Articoli correlati a Elements of geometry of Euclid
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1808 Excerpt: ...since Ep.fr are any equimultiples whatever of jtK, Fl; and gS, hT are any equimultiples whatever of Cm, Hn; Ek. will have the fame ratio to en, that Fl has to Hn (V. Des. 5.) Q. E.D. PROP. VI. Theorem. If four magnitude be proportional, and the first be greater than the second, the third will' also be greater than the fourth; and if equal, equal; and if less, less. K ' 1 Si » A, ( Ct f X-i. I F, Hi t" Let A have to b the fame ratio that c has to D; then if A be greater than B, C will also be greater than D; and if equal, equal; and if less, less. For, of a and c take any equimultiples E and G, and f B and D the fame equimultiples f and H. Then, because a is to B, as c is to D by Hyp.), if E be greater than f, g will also be greater than H; and if equal, equal; and if less, less (V. Des. 5.) And, since E, F, g, h are the fame multiples of A, B, ', d, each of each, these last magnitudes will also observe the fame agreement of equality, excess, or desect with their equimultiples. If, therefore, a be greater than B, c will also be greater than D; and if equal, equal; and if less, less. Q. £. D. PROP. PROP. VII. Theorem. If four magnitudes be proportional, they will be proportional also when taken inversely.. ai--1 Hi ' r A,. Ci 1 Bi ' Si 1 V. 1 F, 1 If a has to B the-fame ratio that c has to D; then, in versely, b will have to a the fame ratio that D has to c. For, of B and n take any equimultiples whatever E and T; and of A and c any equimultiples whatever g and h: Then, fince a is to b as c is to D by Hyp.), and g, H are equimultiples of a, c, and e, f of B, D by Const.), if g be greater than e, h will be greater than r; and if equal, equal; and if less, less (V. Des. 5.) And, because g has with E the fame agreement of equality, excess, or dese...
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