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 given, wherefore (2. dat.) DE, and ED B C -| 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. Ir 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 E the ratio of AB to CD, the ratio of AE to CF, which is the same (12. 5.) F * 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 than the ratio of CD to AB: first, let the ratio of BE to DF A B G E 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; C D F 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. Ir 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. Because AE, CF are each of them given, their ratio is given (1. dat.): and if this Α E D B ratio be the same with the ratio of AB to C F 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. EG 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 is given, and CF is given, wherefore (2. dat.) AG is A 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 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 F D EB above a given magnitude EG, has a given ratio to FD. In the same manner the other case is demonstrated. Ir 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 is given; and EA is given, therefore E the whole EG is given: and because as AB to CD, so is AG to CF, and A G B so is (19. 5.) the remainder GB to C the remainder FD; the ratio of GB F D 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. Ir 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. GA B E 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 (2. dat.); and BE is given; the whole GE is therefore given; and because as AB to CD, so is GB to FD, and so is (19. 5.) GA to FC; the ratio of GA to FC is F given: and AE together with GA is * See Note. C D 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. Ir 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, and let CD be taken from the given magnitude CF: the remainder EB is given, together with the magnitude to which the other remainder DF has a given ratio. A C EB D F G Because the ratio of AB to CD is given, make as AB to CD, so AG to CF: the ratio of AG to CF is therefore given, and CF is given, wherefore (2. dat.) AG is given; and AE is given, and therefore the remainder EG is given; and because as AB to CD so is AG to CF: and so is (19. 5.) the remainder BG to the remainder DF; the ratio of BG to DF is given and EB together with BG is given, because EG is given: therefore the remainder EB together with BG, to which DF the other remainder has a given ratio, is given. The second part is plain from this and prop. 15. Ir from two given magnitudes there be taken magnitudes which have a given ratio to one another, 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 AB, CD be two given magnitudes, and from them let the magnitudes AE, CF, which have a given ratio to one another, be taken; the remainders EB, FD 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. Because AB, CD are each of them A given, the ratio of AB to CD is given: and if this ratio be the same with the ratio of AE to CF, then the remainder EB has (19. 5.) the same given ratio to the remainder FD. C F D E B • See Note. EG B But if the ratio of AB to CD be not the same with the ratio of AE to CF, it is either greater than it, 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 AE to CF, so make AG to CD; therefore the ratio of AG to CD is given, because the ratio of AE to CF is given; and CD is given, wherefore (2. dat.) AG is given; and because the ratio of AB to CD is greater than the ratio of (AE to CF, that is, than the ratio of) AG to A CD; AB is greater (10. 5.) than AG: and AB, AG are given; therefore the remainder BG is given: and because C as AE to CF, so is AG to CD, and so is (19. 5.) EG to FD; the ratio of EG to FE is given: and GB is given; therefore EG, the excess of EB above a given magnitude GB, has a given ratio to FD. The other case is shown in the same way. F D If there be three magnitudes, the first of which has a given ratio to the second, and the excess of the second above a given magnitude has a given ratio to the third; the excess of the first above a given magnitude shall also have a given ratio to the third.* Let AB, CD, E, be the three magnitudes of which AB has a given ratio to CD; and the excess of CD above a given magnitude has a given ratio to E: the excess of AB above a given magnitude has a given ratio to E. Let CF be the given magnitude, the excess of CD above which, viz. FD has a given ratio to E: and because the ratio of AB to CD is given, as AB to CD, so make AG to A CF; therefore the ratio of AG to CF is given; and CF is given, wherefore (2. dat.) AG is given: and because as AB to CD, so is AG G to CF, and so is (19. 5.) GB to FD; the ratio of GB to FD is given. And the ratio of FD to E is given, wherefore (9. dat.) the ratio of GB to E is given, and AG is given; therefore GB, the excess of AB above a given magni- B D tude AG, has a given ratio to E. C F E COR. 1. And if the first have a given ratio to the second, and the excess of the first above a given magnitude have a given ratio to the third; the excess of the second above a given magnitude shall have a given ratio to the third. For, if the second be called the first, and the first the second, this corollary will be the same with the proposition. COR. 2. Also, if the first have a given ratio to the second, and the excess of the third above a given magnitude have also a given * See Note. |
