b 2. def. found, let this be the ratio of the given magnitudes DE, EF: And because the given magnitude AB has to BC the given ratio of DE to EF, if to c 2. dat. d 4. dat. 8. A D C B F E DE, EF, AB a fourth propor- In the same manner, and with the like limitation, if the difference AC of two magnitudes AB, BC, which have a given ratio, be given; each of the magnitudes AB, BC is given. PROP. IX. Magnitudes which have given ratios to the same magnitude, have also a given ratio to one another. Let A, C have each of them a given ratio to B; A has a given ratio to C. Because the ratio of A to B is given, a ratio which is the a 2. def. same to it may be found; let this be the ratio of the given magnitudes D, E: And because the ratio of B to C is given, a ratio which is the same with it may be found 2; let this be the ratio of the given magnitudes АВСДЕН F, G: To F, G, E, find a fourth PROP. X. If two or more magnitudes have given ratios to one another, and if they have given ratios, though they be not the same, to some other magnitudes; these other magnitudes shall also have given ratios to one another. Let two or more magnitudes A, B, C have given ratios to one another; and let them have given ratios though they be not the same, to some other magnitudes D, E, F: The magnitudes D, E, F have given ratios to one another. Because the ratio of A to B is given, and likewise the ratio of A to D; therefore the ratio of D to B is given2: but the ratio of B to E is given, therefore the ratio of D to E is given, and because the ratio of B to A D B E F C C is given, and also the ratio of B to E; the ratio of E to C is given: And the ratio of C to F is given; wherefore the ratio of E to F is given: D, E, F have therefore given ratios to one another, PROP. XI. If two magnitudes have each of them a given ratio to another magnitude, both of them together shall have a given ratio to that other. Let the magnitudes AB, BC have a given ratio to the magnitude D: AC has a given ratio to the same D. A 9. a 9. dat. 22. a 9. dat. B C b 7. dat. Because AB, BC have each of them a given ratio to D, the ratio of AB to BC is given 2: And, by composition, the ratio of AC to CB is given: But the ratio of BC to D is given ; therefore the ratio of AC to D is given. D c hyp. PROP. XII. If the whole has to the whole a given ratio, and the parts have to the parts given ratios, but not the same ratios: Every one of them, whole or part, shall have to every one a given ratio. Let the whole AB have a given ratio to the whole CD, and the parts AE, EB have given ratios, but not the same ratios to the parts CF, FD: Every one shall have to every one, whole or part, a given ratio. A E C F G D B Because the ratio of AE to CF is given, as AE to CF, so make AB to CG; the ratio therefore of AB to CG is given; wherefore the ratio of the remainder EB to the remainder FG is given, because it is the same with the ratio of AB to CG: And, by hypothesis, the ratio of EB to FD is given, wherefore the ratio of FD to FG is given; and, by conversion, the ratio of FD to DG is given: And because AB has to each of the magnitudes CD, CG a given ratio, the ratio of CD to CG is given; and therefore the ratio of CD to DG is given: but the ratio of GD to DF is given, wherefore the ratio of CD to DF is given, and consequently d the ratio of CF to FD is given; but the ratio of CF to AE is given, as also the ratio of FD to EB; wherefore the ratio of AE to EB is given; as also the ratio of AB to each of them f: The ratio therefore of every one to every one is given. See N. 24. a 2. def. PROP. XIII. If the first of three proportional straight lines has a given ratio to the third, the first shall also have a given ratio to the second. Let A, B, C, be three proportional straight lines, that is, as A to B, so is B to C; if A has to C a given ratio, A shall also have to B a given ratio. Because the ratio of A to C is given, a ratio which is the same with it may be found a; let this be the ratio of the given b 15. 6. straight lines D, E; and between D and E find a mean b e d 1. def. e 2. cor. 20. 6. f 11. 5. 1 proportional F; therefore the rectangle contained by D and c 17. 6. E is equal to the square of F, and the rectangle D, E is given, because its sides D, E are given, wherefore the square of F, and the straight line F is given: And because as A is to C, so is D to E; but as A to C, so is the square of A to the square of B, and as D to E, so is the square of D to the square of F: Therefore the square of A is to the square of B, as the square of D to the square of F: As therefore & the straight line A to the straight line B, so is the straight line D to the straight line F: Therefore the ratio of A to B is given because the ratio of the given straight lines D, F, which is the same with it, has been found. a PROP. XIV. A A B C FE g 22. 6. D a 2. def. A. If a magnitude, together with a given magnitude, See N. has a given ratio to another magnitude; the excess of this other magnitude above a given magnitude has a given ratio to the first magnitude: And if the excess of a magnitude above a given magnitude has a given ratio to another magnitude; this other magnitude, together with a given magnitude, has a given ratio to the first magnitude. Let the magnitude AB, together with the given magnitude BE, that is, AE, have a given ratio to the magnitude CD; the excess of CD above a given magnitude has a given ratio to AB. Because the ratio of AE to CD is given, as AE to CD, so make BE to FD; therefore the ratio of BE to FD is given, and BE is given; wherefore FD A B E a 2. dat. CF, as AE to CD: But the ratio Next, Let the excess of the magnitude AB above the given magnitude BE, that is, let AE have a given ratio to the mag nitude CD; CD together with a given magnitude has a given ratio to AB. Because the ratio of AE to CD is given, as AE to CD, so make BE to FD; therefore the ratio of BE to FD is given, and BE is A a 2. dat. given, wherefore c 12. 5. See N. a 2. dat. B. FD is given: And because as AE to CD, so is BE to FD; AB is to CF, as AE to CD: But the C E B DF ratio of AE to CD is given, therefore the ratio of AB to CF is given: that is, CF which is equal to CD, together with the given magnitude DF, has a given ratio to AB. PROP. XV. If a magnitude, together with that to which another magnitude has a given ratio, be given; the sum of this other, and that to which the first magnitude has a given ratio, is given. Let AB, CD be two magnitudes, of which AB together with BE to which CD has a given ratio, is given: CD is given together with that magnitude to which AB has a given ratio. A B Because the ratio of CD to BE is given, as BE to CD, so make AE to FD; therefore the ratio of AE to FD is given, and AE is given, wherefore a FD is given And because as BE to b Cor. 19.5. CD, so is AE to FD: AB is to FC, as BE to CD: And the ratio 10. of BE to CD is given, wherefore E F C D the ratio of AB to FC is given: And FD is given, that is, CD together with FC to which AB has a given ratio, is given. PROP. XVI. See N. If the excess of a magnitude above a given magnitude has a given ratio to another magnitude; the excess of both together above a given magnitude shall have to that other a given ratio: And if the excess of two magnitudes together above a given magnitude, has to one of them a given ratio; either the excess of the other above a given magnitude has to that one a given ratio; or the other is given together with the magnitude to which that one has a given ratio. |