tude CD; CD together with a given magnitude has a given ratio to AB. A Because the ratio of AE to CI) 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 a. 2. Dat. FD is given a. and because as AE to CD, fo is BE to FD, AB is to CF, as AE to C c. 12. 5. See N. B. C E B D F CD. but the ratio of AE to CD is given, PROP. XV. a magnitude together with that to which another magnitude has a given ratio, be given; this other is given together with that to which the firft magnitude has a given ratio. 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 2. 2. Dat. given, wherefore a FD is given. and be- A caufe as BE to CD, fo is AE to FD; B E C D b.Cor.19.5. AB is b to FC, as BE to CD. and the F ratio of BE to CD is given, wherefore 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. See N. IO. a PROP. XVI. F the excefs 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. Let the excess of the magnitude AB above a given magnitude, have a given ratio to the magnitude BC; the excefs of AC, both of them together, above a given magnitude, has a given ratio to BC. Let AD be the given magnitude the excefs of AB above which, viz. DB, has a given ratio to BC. and because DB has a given ratio. to BC, the ratio of DC to CB is A DB C given, and AD is given; therefore DC, the excefs of AC above a. 7. Dat. the given magnitude AD, has a given ratio to BC. Next, let the excefs of two magnitudes AB, BC together above a given magnitude have to one of them BC a given ratio; either the excefs of the other of them AB a A DBEC bove a given magnitude fhall have to BC a given ratio; or AB is given together with the magnitude to which BC has a given ratio. Let AD be the given magnitude, and first fet it be less than AB; and because DC the excess of AC above AD has a given ratio to BC, DB has a given ratio to BC; that is DB, the excefs b. Cor. 6. of AB above the given magnitude AD, has a given ratio to BC. Dat. But let the given magnitude be greater than AB, and make AE equal to it; and becaufe EC, the excefs of AC above AE, has to BC a given ratio, BC has a given ratio to BE; and be- c. 6. Dat. cause AE is given, AB together with BE to which BC has a given ratio, is given. PROP. XVII. the excess of a magnitude above a given magnitude See N. has a given ratio to another magnitude; the excefs of the fame firft magnitude above a given magnitude, fhall have a given ratio to both the magnitudes together. and if the excess of either of two magnitudes above a given magnitude has a given ratio to both magnitudes together; the excess of the fame above a given magni tude shall have a given ratio to the other. Let the excefs of the magnitude AB above a given magnitude have a given ratio to the magnitude BC; the excefs of AB above a given magnitude has a given ratio to AC. A a Let AD be the given magnitude; and because DB, the excess of AB above AD, has a given ratio to BC; the ratio of DC to DB is a. 7. Dat. given a. make the ratio of AD to DE the fame with this ratio; therefore the ratio of AD to DE is b. 2. Dat. c. 12. 5. given. and AD is given, wherefore A ED B C c b DE, and the remainder AE are Next, let the excefs of AB above a given magnitude have a given ratio to AB and BC together, that is to AC; the excess of AB above a given magnitude has a given ratio to BC. Let AE be the given magnitude; and because EB the excefs of AB above AE has to AC a given ratio, as AC to EB, so make d. 6. Dat. AD to DE; therefore the ratio of AD to DE is given, as alfo d e. 19. 5. the ratio of AD to AE. and AE is given, wherefore ↳ AD is given. and because as the whole, AC, to the whole, EB, fo is AD to DE; the remainder DC is to the remainder DB, as AC to EB; and the ratio of AC to EB is given, wherefore the ratio of DC f. Cor. 6. to DB is given, as alfof the ratio of DB to BC. and AD is given, therefore DB, the excess of AB above the given magnitude AD, has a given ratio to BC. Dat. 14. a. 1. Dat. PROP. XVIII. F to each of two magnitudes, which have a given ratio to one another, a given magnitude be added the wholes fhall either have a given ratio to one another, or the excess of one of them above a given magnitude fhall have a given ratio to the other. Let the two magnitudes AB, CD have a given ratio to one another, and to AB let the given magnitude BE be added, and the given magnitude DF to CD. the wholes AE, CF 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 BE, DF are each of them given, their ratio is given a. 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 AB to CD, or, by inverfion, the ratio of DF to BE is greater than the ratio of CD 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 c C G E D F c. 2. Dat. d BG is given. and because 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. 5+ because as AB to CD, fo 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 becaufe 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 cafe is demonftrated in the fame manner. PROP. XIX. F from each of two magnitudes, which have a given ratio to one another, a given magnitude be taken; 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 AE be taken, and from CD, the given magnitude CF. the remainders EB, FD fhall either have a given ratio to one another, or the excefs of one of them above a given magnitude shall have a gi ven ratio to the other. A E B F D a. 1. Dat. Because AE, CF are each of C them given, their ratio is given a; and if this ratio be the fame with the ratio of AB to CD, the ratio of the remainder EB to the re b. 19. 5. C. 2. Dat. d. 10. 5. 16. mainder FD, which is the fame b with the given ratio of AB to CD, fhall be given. A E G B F D 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 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 CF is given, wherefore AG is given. and because the ratio of AB to CD, C that is the ratio of AG to CF, is greater than the ratio of AE 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 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 excefs of EB above the given magnitude EG, has a given ratio to FD. in the fame manner the other cafe is demonftrated. b PROP. XX. F to one of two magnitudes which have a given ratio IF t from the other a given magnitude be taken; the excess of the fum above a given magnitude fhall 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 excefs of the fum 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, so AG to CF. therefore the ratio of AG to CF is given, and CF is a. 2. Dat. given, wherefore a AG is given; E and EA is given, therefore the whole EG is given. and because C h. 19. 5. as AB to CD, fo is AG to CF, A G B F D remainder FD; the ratio of GB to FD is given. and EG is given, |