0.5

1 cm

GEOMETRICAL CONSTRUCTIONS—DRAWING OF ELEMENTARY FORMS.

S3

tions of ellipses. It is employed in striking out vaults,

and stone-bridges.

An instrument has been prepared by Mr. James Finney,

for describing ellipses, represented in (Fig. 150). The

pen A is made with a socket, adjustable upon a graduated

radius lever fixed on the upright spindle B, finished with

a milled head H, by which it is turned. The spindle

revolves in a brass block N, fitted into a slot in the plat-

CFig. 150.)—Scale Half Size.

Finney’s ELLiPTOGB.AfH.

form M, in which it is capable of sliding longitudinally.

The spindle is adjusted horizontally in a circular disc K,

by the screw R, to regulate the eccentricity; and the disc

is let into the socket L, made with parallel sides to work

between the guides h, h. The whole apparatus is sus-

tained by and screwed to the block 0. On turning the

spindle, which carries the pen with it, the disc rotates ;

and as the nut is confined to the straight path of the slot,

the disc must have a transverse movement in combination

with its revolving movement, carrying with it the frame

L between the guides h, h. By thus combining a longi-

(Fig. 151.)

E

tudinai with a circular motion of the pen, an ellipse is

formed, of which the length is equal to twice A C, and

the breadth equal to twice A B. (Fig. 151) illustrates the

geometrical character of these motions.

The Parabola.

'TIE

-AT>

\

Definition.—A parabola (Fig. 152), has every point in

its curve equally distant from the directrix, E N, and

focus, F.

Problem LIY.—To construct a parabola, when the

focus and the directrix are given.

1st Method.—Through focus, F, draw axis, A B, perpen-

(Fig. 152) ( dicular to directrix, E N, and

bisect A F at e. Then e is vertex

of the curve. Set off distances,

Jc A C, A D, A E, from A to E, and

i also from A to N; draw lines from

-Ya---A focus, F, to points, C, D, E, &c.;

-j through points, C, D, E, &c., draw

parallels to axis, A B; bisect lines

from F to E N, by perpendiculars

which will meet parallels in the points, G, H, I; through

which, and vertex, e, one side of curve is to be

drawn. Transfer distances, E I,

D H, C G, to parallels on the

other side of axis, and so determine

points through which the second

half of the curve is to be drawn.

2d Method.—Place a straight-

edge to the directrix, E N, and

apply to it a square, L, E, G.

Fasten to the end, G, one end of

a thread or cord, equal in length

to the edge, E G; and fix the

other end to the focus, F; slide the square steadily along

the straight-edge, holding the cord taut against the edge of

S3

tions of ellipses. It is employed in striking out vaults,

and stone-bridges.

An instrument has been prepared by Mr. James Finney,

for describing ellipses, represented in (Fig. 150). The

pen A is made with a socket, adjustable upon a graduated

radius lever fixed on the upright spindle B, finished with

a milled head H, by which it is turned. The spindle

revolves in a brass block N, fitted into a slot in the plat-

CFig. 150.)—Scale Half Size.

Finney’s ELLiPTOGB.AfH.

form M, in which it is capable of sliding longitudinally.

The spindle is adjusted horizontally in a circular disc K,

by the screw R, to regulate the eccentricity; and the disc

is let into the socket L, made with parallel sides to work

between the guides h, h. The whole apparatus is sus-

tained by and screwed to the block 0. On turning the

spindle, which carries the pen with it, the disc rotates ;

and as the nut is confined to the straight path of the slot,

the disc must have a transverse movement in combination

with its revolving movement, carrying with it the frame

L between the guides h, h. By thus combining a longi-

(Fig. 151.)

E

tudinai with a circular motion of the pen, an ellipse is

formed, of which the length is equal to twice A C, and

the breadth equal to twice A B. (Fig. 151) illustrates the

geometrical character of these motions.

The Parabola.

'TIE

-AT>

\

Definition.—A parabola (Fig. 152), has every point in

its curve equally distant from the directrix, E N, and

focus, F.

Problem LIY.—To construct a parabola, when the

focus and the directrix are given.

1st Method.—Through focus, F, draw axis, A B, perpen-

(Fig. 152) ( dicular to directrix, E N, and

bisect A F at e. Then e is vertex

of the curve. Set off distances,

Jc A C, A D, A E, from A to E, and

i also from A to N; draw lines from

-Ya---A focus, F, to points, C, D, E, &c.;

-j through points, C, D, E, &c., draw

parallels to axis, A B; bisect lines

from F to E N, by perpendiculars

which will meet parallels in the points, G, H, I; through

which, and vertex, e, one side of curve is to be

drawn. Transfer distances, E I,

D H, C G, to parallels on the

other side of axis, and so determine

points through which the second

half of the curve is to be drawn.

2d Method.—Place a straight-

edge to the directrix, E N, and

apply to it a square, L, E, G.

Fasten to the end, G, one end of

a thread or cord, equal in length

to the edge, E G; and fix the

other end to the focus, F; slide the square steadily along

the straight-edge, holding the cord taut against the edge of