Solid parallelepipeds contained by parallelograms equiangular to one another, each to each, that is, of which the solid angles are equal, each to each, have to one another the ratio compounded of the ratios of their sides. The The Thirteen Books of Euclid's Elements - Page 347by Euclid, Johan Ludvig Heiberg - 1908Full view - About this book
| Euclid, Micaiah John Muller Hill - Euclid's Elements - 1900 - 143 pages
...ratio AB : AP is the ratio compounded of AB : AE and AD : AG. Art. 147. COROLLARIES. (1) Two rectangles **have to one another the ratio compounded of the ratios of their sides.** From (1) and the definition of duplicate ratio (Art. 129) it follows that 88 EUCLID, BOOKS V. AND VI.... | |
| M. J. M. HILL - 1900
...ratio AB : AP is the ratio compounded of AB-.AE and AD : AG. Art. 147. COROLLARIES. (1) Two rectangles **have to one another the ratio compounded of the ratios of their sides.** From (1) and the definition of duplicate ratio (Art. 129) it follows that Art. 148. EXAMPLE 69. The... | |
| David C. Lindberg - Science - 1980 - 549 pages
...duplicate of ratio (A,B), and (A,D) its triplicate (V. defs. 9, 10), and that "equiangular parallelograms **have to one another the ratio compounded of the ratios of their sides"** (VI.23), he did not use or develop the concept further. 72 Mathematical astronomers found more use... | |
| Sir Thomas Little Heath - Mathematics - 1981 - 446 pages
...compounded in VL 23; the corresponding proposition to VI. 23 then follows (5), namely, that plane numbers **have to one another the ratio compounded of the ratios of their sides.** Propositions 8—10 deal with the interpolation of geometric means between numbers. If a: b = a :f,... | |
| Peter M. Engelfriet - Mathematics - 1998 - 488 pages
...alternative to supplying the lemma (II, pp. 242-6). 23l¿N¿th¿*fl¿ Heath Equiangular parallelograms **have to one another the ratio compounded of the ratios of their sides. (The** ratio of two equiangular parallelograms: connect together the two ratios of each of the two sides of... | |
| Douglas M. Jesseph - Mathematics - 2000 - 419 pages
...following; "A ratio is that relation or habitude of homo18. This asserts, "Equiangular parallelograms **have to one another the ratio compounded of the ratios of their sides."** Euclid does not actually define the term compound ratio, but from the context of the proof in Elements... | |
| Douglas M. Jesseph - Mathematics - 2000 - 419 pages
...following: "A ratio is that relation or habitude of homo18. This asserts, "Equiangular parallelograms **have to one another the ratio compounded of the ratios of their sides."** Euclid does not actually define the term compound ratio, but from the context of the proof in Elements... | |
| Archimedes, Sir Thomas Little Heath - Mathematics - 2002 - 377 pages
...cylinder passes through P. Proposition 10. It was proved by the earlier geometers that any two cones **have to one another the ratio compounded of the ratios of their** bases and of their heights. The same method of proof will show that any segments of cones have to one... | |
| C. Sasaki - Mathematics - 2003 - 496 pages
...23, which we have cited above (p. 2i0), had its arithmetical counterpart VIII, 5: “Plane numbers **have to one another the ratio compounded [of the ratios] of their sides.”** In his excellent monograph titled Philosophy of Mathematics and Deductive Structure in Euclid's Elements,... | |
| Jacob Bernoulli, Edith Dudley Sylla - Mathematics - 2006 - 430 pages
...Compounding of ratios appears in Euclid, Elements, Book VI, Proposition 23, "Equiangular parallelograms **have to one another the ratio compounded of the ratios of their sides."** See Edith Sylla, "Compounding Ratios: Bradwardine, Oresme, and the first edition of Newton's Principia,"... | |
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