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For, let the circumferences ABCD, abed, be divided into quadrants by the radii OaA, O&B, OcC, Oa*D, then the quadrants AB, ab, will be similar arcs; therefore, OA :Oa:: AB : ab wherefore, OA :0a:: 4AB : lab.

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PRACTICAL GEOMETRY.

PROBLEM 1.

174. To make an angle at a given point, E, {fig. 35, pl. I,) in a straight line, DE, equal to a given angle ABC.

From the centre B, with any radius, describe an arc gh, cutting BA at g, and BC at h; from the point E, with the same radius, describe an arc, ik, cutting ED at i: make ik equal to gh, and through the point k draw EF: then the angle DEF will be equal to the given angle ABC.

Problem 2.

175. To bisect a given angle ABC [fig. 36, pl. I).

From BA and BC cut off Be and Bf, equal to each other; from the points e and/ as centres, with any radius greater than the distance ef, describe arcs, cutting each other at G, and join BG, which will bisect the angle ABC, as required.

Problem 3.

176. Through a given point #, {fig.37.pl. I,) to draw a straight line parallel to a given straight line, AB.

From g draw ge, to cut AB at any angle in the point e: in AB take any other point/; make the angle Bfh equal to /eg, and make f h equal to eg, and through the points g and h draw the line CD; then CD will pass through g parallel to AB, as required.

Problem 4.

177. At a given distance, parallel to a given straight line, AB, (Ar. 38, pl. I,) to draw a straight line, CD.

In the given straight line AB, take any two points, e and f; and, from the centres e andyj with the given distance, describe arcs at p and q; draw the line CD, to touch the arcs p and q; then CD will be parallel to AB, at the distance required.

Problem 5.

178. To bisect a given straight line, CD, [fig. 39, pl. I,) by a perpendicular. From the points C and D, with any distance greater than the half of CD,

describe arcs cutting each other in A and B: join AB, and this line will bisect AB perpendicularly.

Problem 6.

179. From a given point C, (fig. 40, pl. I,) in a given straight line, AB, to erect a perpendicular.

In the straight line, AB, take any two points, e and f, equally distant from C: from the points e andy, with any equal radius, greater than the half of ef, describe arcs cutting each other at D, and draw CD, which will be perpendicular to AB.

Problem 7.

180. From a given point, B, {fig. I, pl. II,) at the extremity of a given straight line, AB, to draw a perpendicular.

Take any point, E, above the line AB, and, with the radius BE, describe the arc BC, cutting AB in d: draw the straight line dEC, and join BC, which will be the perpendicular required.

Problem 8.

181. From a given point C, {fig. 2, pl. II,) to let fall a perpendicular to a given straight line, AB.

From the point C, with any radius greater than the distance of AB, describe an arc cutting AB at e and/; from the points e and /, 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, Mg. 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, (V. 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, Chg. 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 true, 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. 14i,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.

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