| Euclid, Isaac Todhunter - Euclid's Elements - 1883 - 428 pages
...it has to AB. [V. Dtfnitien 7. Wherefore, of unequal magnitudes &c. QED LKD PROPOSITION 9. THEOREM. **Magnitudes which have the same ratio to the same magnitude,...magnitude has the same ratio, are equal to one another.** First, let A and B have the same ratio to C: A shall be equal to B. For, if A is not equal to B, one... | |
| Euclides, James Hamblin Smith - 1883 - 376 pages
...mqA, .'. D has to B a greater ratio than D has to A. V. Def. 7. QED PROPOSITION VIII. (Eucl. v. 9.) **Magnitudes, which have the same ratio to the same...another ; and those, to which the same magnitude has the** so/me ratio, are equal to one another. Let A and B have the same ratio to C. Then must A = B. For if... | |
| Euclides - 1884 - 434 pages
...A by m, but does not exceed the multiple of A + B by m ; .-. C: A is greater than C : A + BV Def. 9 **Magnitudes which have the same ratio to the same magnitude...magnitude has the same ratio are equal to one another.** First let A : C = B : C : it is required to prove A = B. For if A be greater than B, then A : C is... | |
| Edward Mann Langley, W. Seys Phillips - 1890 - 538 pages
...nC>mB, while nC<mh, which is contrary to the hypothesis. Similarly A cannot be less than B. Hence : — **(1) Magnitudes which have the same ratio to the same magnitude are equal to one another.** (2) Magnitudes to which the same magnitude has the same ratio are equal to one another. PROPOSITION... | |
| Euclid - Geometry - 1890 - 442 pages
...when in (A + M) > (n + i) B, mA< (n + i) B; .-. A + M : B > A : B, .-. also B : A + M < B : A. 2">, **magnitudes which have the same ratio to the same magnitude are equal.** Let A, B, C be three magnitudes of the same kind, such that A : C = B : C. Then, by i°, if A > B,... | |
| Euclid - Geometry - 1892 - 460 pages
...magnitude has the same ratio to equal magnitudes. For if A = B, then C : A = C : B. PROPOSITION 6. (i) **Magnitudes which have the same ratio to the same magnitude are equal to one another.** That IB, if A : C = B : C. then A = B. (ii) Those magnitudes to which the same magnitude has the same... | |
| Irving Stringham - Algebra - 1893 - 164 pages
...because nC > mR while nC is either < m A, or at most = m A, .-. C : B > C : A. (Def. 7.) PROPOSITION 6. **"Magnitudes which have the same ratio to the same...magnitude has the same ratio are equal to one another.** ' ' That is, A, B, C being three magnitudes of the same kind ; if A : C : : B : C, then A = B. and... | |
| Joseph Battell - Force and energy - 1903 - 722 pages
...— will hold a greater number of pints than quarts. No demonstration necessary. PROPOSITION IX. ' **Magnitudes which have the same ratio to the same magnitude...magnitude has the same ratio are equal to one another.'** '• Still another proposition that needs not demonstration. PROPOSITION X. "Ratio being the number... | |
| Euclid - Euclid's Elements - 1904 - 488 pages
...taking m and n as before, nC > mB, while nC is not > mA ; .: C:B>C:A. Def. 1. PROPOSITION 6. Mar/nitndes **which have the same ratio to the same magnitude are...one, another ; and those to which the same magnitude** hax the same ratio are equal to one another. First, let A : C : : B : C ; • then shall A = B. For... | |
| Cora Lenore Williams - Geometry - 1905 - 122 pages
...same magnitude has a greater ratio to the less of two magnitudes than it has to the greater. Prop. 26. **Magnitudes which have the same ratio to the same magnitude...magnitude has the same ratio are equal to one another.** Prop. 27. That magnitude which has a greater ratio than another has to the same magnitude is the greater... | |
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