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. 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.) A AG is given; and AE is given, E B G and therefore the remainder EG (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. second part is plain from this and prop. 15. The IF 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 A them given, the ratio of AB to CD is given and if this ratio F D E B be the same with the ratio of C AE to CF, then the remainder EB has (19. 5.) the same given ratio to the remainder FD. 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 • See Note. E G B given; and because the ratio of AB to CD is greater than the ratio of (AE to CF, that is, than the A ratio of) AG to CD; AB is greater (10. 5.) than AG: and AB, AG are given; therefore the remainder BG C is given and because as AE to CF, . F D 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. 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; C F B D 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. * See Note. 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 ratio to the second, the same excess shall have a given ratio to the first; as is evident from the 9th dat. If there be three magnitudes, the excess of the first whereof above a given magnitude has a given ratio to the second; and the excess of the third above a given magnitude has a given ratio to the same second: the first shall either have a given ratio to the third, or the excess of one of them above a given magnitude shall have a given ratio to the other. Let AB, C, DE be three magnitudes, and let the excesses of each of the two AB, DE above given magnitudes have given ratios to C; AB, DE 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. F DI Let FB, the excess of AB above a given magnitude AF, have a given ratio to C; and let GE, the excess A of DE above the given magnitude DG, have a given ratio to C; and because FB, GE have each of them a given ratio to C, they have a given ratio (9. dat.) to one another. But to FB, GE the given magnitudes AF, DG are added; therefore (18. dat.) the whole magnitudes AB, DE have either a given ratio to one another, or the excess of one of them above a given magnitude has a given ratio to the other. B C E If there be three magnitudes, the excesses of one of which above given magnitudes have given ratios to the other two magnitudes; these two 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, EF be three magnitudes, and let GD the excess of one of them CD above the given magnitude CG have a given ratio to AB; and also let KD the excess of the same CD above the given magnitude CK have a given ratio to EF: either AB has a given ratio to EF, or the excess of one of them above a given magnitude has a given ratio to the other. H Because GD has a given ratio to AB, as GD to AB, so make CG to HA; therefore the ratio of CG to HA is given : and CG is given, wherefore (2. dat.) HA is given; and because as GD to AB, so is CG to HA, and so is (12. 5.) CD to HB; the ratio of CD to HB is given: also because KD has a given ratio to EF, as KD to EF, so make CK to LE: therefore the ratio of CK to LE is given; and CK is given, wherefore LE (2. dat.) is given: and because as KD to EF, so is CK to LE, and so (12. 5.) is CD to LF; the ratio of CD to LF is given but the ratio of CD to HB is given, wherefore (9. dat.) the ratio of HB to LF is. given and from HB, LF the given magnitudes HA, LE being taken, the remainders AB, EF shall 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 (19. dat.). : Another Demonstration. C A G E K B D F Let AB, C, DE be three magnitudes, and let the excesses of one of them C above given magnitudes have given ratios to AB and DE; either AB, DE 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. F A G D Because the excess of C above a given magnitude has a given ratio to AB; therefore (14. dat.) AB together with a given magnitude has a given ratio to C: let this given magnitude be AF, wherefore FB has a given. ratio to C: also because the excess of C above a given magnitude has a given ratio to DE; therefore (14. dat.) DE together with a given magnitude has a given ratio to C: let this given magnitude be DG, wherefore GE has a given ratio to C: and FB has a given ratio to C, therefore (9. dat.) the ratio of FB to GE is given and from FB, GE the given magnitudes AF, DG being taken, the remainders AB, DE 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 (19. dat.). B C E If there be three magnitudes, the excess of the first of which above a given magnitude has a given ratio to the second; and the excess of the second above a given magnitude has also a given ratio to the third; the excess of the first above a given magnitude shall have a given ratio to the third. Let AB, CD, E be three magnitudes, the excess of the first of which AB above the given magnitude AG, viz. GB, has a given ratio to CD; and FD the excess of CD above the given magnitude CF, has a given ratio to E: the excess of AB above a given magnitude has a given ratio to E. A G F Because the ratio of GB to CD is given, as GB to CD, so make GH to CF: therefore the ratio of GH to CF is given; and CF is given, wherefore (2. dat.) GH is given: and AG is given, wherefore the whole AH is given: and because as GB to CD, so is GH to CF, and so is (19. 5.) the remainder HB to the remainder FD; the ratio of HB to FD is given: and the ratio of FD to E is given, wherefore (9. dat.) the ratio of HB to E is given: and AH is given; therefore HB, the excess of AB above a given magnitude AH, has a given ratio to E. "Otherwise, H B Ꭰ E Let AB, C, D, be three magnitudes, the excess EB of the first of which AB above the given magnitude AE has a given ratio to C, and the excess of C above a given magnitude has a given ratio to D: the excess A of AB above a given magnitude has a given ratio to D. E Because EB has a given ratio to C, and the excess of C above a given magnitude has F a given ratio to D; therefore (24. dat.) the excess of EB above a given magnitude has a given ratio to D: let this given magnitude be B therefore FB, the excess of EB above EF; EF, has a given ratio to D: and AF is given, because AE, EF |