found to the sum of the two magnitudes by which the given ratio is exhibited, one of them, and the given magnitude; each of the parts is given.* C B F E Let the given magnitude AB be divided into the parts AC, CB which have a given ratio to one another; if a fourth proportional can be found to the above named magnitudes; AC and CB are each of them given. A Because the ratio of AC to CB is given, the ratio of AB to BC is given (7. dat.); therefore a ratio which is the same with it D can be found (2. def.); let this be the ratio of the given magnitudes, DE, EF: and because the given magnitude AB has to BC the given ratio of DE to EF, if unto DE, EF, AB a fourth proportional can be found, this which is BC is given (2. dat.); D and because AB is given, the other part AC is given (4. dat.) A C B FE In the same manner, and with the like limitation, if the difference AC of two magnitudes AB, BC which have a given ratio be given; each of the magnitudes AB, BC is given. MAGNITUDES which have given ratios to the same magnitude, have also a given ratio to one another. Let A, C have each of them a given ratio to B; A has a given ratio to C. Because the ratio of A to B is given, a ratio which is the same to it may be found (2. def.); let this be the ratio of the given magnitudes D, E: and because the ratio of B to C is given, a ratio which is the same with it may be found (2. def.); let this be the ratio of the given magnitudes F, G: to F, G, E find a fourth proportional H, if it can be done; and because as A is to B, so is D to E; and as B to C, so is (F to G, and so is) E to H; ex æquali, as A to C, so is D to H: therefore the ratio of A to C is given (2. def.) because the ratio of the given magnitudes D and H, which is the same with it, has been found: but if a fourth proportional to F, G, E cannot be found, then it Α B E H F G can only be said that the ratio of A to C is compounded of the ratios of A to B, and B to C, that is, of the given ratios of D to E, and F to G. See Note. Ir two or more magnitudes have given ratios to one another, and if they have given ratios, though they be not the same, to some other magnitudes; these other magnitudes shall also have given ratios to one another. Let two or more maguitudes A, B, C have given ratios to one another; and let them have given ratios, though they be not the same, to some other magnitudes D, E, F; the magnitudes D, E, F have given ratios to one another. A Because the ratio of A to B is given, and likewise the ratio of A to D; therefore the ratio of D to B is given (9. dat.): but the ratio of B to E is given, therefore (9. dat.) Bthe ratio of D to E is given: and be- C cause the ratio of B to C is given, D E F and also the ratio of B to E; the ratio of E to C is given (9. dat.) : and the ratio of C to F is given; wherefore the ratio of E to Fis given; D, E, F have therefore given ratios to one another. Ir two magnitudes have each of them a given ratio to another magnitude, both of them together shall have a given ratio to that other. Let the magnitudes AB, BC have a given ratio to the magnitude D; AC has a given ratio to the same D. Because AB, BC have each of them A a given ratio to D, the ratio of AB to D B Ir the whole have to the whole a given ratio, and the parts have to the parts given, 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, 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 FG is * Sco Notc. Α E C F G D B given, because it is the same (19. 5.) with the ratio of AB to CG: and the ratio of EB to FD is given, wherefore the ratio of FD to FG is given (9. dat.); and by conversion, the ratio of FD to DG is given (6. dat.): and because AB has to each of the magnitudes CD, CG a given ratio, the ratio of GD to CG is given (9. dat.); and therefore (6. dat.) the ratio of CD to DG is given but the ratio of GD to DF is given, wherefore (9. dat.) the ratio of CD to DF is given, and consequently (cor. 6. dat.) the ratio of CF to FD is given; but the ratio of CF to AE is given, as also the ratio of FD to EB, wherefore (10. dat.) the ratio of AE to EB is given; as also the ratio of AB to each of them (7. dat.): the ratio therefore of one to every one is given. Ir 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 same with it may be found (2. def.); let this be the ratio of the given straight lines D, E; and between D and E find a (13. 6.) mean proportional F; therefore the rectangle contained by D and E is equal to the square of F, and the rectangle D, E is given, because its sides D, E are given; wherefore the square of F, and 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 (2. cor. 20. 6.) the square of A to the square of B; and as D to E, so is (2. cor. 20. 6.) the square of D to the square of F: therefore the square (11. 5.) of A is to the square of B, as the square of D to the square of F: as therefore (22. 6.) the straight line 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 (2. def.), because the ratio of the given straight lines D, F, which is the same with it, has been found. PROP. XIV. A D F E A. Ir a magnitude together with a given magnitude 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 See Note. with a given magnitude has a given ratio to the first magnitude.* Let the magnitude AB together with the given magnitude BE, that is, AE, have a given ratio to the magnitude CD; the excess of CD above a given magnitude has a given ratio to AB. A B E 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 given (2. dat.) and because as AE to CD, so is BE to FD, the remainder AB is (19. 5.) to the remainder CF, as AE C 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. F D Next, Let the excess of the magnitude AB above the given magnitude BE, that is, let AE have a given ratio to the magnitude CD: CD together with a given magnitude has a given ratio to AB. Α 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 given (2. dat.). And because as AE to CD, so is BE to FD, C AB is to CF, as (12. 5.) AE to CD: but E B D F the ratio of 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. Ir 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, and AE is given, wherefore (2. dat.) FD is given: A and because as BE to CD, so is AE to FD: AB is (cor. 19. 5.) to FC, as BE to CD and the ratio of BE to CD is given, wherefore the ratio of AB to FC is F BE C D given: and FD is given, that is CD together with FC, to which AB has a given ratio, is given. See Note. Ir 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. A D B C Let AD be the given magnitude, the excess 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 given (7. dat.), and 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 A D BE C of them BC a given ratio; 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 (cor. 6. dat.) a given ratio to BC; that is, DB, 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 AE equal to it; and because EC, the excess of AC above AE, has to BC a given ratio, BC has (6. dat.) a given ratio to BE; and because AE is given, AB together with BE, to which BC has a given ratio, is given. Ir the excess of a magnitude above a given magnitude have 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 have a given ratio to both magnitudes together; the excess of the same above a given magnitude shall have a given ratio to the other.* * See Note. |
