the ratio of AB to FC is given: and FD is given, that is, CD together with FC, to which AB has a given ratio, is given. If the excess of a magnitude above a given magnitude have See N. 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, have 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 be- A cause DB has a given ra tio to BC, the ratio of DC to CB is given*, and AD is 7 Dat. 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 magni tude, have to one of them A D BEC 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* a given ratio * Cor. 6. to BC; that is, DB the excess of AB above the given Dat. 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 given ratio to BE; and because AE is given, AB together with BE to which BC has a given ratio is given. a * 6 Dat. PROP. XVII. 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. A EDB C 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 *; 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* DE and the remainder AE are given. And because as DC to DB, so 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 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 * the ratio of AD to AE: and AE is given, wherefore* AD is given: and because, as the whole AC, to the whole EB, so is AD to DE, the remainder DC is * 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 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. PROP. XVIII. 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. 14. Because BE, DF, are each of them given, their ratio* *1 Dat. is given, and if this ratio be the same with the ratio of AB to CD, the ratio of AE to CF, which is the same* with the given ratio of AB to CD, shall AB be given. But if the ratio of BE to DF be not the same with the ratio of CD F E A B G E C D F * 12. 5. 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 of BE to DF be greater than the ratio of AB to CD; and as AB to CD, so make BG to DF; therefore the ratio of BG to DF is given; and DF is given, therefore* BG is * 2 Dat. 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 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. PROP. XIX. 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 an * 12.5. 15.. * 1 Dat. * 19.5. * 2 Dat. * 10. 5. * 19.5% 16. other, 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 magnitude shall have a given ratio to the other. Because AE, CF each of them given, their are A E B C F D ratio is given *; and if this ratio be the same with_the But if the ratio of AB to CD be not the same with A E GB C F D * than the ratio of AE to CF; AG is greater 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 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. PROP. XX. 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 FĎ. 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* AGE A is given: and EA is given, therefore the whole EG is given and because as C AB to CD, so is AG to * 2 Dat. GB F D CF, and so is the remainder GB to the remainder 19. 5. 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 See N. 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 given*; and BE is given, the whole GE is therefore given: and because as AB to CD, so is GB to FD, and so is GA to FC; the ratio of GA to FC is G *2 Dat. A BE F D * 19.5. given: and AE together with GA is given, because GE is given; therefore, the sum AE, together with |