proportional F; therefore the rectangle contained by D and c2 cor. 20. 6. square of A to the square of B; and as D to E, so is c the square of D to the square of F: А d 11. 5. therefore the square d of A is to the square of B, as the square of D to the square of F: D F E as thereforee the straight line A to the straight 1 e 22. 6. line B, so is the straight line D to the straight a 2. def line F: therefore the ratio of A to B is given a, because the ratio of the given straight lines D, F which is the same with it has been found. IF a magnitude together with a given magnitude See N. has a 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 has 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 AĚ, 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; wherefore FD is gi A B E ven a : and because as AE to CD, so a 2. dat. is BE to FD, the remainder AB is b b 19. 5. to the remainder CF, as AF to CD: C F D but the ratio of AE to CD is given, -1 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 AE have a given ratio to the mag a 2. dat. nitude 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 given, and BE is given, A E B wherefore FD is given a. And because as AE to CD, so is BE to FD, AB is to CF, as c AE to CD: but the ratio of C DF AE to CD is given, therefore the ratio of AB to CF is given : that is, CF which is equal to CD, together with the given magnitude DF, has a given ratio to AB. c 12. 5. B PROP. XV. See Note. IF a magnitude, together with that to which another 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. а 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 AL 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 á 2 dat. AE is given, wherefore a FD is given : A B E and because as BE to CD, so is AE to - the ratio of BE to CD is given, where- F C D See Note. IF the excess of a magnitude above a given magni tude, 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 ration, 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 excess of AC, both of them together, above the given magnitude, has a given ratio to BC. Let AD be the given magnitude, the excess of AB above which, viz. DB has a given ratio A D B С to BC: and because DB has a given ratio to BC, the ratio of DC to CB is given a, and AD is given ; therefore DC, the excess of a 7. dat. AC above the given magnitude AD, has a given ratio to BC. Next, let the excess of two magnitudes AB, BC together, above a given magnitude, have to A D B E C one of them BC a given ratio; ei -ther the excess of the other of them AB above the given magnitude shall 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 excess AC above AD has a given ratio to BC, DB has b a given ratio to BC; that is, DB the excess 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 At. equal to it; and because EC, the excess of AC above AE, has to BC a given ratio, BC hasc a given ratio to BE ; and be-c 6. dat. cause Ar, is given, AB together with BE, to which BC has a given ratio, is given. IF the excess of a magnitude above a given magni- See Note: tude has a given ratio to another magnitude ; the excess of the same first magnitude above a given magnitude, shall 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 same above a given magnitude shall have a given ratio to a th other Let the excess of the magnitude AB above a given magnitude have a given ratio to the magnitude BC; the excess of AB above a given magnitude has a given ratio to AC. Let AD be the given magnitude ; and because DB, the ex cess of AB above AD, has a given ratio to BC; the ratio of DC a 7. dat. to DB is given a ; make the ratio of AD to DE the same with this ratio ; therefore the ratio of A E D B C AD to DE is given: and AD is -1 b 2. dat. given, whereforeb Di., and the remainder AE are given: and because as DC to DB, so is AD c 12. 5. to DL, AC isc 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 AC 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 hare 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 excess 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 alsod the ratio of AD to AE: and AE is given, whereforeb AD is given : and because, as the whole AC, to the whole EB, so is e 19. 5. AD to DE, the remainder DC ise to the remainder DB, as AC to EB; and the ratio of AC to EB is given; wherefore the ratio f Cor. 6. of DC to DB is given, as alsof the ratio of DB to BC: and AD dat, is given ; therefore DB, the excess of AB above a given mag nitude AD, has a given ratio to BC. IF to each of two magnitudes, which have a given ratio to one another, a given magnitude be added; the whole 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 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 1. dlat. above a given magnitude has a given ratio to the other a. Because BE, DF are each of them given, their ratio is gived, and if this ratio be the same with A B E the ratio of AB to CD, the ratio of -1 AE to CF, which is the same b with C D F b 12. 5. the given ratio of AB to CD, shall 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 А B G E the ratio of BE to DF be greater than the ratio of AB to CD; and as с D F AB to CD, so make BG to DF; therefore the ratio of BG to DF is given ; and DF is given, thereforec BG is given : and because c 2. dat. BE has a greater ratio to DF than (AB to CD, that is, than) BG to DF, BE is greaterd than BG; and because as AB to CD, d 10. so is BG to DF; therefore AG isb 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 a given magnitude GE, has a given ratio to CF. The other case is demonstrated in the same manner. IF from each of two magnitudes, which have a given ratio to one another, a given magnitude be taken, 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 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 A B C F D Because AE, CF are each of them given, their ratio is givena: a 1. dat. and if this ratio be the same with the ratio of AB to CD, the |