given, because it is the same (19. 5.) with the ratio of AB to CG: and the ratio of EB to FD is given, where A E B fore 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 С F G D 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. 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 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. A B 20. 6.) the square of D to the square of F: therefore the square (11. 5.) of A is to the square of B, DF E 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. 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. 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 A B E (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 F D 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 magnitude 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 A E B 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 D F AB is to CF, as (12. 5.) AE to CD: but 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 B C D wherefore 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. * See Note. 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 A D B BC: and because DB, has a given ra -tio 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 B E 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.* * See Note. 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 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 A E D B C 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, th remainder DC is (19. 5.) to the remainder DB, as AC to EB; and the ratio of AC tó 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 the ratio of AB to CD, the ratio of AE to CF, which is the same (12. 5.) with the given ratio of AB to CD, CD shall be given. * See Note. 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 tha the ratio of CD to AB: first, let the ratio of BE to DF A B G be greater than the ratio of AB to CD; 11 and as AB to CD, so make BG to DF; therefore the ratio of BG to DF is given; C D 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. PROP. XIX. 15. 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 magnitude shall have a given ratio to the other. A E B Because AE, CF are each of them given, their ratio is given (1. dat.): and if this ratio be the same with the ratio of AB to CF D 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. But if the ratio of AB to CD be not the same with the ratio of AE to CF, either it 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 is given, and CF is given, wherefore (2. dat.) AG is A E G B given : and because the ratio of AB to --CD, that is, the ratio of AG to CF, is greater than the ratio of AE to CF; C F D 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 |