* E. 5. as DF to FE; and by conversion *, AC is to AB, as DF to DE: wherefore because AC is to each of the magnitudes AB, BC, as DF to each of the others DE, EF; the ratio of AC to each of the magnitudes AB, BC, is given *. • 2 Def. 7. PROP. VIII. See N. If a given magnitude be divided into two parts which have a given ratio to one another, and if a fourth proportional can be found to the sum of the two magnitudes by which the given ratio is exhibited, one of them, and the given magnitude ; each of the parts is given. Ç_B E *7 Dat. * 2 Def. Let the given magnitude AB be divided into the A D F E each of them given. Because the ratio of AC to CB is given, the ratio of AB to BC is given *, therefore a ratio, which is the same with it, can be 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 unto DE, EF, AB, a fourth proportional can be found, this which is BC is given *; and because AB is given, the other part AC is given *. 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 magni tude, 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 same to it may be found *; let this be the ratio * 2 Def, * 2 Def. of the given magnitudes D, E: and because the ratio A B C D E H a fourth proportional to F, G, E, cannot be found, then it can only be said that the ratio of A to C is compounded of the ratios of A to B, and B to C, that is, of the given ratios of D to E, and F to G. F G 1 9 * 2 Def. 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 D to B is given *; but the ratio of B to E is B E * 9 Dat. given; therefore * the ratio of D to E is C F given: and because the ratio of B to C is given, and also the ratio of B to E; the ratio of E to C is given *: and the ratio of C *9 Dat. to F is given ; wherefore the ratio of E to F is given ; D, E, F, have therefore given ratios to one another. * 9 Dat. 22. PROP. XI. magnitude, both of them together shall have a given ratio Let the magnitudes AB, BC, have a given ratio to the magnitude D, AC has a given ratio to the same D. Because AB, BC, have each of them a given ratio to A B C D, the ratio of AB to BC is given *: and by composition, the ratio of AC to CB is D given * : but the ratio of BC to D is given : therefore * the ratio of AC to D is given. PROP. XII. 9 Dat. 7 Dat. * 9 Dat. 23. See N. * 19. 5. If the whole have to the whole a given ratio, and the parts have to the parts given, 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, but not the same, ratios to the parts CF, FD: every one shall have to every one, whole or part, a given ratio. 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 the ratio of EB A E B to FD is given, wherefore the ratio of FD to FG is C F G D 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 * 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 * 9 Dat. 6 Dat. 9 Dat. * 6 Dat. 9 Dat. * Cor. 6. Dat. * 10 Dat. 7 Dat. AB to each of them *. The ratio therefore of every one to every one is given. If the first of three proportional straight lines have a given See N. 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*: • let this be the * 2 Def. ratio of the given straight lines D, E; and between D and E find a* mean proportional F; therefore the * 13. 6. rectangle contained by D and 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 * 2 Cor. 20. to the square of B; and as D to E, so A B C is * the square of D to the square of F: * 2 Cor. 20. therefore the square* of A is to the D F E square of B, as the of D to the square square of F: as therefore* the straight * 22, 6. 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 * * 2 Def. because the ratio of the given straight lines D, F, which is the same with it, has been found. 6. 6. A. PROP. XIV. If a magnitude, together with a given magnitude, have a See N. 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 have 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; * 2 Dat. wherefore FD is given *: and А Β Ε с FD as AF to CD: but the ratio of AE to CD is given; therefore the ratio of AB to CF is given; that is, CF, the excess of CD above the given magnitude FD, has a given ratio to AB. Next, let the excess of the magnitude AB above the given magnitude BE, that is, let A E have a given ratio to the magnitude 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 А. Ε Β given, and BE is given; where* 2 Dat. fore FD is given *: and because, as AE to CD, so is BE to FD, C DF * 12. 5. AB is to CF, as* AE to CD: but the ratio of AE to CD is given, therefore the ratio of PROP. XV. 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. 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, * 2 Dat. wherefore * FD is given: and А Β Ε because as BE to CD, so is * Cor. 19.5. AE to FD: AB is * to FC, as BE to CD: and the ratio of F Ç D B. |