0.5

1 cm

112

THE ENGINEER AND MACHINIST’S DRAWING-BOOK.

ence. In the horizontal wheel the centre lines and the

width of the roots of the teeth are drawn on the plan,

and visual rays drawn from these, give the divisions on

the picture-line, from which perpendiculars are drawn

to intersect the middle perspective circle. The visual

rays of the centres of the teeth, and their corresponding

perpendiculars only, are shown. The hither ends of the

teeth may be sketched in; but if the student cannot do

this properly, it will be necessary to draw the lines bound-

ing the upper surfaces of the teeth, all on the plan; and

by visual rays to find their perspective widths in the same

way as the widths of their roots.

The divisions of the teeth of the vertical wheels are

drawn by another method, which, as we have said, is

equally applicable to that of the horizontal wheel. The

line F F on the ground plan of the right hand vertical

bevel-wheel, is also the horizontal projection of the side of

its circumscribing square. This line intersects the picture-

line at the mark X. and consequently if a perpendicular be

drawn from x, and transferred to the perspective ground-

line, and the divisions of the teeth taken on a correspond-

ing vertical line on the geometrical drawing are set up

on this, then lines from these divisions drawn to the

right hand vanishing point will intersect the perspective

circle, and will give the divisions on it. In this, as in the

horizontal wheel, the centres of the teeth only are shown,

and the remark as to the means of completing the repre-

sentation of the teeth of the horizontal wheels, applies

equally to the completion of the representation of the teeth

in the vertical wheel. The teeth of the left hand vertical

wheel are formed by drawing from the divisions of the

right hand wheel, lines to the left hand vanishing point, to

intersect the perspective circles of the left hand wheel.

Part of the shaft, with its clutch and handle, are also

shown in this drawing.

Plate 70, figs. 4 A and 4 B (Perspective Lesson, Part

Fourth). To the plan are now added the details of the

clutch and the ribs, and other details of the wheels. The

circles of the clutch are obtained in the same manner as

the circles of the wheels. The ribs of the horizontal

wheel may be obtained by producing the original lines on

the plan, to meet the picture-line, as at G G, and trans-

ferring them to the perspective at g g, and finding the per-

spective of the square which their intersection forms at li h.

It may be almost unnecessary to repeat that the circles of

the shafts, and annular openings in the faces of the wheels,

are obtained from their circumscribing squares.

Plate 71, figs. 5 A and 5 B (Perspective Lesson, Part

Fifth). The plumber-blocks are now added to the plan.

All the directions necessary for finding the perspectives of

any points have been given in the previous stages of this

lesson, and if the student has carefully followed the pro-

cess, he will have no difficulty in finding the perspectives

of the plumber-blocks, and consequently completing the

entire figure. The pencil lines of construction being obli-

terated, we have a perfect perspective representation in

outline; and the figure thus completed, is given, fully

shaded, in the engraved title to the work.

SECTION II.

Isometric Projection.

This is a conventional manner of representing an object,

in which it has somewhat the appearance of a perspective

drawing, with the advantage of the lines situated in the

three visible planes at right angles to each other, retain-

ing their exact dimensions. For the representation of

such objects, therefore, as have their principal parts in

planes at right angles to each other, this kind of projection

is particularly well adapted. The name isometrical was

given to this projection by Professor Farish, of Cambridge.

The principle of isometric representation consists in

selecting for the plane of the projection, one equally in-

clined to three principal axes, at right angles to each

other, so that all straight lines coincident with or parallel

to these axes, are drawn in projection to the same scale.

The axes are called isometric axes, and all lines parallel to

them are called isometric lines. The planes containing

the isometric axes are isometric planes; the point in the

object projected, assumed as the origin of the axes, is

called the regulating point.

(Fig. 23i.) If any 0f flie solid angles of a

cube (Fig. 231) be made the regulat-

ing point, and the three lines which

meet in it the isometric axes, then

it may be demonstrated, that the

plane of projection to be such that

these axes will make equal angles

with it, must be at right angles to

that diagonal of the cube which

passes through the regulating point.

The projection of the cube will therefore be as A B C D

E F G in the figure.

Let r B, s D (No. 1, Fig. 232) be the side of a cube,

and r D the diagonal of the side, produce s D, and make

C D equal to the diagonal r D, complete the parallelogram

C A, B D, and draw its diagonal A D, which is then the

(Fig. 232.)

.agonal of a cube, of which r B D s is the side, and which

represented in plan in (No. 2), A'B C F. Through D,

THE ENGINEER AND MACHINIST’S DRAWING-BOOK.

ence. In the horizontal wheel the centre lines and the

width of the roots of the teeth are drawn on the plan,

and visual rays drawn from these, give the divisions on

the picture-line, from which perpendiculars are drawn

to intersect the middle perspective circle. The visual

rays of the centres of the teeth, and their corresponding

perpendiculars only, are shown. The hither ends of the

teeth may be sketched in; but if the student cannot do

this properly, it will be necessary to draw the lines bound-

ing the upper surfaces of the teeth, all on the plan; and

by visual rays to find their perspective widths in the same

way as the widths of their roots.

The divisions of the teeth of the vertical wheels are

drawn by another method, which, as we have said, is

equally applicable to that of the horizontal wheel. The

line F F on the ground plan of the right hand vertical

bevel-wheel, is also the horizontal projection of the side of

its circumscribing square. This line intersects the picture-

line at the mark X. and consequently if a perpendicular be

drawn from x, and transferred to the perspective ground-

line, and the divisions of the teeth taken on a correspond-

ing vertical line on the geometrical drawing are set up

on this, then lines from these divisions drawn to the

right hand vanishing point will intersect the perspective

circle, and will give the divisions on it. In this, as in the

horizontal wheel, the centres of the teeth only are shown,

and the remark as to the means of completing the repre-

sentation of the teeth of the horizontal wheels, applies

equally to the completion of the representation of the teeth

in the vertical wheel. The teeth of the left hand vertical

wheel are formed by drawing from the divisions of the

right hand wheel, lines to the left hand vanishing point, to

intersect the perspective circles of the left hand wheel.

Part of the shaft, with its clutch and handle, are also

shown in this drawing.

Plate 70, figs. 4 A and 4 B (Perspective Lesson, Part

Fourth). To the plan are now added the details of the

clutch and the ribs, and other details of the wheels. The

circles of the clutch are obtained in the same manner as

the circles of the wheels. The ribs of the horizontal

wheel may be obtained by producing the original lines on

the plan, to meet the picture-line, as at G G, and trans-

ferring them to the perspective at g g, and finding the per-

spective of the square which their intersection forms at li h.

It may be almost unnecessary to repeat that the circles of

the shafts, and annular openings in the faces of the wheels,

are obtained from their circumscribing squares.

Plate 71, figs. 5 A and 5 B (Perspective Lesson, Part

Fifth). The plumber-blocks are now added to the plan.

All the directions necessary for finding the perspectives of

any points have been given in the previous stages of this

lesson, and if the student has carefully followed the pro-

cess, he will have no difficulty in finding the perspectives

of the plumber-blocks, and consequently completing the

entire figure. The pencil lines of construction being obli-

terated, we have a perfect perspective representation in

outline; and the figure thus completed, is given, fully

shaded, in the engraved title to the work.

SECTION II.

Isometric Projection.

This is a conventional manner of representing an object,

in which it has somewhat the appearance of a perspective

drawing, with the advantage of the lines situated in the

three visible planes at right angles to each other, retain-

ing their exact dimensions. For the representation of

such objects, therefore, as have their principal parts in

planes at right angles to each other, this kind of projection

is particularly well adapted. The name isometrical was

given to this projection by Professor Farish, of Cambridge.

The principle of isometric representation consists in

selecting for the plane of the projection, one equally in-

clined to three principal axes, at right angles to each

other, so that all straight lines coincident with or parallel

to these axes, are drawn in projection to the same scale.

The axes are called isometric axes, and all lines parallel to

them are called isometric lines. The planes containing

the isometric axes are isometric planes; the point in the

object projected, assumed as the origin of the axes, is

called the regulating point.

(Fig. 23i.) If any 0f flie solid angles of a

cube (Fig. 231) be made the regulat-

ing point, and the three lines which

meet in it the isometric axes, then

it may be demonstrated, that the

plane of projection to be such that

these axes will make equal angles

with it, must be at right angles to

that diagonal of the cube which

passes through the regulating point.

The projection of the cube will therefore be as A B C D

E F G in the figure.

Let r B, s D (No. 1, Fig. 232) be the side of a cube,

and r D the diagonal of the side, produce s D, and make

C D equal to the diagonal r D, complete the parallelogram

C A, B D, and draw its diagonal A D, which is then the

(Fig. 232.)

.agonal of a cube, of which r B D s is the side, and which

represented in plan in (No. 2), A'B C F. Through D,