be given. But if the ratio of BE to DF be not the same with the ratio of AB to CD; either it is greater than the ratio of AB to CD, or, by inversion, the ratio of DF to BE is greater than the ratio of CD to AB. first, let the ratio of BE to DF be greater than the ratio of AB A B G E to CD; and as AB to CD, fo make BG to DF; therefore the ratio of C D F BG to DF is given; and DF is given, therefore · BG is given. and because BE has a greater ratio to DF C. 2. Dat. than (AB to CD, that is than) BG to DF, BE is greater than BG, d. 19. $. and because as AB to CD, so is BG to DF, therefore AG is b to CF, as AB to CD. but the ratio of AB to CD is given, wherefore the ratio of AG to CF is given; and because BE, BG are each of them given, GE is given. therefore AG, the excess of AE above the given magnitude GE has a given ratio to CF. the other case is demonstrated in the same manuer. IF PRO P. XIX. ratio to one another, a given magnitude be taken; the remainders shall either have a given rario to one another, or the excess of one of them above a given magnitude, shall have a given ratio to the other. Let the magnitudes AB, CD have a given ratio to one another, and from AB let the given magnitude AE be taken, and from CD, the given magnitude CF. the remainders EB, FD shall either have a given ratio to one another, or the excess of one of them above a given magnitude shall have a given ratio to the other. A Б Because AE, CF are each of them given, their ratio is given ; C F D and if this ratio be the same with the ratio of AB to CD, the ratio of the remainder EB to the reАз 2 mainder b. 19.5. mainder FD, which is the same b with the given ratio of AB to CD, shall be given. But if the ratio of AB to CD be not the same with the ratio of AE to CF, either it is greater than the ratio of AE to CF, or, by inversion, the ratio of CD to AB is greater than the ratio of CF to AE. first, let the ratio of AB to CD be greater than the ratio of AE to CF, and as AB to CD, fo make AG to CF; therefore the ratio of AG to CF is given, and c. 6. Dat. CF is given, wherefore C AG is A EG B given. and because the ratio of to CF, is greater than the ratio of d. 10. 5. A E to CF; AG is greater than AE. and AG, AE are given, therefore the remainder EG is given. and as AB to CD, fo is AG to CF, and so is b the remainder GB to the remainder FD; and the ratio of AB to CD is given, wherefore the ratio of GB to FD is given; therefore GB, the excess of EB above the given magnitude EC, has a given ratio to FD. in the same manner the other case is demonstrated. FD 16. F P R O P. XX. one another, a given magnitude be added, and from the other a given magnitude be taken ; the excess of the fum above a given magnitude shall have a given ratio to the remainder, Let the two magnitudes AB, CD have a given ratio to one another, and to AB let the given magnitude EA be added, and from CD let the given magnitude CF be taken; the excess of the sum EB above a given magnitude has a given ratio to the remainder FD. Because the ratio of AB to CD is given, make as AB to CD, lo AG to CF, therefore the ratio of AG to CF is given, and CH is a. 3. Dat. given, wherefore & AG is given; and E A is given, therefore the E A А. с F D 8, 19.5. so is b the remainder GB to the remainder FD; the ratio of GB to FD is given. and EG is given, therefore therefore GB, the excess of the sum EB above the given magnitude EG, has a given ratio to the remainder FD. IE a given magnitude be added to one of them, and the other be taken from a given magnitude; the sum together with the magnitude to which the remainder has a given ratio, is given. and the remainder is given together with the magnitude to which the sum has a given ratio. Let the two magnitudes AB, CD have a given ratio to one another; and to AB let the given magnitude BE be added, and let CD be taken from the given ́ magnitude FD, the sum AE is given together with the magnitude to which the remainder FC has a given ratio, Because the ratio of AB to CD is given, make as AB to CD, fo GB to FD. therefore the ratio of GB to FD is given, and FD is given, wherefore GB is given ; and BE is given, the whole GE is G A B E therefore given. and because as c AB to CD, so is GB to FD, and so G is b GA to FC; the ratio of GA 10 b. 19. 5. FC is given. and AE together with GA is given, because GE is given ; therefore the sum AE together with GA to which the remainder FC has a given ratio, is given. the fecond part is manifest a. 2. Dat. from Prop. 15 D. PRO P. XXII. two magnitudes have a given ratio to one another, if Sec N. from one of them a given magnitude be taken, and the other be taken from a given magnitude; each of the remainders is given together with the magnitude to which the other remainder has a given ratio. Let the two magnitudes AB, CD have a given ratio to one another, and from AB let the given magnitude AE be taken, and let CD Аа 3 CD be taken from the given magnitude CF; the remainder EB is given together with the magnitude to which the other remainder DF has a given ratio. Because the ratio of AB to CD is given, make as AB to CD, so AG to CF. the ratio of AG to CF is therefore given, and CF is . 2. Dat. given, wherefore · AG is given; and AE is given, and therefore the A G remainder EG is given. and because as AB to CD, so is AG to с b. 19. 5. CF, and so is b the remainder BG to the remainder DF; the ratio of BG to DF is given. and EB together with BG is given, because EG is given, therefore the remainder EB together with BG to which DF the other remainder has a given ratio is given. the second part is plain from Prop. 15. 20. See N. PRO P. XXIII. tudes which have a given ratio to one another, the remainders shall either have a given ratio to one another, or the excess of one of them above a given magnitude shall have a given ratio to the other. Let AB, CD be two given magnitudes, and from them let the magnitudes AE, CF which have a given ratio to one another be taken; the remainders EB, FD either have a given ratio to one another, or the excess of one of them above a given magnitude has a given ratio to the other. Because AB, CD are each of A E B them given, the ratio of AB to CD is given. and if this ratio be C F D the same with the ratio of AE to 2. 19. 5. CF, then the remainder EB has a the same given ratio to the remain der FD. But if the ratio of AB to CD be not the same with the ratio of AE to CF, it is either greater than it, or, by inversion, the ratio of CD to AB is greater than the ratio CF to AE. first, let the ratio of AB to CD be greater than the ratio of AE to CF; and as AE to CF, fo make AG to CD, therefore the ratio of AG to CD is given, because the ratio of AE to CF is given; and CD is given, where fore C. 1O. S fore b AG is given; and because the ratio of AB to CD is greater b. 2. Dät. than the ratio of (AE to CF, that is, than the ratio of) AG to CD; A GB AB is greater than AG. and C F D AB, AG are given, therefore the remainder BG is given. and because as AE to CF, fo is AG to CD, and so is * EG to FD; the 4. 19. s. ratio of EG to FD is given, and GB is given, therefore EG the excess of EB above the given magnitude GB, has a given ratio to FD. the other case is shewn in the same way. PRO P. XXIV. 13: F there be three magnitudes, the first of which has a See N. given ratio to the second, and the excess of the second above a given magnitude has a given ratio to the third; the excess of the first above a given magnitude shall also have a given ratio to the third. a. 2. Dat. b. 19. 5. Let AB, CD, E be three magnitudes, of which AB has a given ratio to CD; and the excess of CD above a given magnitude has a given ratio to E. the excess of AB above a given magnitude has a given ratio to E. Let CF be the given magnitude the excess of CD above which, viz. FD has a given ratio to E, and because the ratio of AB to CD is given, as AB to CD so make AG to CF; therefore the ratio of AG to CF is given; and CF is given, wherefore * AG is given and because as C AB to CD, so is AG to CF, and fo is b GB 10 FD; the ratio of GB to FD is given. and the ratio of FD to E is given, wherefore < the ratio of GB to E is given. and AG is given, therefore GB the excess of AB above the given magnitude AG has a given ratio to E. B D E Cor. 1. And if the first has a given ratio to the fecond, and the excefs of the first above a given magnitude has a given ratio to the third; the excess of the second above a given magnitude ihall have a given ratio to the third. for if the second be called the first, and the first the second, this Corollary will be the fune with the Proposition. COR. c. 9. Dat. A a 4 |