See N. If the whole has to the whole a given ratio, and the parts have to the parts given ratios, but not the same ratios : Every one of them, whole or part, shall have to every one a given ratio. Let the whole AB have a given ratio to the whole CD, and the parts AE, EB have given ratios, but not the same ratios to the parts CF, FD: Every one shall have to every one, whole or part, a given ratio. Because the ratio of AE to CF is given, as AE to CF, so make AB to CG; the ratio therefore of AB to CG is given; wherefore the ratio of the remainder EB to the remainder a 19. 5. FG is given, because it is the same with the ratio of AB to CG: And, by hypothesis, the B wherefore the ratio of FD to b 9. dat. FG is given ; and, by con с F G D version, the ratio of FD to DG c 6. dat. is given : And because AB has to each of the magnitudes CD, CG a given ratio, the ratio of CD to CG is given b; and therefore the ratio of CD to DG is given : but the ratio of GD to DF is given, whered cor. 6. fore b the ratio of CD to DF is given, and consequently d the ratio of CF to FD is given; but the ratio of CF to AE is e 10. dat. given, as also the ratio of FD to EB; wherefore e the ratio of AE to EB is given; as also the ratio of AB to each of them f; dat. f 7. dat. See N. If the first of three proportional straight lines has a given ratio to the third, the first shall also have a given ratio to the second. Let A, B, C, be three proportional straight lines, that is, as A to B, so is B to C; if A has to C a given ratio, A shall also have to B a given ratio. Because the ratio of A to C is given, a ratio which is the a 2. def. same with it may be found a ; let this be the ratio of the given b 15. 6. straight lines D, E; and between D and E find ab mean proportional F; therefore the rectangle contained by D and c 17.6. E is equal to the square of F, and the rectangle D, E is given, because its sides D, E are given, wherefored the square of F, and dl. def. the straight line F is given: And because as A is to C, so is D to E; but as A to C, so is e the square of A to the square of B, and e 2. cor. as D to E, so ise the square of D to the 20. 6. square of F: Therefore f the square of A is A B C fll. 5. to the square of B, as the square of D to the square of F: As therefore the straight line D F E A to the straight line B, so is the straight line D to the straight line F: Therefore the ratio of A to B is given a because the ratio a 2. def. of the given straight lines D, F, which is the same with it, has been found. g 22. 6. PROP. XIV. 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, A E, 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 A B E is given a : And because as AE to CD, so is BE to FD, the re С mainder AB is b to the remainder F D CF, as A E to CD: But the ratio of AE to CD is given; 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. b 19. 5. # nitude CD; CD together with a given magnitude has a given Because the ratio of A E to CD is given, as A E to CD, so E B a 2. dat. given, wherefore FD is given a : And с DF because as AE to CD, so is BE to FD; F c 12. 5. AB is to CF, as AE to CD: But the ratio of AE to CD is given, therefore the ratio of AB to CF See N. 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. 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, a 2. dat. and AE is given, wherefore a FD A B E F с D See N. If the excess 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 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 to BC: And because DB A D B C has a given ratio to BC, the ratio of DC to CB is given, and a 7. dat. AD is given ; therefore DC, the excess of 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 one of them BC a given ra- A DB EC tio ; either 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 of AC above AD has a given ratio to BC, DB has a given ratio to BC; that is, DB b Cor. 6. the excess of AB above the given magnitude AD has a given ratio to BC. But let the given magnitude be greater than AB, and make A E equal to it; and because EC, the excess of AC above AE, has to BC a given ratio, BC has a given ratio to BE; c 6. dat. and because AE is given, AB together with BE to which BC has a given ratio, is given. dat. و If the excess of a magnitude above a given magni. See N. 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 the other. Let the excess of the magnitude AB above a given magni. tude 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 a 7. dat. DC to DB is given a : Make the ratio of AD to DE the same with this ratio ; therefore the ratio of AD to DE is given ; A E DB C b 2. dat. and AD is given, wherefore b DE and the remainder AE are c 12. 5. given : And because as DC to DB, so is AD to DE, AC is c 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 A E, 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. Let AE be the given magnitude ; and because EB the excess of AB above A E 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 also d the ratio of AD to AE: And AE is given, where fore 6 AD is given : And because, as the whole AC, to the e 19. 5. whole EB, so is AD to DE, the remainder DC is e 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 excess of AB above a given magnitude AD, has a given ratio to BC. dat. If to each of two magnitudes, which have a given ratio to one another, a given magnitude be added ; the wholes 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 two magnitudes AB, CD have a given ratio to one another, and to A B 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 a l. dat other a. Because BE, DF are each of them given, their ratio is given, |