In the EB above a given magnitude EG, has a given ratio to FD. 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 given; and EA is given, therefore E A G B the whole EG is given : and because -as AB to CD, so is AG to CF, and so is (19. 5.) the remainder GB to C F D 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 given (2. dat.); and BE is given; the whole GE is G A B therefore given; and because as AB to --- -t. CD, so is GB to FD, and so is (19. 5.) GA to FC; the ratio of GA to FC is F С D given: and AE together with GA is * See Note. 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, 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.) AG is given; and A G AE is given, and therefore the remainder EG is given; and because as AB to CD so is AG to CF: and so с D F 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. 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 them A B 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 C F D EB has (19. 5.) the same given ratio to the remainder FD. * See Note. 20 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 E G B CD; AB is greater (10. 5.) than AG: and AB, AG are given; therefore the remainder BG is given: and because C F D 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. 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 С given: and because as AB to CD, so is AG G to CF, and so is (19. 5.) GB to FD; the ratio F 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 E tude AG, has a given ratio to 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. ratio to the second, the same excess shall have a given ratio to the first; as is evident from thd 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. Let FB, the excess of AB above a given magnitude AF, have å 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 F D a given ratio (9. dat.) to one another. But to FB, GE the given magnitudes AF, DG are Gadded; 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 B С given magnitude has a given ratio to the other. 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. 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 H of CK to LE is given; and CK is given, wherefore LE (2. dat.) is given: and because L as KD to EF, so is CK to LE, and so (12. A 5.) is CD to LF; the ratio of CD to LF is given: but the ratio of CD to HB is given, K wherefore (9. dat.) the ratio of HB to LF is given: and from HB, LF the given magnitudes HA, LE being taken, the remainders A D F 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. 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. 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 F 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; A therefore (14. dat.) DE together with a given magnitude has a given ratio to C: let this given magnitude be DG, wherefore GE has B 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.). 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. Because the ratio of GB to CD is given, as GB to CD, so make |