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. 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 A 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. If 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.* 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. * See Note. A ED BC 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 given (7. dat.): make the ratio of AD to DE the same with this ratio; therefore the ratio of AD to DE is given: and AD is given, wherefore (2. dat.) DE, and the remainder AE are given and because as DC to DB, so is AD to DE, AC is (12. 5.) 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 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 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, as also (6. dat.) the ratio of AD to AE: and AE is given, wherefore (2. dat.) AD is given: and because, as the whole AC, to the whole EB, so is AD to DE, the remainder DC is (19. 5.) to the remainder DB, as AC to EB; and the ratio of AC to EB is given; wherefore the ratio of DC to DB is given, as also (cor. 6. dat.) 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. 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 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 (1. dat.). Because BE, DF are each of them given, their ratio is given, and if this ratio be the same with A B with the given ratio of AB to CD, C D F G D F E E 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 the ratio A B of BE tó DF be greater than the ratio of AB to CD; and as AB to CD, so make BG to DF; therefore the ratio C of BG to DF is given; and DF is given, therefore (2. dat.) 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 (10. 5.) than BG; and because as AB to CD, so is BG to DF; therefore AG is (12. 5.) 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 mag- A nitude shall have a given ratio to the other. E Because AE, CF are each of them C F given, their ratio is given (1. dat.): D B and if this ratio be the same with the ratio of AB to CD, the ratio of the remainder EB to the remainder FD, which is the same (19. 5.) with the given ratio of AB to CD, shall be given. E G F D B 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 inversion, the ratio of 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, so make AG to CF; therefore the ratio of AG to CF A is given, and CF is given, wherefore (2. dat.) AG is given: and because the ratio of AB to CD, that is, the C ratio of AG to CF, is greater than the ratio of AE to CF; AG is greater (10. 5.) than AE and AG, AE are given, therefore the remainder EG is given; and as AB to CD, so is AG to CF, and so is (19. 5.) 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 excess of EB above a given magnitude EG, has a given ratio to FD. In the same manner the other case is demonstrated. If 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 sum above a given magnitude shall 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 excess of the sum 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 given, wherefore (2. dat.) AG E A G B to CF, and so is (19. 5.) the remain der GB to the remainder FD; the ratio of GB to FD is given, and EG is given, therefore GB, the excess of the sum EB above the given magnitude EG, has a given ratio to the remainder FD. IF two magnitudes have a given ratio to one another, if a given magnitude be added to one of them, and the other be taken from a given magnitude; the sum, 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 sum 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 sum 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, so GB to FD: therefore the ratio of GB to FD is given, and FD is given, wherefore GB is GA given (2. dat.); and BE is given; the B E whole GE is therefore given and because as AB to CD, so is GB to F C FD, and so is (19. 5.) GA to FC; D the ratio of GA to FC is given and AE together with GA is given, because GE is given; therefore the sum AE together with GA, to which the remainder FC has a given ratio, is given. The second part is manifest from prop. 15. If two magnitudes have a given ratio to one another, if 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, * See Notes. |