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PROP. I. BOOK X. LEMMA.
F there be two unequal magnitudes, from the greater be taken
ü part greater than its half, and from the remainder a part ta. ken greater than its half ; this may be done till the magnitude re* maining 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 less than the assigned 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 less 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.
PROP. 11. THEOR.
IRCLES are to each other as the squares of their diame.
Let ABCD, EFGH, be circles, whose diameters are BD, FH; then, as the square of BD is to the square of FH, so is the circle ABCD to the circle EFGH. If not, the circle ABCD will be to some figure either less or greater than the circle EFGH.
First, let it be to a figure S, less than the circle EFGH, in which inscribe the square 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 square EFGH will be half the square described about the circle ; but the circle is less than the square described about it; therefore the square EFGH is greater than half the circle. Let the circumferences EF, FG, GH, HE, be bisedted in the points K, L, M, N, and join EK, KF, FL, LG,