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I

PROP. I. BOOK X. LEMMA.

F there be two unequal magnitudes, from the greater be taken a part greater than its half, and from the remainder a part taken greater than its half; this may be done till the magnitude`remaining be less than any proposed magnitude.

Let AB and C be two unequal magnitudes, of which AB is the greater; from AB let a part BH be taken greater than the half, and from the remainder AH a part KH greater than its half; and so on, till the remaining magnitude, which let be AK, be lefs than the affigned magnitude C. Let C be multiplied till it become greater than AB, which let be DE, and divide it into the parts DF, FG, GE, each equal to C. Then, because DE is greater than AB, and the part EG taken from it lefs than the half thereof, and the part BH greater than the half of AB, there remains DG greater than AH. Again, because GD is greater than HA, and GF, half of GD, is taken from it; and if from HA be taken HK greater than the half of HA, there will remain FD greater than KA; but FD is equal to C; therefore KA is less than C. Which was required.

Book XII

PROP. II. THEOR.

IRCLES are to each other as the fquares of their diame

C/

ters.

Let ABCD, EFGH, be circles, whofe diameters are BD, FH; then, as the fquare of BD is to the fquare of FH, fo is the circle ABCD to the circle EFGH. If not, the circle ABCD will be to fome figure either lefs or greater than the circle EFGH.

First, let it be to a figure S, lefs than the circle EFGH, in which infcribe the fquare EFGH, which will be greater than half the circle. For, if tangents are drawn to the circle, thro' the points E, F, G, H, the fquare EFGH will be half the fquare defcribed about the circle; but the circle is lefs than the fquare described about it; therefore the fquare EFGH is greater than half the circle. Let the circumferences EF, FG, GH, HE, be bifected in the points K, L, M, N, and join EK, KF, FL, LG,

GM,

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