Chap. VIII.

GEOMETRY.

88 r

Fig. 1177.

7b reduce the Area of a Square to that of a Circle , divide oneside of the square, as C D, into two equal parts in E: at thispoint E, draw E F perpendicular to D C, half the length of D E ;from the point F as a centre, with the radius FD, describe thecircle DC G, which will be equal to the square ABCD.

Similar four-sided figures, or of any number of sides, areto one another in the duplicate ratio of their homologoussides, and universally similar rectilineal figures are to oneanother in the duplicate ratio of their homologous sides, andaccording to cor. 2. attached to the 20th prop, of Euclid’sSixth Book, if three straight lines be proportionals, as the thirdis to the third, so is any rectilineal figure upon the first, to asimilar and similarly described rectilineal figure upon the second :and the 6ame geometrician teaches us in the 9th prop, of hisFifth Book, that magnitudes which have the same ratio tothe same magnitude are equal to one another, and those towhich the same magnitude has the same ratio are equal toone another.

7b reduce a circle to an Oval on a A

given Length , as the circle ABCD,to an oval, whose greatest diametershall be equal to E F; draw a rightline G H, and set off the givenlength E F, from G to I; at thepoint I elevate a perpendicular I K,of the same length as the diameterof the circle D B: draw the rightline G K, and bisect it in L ; fromL draw the perpendicular LM;from the point N, where it cutsG I, with the radius GN, describea semicircle GKO, and lO willbe the shortest diameter of the oval :then draw the two diameters of theoval at right angles to each other;take the given length E F with athread, fold it in half, and place thetwo extremities on the diameterQR, at the points Y and X, equi-distant from V; so that the fold orangle of the thread may coincidewith the point S, the width of therequired oval; so that by movinga pencil through the points S, R, Tcircle ABCD.

7b unite several Figuresinto One , and increasing theContent of others. — To re-duce several figures into atriangle whose height shallbe equal to a given height,as that of the triangleABC, and the squareD E F G : draw* a rightline H I, and on this lineat the point H, make suchan angle as it is desired thetriangle should have, asIHK: take the heightAT of the triangle ABC,and draw a line RS parallelto H I; at the distance A T,set off the line CB onH I, from H to M: reducethe iT^ re DEFG > to a triangle DNG.and make the triangle DNG equal in heightto B . C ’ 3515 ‘I* ‘"“"gle O P G ; set off the hise G P on H I from M to Q, and draw1. ti; the triangle LQH will be equal to the square D E F G, and the triangle ABC.

Fig. 1179.

Fig. 1178.

and Q, you will describe an oval equal to the given

Fig. 1180.

Fig. 1181.