A B But if the ratio of BE to DF be not the fame with the ratio of AB to CD; either it is greater than the ratio of AD to CD, or, by inverfon, the ratio of DF to BE is greater than the ratio of CI) to AB. firft, let the ratio of BE to DF be greater than the ratio of AB to CD; and as AB to CD, fo make BG to DF; therefore the ratio of BG to DF is given; and DF is given, therefore b. 2 GE C D c. 1. Eat. BG is given. and becaufe BE has a greater ratio to DF than (AB to CD, that is than) BG to DF, BE is greater than BG. and d. 10. becaufe as AB to CD, fo is BG to DF, therefore AG is 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 excefs of AE above the given magnitude GE has a given ratio to CF. the other cafe is demonftrated in the fame manner. IF PROP. XIX. F from cach of two magnitudes, which have a given ratio to one another, a given magnitude be tal.en; the remainders fhall either have a given ratio to one another, or the excefs of one of them above a given magnitude, fhall 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 AB be taken, and from CD, the given magnitude CF. the remainders EB, FD fhall either ave a given ratio to one another, or the excels of one of them above a given magnitude fhall have a gi ven ratio to the other. Becaute AE, CF are each of them given, their ratio is given *; and if this ratio be the fame with A E B D CF the ratio of AB to CD, the ratio of the remainder EB to the re b. 19. 5. mainder FD, which is the fame b with the given ratio of AB to CD, fhall be given. But if the ratio of AB to CD be not the fame with the ratio of AE to CF, either it is greater than the ratio of AE to CF, or, by inverfion, the ratio of CD to AB is greater than the ratio of CF to AE. firft, let the ratio of AB to CD be greater than the rado of AE to CF, and as AB to CD, fo make AG to CF; therefore the d. 10. 5. a 6. B 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, d fo is 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 exccfs of EB above the given magnitude 13, has a given ratio to FD. in the fame manner the other cafe is demonftrated. IF PROP. XX. F to one of two magnitudes which have a given ratio to one another, a given magnitude be added, and from the other a given magnitude be taken; the excefs of the fum above a given magnitude fhall have 2 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 excefs of the fam 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, fo AG to CF. therefore the ratio of AG to CF is given, and CF is a. 2. Dat. given, wherefore AG is given; E A and EA is given, therefore the C GB remainder FD; the ratio of GB to FD is given. and EG is given, therefore GB, the excefs of the fum EB above the given magnitude EG, has a given ratio to the remainder FD. IF two magnitudes have a given ratio to one another, if see N. a given magnitude be added to one of them, and the other be taken from a given magnitude; the fum 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 fum has a given ratio. Let the two magnitudes AB, CD have a given ratio to one another; and to AB let the given magnitude EE be added, and . let CD be taken from the given magnitude FD. the fum 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 ;G A B E a. 2. Dat. E C and BE is given, the whole GE is b. 19. 5. F two magnitudes have a given ratio to one another, See N. if 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 anoher, and from AB let the given magnitude AE be taken, and let 314 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, fo AG to CF. the ratio of AG to CF is therefore given, and CF is a. 2. Dat. given, wherefore AG is given; A DF and AE is given, and therefore the PROP. XXIII. F from two given magnitudes there be taken magnitudes which have a given ratio to one another, the remainders fhall either have a given ratio to one another, or the excefs of one of them above a given magnitude fhall have a given ratio to the other. Let AB, CD be two given magnitudes, and from them let the marmitudes 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 excels of one of them above a given magnitude has a given ratio to the other. Decause AB, CD are each of A B F D CF, then the remainder EB has the fame given ratio to the remainder FD. But if the ratio of AB to CD be not the fame with the ratio of AE to CF, it is either greater than it, or, by inverfion, the ratio of CD to AB is greater than the ratio CF to AE. fift, 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, becaufe the ratio of AE to CF is given; and CD is given, where E GB F D C. 10. S. fore AG is given; and because the ratio of AB to CD is greater b. a. Dat. then the ratio of (AE to CF, that A is, than the ratio of) AG to CD; AB is greater than AG. and AB, AG are given, therefore the remainder BG is given. and because ,C as AE to CF, fo is AG to CD, and fo is EG to FD; the ratio a. 19. 5. of EG to FD is given. and GB is given, therefore EG the excefs of EB above the given magnitude GB, has a given ratio to FD. the other cafe is fhewn in the fame way. IF F there be three magnitudes, the first of which has a Sce N. given ratio to the fecond, and the excefs of the fecond above a given magnitude has a given ratio to the third; the excefs of the firft above a given magnitude fhall also have a given ratio to the third. Let AB, CD, E be three magnitudes, of which AB has a given ratio to CD; and the excefs of CD above a given magnitude has a given ratio to E. the excefs of AB above a given magnitude has a given ratio to E. Let CF be the given magnitude the excefs 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 fo make AG to CF; there- A fore the ratio of AG to CF is given; and CF is given, wherefore AG is given. and becaufe as AB to CD, fo is AG to CF, and fo is . GB to 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. c BDE COR. 1. And if the firft has a given ratio to the fecond, and the excefs of the firft above a given magnitude has a given ratio to the third; the excefs of the fecond above a given magnitude shall have a given ratio to the third. for if the fecond be called the first, and the firft the fecond, this Corollary will be the fama with the Propofition. a. 2. Dat. b. 19. 5. c. 9. Dat, |