The Elements of Euclid, Containing the First Six, and the Eleventh and Twelfth Books; Chiefly from the Text of Dr. Simson, with the Planes in the Elev - Tapa blanda

Sinopsis

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1827 edition. Excerpt: ...ifG L, then K N; if equal, equal; if less, less. 21.1. Now G, K, are equimults. of A, D, and L, N, are equimults. of C, F,.-. A: C:: D: F. Secondly--Let there be four mags. A, B, C, D, and four others E, F, G, H, which taken two and two in cross order, have the same ratio, viz. A:B::G:H; B:C:: F: G, and C: D:: E: F. Then shall A: D:: E: H. For, v A, B, C are three mags. and F, G, H, are three others, which taken two and two in cross order, have the same ratio;.-. by 1st case, A: C:: F: H; but C: D:: E: F, /. by 1st case A: D:: E: H. And so on, whatever be the number of mags. Therefore, if there be any number, &c. &c. Q. E. 0. PROP. XXIV.--Theorem. If the first has to the second the same ratio which the third has to the fourth; and the fifth to the second the same which the sixth has to the fourth; the first and fifth together shall have to the second, the same ratio which the third and sixth together have to the fourth. DE,3rd,: F,4th, and let BG,: F, 4th; then AG, 1st, + 5th,: F,4th. Therefore, if the first, &c. &c. Q. E. D. Cor. 1. If the same hypothesis be made as in the proposition, the excess of the first and fifth shall be to the second, as the excess of the third and sixth to the fourth. The demonstration of this is the same with that of the proposition, if division be used instead of composition. Cor. 2. The proposition holds true of two ranks of magnitudes, whatever be their number, of which each of the first rank has to the second magnitude the same ratio that the corresponding one of the second rank has to a fourth magnitude; as is manifest. PROP. XXV.--theorem. If four magnitudes of the same kind are proportionals, the greatest and least of them together are greater than the other two together. Let the four...

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