From the point C, with any radius greater than the distance of AB, describe an arc cutting AB at e and f; from the points e and f, as centres, with any equal radius greater than the half of A, describe arcs cutting each other in D, and draw CD, which will be the perpendicular required.

PROBLEM 9.

182. To describe the segment of a circle, which shall have a given length or chord, AB, (fig. 3, pl. II,) and a given breadth, or versed sine, CD.*

By problem 2, bisect the straight line AB, by a perpendicular CE; from the point D, where the perpendicular cuts the chord AB, make DC equal to the breadth, or versed sine: join AC; and, by problem 1, make the angle CAE equal to the angle ACE: from E, as a centre, with the radius EA or EC, describe the arc ACB, which will be the segment required.

PROBLEM 10.

183. Through three given points, A, B, C, (fig. 4, pl. II,) to describe the circumference of a circle.

Join AB, BC; and, by problem 2, bisect each of the lines AB and BC by a perpendicular, and let the perpendiculars meet each other in I: from the centre I, with the distance IA, IB, or IC, describe the circle ABC, which is that required.

PROBLEM 11.

184. Upon a given straight line, AB, (fig. 5, pl. II,) to describe an equilateral triangle.

From the centres A and B, with the radius AB, describe arcs cutting each other at C. Join AC and BC; then ABC will be the equilateral triangle required.

PROBLEM 12.

185. Upon a given straight line, AB, (fig. 6, pl. II,) to describe a square. From the point B, by problem 5, draw BC perpendicular to AB; make BC equal to AB: from the points A and C, as centres, with a radius equal to AB

The meaning of sine, versed sine, &c. is given in Trigonometry, hereafter.

or B, describe arcs cutting each other in D, and join AD and DC; then, ABCD is the square required.

PROBLEM 13.

186. Upon a given straight line, AB, ( figures 7 and 8, pl. II,) to describe a regular polygon of any number of sides.

Produce the side AB to P, and on AP, from the centre B, describe a semicircle ACP; divide the semi-circumference ACP into as many equal parts as the number of sides intended; through the second division, from P, draw the line BC; bisect AB and BC by perpendiculars cutting each other in S; from S, with the radius AS, BS, or CS, describe a circle ABCDE, then carry the side AB or BC round the remaining part of the arc, which will be found to contain the remaining sides of the number required.

Figure 7 is an example of a pentagon. Figure 8 is an example of a hexagon: but, in this figure, we need not proceed by the general method; we have only to make a radius of the given side AB; and take the points A and B as centres; and form the arcs AG and BG, and strike a circle with the radius GA or GB, which will contain the side AB six times.

PROBLEM 14.

187. In a given square, ABCD, (fig. 9, pl. II,) to inscribe a regular octagon, so that four alternate sides of the octagon may coincide with four sides of the square.

Draw the diagonals AC and BD, cutting each other in S; on the sides of the square make AL, AF, BE, BH; CG, CK; and DI, DM, each equal to half the diagonal; join ME, FG, HI, KL; then will FGHIKLMEF be the octagon required.

PROBLEM 15.

188. In a given triangle ABC, (fig. 10, pl. II,) to inscribe a circle.

Bisect any two angles, A and B, by the straight lines AE and BE, and the point E, the intersection of these two lines, will be the centre of the inscribed circle: draw ED perpendicular to AB, cutting AB in D; from E, with the

radius ED, describe the circle DFG, which will be inscribed in the triangle ABC, as required.

PROBLEM 16.

189. A circle, DEF, (fig. 11, pl. II,) and a line AB, touching it, being given, to find the point of contact.

From the centre C draw the perpendicular CD, cutting AB in D, which is the point of contact required.

PROBLEM 17.

190. Two straight lines, AB, BC, (fig. 12, pl. II,) forming any angle, being given, to describe a circle to touch each of these lines at a given point, A, in one of them.

Make BC equal to BA, and draw AD perpendicular to AB, and CD perpendicular to BC; from the point of intersection D, with the radius DA or DC, describe the circle ACE, which is that required.

PROBLEM 18.

191. In a given circle, ABCD, (fig. 13, pl. II,) to inscribe a square. Draw the diameters AC and BD at right angles, and join AB, BC, CD, DA-; then ABCD will be the square required.

PROBLEM 19.

192. To describe a segment, ABC, of a circle, by means of an angle.

Let AC (fig. 14, pl. II,) be the length or chord, and DB the versed sine. Join BA and BC; produce BA to E, and BC to F, making BE and BF of any length, not less than the chord AC. Prepare two straight edges, BE and BF, and fasten them together at the angle B, so that their outer edges may form the angle ABC; and, to keep them to the extent, fix another slip, GH, to each straight edge at G and H. Bring the angular point B to A, then move the angle thus formed by the straight edges, so that the edge BE may always move upon the point A, and the edge BF upon the point C; then if, during the time of moving, a pencil be held to the angular point B, and the point to trace over the plane, the segment of a circle will be described.

ANOTHER METHOD.

193. Let AC (fig. 15, pl. II,) be the length or chord, and BD the versed sine. Join AB, and draw BE parallel to AC, making BE of any length, not less than AB. Form a triangular piece of wood, ABE: bring the angular point B, of the triangle, to the point A; and move the triangle, so that the side BA may slide upon A, and the side BE upon B: then if, during the motion, a pencil be held at the angular point B, with its point tracing over the plane, the arc AB will be described by the point of the pencil. The arc AB being described, the arc BC will be described in a similar manner; and, consequently, the whole segment of the circle, as required to be done.

PROBLEM 20.

194. Between two straight lines, E and F, (fig. 1, pl. III,) to find a mean proportional.

Draw the straight line AB. Make AC equal to E, and CB equal to F. Upon AB, as a diameter, describe the semi-circle ADB : from the point C draw CD, perpendicular to AB, and CD will be the mean proportional required.

PROBLEM 21.

195. To find a straight line equal in length, nearly, to the arc of a circle. Let ABC, (fig. 2, pl. III,) be the given arc. Join AC, which prolong to F. Bisect the arc ABC in B, and make AE equal to twice AB. Divide CE into three equal parts, and set one of them off from E to F; then the straight line AF is nearly equal in length to the arc ABC.

PROBLEM 22.

196. To describe a triangle, of which the three sides shall be equal to three given straight lines, provided that any two of them are greater than the third. Let D, E, F, (fig. 3, pl. III,) be the three given straight lines. Draw AB, and make AB equal to the straight line D. From the point A, with the dis

tance of the line F, describe an arc; and from the point B, with the extent of

the line CE, describe another arc, cutting the former at C, and join AC and BC: then is ABC the triangle required.

In this manner a triangle may be made equal to another given triangle ;

for this is only making the sides of the triangle equal to those of the given

triangle.

PROBLEM 23.

197. To describe a trapezium equal and similar to a given trapezium.

Let it be required to describe a trapezium equal and similar to the given trapezium, ABCD (fig. 4, pl. III).

Draw the straight line, FG, fig. 5, and make FG equal to BC: upon FG describe the triangle FGH, equal to the triangle BCD; and, upon FH, describe the triangle FHE, equal to the triangle BDA, and the whole figure, EFGH, will be equal and similar to the figure ABCD.

PROBLEM 24.

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

It is required to make a rectangle equal to the given triangle, ABC (fig. 6, pl. III).

Draw CF perpendicular to AB, cutting AB in F. Divide CF into two equal parts in the point G. Through G draw DE parallel to AB, and draw AD and BE perpendicular to AB; then the rectangle ABED will be equal to the triangle ABC, as required to be done.

PROBLEM 25.

199. To make a square equal to a given rectangle.

Let it be required to make a square equal to the given rectangle, ABCD, (fig. 7, pl. III).

Produce the side AB of the rectangle to E, and make BE equal to BC. Draw BG perpendicular to AE; and, on AE, as a diameter, describe the semicircle AGE; and, on the straight line, BG, describe the square BGFH; which is the thing required to be done.

We now see that a triangle may be reduced to a rectangle, and a rectangle may be reduced to a square; therefore a triangle may be reduced to a square.