the rays of

off

obliquely to the surface of the square, and a cross-

section of the rays

is

made perpendicular

to the central ray, the sec-

tion will present a true picture of the apparent form of in its oblique

formed

in the

position;

it

will

by employment

of

of

making these

models

its

mode

of

oblique position.

facts apparent.

One method

in a conical or pyramidal

obliquely on several bases, showing cross-sections. tion to this

the square

be exactly similar to the image

eye by the rays from the square in

There are several ways is,

and

experiment and proof

is

The

form

built

only objec-

in the cost of the models,

which are difficult of construction.

An

easier

method

is,

to set

up a plate

central ray, and, looking through

any

object, to trace

upon the

it

of glass perpendicular to the

at right angles to its surface

glass with a

common

of soap, the outline of the object, with the

head

pencil, or

upon

one made

in a fixed position.

MODEL AND OBJECT DRAWING,

32

The

outline

on the glass

will

be a true picture of the object.

The

glass will be a cross-section of the bundle of rays from the object (Fig. 12).

Thus, the picture of the plane

——

abed will

be accurately traced on

the transparent -/r^-

at

T

P.

plane

tw'e of an objeet

by tracing

its

transparent plane perpendicular to the central ray

a cross-section of the rays from

may

Hence, we

this general principle

may

interposed

:

A

state

true pie-

be obtained

apparent form on a

from

the object, or by

the object perpendicular to the central ray.

THE DRAWING OF THE RECTANGLE OR OF THE SQUARE.

The drawing special interest,

of the rectangle or the

which the

stu-

dent would do well to consider,

and to

master completely, in

order to

make

all

square presents a few points of

Tig, 13

the drawing of

rectangles easy and sure. First,

and

c

d

when two

(in this

and lower

sides,

case, the

sides), of a

ab

upper

square or

a rectangle are perpendicular to

the

central

them, dc,

2it

ray,

but

one

of

a greater distance

from the eye than the other, as

in Fig.

13,

then the two lines which are perpendicular to

c.r..

THE RECTANGLE OR THE SQUARE. a b and dc^ are seen to be

i.e.,

parallel

but,

;

33

they are

since

un-

equally distant from the eye, the nearer line, ah, will appear to be

longer than dc.

Thus, in Fig.

be seen

will

which

14,

ab

is

to cross

the plan of the above, the rays from c

2X

c'

d

d' ; so that, relatively to ab, c

appear to be only as long as c" d" on the transparent plane.

examine the image formed on

——

Fig.is^n

!

/ 2

J-

^"

(Fig. 15)

\

ab

\^

;

we

P,

a

d and

find that

it

c' d'\

d

\s

0! b' c" d''

It will

and proceed

in

in the

it

and

of

and

a! d'\

.

be

be seen, there-

b c will appear to be convergent lines, seem-

ing to approach each other as they recede from the eye.

In the same way

2

thus, the figure of the rectangle will

four points on any two receding lines, as above,

we

to

i

is

;

will

If

consists of the

apparent height,

viz.,

the apparent length of a

given in the figure fore, that the lines

:

the apparent length of ^r^

d b' ;

of be, b' c"

\'

T

elements

following

d

we

same method

may be proved

By assuming

could construct a rectangle to

that

show the convergence.

all

receding parallel

lines,

whatever position, seem to converge or incline to each other as

they recede, and would, therefore, the

same

In

point.

if

extended

sufficiently,

meet

in

all

cases this will appear from

„_.. ^„ 16 msi

t

the fact that the distance

between them, which line

of

a certain

seems to diminish as

it

a

in length

becomes more

Thus, in Fig. 16

is

length,

distant.

let

E

represent the position of the eye, and

i,

2, 3,

the positions of three equal lines in the same plane with the eye and

MODEL AND OBJECT DRAWING,

34

Let the

with each other. twice,

Draw

and

it

is

be at a certain distance, 2 distance of

lines

with

Thus, 2

T P, we

it

11',

because

is

it

;

and

3 3' will

its

three times as far from the eye.

distance from the eye.

the position of

\\ they

i

because

\\

i

appear to be one-

follows that the apparent length of a line

portional to to

the eye.

have their relative apparent

shall

appear to be one-half as long as

2' will

twice the distance from the eye

Hence

11' from

2' at

from the extremities of each of these lines to E, and,

third as long as

up

i i'

three times, the

3 3' at

at their intersection

lengths.

line

and

If 2 2'

inversely pro-

is

3 3'

would appear to be

were moved the same

of

length.

We First,

have thus obtained these additional general principles

Equal magnitudes appear equal

Equal magnitudes appear unequal

at equal distances

at unequal distances

;

:

viz..

Second,

;

and,

Third,

Equal magnitudes appear inversely proportional to their distances. These principles determine the convergence of all parallel lines

as

they recede from the eye.

THE APPARENT FORMS OF ANGLES, Place a square plane in such a position that equally distant from that, in

as

this

they really are

position

a

the eye, as in Fig.

position,

b' c d' ,

;

but

abed.

It is

evident,

the angles will appear to be right angles,

all

so as

ij,

the angles are

all

if

the

plane

to bring

is

revolved

appearance will be at once changed, and

have been apparently destroyed.

about c

d

a b into the position of all

into the a' b' ,

the right angles will

Thus, the angles at a' and

appear to have been opened, while those at

c'

the

and

d

will

b' will

appear to

THE APPARENT FORMS OF ANGLES. have been partly closed. c'

If

the revolution of the plane about the line

were continued, the process

d'

other set would go on until

guished

;

the points

and

a!

opening one set and closing the

the angles would appear to be extin-

all

b'

of

coming

into the

same

and the whole plane assuming the appearance

eye,

35

line

with the

a

of

straight

line.

Now, b' in

since the angles at

d and

the oblique position appear to

be opened more than right angles,

and since rays from the angle ^are

more oblique than the angles at

c'

and

closed, considering

on

p.

30 we

at b',

d!

and since

appear partly

what was shown

may deduce

ing general statements

the follow:



Whenever a rectangular plane seen obliquely, the nearest

and

is

the

farthest angles appear obtuse, the latter being the more obtuse

;

and

the tzvo intermediate angles appear always acute.

This rule

will

of the square,

apply to every possible position of the rectangle and

which

is

only a particular case of the rectangle.

As

rectangular (solids) volumes are drawn by representing their separate faces,

and as each face must be solved or read by

itself,

as well as

with reference to the others, the principles above stated go far to enable the student to represent accurately rectangular

There

we have

is,

however, one other deduction which

three

solids.

may be

noticed.

rectangular planes in an oblique position,

as,

If

for

instance, the three sides of a cube forming one solid angle, there will

MODEL AND OBJECT DRAWING.

36

This

appear to be three obtuse angles about that point. be the case Fig.

when three

always

will

are visible

sides

there

:

19

can never be a combination of one right and two obtuse angles, or of one acute and two obtuse angles but the three angles about that nearest point of the

cube must always be obtuse, as

The advantage by every teacher,

of

as

it

in Fig.

this rule will offers at

doubtful points where

19.

be appreciated

once a test for

many

the eye alone might not be

able to detect the error.

THE DRAWING OF THE CUBE. Definition First, place

two

:

The CUBE

is

a volume bounded by six equal squares.

the cube on a horizontal plane directly in front, with the

side-lines of the front square equally distant

top of the square being a

little

from the eye

the front and the top of the cube will be seen (Fig. 20). tion the front face of the cube

is

we should draw

the

In this posi-

usually drawn as a

square, with the side-lines vertical, for the

that

;

nearer than the bottom, so that only

same reason

the sides of a house vertical, and

not converging as they recede upward.

We

should

then ascertain by observation, on the pencil held at arm's length in a vertical position, corresponding with ac, the

measurement

upper face

of the apparent height of

of the cube.

as the front face.

Let us suppose

it

the

to be one-fourth as high

Divide one vertical side of the front face into four

equal parts, and place one

of

these parts above the line a

b,

and

THE DRAWING OF THE CUBE, draw ef

much on a

length,

indefinite

of

ef appears

shorter

to be than

and draw dotted

b,

vertical

the lines a e and bfmdiy

f:

parallel

a

and mark

b,

lines

apparent length

its

from these points to

now be drawn, and

Next place the cube so that three

Next observe how

ab.

to

37

the figure

sides will be visible

;

and

e

complete.

is

the model

resting on a horizontal plane, showing the front, right side, and

still

top (Fig.

The

21).

to be

first line

drawn

the measure of every other

the nearest vertical, a

is

The second

line.

must be placed by observing

its

line,

This line

b.

is

a Cy

position in the model,

degree of inclination to an imaginary horizontal

its

through

line

a,

standard line a

Then the

and

length

its

compared with the

b.

third line, af^ should be read from the

model, as to position, inclination, and length, in a similar manner.

We

have now one line in each of the three sets

of parallels to be drawn.

Since every other line in the model

is

parallel

to

one of these

three, therefore the three lines are the ruling lines of the drawing.

We

should next observe

representing

with a

b,

it

in

its

and draw

if

^<3f is

true proportion

it.

is

:

Connect b with

the convergence of the lines it

shorter than ab^ and,

then draw b e.

fg and eg has

if so,

d.

By drawing

how much,

Compare b

d and

been determined

;

fe b

e,

so that

only necessary that they should have the same degree of conver-

gence, as

the lines

are

respectively parallel

complete the drawing of the model.

If correctly

first,

three obtuse angles about the point a;

d, g,

and

e will also

to

each.

These

lines

drawn, there will be,

second, the angles at

appear obtuse, and more obtuse than the angles

MODEL AND OBJECT DRAWING,

38 in

their respective planes at

a;

third, the

remaining angles

will

be

acute. third position of the

The

appear about equal.

cube

one in which the three faces

is

will

Place the cube on an inclined plane, or put some-

thing under the back corner, so that there will be no vertical lines in the model. In this position let a be the nearest point

which seems to be nearest vertical third,

a

dj

comparing the

two

last

;

:

draw

the line a

first

then the line ^ ^ to the

left

;

by

and

lines with the first to obtain their

lengths

different

(Fig.

Having

22).

ob-

tained the positions and lengths of these

Fig, 22 ^""~;;^S<^'

three lines,

only remains to draw the

it

other six lines with the proper convergence,

which must be noted from the model

There lines

will be,

when complete,

itself.

three sets of

each set converging to a different

;

point.

Let us observe, again, that about the point a

we have

each face

is

three obtuse angles, and that the opposite angle on

more obtuse than the angle

that the angles at

c,

d,

and b are

all

in the

acute.

same plane

There

is

at a,

and

one other rule

very useful in the criticism of drawings by pupils deducible from this case;

viz..

Take the two

faces

A

and B, and

call c

e^

the side-lines of the two faces, a b being the dividing line side-lines will i.e.,

:

Now

and with the

take the two faces side-lines

direction opposite to the other

C and

ab

then these

converge in a direction opposite to the other face

downwards.

line a d,

df, and

C

;

B, with the dividing

bf and eg. They will converge face A; i.e., to the right.

in a

THE HEXAGON AND THE HEXAGONAL In the same way the side-lines of the two faces b

Cy

and dg, to the

i.e.,

will

A

PRISM. and C,

39

i.e.,

converge in a direction opposite to the other face

Hence the

left.

rule

:

In drawing any rectangular

a

c,

B

solidj

three faces being visible^ the side-lines of any two faces will seem to con-

verge in a directio7t opposite

to

the third visible face.

be seen

It will

that the third visible face always indicates the ends of the lines nearest

the eye.

THE METHOD OF DRAWING THE HEXAGON AND THE HEXAGONAL PRISM. In drawing the hexagon and the hexagonal prism and the pyramid,

we have

consider the elements of the hexagon as a geometrical

first to

quantity.

Describe a

circle, and,

with the radius from each end of

the horizontal diameter as a center, cut the circumference in points

By

above and below. six equal

arcs

figure of the

:

means the circumference

this

drawing the chords

hexagon

(Fig.

23).

these arcs,

of

Draw

divided into

is

we complete

radial lines

the

from the outer

angles to the center, thus dividing the hexmg. 23

agon into

six equal equilateral triangles, all

having their inner angles at the center of the hexagon (Fig.

If

24).

we draw

the

alti-

tudes of the two triangles having the com-

mon

base a

0,

we

shall

dividing the base a for

it

is

evident

have the line b

into

that

the

f

two equal parts altitude

of

;

an

equilateral triangle will always bisect the base. of the

two triangles having the common base

have the

line c

e,

dividing the base

d

into

Again,

d

if

the altitudes

are drawn,

two equal

we

parts.

shall

Since

MODEL AND OBJECT DRAWING,

40 a

and

o

d are

equal,

equal parts, which

we

plain that the diameter

it is

number

will

i,

2, 3, 4,

divided into four

is

beginning

at the left.

Let us now turn the hexagon into a Fig.

posi-

24 b

tion oblique to the eye, so that the point a will

be nearer

eye than the point

to the

be seen that the points

c

and

d:

it

will

appear

e will

nearer to each other than b and /, because the line

bf is

Hence

nearer to the eye than c e (Fig.

be and

the two lines

parallel to

a

d, will

f

e^

25).

which are

appear to converge

also,

:

the four geometrically equal parts of the diameter,

being

unequal distances from the eye,

at

unequal longest

;

;

the nearest part,

and

2,

i,

the next in

and 4 the shortest

of

will

Again,

all.

visible in

an oblique position.

appear

appear to be the

length

have the hexagonal prism before

will

the next

us suppose

let

us,

We

3,

;

we

with one end first

read from

the model the central rectangle b c ef; that

is,

we

observe these four lines, and draw them in their relative positions

and

Thus, as 3

relations.

^

and

mg. 26

fe

converge upwards, supposing the eye to be a

little

above the model, we have the central

angle

beef drawn

Draw

the diagonals b e and

in its true position

ef

:

(Fig.

they will

rect26).

cross

each other in O, the true center of the rectangle.

Now two

draw the diameter through O,

lines b e

and

verge at the same point with them.

f e ; that We find

is,

parallel to the

so that

that

it

will

we have

con-

the two

THE HEXAGON AND THE HEXAGONAL central divisions of the diameter, 2 and

portional lengths

;

and 2

these two divisions, the diameter;

:

we have

Comparing

3.

the ratio between the several divisions of

will

so that we can point

off

i

to

be longer than

appear to be longer than

the

first

and the

d on

the diameter,

and

last divisions of

we have

by

3,

2,

diameter by observing the ratio of the two middle divisions. thus placed the points a

41

represented in their pro-

3,

appear to be longer than

by as much as 2 appears

same proportion

exactly the

than 4

for,

will

PRISM.

3

the

Having

only to draw the

adjacent sides to complete the apparent form of the hexagon in this position. It will

be seen,

that, to

only necessary to read and follows

rest

model

:

draw the hexagon from the model, draw the central rectangle

without

necessarily,

any further

;

and

examination

it

is

all

the

of

the

and, provided these four lines of this rectangle are correctly

located, the

whole hexagon

is

easily represented in its true propor-

tions.

Any two but is

it is

opposite sides

may be taken

for the ends of the rectangle,

usually best to choose the upper and the lower (when there

The

an upper and a lower).

care, allowing

no error

four lines must be drawn with great

of observation or of execution to occur

;

since

the rest of the hexagon depends upon them.

This position

analysis of

covers

hexagon.

the

every

Let

us

suppose

that one of the possible positions of the central

rectangle

is

-Frg.

conceivable

27

\

/

A^

^,1

represented by the figure

d c ef, <5/and c e being the longer lines (Fig. 27).

cutting each other at

0,

Draw

the center of the rectangle.

the diagonals

Through

point draw the diameter as before, parallel to the ends be and

this

f

e.

MODEL AND OBJECT DRAWING.

42

We

then have the two central divisions 2 and

shall

Laying

(Fig. 28).

off

the points a and

3,

giving the ratio

on the diameter, so as to

d,

give the four divisions of the diameter in their Tig.

28

diminishing ratio from four lines a is

b,

c d, af,

rectangle

in

diagonals to ascertain the central point

the

b

Cy

if

the hexagon

we have

the central

position

(Fig. 29)

the diameter as before, parallel to the

fe and

d e^ and

and

completed.

In the same way,

lines

draw the other

to 4,

i

Fig.

b c e f,

draw the

and through

;

o

draw

29

which, in this case,

have but slight convergence next, lay :

off

the points a and

d,

as before,

and

ab c d ef. positions may be

then complete the hexagon

These several

^

others

all

beins: ^ referable to the same.

30

Fig.

regarded as typical,

Let us now suppose we have before us the

X

""---^I n^

""^--^^^ ^

hexagonal prism standing on one of

P

la

the upper base being visible

:

its

bases,

we should draw

the nearest line of that visible base a b (Fig.

Next, by observation, determine the posi-

30).

tion of a

angle, case,

the

c,

the nearest side of the central rect-

and compare

it

is

to

a b and

^^

c d.

Now,

by the same

c

b).

d and 7i,

(in this

Determine ^

^ in

the diagonals

draw the diameter

^^ is longer than and nf shorter than

since

ratio,

length with a b

same way, and draw

through the center

than

its

two-thirds of a

make

:

parallel

eg- longer

n by the same ratio.

THE HEXAGON AND THE HEXAGONAL and complete the hexagon.

Make / h

ging with bf,

and is

f d.

to a b

parallel c

e,

and a

The amount

;

jk

d,

of

Draw the hi conver-

PRISM.

the prism.

vertical lines of

Fig.

with a e

43

31

convergence

be determined by observation.

to

Let us next suppose the hexagonal prism placed in a position oblique to the eye, and inclined

;

a b representing

the nearest line of the central rectangle of the visible base

(Fig.

31)

:

observe and draw the two side-lines of the same rectangle, join

c

d ; drawing

find the center,

ac and

b d,

and

the diagonals,

we

through which, as before, draw the diameter paralthe lines a b and c

to

lel

We

fix

the points e and

d.

/ in

due proportion from the two central line,

divisions

of

the same

and complete the hexagon.

Observing the inclination of .

the side-lines of the prism, draw

them

in

with the

the

correct

position

proper convergence.

I

Next, draw the visible lines of .

the invisible base, converging

with their respective parallels of the visible base,

g i with

a

e.

It will

be seen that there

will

gh

with a

by

be four systems of con-

MODEL AND OBJECT DRAWING,

44 verging

lines,

system,

^^ of

ad

nal

and that u b may be taken for the

A

fifth

hexagon

is

system would be indicated by a

c

and

Following the method here indicated, the

not essential.

it is

first

the second, the diagonal b c oi the third, and the diago-

oi the fourth.

b dy but

the

initial line of

an easy subject to draw in

all

possible positions (Fig. 32).

THE CIRCLE.

A

seen in various positions, in whole or in part, appears to

circle

the eye as a circle, a straight line, an ellipse, a parabola, or as a hyperbola

;

that

First,

is,

A

mathematically speaking, as one of the conic sections.

circle is

seen as a true circle

from the plane of the

when Tig.

__^ —

P

S3



J ^L—

""^^

when the

central ray of light

circle is perpendicular to that plane it

with the plkne.

to the right

Thus,

^^"^^ -^b

p

33),

and the central ray of

n form a right angle on

for

if

we

from the

circle.

Then

we

central ray,

to the eye, shall

left,

a b repre-

all

light

from

sides with

the circle will appear as a true circle

cut the rays of light which circle

let

and

sent the side view of a circle (Fig.

_^_

the plane of the

is,

forms a right angle above

and below, and

--^

that

;

by a plane

come at

in the

;

form of a cone,

//, perpendicular

to the

have a section of the cone of rays parallel to

the base of the cone, consequently a sub-section, and therefore similar to the base, that is a circle.

Second,

A

circle is

seen as a straight line

when the

rays of light

proceeding from the circle to the eye mov^e in the direction of the plane of the circle.

Let a b and

a' b' , Fig.

34 and Fig. 35, represent the side

THE view of a

circle,

Then

circle.

CIRCLE.

with the eye placed in the direction of the plane of the

the circle would appear as a straight

figure the circle

is

and

in a vertical,

eye

The

will

In the upper

line.

in Fig.

the lower figure in a horizontal, position.

45

34

rays from the circle to the

be in a single plane

:

no part

of

the upper or under surface, or right or surface,

left

gives

rays

to

the

eye.

Hence, a section of the plane of rays

PP

and P'P' would be a straight

that

is,

the

circle,

thus

would

seen,

have the appearance of a straight Third,

A

circle is

at

line

line.

seen as an ellipse

when

ceeds obliquely from the plane to the eye. side view of a circle with the eye at E,

the plane of the circle (Fig.

36).

will

onstration

Thus,

Then the

constructing of

The

;

to

on the

proof of this

but a real ocular dem-

wood an oblique cone on

a circular base, IFig.Se

a b represent a

figure of the circle, ellipse.

be better illustrated farther on

may be had by

let

and the central ray oblique

plane of section, P P, will appear as a true

theorem

the ray of light pro-

making a

corresponding to P

P.

cross-section

We

may

note

here, however, that, as the obliquity of

ray with

the central

the plane

of the circle increases, the diameter,

more oblique the

more foreshortened, and

angles to this ray will

not

that

to

this

ray,

becomes

the one which remains at right

be foreshortened at

all.

these two diameters are at right angles to each other,

Hence, since it

will

be

evi-

MODEL AND OBJECT DRAWING.

46

becomes flattened

dent, that apparently the circle of its diameters, as in Fig.

one

plane of the eye,

a

^,

the same

A A,

37

in the direction of

the diameter in the vertical

when revolved

into its oblique position

to the central ray Tig.

E

The

c.

37 a;,

-

the plane

l\

apparent

P

shows their relative lengths A' A' without P,

foreshortening,

oblique

'^^--^J

We

and

a^

learn

still

that the perimeter of the ellipse, in

may be made

its

to appear to cover

in

its

foreshortened.

position,

may

a\

further,

examination of a cone, from which a section of the ellipse

ray,

inter-

section of the rays from each, on

by the made,

is

oblique position to the central

and coincide exactly with the

circumference of the circle at right angles to the central ray. Fig. 38. ellipse.

Let "Eab be a cone, and

to fall against the

from the

light

E

Place the eye at

through

:

circumference of the

latter

pass

will

hence

they will appear to coincide.

Now,

if

an

ellipse, in

may be made cle is

in

the former

the section.

It is

a true

circle,

because the rays of

di-

;

rectly

mn

the contour of the ellipse will appear

Fig.

38

P

an oblique position,

to coincide

with a

cir-

a perpendicular position,

reasonable to suppose that a

cle in a position oblique to

it

cir-

the central ray

may be made

to coincide with

the outline of an ellipse at right angles to the central ray. the ellipse seen from the direction of

While seen from the apex

of the

cone

P

In Fig. 38

appears as a perfect ellipse.

e, it

appears as a perfect

circle.

THE That the

CIRCLE.

47

figure of the circle in a position oblique to the central

ray will appear to be a true and symmetrical ellipse

is,

moreover,

evident from the following diagrams.

In order fully to appreciate the nice conditions and relations of the apparent ellipse to the parent

circle,

and several points

of great

subsequent practice shown by these and following

interest in all

grams, careful study and attention to every particular

is

dia-

demanded

of the student.

and 40 represent the

Figs. 39

same position

circle in nearly the

with reference to the eye.

The

first.

Fig. 39,

is

the plan of the

horizontal plane as the circle.

diameter at diameter

U

a'

the

:

the

of

the line

E ^,

^

b' Y.\

to

the apparent

of

eye

the

diameter

apparent

locating the tangential rays a E,

with the eye in the same

shows the place

It

and nearer

to the left,

place

circle,

a

than b'

the real

found

is

by

these are drawn by bisecting

being the center of the circle

:

taking the central point

thus found, as a center, with the half-length of the line as a radius, describe the arc a'

c

U;

it

ence, from which tangential rays a' b'

a

will give the points

may be drawn

must be the apparent diameter, because

it

b'

to the eye.

second, Fig. 40,

revolved a

little

eye, as indicated

from m^

its

so as to

is

and

its

by the diameter m'

;/

position in Fig. 39, and n'

image

is

whether that portion

same

circle,

into a position slightly oblique to the .•

m' has been moved downward

upward from

n.

In this position the upper face of the circle sends eye,

line

circle.

a vertical projection of the

come

The

subtends a larger visual

angle than any other line that can be drawn in the

The

on the circumfer-

formed on the

of the circle to the

its

rays to the

Now we wish to ascertain left of a! U appears to be just

retina.

MODEL AND OBJECT DRAWING,

48

as wide as the larger part, to the right of that line, or

part of the diameter m' this

in'

subtends as large an angle as the part

b' n'

be easily determined by bisecting in the usual way the

may

angle

U

whether that

E ;/,

the whole visual angle subtended by the diameter, and

through the bisecting point 3 draw a line from the eye to the diameter

Fisr.

m' n\ cutting Fig. 39. left of

the

it

in the point b\ the place of the

It seenis, therefore,

the line

circle

SO

to

a! b'

apparent diameter in

that the smaller part of the circle to the

appears to be just as large as the larger part of

the right of

circle in this position is

an

a! b' ,

and that the apparent form of the

ellipse,

with

dividing the ellipse into two equal parts.

a! b' for its

We

may

longer diameter,

say here, that there

THE is

CIRCLE.

in the illustration a slight error,

which

49 will

be noticed farther on,

but which does not, in this case, vitiate the conclusions very much.

Taking the two

lines a'^

b'^

and

i^'

m^'

on the P

P,

the plane of sec-

the longer and the shorter diameter of an ellipse,

tion, as

we

shall

obtain very nearly the apparent form of the circle, as

seen in Fig. 41.

Thus we have the

Tig. 41

^~--

true figure of the apparent form of the circle in this

position,

can obtain

its

—i—-11^

h"

and by the same means we

apparent form in

all

positions intermediate between

that in Fig. 39 and a position at right angles to the central ray.

A true

picture of the circle,

when seen

definition of a true picture given

on

p. 32,

obliquely, according to the

can only be obtained by cut-

ting the cone of rays from the circle to the eye by a plane perpendicular to the central ray or axis of the cone of rays.

than this

may

Although other sections

give ellipses, yet they will not possess the proportions

of the true picture (Fig. 42).

Let

AB

be the vertical projection of

the circle at an angle of 45° to a line drawn from the apparent center

Fis.

42

of

the circle to

E,

the

position

of the eye

:

then

the oblique cone rays

of

be

will

formed upon the base

circular

Now,

B. tions, as

I,

2, 3, 4, 5,

tures of the circle

rays

mn

:

all

A

sec-

perpendicular to the axis, will present true picbut,

if

we take an

oblique section of the cone of

perpendicular to the plane of the circle

A B,

it

is

quite evi-

MODEL AND OBJECT DRAWING.

50

dent that the section can not present a true picture of the

AB

;

because the section

Drawing the section

itself will

rn

n (Fig. 43)

be a

through 180°,

Tig.

position

the point

v! will

cally similar

The

plain that the point

the position

in

m'

is

E in n

base

A B,

about the axis

E x,

m

will

be revolved into the

43

m\ and

be found section

it

circle.

at right angles to the

and then revolving the part of the cone

circle

:

;/

into the position

;;/ ;/,

parallel

;/,

and the

it

will

be a

through 180°, brings

will

A B. Thus, the A B, and geometri-

circle.

revolution of the plane of section, which

to the base,

mn

to the base

be a section parallel to the base

therefore

line

it

into

is

at right angles

a position parallel to the

THE

CIRCLE,

51

same

base,

of a

cone parallel to the base must be similar to the base, and conse-

quently

and shows

at

once that

it

must be a

circle

;

as

sections

all

circles.

It will

not alter the conditions, nor invalidate the conclusions at

whole diagram about the point E, through an angle

to revolve the

A B, will be brought section m n into a vertical

45°, so that the base,

and the plane

of

in the vertical position will still

course, that

all,

be a true

position circle

can not be a picture of the circle

it

of

into a horizontal position,

the section

:

and

;

A B.

it

It

mn

follows, of

may, there-

fore, be asserted that a true picture of the circle in this oblique posi-

tion will be

found by a section

cone of rays, and that ray, will differ,

more

all

at right angles to the central ray of

the

other sections, not at right angles to that

or less, from the true picture, according to their

obliquity to this central ray.

The change

slight error in the illustration

on

p. 48,

which results from the

of the apparent diameter in Fig. 40,

of position in the place

the circle being slightly turned into an oblique position, can corrected ter,

if

now be

This change of position of the apparent diame-

desired.

and the method by which we may ascertain the true position

the apparent diameter of the

of

when it is at any particular angle may be understood by reference to

circle,

of obliquity to the central ray,

Fig. 44.

In Fig. 44

we have

horizontal position

;

the vertical projection, in

the eye being at E'

the plan of the circle

m n,

in the

the eye being at E.

position with reference to the circle,

apparent diameter will be at

an arc of 90°

:

a!

b'.

to the position E''

we have

Now,

if

;/,

of the circle in a

lower figure

we have

With the eye

in this

already seen that the

the eye

is

revolved through

immediately over the center of the

cir-

MODEL AND OBJECT DRAWING.

52 cle, it will

be evident, that, in this position, the apparent diameter can

no longer be will

at a' b\

but that the apparent and the real diameters

occupy one and the same place, and

will

be

identical.

the eye from E'^ back along the arc of 90° towards

Fig.

it

is

If

we move

former position.

44

evident that the position of the apparent diameter will recede

from the position of the of b\

its

real

when the eye has returned

diameter until

it

reaches the position

to the position E', thus passing over

the entire space between a b and a'

b'.

Let us now see

if

we can

THE

CIRCLE.

53

determine the position of the apparent diameter when the eye

any particular point on

is

at

this arc.

In the passage of the eye over the arc from E' to

d;

vertically over every point of the radial line YJ

moves

E'', it

and,

when

it

has

passed vertically over every point of the radial line E' c\ the apparent

diameter of the circle has receded from to the position of the real diameter a

extreme position

its

Hence

b.

it

follows, that,

c^ b'

when

the eye has passed vertically over any particular portion of the line

E' c\ the apparent diameter will have passed over the same proportion of

the line

x c,

the difference between the extreme position of the

We

apparent diameter and the real diameter in the plan.* fore, find

the position of the apparent diameter with the eye at any

given point

from

E^'^,

may, there-

E'^' in

E' E^' by drawing to E^

the arc

the assumed position of the eye on the arc E'

line will divide the line E'

dividing the line

xc

c'

into

two

parts, E'

D

and

into similar proportional parts,

a vertical line

c^

E'^,

This

to D.

D /.

Then, by

we can determine

the position of the apparent diameter with the eye at the given point.

To draw a

make

divide line

xc

into proportionals similar to the divisions of E'r',

from E' to YJ\ produce

x' c" equal

and

x c.

parallel to

b' a!

so as to cut E' YJ' in

From D,

E' c, draw a line to YJ' cutting x^ c" proportionally to E' o".

(See "Robinson's Geometry," Bk.

By drawing the point in

;r

^

a parallel to E''

c'

from

2, 0''

Theo.

ter of the circle in' n' or

This method

is

m

71,

true for

a'^ b'\

in the point

m

0,

we

shall

have

the apparent diameE^''.

other positions of the eye on the arc

* This method of determining the apparent diameter trigonometric principles.

xc

with the eye at the point all

c'

17, et seq.)

cutting

through which we can draw

x\ and

the point of division on

is

given, without entering

upon

MODEL AND OBJECT DRAWING.

54 E'

since 1^"

E'',

with one on

is

any point

in

p. 48, all error,

Hence, by combining

it.

however

may be

slight,

method

this

eliminated from

that problem.

The

correctness of the foregoing solution

another way.

It is

evident,

when

the eye

may be at

is

E

tested also in

E', that

if

two

planes are drawn through the eye, and tangent to the circle on opposite

and perpendicular

sides,

to the plane of the circle, the planes will cut

each other in a vertical line passing through the eye, and will be tangent to the circle at the extremities of the apparent diameter a! b';

E

tangent lines vertical line

b'

and

drawn through

It is also evident, that,

be tangent to the

Now,

at

tion of these

extended.

E'

E'^,

would be the

would be

two planes would diameter a

in a horizontal line passing

if

line until

be the trace of

E'',

b,

and

through

the intersec-

extended, would cut the plane of the circle

Thus, draw through the point

E'^'

a line tangent to the arc

/ E'

cuts the line

it

extended in

the line of intersection of the

passing through the eye at

and tangent

E''',

extremities of the apparent diameter

y on

line of their intersection.

at YJ\ these

is

any intermediate points along the arc E'

and extend the

this will

E

when the eye

two planes,

as the

would be the traces of these planes, and a

circle at the extremities of the real

that their intersection E'^.

E a'

to the horizontal plane at J,

the circle, by bisecting the line J

c,

a'^

b"

.

For

to if

we

J':

two planes

the circle at the project the point

and then draw from J tangents to and drawing an arc from its central

point with the half-length of the line as a radius, cutting the circumference, the arc will pass of

through the two points

a'^ b'\

the extremities

the apparent diameter, thus showing that the two planes drawn

tangent to the to the

arc E'

circle,

E'', at

and intersecting

the point

E''',

at the eye in a line tangent

the place of the eye, will also be

THE

CIRCLE,

55

tangent to the circle at the extremities of the apparent diameter

a

f?

.

Fourthy A.

When

a part only of a circle from a point somewhere

:m0,48

in a straight line

center

is

drawn perpendicular

seen through a plane parallel to this line (Fig.

be a vertical projection of the ter c\

to the plane of the circle at its

Then we have

circle

with the eye

at

45).

Let

mn

E, over the cen-

a cone of rays on a circle as a base, with

E

as

MODEL AND OBJECT DRAWING,

56

Cut

the apex.

cone by a plane,

this

Y parallel to the axis of the cone.

rays from the circumference of the circle on this plane will trace

The

The

true curve, as seen from the point E.

its

hyperbola^ since

from geometry we learn that

parallel to the axis will

be hyperbolas.

The

section will be a true all

sections of a cone

true form of the curve

projected on the horizontal plane between the points J

is

straight line J

by laying P' \"

E

7t,

P

The elements

as a base.

P on the

of the curve are obtained

on J P from the points p p p and the vertical distances etc., each on its respective radial lines C i, C 2, C 3, etc.

off

Y 2'\

,

B. to

]'

When

a part of a circle

m n being the circle

upon the plane

V,

S'

is

seen through a plane S'

as seen from

E, will present the form of the

a section of the cone of rays parallel to one

parabola^ because

it

side of the cone.

From geometry we know

is

The development

parabolas.

V parallel

as before, then that part of the circle traced

of the

that

curve in

its

all

such sections are

true form

is

seen in

V S V while the dotted curved line between V and V just to the left of V S V is the horizontal projection of the curve of section, and not its true form. The curve V S V is obtained by throwing down the

full line

from S'

V

;

all

the elements or normals of the curve from

upon the horizontal plane, each

upon

nt n,

its

V as

a center

and then projecting them upon the horizontal

own normal drawn from

V V.

All possible forms of the circle as seen in various positions are referable to their places

some one

among

of the conic sections, all consequently taking

the absolute mathematical figures.

Let the student

thoroughly master these forms, and trust to no methods not referable to fixed geometric formulas.

METHOD OF DRAWING CIRCULAR

OBJECTS.

57

METHOD OF DRAWING CIRCULAR OBJECTS. The

application of the principles already developed relating to the

circle will

be found necessary whenever the student attempts to draw

any circular der, cone,

object, or objects

will

it

combinations of

to

Let us take

circles.

cylinder, as one of the simplest volumes, having

There are eight the cylinder.

of

be necessary to consider several other facts and princi-

applied

ples, as

many

But, in order to deal successfully with

multitude of objects.

them,

having circular bases, such as the cylin-

frustum of a cone, vases, cups, saucers, wheels, and a great

As

first

the

two circular bases.

rules applicable to the dimensions

and positions

of

the same rules apply with some slight modifications

to all objects having

two opposite

circular bases, as vases, goblets, etc.,

they are in an eminent degree generic, and consequently important.

We

now

will

consider several facts relating to the cylinder, and see

what deductions we can draw from them. Firsts eye^ both

An

When

the two bases of a cylijider are equally distant

are invisible (Fig. 46).

from -p.^

the

^g

apparent exception to this rule would be found

by taking a cylinder Placing

dollar.

it

of

the dimensions

of

a silver

so as to be seen by both eyes, both

bases would be visible, the one to one eye, and the opposite to the other

;

but the rule requires that

we should

look with one eye only, in

which case the exception vanishes. Second, The visible base of a cylinder

is

always nearer

to

the eye

than the invisible base. Thirdy The visible base is always apparently longer than the invisible base.

MODEL AND OBJECT DRAWING,

58

Fourth, The invisible base

is

always wider in proportion

to its

length

than the visible base.

The last two rules may be longer and narrower, and the

stated thus

:

The

invisible base is

visible base is

always

always shorter and pro-

portionately wider.

The longer diameters of the ellipses, which represent the bases of a cylinder, are always perpendicular to the axis of the cylinFifth,

der.

The shorter diameters of the

Sixth,

always coincident

ellipses are

with the axis of the cylinder. Seventh, The side-lines of a cylinder always appear to converge in the direction of the invisible base.

Eighth,

When a

cylinder is in a vertical positio7i, the plane of

delineation is supposed to be vertical also vertical

and parallel, and of

with the general practice in

;

and

the side-lines are

drawn

course without convergence, in accordance all

architectural subjects.

In illustration of this last statement, reference

may be made

to

Geometrical Perspective, where all regular polygons which are parallel to the

In

picture plane are represented, in the picture, by regular polygons.

all

architectural subjects, the plane of delineation

posed to be

To

illustrate the third rule, that the ellipse representing the visible

invisible base,

is

is

the

always sup-

vertical.

base of a cylinder

der

is

is

always longer than the ellipse representing the

we have only

a constant quantity,

to consider that the diameter of the cylin-

and therefore the same

same constant quantity

at

at either end.

unequal distances from the eye, the

nearer end must appear the longer (see illustration on Fig. 47.

Let a

c

If it

represent the axis of a cylinder, b

d

p.

33), as in

the nearer, and

METHOD OF DRAWING CIRCULAR ef

the farther diameter

then b

:

d

will

OBJECTS,

59

be longer than ef^ because a

nearer line appears longer than an equal line more distant.

The 47

Tig.

rule that the invisible base

wider in proportion to visible base,

will

is

length than the

its

be readily understood by

observing the following diagram,

where and a allel

E

Fig.

48,

represents the position of the eye, bf, eg,

e^

always

circles

;

d h,

and

the lines a

four equal and pare,

b f, etc.,

showing

the actual width from front to back.

drawing rays from each the eye,

in

is

it

the

same plane

appears as a straight line

width

;

eg

while

appear

dh

all.

For,

widest

of

we

in-

if

terpose

the

parent

plane

width

the circle

the eye

bf

will

;

and consequently

it

have some apparent

^^s-As Y"^^^^

/"-'«.

trans-

the relative ent

:

as

will

wider,

still

and

of these circles to

evident that the circle a e will have no apparent width,

it is

because

By

T

P, \1

apparof

c'

1

the /

several

circles

will

be expressed by the distances

and

d'

Ji!y

U f\ of

g\ which e'

d' Ji! is nearly twice as

ent width,

is

long as

b'

f

;

and dh,

the most distant from the eye

:

of

which

d' h' is the appar-

hence the

rule.

MODEL AND OBJECT DRAWING,

6o It will

be seen from the diagram that the rays of light come more

directly

from the surface of the

others.

The same

circle

dh

than from either of the

holds good for every invisible base of a cylinder as

compared with the

visible base in

any possible

The

position.

rule,

that the longer diameters of the ellipses are always perpendicular to

the axis,

may be made

clear to the pupil

by walking around a

circle

and observing --/J^

its

apparent

greatest

Let C be

length.

the center of the

ms'^9

circle,

and

E E^ E'',

Fig. 49, represent

/'

/ / '

/

/

y

^^-'^

^^^^

the successive positions of the eye the lon2:er :

diameters or major

^^,-^'

axes of the ellipses

^^''''

^'

join the tan-

will

rays

gential

;

i.e.,

Ef>

from diameter will be at a at

ef: and

in

circle is the

b,

from E' the longer diameter will be

This

is

it

will

from

E'',

it is

perpendicular to the plane in which they

a matter that should be determined by observation, by walk-

keeping be for

at c d,

base will appear to be perpendicular to these diameters at

ing round the circle and noticing follow,

the longer

each case the apparent axis of the cylinder of which this

their central points, because are.

E

all

its

how

the apparent diameter seems to

position perpendicular to the central ray.

possible positions in which the cylinder

may be

And

so

placed.

METHOD OF DRAWING CIRCULAR The if

6i

sixth rule, that the shorter diameters are coincident with the

may be

apparent axis of the cylinder, that,

OBJECTS.

the longer diameter

The

longer diameter.

axis,

because

it

perpendicular to the

is

truth of the proposition should be confirmed

by observing the cylinder That the

perpendicular to the axis, the shorter

is

must be coincident with the

readily understood from the fact,

in various positions.

side-lines of a cylinder will always

direction of the invisible base,

is

appear to converge in the

evident from the

fact, that,

the apparent

diameters of the cylinder being geometrically equal, the more distant will

appear the shorter

apparently shorter

two bases

The

will

;

hence, as

:

and the

lines

we have

seen, the invisible base

is

connecting the extremities of the

appear to converge in the direction of the invisible base.

eighth rule, in regard to harmonizing the fundamental princi-

ples of model-drawing with architectural methods, lines are

drawn

vertical in the picture

that the principles already laid

we

:

where

all

vertical

attention to the fact,

call

down have no

reference to merely

vertical or horizontal positions, but simply relate to absolute relations

of the object to the eye in all possible positions, with the plane of

section of the rays, that

is,

the plane of perspective, perpendicular

to the central ray of light.

In architectural methods the plane of section

is

supposed

to

be

parallel to the vertical lines in the object; and, of course, the central

ray would be supposed to be horizontal. really the case.

This point presents no

This would not always be difficulty to the

makes himself thoroughly acquainted with the It

student

who

principles here deduced.

should be observed here, that the differences in the lengths or

breadths of the two bases of a cylinder are inversely proportional to

the distance of the eye

:

thus,

if

the eye

is at

an

infinite distance, there

MODEL AND OBJECT DRAWING.

62

would be no apparent difference ellipses

representing the bases, because the rays of light would be So,

practically parallel. little

;

same

the length and breadth of the

in

and,

when the

the convergence of

It follows,

the distance is

is

great, the difference is

the difference

little,

same general

principle, for the

regard to

when

distance

reasons,

The

great.

is

be observed in

will

lines.

from the above statement

drawing of

of fact, that every

models, and every picture, can be best seen from one particular point,

and

will

Hence

appear accurate from no other point of view.

follows, as

a matter of necessity, that

it

the spectator, at the proper

distance from the drawing, should place his eye at the point from

which

all

the lines can be seen in their true proportion.

Having deduced our principles and

now

rules, let us

place the

cylinder in a vertical position, with the upper base visible.

the apparent axis,

^—

1^

Tig.

it

or, as

we may

always being understood that

real axis of the cylinder.

SO

Fig. 50,

is

drawn

to

into

B

parts,

and on either side

of

of

axis.

Let us suppose

measure

:

quarter oi c

of the ellipse with its length

found to be four times as long as

d into

d

into

it

is

two equal parts by a dot

halves by a dot at

2.

AB

similarly,

off horizontally

compare the apparent width

therefore, one-half oi c

A B,

Divide the axis

and the upper half

A

the

any length

A

it is

mean

"

the upper base

quarter-length thus obtained, on a line perpendicular to

this case,

the axis;

not

in a vertical position of

be one-half of the

two equal

it,

we do

with the height of the cylinder. it

^'

In this case the axis

Compare the length

desired.

call

draw

First

B.

suppose, in

wide. at

a

Next

i,

Divide,

and the

Place this eighth above and

ME2HOD OF DRAWING CIRCULAR below the middle point of the quarter required (Fig.

diameter of the

The lower

51).

observing, however, that to. its

of

be found

must be wider

it

length (which, in this case,

if

only

is

e ^< Fig.

51

in proportion

the same as that

width of the lower

Rule

(see

the width of the upper ellipse

of its length, the

of the shorter

same way

in the

the upper) than the upper ellipse

Thus,

It

the curve.

ellipse should

63

two eighths making the

line c d, the

Thus we have the length

ellipse, as well as its position.

remains to draw

OBJECTS.

is

ellipse

more than one-fourth

8).

one-fourth

must be

of its length.

The whole

of

the lower ellipse should be indicated, the farther half

by a dotted or shadowed

Fig. 5U

two

to the

ellipses,

line only (Fig. 52).

may be drawn

Finally the side-lines

as tangents

thus completing the drawing

of the model.

now, the cylinder

If, -H-+-

comparing with which

its

placed on

observe the apparent position of the Fig.

53).

it

axis.

To do

S3

axis

any length

the nearer ellipse

comparison with the

axis,

make an

Draw, as the

in this position, the line a b oi

how long

to

so

we must

first

suppose

will

its side,

it

direction with the horizontal,

we

angle of 20° (Fig.

observe

is

appears in an oblique position,

that

this,

is

:

in

hold

the pencil at arm's length at right angles to

a line drawn from the eye to the center of the cylinder, perpendicular to its axis, so that

it

corresponds to the longer

MODEL AND OBJECT DRAWING.

64 diameter of the

and determine

ellipse,

its

length by moving the thumb

the pencil towards the end.

of

along on the side

Having thus

obtained the apparent length of the nearest ellipse, turn the hand,

keeping the pencil at right angles to the central ray

till

it

coincides

with the axis of the cylinder, with which compare the length of the

We

ellipse.

point c to of

the

suppose

will

mark

this

it

then

:

\^

above and below

each

a,

by means

a mark

off

at a distance equal to the

find,

Suppose

to be one-third of the longer diameter.

dn

into three equal parts

d and n

the points

Proceed to it

by points

i

and

Since the shorter diame-

2.

the cylinder, produce the axis, upon

which mark the points

and w, each half a third from

curve of the ellipse through the four points

Next ascertain the length determine

it.

Do

thorough knowledge

seems to

be,

of the invisible ellipse

it.

Put

Make

it

guess at

obtained.

and then proceed

Then draw

the

must be

less

0.

;

it

measurement with the pencil

not is

dm n

This gives

a.

the position and the length of the shorter diameter.

:

c.

Divide, therefore,

axis of

than that of the visible

respec-

half of a

of the pencil as before, the shorter diameter.

ter coincides with the

so

Place the

the length of the longer diameter

Divide a c into two equal parts, and, drawing a

ellipse.

line perpendicular to ab, 2X

tively

to be two-thirds of the axis.

ac

as before will

aside guess-work until as

much

shorter as

to estimate the shorter

the same by observing the half-ellipse which

is visible.

it

diameter of

When

these

points have been determined, complete the ellipse, drawing the whole curve, the invisible half with a dotted line.

ure of the cylinder in

its

Lastly complete the

oblique position by drawing

fig-

the sides tangent

to the ellipses.

The foregoing

explanations and principles will enable the student

METHOD OF DRAWING CIRCULAR attentive to

them

to

draw the cylinder

in

OBJECTS.

65

any possible position

may

it

Let no accident of position or relation trick you out of

be placed.

your knowledge

of principles

and

facts.

There are many necessary modifications

when we come

to

draw

vases.

The same

above principles

of the

general laws prevail, but

they are modified in their application.

For instance, the bases may not have the same actual diameters as in the case of the cylinder.

and masrnitude ^ Thus, of

The same

law, however, as to position

exists. 'Fig.

a vase, are unequal, the lower being

the rule applies ately shorter

base (Fig.

S4

the bases, or the circles at the top and bottom

if

and the invisible

;

will

the larger,

appear propoi-tion-

and proportionately wider than the

The same

54).

principle will also hold

all

the minor bands of ornament,

as

you move

in

such there

if

still

visible

and

;

!

good for Thus,

are.

the direction of the invisible base,

appear proportionately shorter and wider

/

all

ellipses

this is true

must

whether

they are actually larger or smaller ellipses than the visible one.

Other applications Fig.ss

/K

/ \ A;^^£~V

Take the case

i_

^/l

I

_

of this

law are found in

drawing the cone, and some bands on vases.

'\

\-^_J___^-^

tions of a cone.

two

of It is

parallel

sec-

circles,

evident that the ellipse

43, representing the upper circle in Fig.

55, will

be proportionately longer and narrower than the

6

lower if

ellipse,

i

2,

according to the rule

the top of the cone were removed,

Now, from the nature

of the cone,

half of the curve of the ellipse

if

it

would be the

we may be the eye

is

;

because,

visible

able to see

one.

more than

considerably above

its

MODEL AND OBJECT DRAWING,

66

we

plane; as in Fig. 56

the line

tangent to the the

find that

of the surface of the

we

see

more than

of the curve of

to

It

be greater

is

we

possible that

at the

sides than

on account of the obliquity of the surface of the band Fig.

at the

57 Fig.

Fig.

apex are

parallel ellipses, as in Fig. 57,

half the ellipses.

may appear

cone in front of

lines to the

and also much more than half

ellipse,

the width of the band in front,

all

So when we have two

ellipse.

may

see

which joins the points where the

2,

i

58

66

front or middle point tending to foreshorten

its

width

at that point

while the width at the sides will not appear to be foreshortened at

Take, again, the rim of a bowl, as Fig.

may appear

58.

The width

all.

of the rim

greatest at the sides, nothing at the back, and interme-

diate at the front, or as wide or wider at the front according to the

angle of obliquity,

cone with

its

apex

if

it

happens to be a portion of the surface of a

at a.

Quite an opposite modification would occur in the case of a surface-

band on the sides and

of a vase or

bowl seen below the eye, as in Figs. 59

60.

In this case the band a b would seem to be widest at the front, gradually tapering towards the sides, as

because the band

is

shown

in the figure.

This

is

practically on a section of a cone, the slant height

METHOD OF DRAWING CIRCULAR of

which

is

OBJECTS.

67

very obUque to the central ray, the opposite of the condi-

tion in the rim of the bowl. Fig.

Another very important application

59

forms

parent

circles

of

of the ap-

found

is

Fig' GO

drawing of rims and hoops,

in the

bands.

may have

a vessel, as

The rim would at the sides its thickness

in the

least, as

front

thickness

and back, the reverse

would

be

we

Fig.

61.

in

and from sides

sides,

the line expressing

would be true ; and the

rims,

would not appear

to

back.

to be foreshortened

thickness would be at right

its

angles to the rays of light to the eye at the

to

in this case present

a varying quantity from front to

Thus,

As

or raised

but,

:

Flg.Gl

of this

lines expressing the

proportionately

fore-

shortened, provided the inner and the outer ellipses

front

were

in the

thickness,

same plane

;

but the

being nearer to the

eye,

would appear greater than the thickness

at

the back.

The

principle will be at once seen

we

if

consider the rim to be one-quarter of the

diameter across the top of the vessel (Fig. Fig.

€9

62).

Then we

from the ends represented

shall of

by

have to take a quarter

each diameter of the circle the

larger

ellipse,

through these points draw the curve,

and b

2,

on the diameter a

b,

will

be real quarters of the line

;

and a

i

but on

MODEL AND OBJECT DRAWING.

68 the diameter c

d,

the real quarters being at unequal distances from the

much

eye, the farthest quarter will appear

The

one.

trouble, but those

The

error.

precise difficulty in this division will be hereafter considered.

first

circle,

be presented to

parts, that there will

the eye a series of diminishing quantities, the

or nearest of which

appear to be the largest, and the farthest will appear to be the

smallest

than

nearest

placed without

on the shorter diameter are more hable to

was divided into four equal

will

may be

be readily understood, since the shorter diameter of the

It will

c d,

smaller than the

quarter-points on the long diameter

d 4.

;

so that

Hence

we should have a i=^ the thickness of

all

2,

while c 3 would be greater

rims having the faces at right

angles to the axis appears greatest at the sides or at the ends of the

major axis of the

on the back.

and the rims appear thicker

ellipse,

Thus we have the

rule for rims.

The

ness of a rim at the ends of the short diameter bears the to the thichtess at the

than

appare7it thick-

same

p7'oportion

ends of the long diameter as exists betzveen the lo7ig

Fig.

in front

63

and

the short diameters them-

selves.

The

application of the fore-

going analysis

is

required for a

large class of objects

Take

readily

instance

hoop, for

a

the rule given,

drawn

;

(Fig. 63). :

upper rim

its

by is

but the apparent

new applicadisappear if we draw

varying depth of the hoop from top to bottom requires a tion of

the same analysis.

the five vertical lines, increasing remoteness,

i, i

All difficulty will

2,

is

3, 4,

5,

and note,

that,

by reason

the longest, being nearest

;

of their

while

2

is

METHOD OF DRAWING CIRCULAR shorter than

than

2,

i,

and longer than either

longer than

Thus the

5

;

is

5

i

69

3 is shorter

;

and

5

;

and 4

is

the shortest of the series,

at the greatest distance.

is

it

of the other three

and intermediate between

4,

shorter than 3 and longer than

because

OBJECTS.

five lines

representing the same constant quantity appear

unequal on account of their unequal distances from the eye.

A thoughtless

pupil will always

necessity of thorough

The rim

is

an element which

The

ferences

is

Let

to draw.

be

a!

and

c'

be the center of two

Placing the eye at E,

we

the circles

let

Drawing the outer

in Fig. 64.

rays from the eye to the larger circle, to

some further explanation

and

c

intermediate space between the two circum-

what we wish

be tipped obliquely, as

these particulars, hence the

points.

will require

for its complete comprehension.

concentric circles.

fail in

work on these

or tangential

find the points of

joining these two points by a straight line,

b':

have the position

of the

major axis

on the perspective plane

at

a" b"

of the larger ellipse .

Now,

tangency of the outer rays of the inner

if

we

join

circle d! e\

we

it

;

tangency

we

will

appear

the points shall

From

Fig. 64

nm

and

be obtained

Now

it

all

will its

of

have the

position of the major axis of the inner circle, seen in Fig. 64 on

circle

shall

T

P,

be seen that the foreshortened diameter of the

points and quantities,

in their true proportions

viz.,

no,

Cy

c

r,

on the intersecting plane

construct Fig. 65 by making

jf 0'" r" m'" and

rm,

will

T P.

a'" d!" e'" b'"

the same as the corresponding quantities in Fig. 64.

Draw

the two ellipses in their respective positions, as indicated by

these lines and points, and the true apparent form of the rim will be obtained, as seen through the transparent plane

T P,

from E.

The

MODEL AND OBJECT DRAWING.

70

principle here developed holds

good

in the

apparent forms of

and rims whose

surfaces reside in a single plane,

of the principle

becomes very frequent

It will

wheel

in

in the

all

rings

and the application

drawing of models.

be apparent from the foregoing analysis, that, to draw a

an oblique position, the hub can not be placed

of the ellipse

which represents the

full size of

pushed back of the apparent center

of the

in

the middle

the wheel, but must be

wheel

:

the hub will be somewhat off the center, because

the outer ellipse of it

projects.

If

the

THE DRAWING OF hub

would be another modification

long, there

is

when the

object

in reading the

is

ELLIPSES.

71

the form

of

placed before the draughtsman, there

but,

;

no trouble

is

form by means of the explanation already given.

THE DRAWING OF ELLIPSES. Ellipses seen in various positions appear under several modifications,

some

form,

as

which

of

for

it

is

instance an

important to notice. elliptical

dish,

First,

an

elliptical

seen

obliquely from a point in a plane which contains '""V-

the shorter diameter of the ellipse that

(Fig.

66)

'^X-;

;

m

the eye and the shorter diameter of the

is,

being in the same vertical plane perpen-

ellipse

The

dicular to the longer diameter.

m

ellipse will

^t'ls-

^Q

\J a

appear to diminish in width, according to the degree of obliquity.

Thus, ellipse

;

let

let

E

a 5 be an

ellipse,

n

m

being in the same plane a^ the

be the eye as far above that plane as

m

E.

Then

diameter a b will not appear to be foreshortened, but will appear of full length,

^'-^<-^-.^_ ^^S'^"^

will

its

while the

shorter diameter

"""^---..^

the

appear to

cd be

foreshortened; and, ->.

fe

3'

the nearer the eye is

brought to m, the

shorter will the line

cd

appear; the higher above

ened the

line c

d

becomes.

m

the eye

Again

is

placed, the less foreshort-

(Fig. 6f}, let

;;^

be a point in the

MODEL AND OBJECT DRAWING,

72

extended plane of the ellipse abed, and

above the point m,

the position of the eye

at a distance equal to the line

shorter diameter, will not

the

E

E

Then d e,

;;2.

appear to be foreshortened

;

length being perpendicular to the central ray of light from

But a

the eye.

of foreshortening It will

ellipse,

true

itself to

the longer diameter of the ellipse, one end being

c,

nearer to the eye than the other, becomes foreshortened

m.

its

depending upon the nearness

the amount

;

of the eye to the point

be observed, that to foreshorten the longer diameter of the

the shorter diameter remaining the same, will have the effect

more nearly

of bringing the ellipse It

therefore follows, that

ellipse appears to

just

to the

equal to the

tical dish

for the outline of the

if

circle.

the longer diameter of an

be foreshortened, so as to make shorter

pear to be a perfect It will

form of a

circle,

seem

it

diameter, the ellipse will ap-

and must be so drawn.

be seen, that the apparent form of an

ellip-

might be represented as having a perfect

circle

upper

ellipse.

(See Fig. 6S.)

DRAWING THE TRIANGLE AND TRIANGULAR FRAMES. In drawing a triangle in an oblique position, find

it

is

only necessary to

by observation the apparent inclinations and lengths

lines,

and to place them

in their true positions,

of the three

according to the read-

ing of the same-. But, in

requiring

relation notice,

in

to the triangular frame, there are a

order to secure ready execution

few points

and accurate

work.

Let

abh^

the position of the lower or base front-line of a triangu-

DRAWING THE TRIANGLE AND TRIANGULAR FRAMES. lar

frame standing upon a horizontal plane

the line a

the pencil vertically against the apex of the triangle

the point n

Draw

c.

face of

mined

then

c

a and

c b,

observing

by holding

completing the

Having

the triangular frame. inclinations,

their

b,

noticing where

and thus determine the point

b,

c n,

c,

on a b: compare the length

falls

a

of c n with

First find the

(Fig. 69).

apparent position of the central point of

73

draw

convergent

they are

that

deter-

and a

c e

Fis.69 dy

lines

:

determine the amount of convergence, observe the length of c

a

c in

e,

and draw

the direction of

ters of the

two sides a

draw from each

and

a

b,

it

find the cen-

c by 2X

20i\^

of the

to be one-sixth of that line

i,

direction of c

vergent with

b,

:

line a

b.

divide a n, half of the

into three parts, so that each part will represent the apparent

divisions at

a

lines to the opposite angles.

frame as compared with the

length of an equal third of the line a

angle

and

p,

two points dotted

of these

Let us suppose

convergent with

Now

c.

c

Determine the width

line

de

cab ;

Draw from

2. ;

it

c b, in it

will

i

the line

i

placing the points of these h,

convergent with a

cut the vertical line

the direction of b

will cut

n,

the line

in the direction of b.

These

i

;

en

draw the

in

c,

in the

h: draw kf, con-

line

ap

bisecting the

h in g: draw ^2, convergent with

lines will

complete the right face of

the frame.

Extend

fh

to the point

3,

and draw

3 4,

convergent with

c e

and

From 4 draw and hf, and fix the point 5. From 5 draw the line 5 6 convergent with a b and ^2, completing the inner visible surface of the frame.

a dy fixing the point 4. c b

a dotted line

convergent with

MODEL AND OBJECT DRAWING.

74

The method here given

drawing the triangular frame

for

the method to be pursued in

position will sufficiently indicate

possible positions.

in this

all

other

always important that the student should

It is

determine and keep in mind the different sets of convergent

lines,

always being sure to determine the direction of their convergence.

THE FRAME-CUBE. Construct the outline as in the case of a solid cube, a b being the In the

nearest vertical line (Fig. 70). of this line represents the

frame on the

many right

left side.

left,

one-sixth, divide the line a b into as 2

i

now draw

;

lines,

both to the

from each of these two points, convergent with

d and

a Fie.

how much

apparent width of the vertical piece of the

If it is

equal parts, placing the points

and to the

place determine

first

"70

tively

b d.

:

b

and with a

c,

bf

respec-

draw the diagonals af, ac,

left, will

m,

and

These diagonals, cutting the

drawn from

/,

e

i

and 2 to the right and

determine the points

n,

Oy

of the frame. all

h, k, /,/,

from which complete the

inner squares of the right and

to secure

be,

lines

It will

left faces

be observed,

that,

the varying dimensions of

the framework, only one measurement

need

be

determined

apparent width of the nearest upright standard. nation of this one quantity, as a matter of course,

all

by means

;

viz.,

From

in Xy

the

the determi-

the other remaining dimensions follow of the diagonals,

and

of the converging

DRAWING THE SINGLE

mI

Extend

sets of lines.

to

3,

and ^

>^

CROSS.

75

and draw

to 4,

lines

from

d and a e. Then draw Where these diagonals

3

and from 4 convergent respectively with a

the

diagonals of the upper face of the cube.

cut

the lines from 3 and

4, fix

the angles of the inner square, as in the

case of the two side faces, and complete the square.

Now k,jy

draw from

and

lines

t

and also from frame are

m, and

n^

s lines

convergent with b

Cy

and from

convergent with bf: draw vertical lines from rand/,

and

^, u,

as far as visible.

v,

visible, as, for instance, lines

represented with their

proper

frame-cube will not be found

If

other inner lines of the

from z and

they

The drawing

convergence.

difficult if

j/,

may be of

the

the method here indicated

is

diligently followed.

With at

this

model the danger

some things without

is,

strictly

that a pupil will undertake to guess

observing them, and following the

order and method here laid down.

Such

with a great loss of time, to an entire

efforts will generally lead,

failure.

DRAWING THE SINGLE CROSS. Let

be in either a

it

draw

first

the

squares,

b' c'

d\

vertical, horizontal, or

JFig-.

a b

c dy

ci

d'b" c" d" and

73),

(Figs.

to

an inclined position

;

7i

and

71,

inclose

72,

the

cross in the several positions.

Next draw the

dia

agonals to these squares,

and take the apparent middle division

of

one side of a square equal

MODEL AND OBJECT DRAWING.

76

to the thickness of the arms.

From

these points draw lines through

the squares, parallel to the adjacent sides, cutting the diagonals in points

I,

2, 3,

4

through these points draw two lines parallel

:

converging with, the other two

Tig.

lines,

may be

as the case

;

to,

or

this will

72 Xiff.

73

/

/

/ /

/\ ^io//

/

'~~~/

/;f

/

/

'

-^

v^

/-V7

/

\

/

\

/

7

/,AV /

-
r*

r^~~~~/~~

~y~

-

tp

y // //V 52 /

/

/

/\""'

's^-'<-.L

-r-W

/

\

/ \ \

\

J

/ /

/

0"

•«,,^

'^.

'^^

J:m -~'-'K<^

complete the face of the cross.

Next,

a similar manner, draw

in

lines

from the points

cate

the thickness of the cross, and then complete the drawing by

the lines which

5, 6, 7, 8, 9,

make up

10, parallel or

converging, to indi-

Care must be

the back face of the cross.

taken to draw each line with

its

proper convergence, where there

own system is

of parallels,

and with

its

any.

DRAWING THE DOUBLE CROSS. The double

cross

naturally

comes

after the

should be drawn at least in two positions. shaft upright

and the other two horizontal

:

frame-cube, and

First place

draw,

first,

a

b,

it

it

with one

the nearest

DRAWING THE DOUBLE vertical of the upright shaft (Fig. 74). of either of the horizontal

suppose

to

it

be one-seventh

seven equal parts

draw the

arms

3,

77

Compare the apparent thickness with the line a

of the shaft

then divide the vertical

and from points a and

;

and

lines 2

:

CROSS,

b

Let us

b.

line,

marking the middle

a

into

b,

division,

indicating the positions of the two horizontal

arms, observing the true proportion, taking care to

make

the nearer

longer than the farther arms.

Having drawn the the

3,

and

lines 2

every

inclination of

other line in the drawing will

have been determined when

amount

the

of

convergence of

the other lines has been fixed.

Draw tical

the two remaining verlines,

4 and

5,

and the

apparent form of the square

on the upper end tical shaft

two

:

of the ver-

then construct the

visible sides of the square

at the

In the same

lower end.

manner draw the

visible

ends of the horizontal

shafts,

same

visible lines of the invisible squares of the

and the two It

shafts.

will

be

seen that there will be, in this position of the double cross, two sets of

convergent

will vanish to right.

the

To draw

requires student.

lines,

strict

nine lines in each set

left

this

;

and the other

model

in

this

:

one

set, i\

and

in

2',

set,

i,

3',

4^

2,

3,

etc.,

4,

etc.,

to the

the following position

attention and close observation

on the part of the

MODEL AND OBJECT DRAWING.

^S

The will

we

next position in which

be that

which

in

In this case

in front.

rests

it

we should draw end

position of the arm, one

of

which

one being directly

of its arms,

first

the line a

by

indicating the

The

nearest to us.

is

appears to be nearly vertical, leaning a

will suppose,

Divide ^

suppose this model to be placed

will

upon three

so as to get the central division, as in the last case, observ-

/^

ing that the seven equal divisions of the line a b will present to

eye a series of diminishing quantities from b to

Take the middle seventh

receding.

piece

and make

in

Complete the figure by drawing

all

own

its

own

3, 4, etc.

b, c,

system.

and d are

Observe proportion

etc.

ellipse

lines

first

be seen

It will

belonging to the three different

set,

set of lines,

converging upward,

converging downward to the

third set, converging If

each system.

and each may be considered the leading

;

The

the second

;

The

etc.

3'', 4'',

an

relative

lateral

In this case there will be three sys-

system.

systems of convergence

4',

true

d.

the subordinate lines, each con-

of convergence, with nine lines in

that the lines a

in its

their

is

75).

verging with

tems

these

the

because the line

for the thickness of the

arms

carefully, (Fig.

a,

draw, with their true inclinations, the lines c and

:

we

line,

the right.

little to

downward

left, is

to the right,

is

i\ is

line i, 2,

2',

3',

i", 2'',

the drawing has been accurately made, the curve of

can be drawn through the eight points at the ends of the

The curve may be

four arms.

lightly sketched, as an aid in the con-

struction of the drawing.

Having studied the main principles analysis

of

the geometrical

of

model-drawing from an

conditions under which various forms

appear, these principles must be put into practice

models.

For

this

by the use

of the

purpose each model should be drawn carefully

in

i

DRAWING THE DOUBLE several positions.

able

to

reliance

draw

at

Practice should be kept up sight

any model

in

copies of drawings of models.

poem

in

of

the

student

the

practice

of

is

No

making

much time thrown away. order to learn how to compose one. models is at the present time much

It is

And, although copying pictures

as well copy a

until

79

any possible position.

whatever should be placed upon

One might

CROSS.

only so

MODEL AND OBJECT DRAWING.

8o

many

practiced in

of

our public schools,

I

am

satisfied that the pupils

learn less and less of model-drawing as the

""^Ht;;^

practice continues.

^^[^

\

/

We

have already seen the use of

diagonals

\\

when drawing the frame-cube

the (Fig.

3

\

10

"jS).

Take now the frame-square, and draw

it

/ in several positions,

/s

\

drawing the lines

in the

/

order of

/

/

first six lines, fix

\

/

^A^

--!2

Having placed the

the numerals.

the points 7 and 8 by com-

paring the width of the pieces of the frame

with the line

9

i,

and draw

lines

from them con-

7...

"^ (Fig.

8

verging with 2 and diagonals in points

9,

They

4.

10,

11,

will

cut

the

and 12: these

points determine the inner lines of the frame

Place the points 13 and 14 by drawing lines from 7 and

'jf).

converging with the end-

which give the thick-

lines,

ness

draw

:

with

there

right,

is

any

From

in

and from 14

line

12

on

the up-

in the in-

draw the the

visi-

back side

this will cut the line

or 12,

and

in this set of

13

clined figure, ble

11

same convergence,

the

lines.

the visible

also

from

inner lines

if

^

\

from

11

and give the thickness

on the

inside.

The remaining

lines will follow in their appropriate

DRAWING THE DOUBLE The

places without any difficulty.

CROSS,

8i

success of the drawing will depend

upon following attentively the order here given.

There

another

is

application of the use of diagonals of a rectangle in sketching buildings,

which we may notice here

(Fig. yZ).

Let us suppose we have drawn the vertical rectangle

i,

2,

4,

3,

representing the end of a house, and that the gable, or point of the roof, is vertical

with the real center of the rectangular end

ing the diagonals

and from

where on

with line

i,

6,

we

Find the

roof

;

the angle of the roof will be some-

alti-

by comparison

Tig. 78

and draw to the upper

angles of the end-lines 8 and

As

by draw-

find the real center of the rectangle,

vertical line

this line.

the

tude of

and

5

draw a

it

:

the roof projects

ends, the line of

9.

over the

the ridge can be

drawn, and the projection made as indicated in the drawing.

The

center of the ground-plan, or of the front

of the house, can be obtained in the same way, for the purpose of

cing the front door, or any central feature divided into halves, quarters, and eighths,

Observe always that the intersection any

is

etc.,

by means

of the diagonals,

of the diagonals of a rectangle

position, perspectively represented, gives the real center of the

rectajzgle, it

and these rectangles can be

purpose of placing windovv^s, or other features of the building.

for the

in

;

pla-

and

in the

not the apparent center.

The

place of the chimney,

if

middle of the roof from one end to the other, can be

placed by drawing the diagonals on the roof and through their intersection,

drawing the

roof 8 and 10

:

line

upward convergent with the ends

this line will cut the ridge in the center.

of

the

MODEL AND OBJECT DRAWING.

S2

Place the cube in

two or three positions, and draw

First, in a vertical position,

on a horizontal plane, a

at sight.

it

little

below the

Tig. 80

eye.

each

There set,

will

i, 2, 3,

be two sets of converging

to the left,

and

lines,

with three lines in

to the right (Figs. 79

i', 2', 3',

and

80).

Second, in an oblique position, drawing the lines in the order of the

numbers, observing the three sets of converging 3',

as in the figure above,

and

i^\

2"

3'', ,

lines,

i, 2, 3,

and

i\

2',

converging downward.

Third, place the model in a vertical position, showing the right or left

face

side

narrow in

first

especial care to riS'

wider than

SI

it

(Fig. 81)

true

its

make

:

draw the wide

proportion,

the narrow side no

really appears.

each of

its

i,

2, 3,

this,

to

by means

of the pencil held in the usual

Having drawn the narrow

tion of the lines

To do

remember

compare the horizontal width,

manner, with the length of the line drawn.

taking

the face c

is

face

easily

b,

first vertical

and found the

inclina-

represented by drawing

further boundary-lines converging in their respective sets.

DRAWING THE DOUBLE The

four-sided prism should be

drawn

CROSS.

83

in several different positions,

taking care to note the several

systems of converging

lines

and their directions

(Fig.

%2).

The amount

convergence should

be

in

all

of

cases

determined by

close inspection of the

mod-

els themselves,

the degree

convergence

depending

of

on the distance of the eye.

The

triangular

prism

may

be placed in a variety of positions altitude c

(Fig. 83).

d must

The

be drawn after the base-line a

b,

remembering that the

T^lj.83

nearer half of the base will appear the longer

when the base ^

^

is

a

MODEL AND OBJECT DRAWING.

84

retreating line, and that consequently the altitude c

be beyond the apparent center of a

d must

appear to

b.

Tis. 85 CFig.

87 /CEi— |— :f^

"Fig.

8B

/

h

\

N^-aj---,^^

Place the hexagonal prism on the triangular prism, taking especial care to read according to the

method we have

the sets of converging lines (Fig. 84).

indicated, observing all

DRAWING THE DOUBLE Vases should be drawn with the

axis, find

and widths

CROSS.

85

Beginning

in various positions (Fig. 85).

the proportionate lengths

of the ellipses of the bases,

and

determine the greatest and least diameter,

and the position points

I

and

2,

each on the

of

as

axis,

thus fixing the height of

each.

Observe the use

of section lines at right

angles to the axis, in drawing symmetrical figures

(Figs.

^6 and

87).

As many may

be drawn as desired at equal or unequal distances from each other, provided they are

always at right angles to the axis

be bisected by the

two parts See

<2,

axis.

:

of these section lines will

b^ c, d,

they

will

In other words, the

be equal.

e,f,g, etc., in the illustration.

model

This

drawn

two or three

in

First

tions.

should

upright,

be

posi-

and

then resting upon one side (Fig.

The

^Z).

latter

posi-

tion will try the skill of the

pupil

in

reading

correctly

the apparent form (Fig. 89).

Care should be taken shorten

the

length

proper proportion.

to forein

the

For

this

purpose, compare the apparent length of the axis with the greatest

MODEL AND OBJECT DRAWING.

86 diameter

;

one end of the axis

as

is

the pupil must imagine

invisible,

where on the surface

model he could

of the

place a point that would cover the invisible end

Having drawn the

of the axis.

axis in its ap-

parent proportion with the greatest diameter,

proceed to place the several apparent diameters, as in

the preceding examples.

This model

one

is

ever,

is

on

based

the

any

first

dow7i

of

sketch to

is

see

LIGHT, SHADE, REFLECTED LIGHT, CAST

cations, approaches

of light

is

it

to

drawn

correctly

SHADOW, AND REFLECTIONS. all

its

interesting modifi-

what one might

call

real art,

the subjects hitherto discussed, which pertain to construction. will

be found here, as well as everywhere else in

dealing with absolute law, and that there

There

will

is

art,

that

than

But

we

it

are

no room for guess-work.

be in this department ample chance for the exercise of

close observation, quick apprehension of principles,

and taste

any

:

once.

and shade, with

more nearly

the

after

made, turn the drawing upside if

want of symmetry will thus be seen at

The department

of the outline.

vases,

of

how-

which may be

oval,

forms

these

difficult

Its form,

90).

some parts

slightly modified in

In

more

the

of

ones of the series (Fig.

and

of great care

in execution.

Light, as direct light

we

is

treat

it,

in respect to objects is of

two kinds.

First,

that from the sun or from some other luminous body

LIGHT, SHADE, second, diffused light

is

REFLECTED LIGHT, ETC, which

that

jDcrvades,

in

Sy an

the daytime,

ordinary room. In the

and cast definite

form of

first

shadows

light,

will

limitations,

and

geometrical formulas.

the illuminated surfaces, parts in shade,

mathematical

all

possess

will

have to be dealt with

proportions

Light and shade in this kind

light

of

Geometry, and forms by

regarded as a part of Descriptive

and

in reference to

itself

is

a

separate subject.

But the treatment

of objects in the

the subject of our present inquiry.

diffused light of

objects will be quite different from that of the light,

and yet there

For the purpose left

shoulder to

Take,

first,

is

same objects

of studying these effects,

common

one should

For

this

purpose

other half closed object

let

it

in

he

have

will

front, at a

If

will

be the

it

most

likely

to learn /low to

study attentively the light and shade upon the

ample time must be given

the pupil sitting in this

;

:

with the

sit

the student close one eye, and then with the

in order to see correctly,

:

process

and shade, he

of the student to read light

not be able to see the nicer differences see.

in sun-

of light.

white cube, and place

convenient distance, say six feet or more, from the eye. first effort

is

law pervading both classes of phenomena.

a window, the only source

the

room

a

In this light the appearance of

repeating the effort often, until

all

manner that

is

to this

one, two, or five minutes, to

be seen

is

fully appre-

hended. First, in light,

it

will

be observed that a part of the surface of the cube

and a part

in shade.

Let the student make with

very light sketch of the outline of the object. lines

the right face, which

is

in

is

a pencil a

Shade with

shadow, and then darken

vertical it

with

MODEL AND OBJECT DRAWING.

88

oblique cross-lines in one or two directions, and

fill

open checks

in the

with dots or dashes, to destroy or modify the netted appearance.

Now

be important to note the modifications of the shade on

will

it

this side.

It

should be observed that

the whole face, but

is

it is

not of a uniform depth over

darkest near the front edge, and at the upper

part of the surface in front, at point

A

while

;

it is

lightest at the

and lower part *^

^-s^^^s^-,.^

C

face near

This

light

it

in

part from

from

reflected

plane

rests,

from

(Fig. 91).

last modification

results

the

back

of the

its

on which

and

in

part

contrast with

the darker cast shadow.

The near

the same face

part of

is

dark-

ened by contrast with the high light on the opposite side of the line

along the line

F

A B,

G, and darker at

The

top

lighter at

G

may be

A B.

The

A

illuminated

than

and next

side,

along the line

B,

face will be lightest

and darker along the

line

than at F.

in a

lower light than the right

the position in reference to the light

A F,

at

left

:

to the illuminated face

A D.

It will

it

;

will

side,

according to

be darkest along the line

and lightest next to the dark

be seen that the three faces of the

cube, in this position, present a series of contrasts of light and shade,

along the three lines running to the nearest solid angle, A.

It is

LIGHT, SHADE,

REFLECTED LIGHT, ETC.

exactly in the order of these contrasts that the drawing

express

relief,

shade on the three faces ing

a

as

light

is

made

to

as will be seen hereafter.

be observed, that

It will

89

in Fig.

92 the proportion of light and

exactly reversed

the arrangement

cube,

solid

is

and shade shows that

it

;

and, instead of appear-

of

represents a

half of a hollow cube.

now

In reference

we

to the

cube (Fig.

find that the following facts

observed

:

91),

have been

— On

First,

highest light

an

illuminated

on the nearest part

is

the

plane,

of the

plane.

On

Second,

a semi-illuminated plane, the deepest shade

is

adjacent

to the illuminated plane.

Third,

When

a plane

is

in

shadow, the deepest shade

part of the plane nearest the eye likely

;

on that

and reflected lights would most

appear on the more distant and lower part of the plane.

Fourth,

The

of the object

cast

shadow is darker than the adjacent shaded surface

which cast

it,

and the darkest part

always be nearest to the object casting

The

principles

developed above

rectangular solids, and, with some

which

is

light

of the

will

be found applicable to

modifications, to

and shade may be conveniently studied.

small the object,

if

shadow

all

same

facts

of

light

light, half-light,

shaded surface, affected more or

and cast shadow

:

less

all

objects on

For however

no more than the thousandth of an inch

diameter, there would be the

will

it.

in

and shade, high

by reflected

light

so that an attentive study of the cube in light and

MODEL AND OBJECT DRAWING,

90 shade

will

develop the principles of the whole system of distribution.

Let the student study the cube in light and shade, and draw there

is

nothing more to learn from

For the next example

in light

until

it

it.

and shade,

let

us take the cylinder,

fall

upon

giving us a curved surface (Fig. 93). Place the cylinder so that the light will

over the

it

left

shoulder, and observe the posi-

mj.9. tion

the lights,

of

flected lights,

deepest

the outline

the top, flected

The

comes

the

little

light,

The

bottom.

pears on the

way

in

the

lightest

way

on

by reason less

dark

of reat

half-light,

the

highest light apleft

side,

a

from the outline:

upper base

from

in

the darkest at

is

it

:

and,

re-

and shadow.

shade

right side, a

shades,

at

the

top.

little it

is

The

of the cylinder is in

lightest

next to the

deepest shade on the right side,

and darkest light

at the

back and

left.

In this position neither the highest

nor the deepest shadow occurs at the outline of the model, as

in the cube,

tiguous.

where the highest

light

and the deepest shadow are con-

In natural scenery these contrasts frequently occur in juxta-

position.

we have become acquainted We may now light and shade.

In the study of the cube and cylinder with

many

of the first principles of

LIGHT, SHADE, take up the sphere

;

REFLECTED LIGHT, ETC,

and, since

points of observation,

its

apparent form

we may compare

91

a circle from

is

all

with other objects whose

it

apparent forms are represented by a circle such as the plane circle

and the hollow hemisphere, the cone with the apex toward the eye, a hollow cone with the apex away from the eye, up,

set

sented

in

drawing to

etc.

For

purpose

outline

by a

circles

five

circle,

and

repre-

then,

on the paper, proceed

study and represent the several forms,

with

all

their nice modifications

more useful than the

and

distri-

faithful study of these

objects.

A represents circle

the

flat

shaded surface of a

in nearly uniform tint (Fig. 94).

study of the light and

sphere, as

it

it

m M ,



Nothing can be

butions of light and shade.

the

this

obtainable, these five objects, so that each will be

if

ilii

IP

B,

shade of the

appears to the pupil, with the high light on the upper

lEi^.

95

left-hand side, but a little in from the outline

The deepest shadow

of the circle (Fig. 95). is

seen on the lower right

est near the outline

on the shaded on which

it

side,

rests.

:

side,

but not dark-

reflected light

is

seen

caught up from the plane

The

cast

shadow on the

plane would be, as far as visible, in the form of

an

C 96).

Its

ellipse.

represents the hollow hemisphere (Fig.

shaded surface, in a proper

upon the inner surface by the rim

light, is a cast :

it

shadow thrown

has, therefore,

the disposi-

MODEL AND OBJECT DRAWING.

92

tion of a cast shadow, with reflected light ^

_^.,

which belongs to a shaded

surface.

\

D

represents the cone, with the apex

towards the eye

(Fig.

97).

The deepest

shade will appear under the apex on the

shaded

side, the

highest light being on the

opposite side of the apex.

/

will

It

be ob-

served that near the base, on the light side, it

the base, on the shaded side,

must be it

must be

darkly shaded than under the apex as

we approach

there

is

less

the base from

and

:

shaded

;

and that near

less

so that,

the

difference between

and the shaded side than there apex.

slightly

apex,

the light is

near the

Fix clearly the fact that both the light the shade are focussed 7iear the apex^

on exactly opposite

sides.

E

represents

Notice how tion

of

all

light

the hollow cone (Fig. 98).

the conditions of the distribu-

and shade are reversed from

those in the cone.

i

The means

to be

employed

in

represent-

ing light, shade, and shadow are various selection

may be made

coal-point,

ink,

the

according to the prefer-

ence of the teacher or pupil.

pen and

:

lead-pencil,

We

may

use the

crayon-point, char-

and stump with charcoal or crayon, or the brush with India

ink, or with

any monochrome.

LIGHT, SHADE,

When

lines are used,

REFLECTED LIGHT, ETC,

93

they should be laid as evenly as possible,

and with nice gradation

in

passages of varying depth.

Flat

tint

should be laid with one set of lines running in the same direction

where only one depth

set is used,

required, two or

is

is

it

more

called half-tint (Fig. 99).

sets of lines

may be used

:

If

:

more

the differ-

ent sets should cross each other at an acute angle, as in the illustra-

ill

l!l!il

111

iiii

li!

ill

100

This process

tion, Fig. 100.

be drawn

far at

as often as fine,

is

.±0±

one stroke

convenient.

wiry line

;

is

of the pencil

A

it is

:

rather broad line

and the spaces between

not wider than the lines themselves. for vertical plane surfaces,

The

called hatching.

lines should not

better to is

much

lines should

lift

the pencil

better than a

be uniform, and

Vertical lines are appropriate

and horizontal

lines for horizontal surfaces.

Straight lines should be used for plane, and curved lines for curved, surfaces

;

or both

may be used on

the surface of the cylinder and cone,

MODEL AND OBJECT DRAWING,

94

where the surfaces are both straight and curved, but

in different direc-

tions.

Stippling^ with dots

resorted

to, in

between the

lines in the

order to produce a uniform

effect,

open checks, may be

and

to cause the lines

to blend (Fig. loi).

as

The study of shading should be pursued by drawing and shading many vases^ and other objects as possible (Figs. 102 and 103).

There

is

no danger that the student

too familiar with these objects.

will

draw too many, or become

After each study of a model, a rapid

drawing entirely from memory should be made.

may be compared

When

with the original drawing to test

its

completed,

accuracy.

it

This

REFLECTIONS, practice

is

95

of the greatest value in fixing in the

edge has been acquired

mind whatever knowl-

in the study of objects.

REFLECTIONS. an important element in pictorial

Reflections are

effect, and, in

connection with model-drawing, should receive a passing notice. First,

they are produced by a polished surface taking up the light

and conveying

of an object,

cube,

is

various

to the eye.

it

If

an object, such as a

placed upon a polished table, there will be present modifications

to these, they will

of

shade, and

light,

The

and, in

position of the cube, the

shadow

rests

shadow

cast

be fully reproduced in the reflection

hence the cast shadow

;

the

addition

be reproduced in the reflection of the object,

all

with some exceptions and modifications. ever, never

shadow

all

;

will,

how-

because, in

this

upon the plane

of reflection

:

be modified in proportion to the perfection

will

of the reflecting surface.

A vertical

line reflected

give

a vertical reflection.

104,

a,

with

reflecting plane

than

ay

DE

in the usual

A line

but

o! will is

reflection,

always

Fig.

in

on the

jFis.

104

be shorter

more or if

will

less

measured

way by holding up the

pencil.

inclined to the right, but not to the

front or back, as

more

;

according as the eye

above the plane of

See,

the reflection

as

a!

by a horizontal plane surface

inclined, as

of reflection, as

will

by

at

have

the eye b' :

e

is

and

its

reflection

at a greater distance

/

will

above the plane

have their reflections

/ and /'

MODEL AND OBJECT DRAWING.

96 respectively.

These statements may be

easily verified

by holding a

pencil in various positions against the face of a looking-glass, noticing

the position of the reflection in each case.

m

n

is

In the Fig. 104 the line

perpendicular to the central ray of light from the object to the

eye.

These statements facts

will

guide the student in his observations of the

is

may

There

When

ways.

be no further

will

the modifications under which reflec-

occur. is

another class of reflections with w^hich the student

have to become familiar; and that

more

many

in

apprehended, there

difficulty in its application to all

tions

them

sketching, as he can amplify

in

once the general principle

is,

will

where the objects themselves are which catch up

or less polished, giving reflecting surfaces,

lights

and colors from any illuminated objects near them, producing numerous modifications of

all

is^

The

the lights and shades hitherto noticed.

only law which governs this class of reflections,

indeed,

as,

all

others,

that the angle of reflection is ahvays eqnal to the angle of incide7ice ;

and the position

of

a reflected

light

on a polished object, as on

a polished silver or glazed earthen vase, will be position

of

the object.

the object which

Thus,

with the eye at

E

:

let

is

A, Fig.

let B, at

determined by the

the source of the light reflected by

105,

be a plan of a polished cylinder,

the same distance from the object as E,

be the source of the light reflected from the surface

draw

lines

from

E

and

B

to the point a'

angles with the circumference at that point

minated, because lines drawn from the point equal angles with the arc at that point.

a' will

:

ci

These

to

EB

To

plane perpendicular to the surface of the cylinder. of illumination,

;

E

lines

being in a

find the point ,

making equal

be the point

and B

will

would

also

illu-

make make

REFLECTIONS.

97 This

equal angles with the tangent of the arc at a\

where the point

of light

ter of the cylinder

a

;

but,

is

easily

done

and the eye are equally distant from the cen-

where they are unequally

distant,

it

becomes

problem to find the

difficult

loais of reflection.

(See Appen-

dix A.)

As

these reflections are sub-

jects of observation rather than of construction,

it

will

be

suffi-

cient in this connection merely to

indicate the law which gov-

erns them.

There

is

no class of phenom-

ena more interesting or captivating to the painter than reflections

and reflected

landscape

artist

light.

they

To are

the the

So much

source of some of his most pleasing effects.

is

he depend-

ent upon reflections in water for entertaining the observer of his

works, that a landscape picture without water est

;

is

while a very simple view with water, with

often devoid of interits

multitude of glan-

cing lights and fragmentary shadows, becomes at once pleasing and delightful.

Reflections multiply the quantities which

and harmonic

seem

series

make up

from which the mind derives

to suggest the idea of life

and

activity.

the rhythmical

its

pleasure,

and

APPENDIX. THE FOLLOWING

IS THE SOLUTION OF THE PROBLEM FOR FINDING THE POINT OF REFLECTED LIGHT ON A POLISHED CYLINDER. t

The

following problem depends

for its

surface of a polished cylinder, a point

hght

upon

upon the

finding,

where the angle of incidence from the

be equal to the angle of reflection from the point to the eye.

will

To

solution

find the place of the point of illumination

on a polished

the place of the light and the place of the eye are given

The problem assumes

When

I^t'rsfj

the point of light

When

cylinder,

when



four diiferent forms.

to the axis of the cylinder,

Second,

:

and the eye

and are equally

are in a plane not at right angles

distant from the cylinder.

the plane, in which the eye and light are located,

right angles to the axis of the cylinder,

not at

is

and the points of the eye and

light are

unequally distant from the cylinder.

When

Third,

the eye

and

light are in

a plane perpendicular to the

axis of

the cyhnder, and equally distant from the cylinder.

Fourth,

When

the plane, in which the eye

dicular to the axis of the cylinder,

and

light are located, is

and the point of the eye and

perpenlight are

unequally distant from the cylinder.

The

first

third form

E

and L,

is

and second forms of the problem are not of easy

solution.

The

solved by drawing tangents from the points of the eye and light

to the circumference of the cylinder

ing the angle formed by the two tangents

:

on the near

side,

and by

bisect-

the bisecting hne cutting the center of

the cylinder will also cut the circumference at the point of illumination. 99

MODEL AND OBJECT DRAWING.

100

The it

fourth form of the

problem

Let

E

and

the cylinder

A

L :

not so easy of solution, and

is

manner

can only be solved in the following

:

be the points of the eye and

draw

seems that

it

— light

several concentric circles B,

at

D, G,

unequal distances from etc.

Draw

tangents to

^ppendSx

the circumference of the cylinder

and

to

each of the concentric

pair of tangents intersecting each other in a, b, d, e,f.

circles

;

each

Now, by constructing

a curve passing through these several points of the intersecting pairs of tangents, the curve will cut the circumferences in the points of illumination,

through the center of the cylinder. cylinder in the point of illumination

:

This curve for

we

will

and

shall find that the angles

\

will pass

cut the surface of the

formed by

APPENDIX, lines will

drawn from these points of the curve, intersecting the concentric form equal angles with the circumferences of the

intersection of the curve

the angle of reflection. to these several points will

lOl

make equal

;

circles at the points of

thus showing that the angle of incidence

Or, in other words, the

circles,

lines

is

drawn from

on the circumferences, found by the

angles with tangents drawn through the

equal to

L and E

intersecting curve,

same points; thus

proving that the angles of incidence and reflection are equal, and showing that the point found by the construction of the curve

is

the point of illumination,

the locus of reflection.

Note

i.

— The curve becomes a curve

arrangement of the points through which

Note

2.



If tangents are

side of the cylinder,

drawn

of the fourth degree by virtue of the

it

to the

passes.

same concentric

and the curve extended through the

tangents, the curve will give the points of illumination

cylinders arranged in place of the circles.

circles

on the

far

intersections of the

on the inner

surfaces of

There would seem to be no other

simple solution of this problem that could be worked out visible to the eye.

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