Let the excefs of the magnitude AB above a given magnitude, have a given ratio to the magnitude BC; the excels of AC, both of them together, above the given magnitude, has a given ratio to BC. D B C Let AD be the given magnitude, the excefs of AB above which, viz. DB has a given ratio A to BC: And because DB has a gi"ven ratio to BC, the ratio of DC to CB is given, and AD is given; therefore DC, the excess of a 7. dat. one of them BC a given ratio; ei- A D BEC AB above the 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 let it be less than AB; and because DC the excefs of AC above AD has a given ratio to BC, DB has a given ratio to BC; that is, DB, the ex- b Cor. 6. cefs of AB above the given magnitude AD, has a given ratio to BC. C 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 cause AE is given, AB together with BE to which BC has a given ratio, is given. PROP. XVII. dat. с 6. dat. II. IF F 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 fa given magnitude, fhall have a given ratio to both the magnitudes together. And if the excefs of either of two magnitudes above a given magnitude has a given ratio to both magnitudes together; the excefs of the fame above a given magnitude fhall have a given ratio to the other. Let the excess 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. Let a 7. dat. c 12. 5. Let AD be the given magnitude; and becaufe DB, the excefs of AB above AD, has a given ratio to BC; the ratio of DC to DB is given: Make the ratio of AD to DE the fame with this ratio; therefore the ratio of AD to DE is given: And AD A EDB C b 2. dat. is given, wherefore b DE, and the remainder AE are given: And because as DC to DB, fo is AD to DE, AC is to EB, as DC to DB; and the ratio of DC to DB is given; wherefore the ratio of AC to EB is given: And because the ratio of EB to A€ is given, and that AE is given, therefore EB the excess of AB above the given magnitude AE, has a given ratio to AC. Next, let the excess 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. с Let AE be the given magnitude; and becaufe EB the excess of AB above AE has to AC a given ratio, as AC to EB, so make AD to DE; therefore the ratio of AD to DE is given, d 6. dat. as alfo the ratio of AD to AE: And AE is given, whereforeb AD is given: And becaufe, 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 f Cor. 6. given; wherefore the ratio of DC to DB is given, as also f the ratio of DB to BC: And AD is given; therefore DB, the excefs of AB above a given magnitude AD, has a given ratio to BC. e 19. 5. dat. IF to each of two magnitudes, which have a given ratio to one another, a given magnitude be added; the whole 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 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 a I. dat. above a given magnitude has a given ratio to the other 2. Becaufe BE, DF are each of them given, their ratio is given, and and if this ratio be the fame with the ratio of AB to CD, the ratio of AE to CF, which is the fame with the given ratio of AB to CD, fhall be given. A B C_D A B F E GE b 12. 5. 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: First, 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 BG is given: And because c 2. dat. BE has a greater ratio to DF than (AB to CD, that is, than) d C D F BG to DF, BE is greater than BG: And because as AB to d 10. 5. 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 a given magnitude GE, has a given ratio to CF. The other cafe is demonftrated in the fame manner. IF 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 excess 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 shall either have a given ratio to one another, or the excefs of one of them above a gi-A ven magnitude fhall have a given ratio to the other. Because AE, CF are each of CF E B D and if this ratio be the fame with the ratio of AB to CD, the 15. a I dat. ratio b. 19. 5. c 2. dat. d 10. 5. ratio of the remainder EB to the remainder FD, which is the fame 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 inverfion, the ratio 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 AGA to CF is given, and CF is given,' EG B that is, the ratio of AG to CF, 16. a 2. dat. b 19. 5. I' 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 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, fo AG to CF: Therefore the ratio of AG to CF is given, and CF is given, wherefore AG E A is given; and EA is given, there- GB F D GB to the remainder FD; the ratio of GB to FD is given. bove the given magnitude EG, has a given ratio to the remain. der FD. TF two magnitudes have a given ratio to one another, See N. if 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 BE 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 GBG is given; and BE is given, the whole GE is therefore given: And becaufe as AB to CD, fo is GBF to FD, and fo is b GA to FC; the A BE D ratio of GA to FC is given: And AE together with GA is given, because GE is given; therefore the fum AE together with GA, to which the remainder FC has a given ratio, is given. The fecond part is manifeft from prop. 15. PROP. XXII. a 2. dat. b 19. 5. D. IF F two magnitudes have a given ratio to one another, if see 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 |