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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 SELFINSTRUCTION OF TEACHERS AND ARTSTUDENTS IN THE THEORY AND PRACTICE OF DRAWING FROM OBJECTS
^^4
BY
LUCAS BAKER
tf'
n
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•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
6668
The Law of Rims Demonstrated
The Drawing of
70
Ellipses
71
Drawing the Triangle and Triangular Frames
.
,
.
72
.
The FrameCube
.
Drawing the Single Cross
Drawing the Double Cross Drawing the FrameSquare
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.
artschools
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
artinstruction,
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
artstudent, 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 guesswork
every question
and
not to know,
is
not to be able to draw.
ObjectDrawing has a basis
subject of
can be
;
In this
of
fact
throughout.
mathematical precision pervades the whole settled
by reference
to
fundamental prin
ciples.
Modeldrawing
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
modeldrawing,
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. Modeldrawing 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 readymade 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
Rightangle 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 halfplane of the
by the semicircumference
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 rightangle 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,
shipbuilders,
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 fourinch 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 fourinch square.
We may
call this
shadow the
With the square
in
vertical projection of the square.
the same
position, suppose the
sun directly
overhead: 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
groundline,
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 groundline,
to the vertical plane
groundline,
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 foursided 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 fourinch 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 penknife, 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 halfinch 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 sixtyfour 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 cigarbox,
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 posteriorwall, 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 crosssections. 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 crosssection 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 crosssection 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 onehalf 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
sidelines 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 sidelines 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 onefourth 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 sidelines of the two faces, a b being the dividing line sidelines will i.e.,
:
Now
and with the
take the two faces sidelines
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 sidelines 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 sidelines 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
twothirds 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 sidelines 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 sidelines 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 subsection, 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.
crosssection
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 halflength 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 halflength 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 sidelines 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 sidelines 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
^^sAs 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.
sidelines 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 modeldrawing 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, onehalf oi c
A B,
Divide the axis
and the upper half
A
the
any length
A
it is
mean
"
the upper base
quarterlength thus obtained, on a line perpendicular to
this case,
the axis;
not
in a vertical position of
be onehalf 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 onefourth
8).
onefourth
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 sidelines
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 onethird 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 twothirds of the axis.
ac
as before will
aside guesswork until as
much
shorter as
to estimate the shorter
the same by observing the halfellipse 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 propoition
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 onequarter 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
quarterpoints 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 frontline 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 onesixth 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 FRAMECUBE. 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,
onesixth, 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
framecube 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
rW
/
\
/ \ \
\
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
:
framecube, 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 oneseventh
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
modeldrawing 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 modeldrawing as the
""^Ht;;^
practice continues.
^^[^
\
/
We
have already seen the use of
diagonals
\\
when drawing the framecube
the (Fig.
3
\
10
"jS).
Take now the framesquare, 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 endlines 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 groundplan, 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 boundarylines converging in their respective sets.
DRAWING THE DOUBLE The
foursided 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 baseline 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 guesswork.
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 crosslines 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 semiilluminated 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, halflight,
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
halflight,
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
lefthand 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
coalpoint,
ink,
the
according to the prefer
ence of the teacher or pupil.
pen and
:
leadpencil,
We
may
use the
crayonpoint, 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 halftint (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 modeldrawing, 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 lookingglass, 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.