31. To make a triangle equal to a given quadrilateral, as A B C D.

Prolong the line в A, and draw the line A C; draw also the line D E parallel to a C, and cutting the line BA in E; then draw the line E C, and c E B is the triangle required.

E

D

C

B

32. To form a square nearly equal in area to a given circle.

Let AC BD be the given circle: draw the diameters A B and C D at right angles to each other, bisect the radius d в in e, and draw the line cef; draw also at right angles the lines c n and fr, making each equal to cf; join n r, and n cfr is the square as required.

n

D

Λ

B

d

Note. The line sƒ is equal to one-fourth the circumference of the circle.

33. To inscribe or describe a square within or without a given circle: also to form an octagon from a given square.

to B D, and a d, b c, parallel to a C, which will complete the square equal in breadth to the diameter of the circle.

A

L

G

F

B

I

2. Let A B C D be the given square. Draw the two diagonal lines A C and BD intersecting in o; then with a radius equal to A o, or half the diagonal, and with a as a centre, describe the arc E F, cutting the sides of the square in E and F; then from B as a centre, describe the arc GH, and in the same

E

D

C

K

M

H

manner from C and D describe the arcs I K and L M. Draw the lines L G, F I, H M, and K E, which, with the parts G F, I H, M K, and E L, form the octagon required.

34. To form a square equal to two given squares, or, a circle equal to two given circles.

C

Let A B, A C, equal the sides of the given squares, or diameters of the given circles: make the angle at A a right angle, and draw the line C B, which is the side of a square equal to both the given squares; or bisect the line c B as a diameter, on which describe the circle C B A, which is equal to the two given circles as required.

A

B

35. To draw a right line equal to any given portion of the circumference of a circle.

Let A B C D be a given circle, the whole circum

ference of which is required: draw at right angles the diameters A C, B D, divide the d radius a c into four equal parts, and make c b equal to three of them; draw the tangent Ad parallel to B D, draw the line b D d, then will A d equal one-fourth of the whole circumference ; and if lines be drawn from b,

h

h

a

C

A

through points in the circumference, meeting the line A d, as g g, hh, &c., the corresponding parts will be equal to each other.

36. To draw a spiral with spaces of uniform

distance.

Bisect the height of the spiral, as A B, and divide either half into the number of spaces or revolutions required; then again subdivide any one space into four equal parts, one of which add to half the height of the spiral, and through which draw the line c Dat right angles to a B, thus form

B

C

D

ing the centre of the spiral, around which and equal to one of the subdivisions form a square, its sides being parallel with the lines A B and C D, the angles of which are the centres from whence to describe the various curves; as from 1, with the distance 1 B, describe the curve B D; from 2, with the distance 2 D, describe D A; from 3, with the distance 3 A, describe a c, &c., &c., and from the same centres the spiral may be continued to any extent required.

G. Elevation and Section of an Overshot Water-Wheel constructed by Messrs.
Donkin and Co.

241

H. Details

241

K. Boilers of the "Braganza" steam vessel, by Messrs. Bury, Curtis, and Kennedy
L. Locomotive Boiler

270

271

PRACTICAL GEOMETRY.

GEOMETRY is the science which investigates and demonstrates the properties of lines on surfaces and solids: hence, PRACTICAL GEOMETRY is the method of applying the rules of the science to practical purposes.

1. From any given point, in a straight line, to erect a perpendicular; or, to make a line at right angles with a given line.

On each side of the point a from which the line is to be made, take equal distances, as ▲ b, a c; and from b and c as centres, with any distance greater than b A, or c a, describe arcs cutting each other at d; join a d, which will be the perpendicular required.

b

2. When a perpendicular is to be made at or near the end of a given line.

Take any point n above a b, and with centre n and distance n A, describe a circle, cutting the given line at b; through b and the centre n, draw the diameter b n c, and join C A, which will be the perpendicular required.

A