25. To form a square equal in area to a given

As, or side of the square, then Astx is the square of equal area as required.

26. To form a square equal in area to a given rectangle.

Let the line A B equal the length and breadth of the given rectangle; bisect the line in e, and describe the semicircle A DB; then from

D

A

C

e

B

A with the breadth, or from B with the length of the rectangle, cut the line AB at C, and erect the perpendicular CD, meeting the curve at D, and CD equal a side of the square required.

27. To find the length for a rectangle whose area shall be equal to that of a given square, the breadth of the rectangle being also given.

the line Ag parallel to DF; from the intersection of

the lines at g, draw the line g d parallel to D E, and Ed parallel to Dg; then ED dg is the rectangle as required.

28. To bisect any given triangle.

Suppose ABC the given triangle; bisect one of its sides, as AB in e, from which describe the semicircle Ar B; bisect the same in r, and from B, with the distance Br, cut the diameter AB in v; draw the line vy parallel to AC, which will bisect the triangle as required.

C

A

B

29. To describe a circle of greatest diameter in a given triangle.

Bisect the angles A

and B, and draw the intersecting lines AD, BD, cutting each other in D; then from D as centre, with the distance or radii

A

DC, describe the circle Cef, as required.

B

30. To form a rectangle of greatest surface, in a given triangle.

Let A B C be the given triangle; bisect any two of its sides, as A B, BC, in e and d; draw the line ed; also at right angles with the line ed, draw the lines ep, dp, and eppd is the rectangle required.

p

с

Р

d

A

e

B

31. To make a rectangle equal to a given triangle.

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

Prolong the line B A, and draw the line AC; draw also the line DE parallel to A C, and cutting the line BA in E; then draw the line EC, and C E B is the triangle required.

E

D

B

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

Let ACBD be the given circle; draw the diameters AB and CD, bisect the radius dB in e, and draw the line Cef; draw also at right angles the lines Cn and fr, making each equal to Cf; join nr, and n Cfr is the square as required.

n

A

D

B

d

a

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

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

Describe a circle, as A B C D, in diameter equal to

d

6

5

с

the extremity of the angles, or breadth of square required; draw at right angles the diameters A CBD, and within the circle draw the lines A 1 B, B1C, C1D, and D1 A, meeting the diameters A B C and D, which completes the inscribed square;—again, with the distance 1n from ABC and D, as centres, describe small arcs; draw lines from each as tan- n gents, meeting the diameters at ABC and D; which will complete the square, equal in breadth to the circle's diameter.

7

8

a

4

3

B

1

2

b

Then from the extremity of the angles a b c and d, as centres with the distance ao, bo, &c., cut the sides of the square in 1, 2, 3, &c.; draw the lines 23, 45, 67, 81; which completes the octagon as required.

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

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 c line C B, which is the side of a square equal to both the given squares; or bisect the line CB as a diameter, on which describe the circle C B, and it is equal to the two given circles as required.

A

B

36. 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 radius a C into four equal parts, and make Cb equal to three of them; draw the tangent Ad parallel to B D, draw the line b Dd, then will Ad equal one-fourth of the

whole circumference; and if lines be drawn from b, through points in the circumference, meeting the line Ad, as gg, hh, &c., the corresponding parts will be equal to each other.

37. To draw a spiral with spaces of uniform dis

tance.

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 CD at right angles to A B, thus form

C

B

A

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 CD, 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 BD; from 2, with the distance 2 D, describe DA; 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.