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, fo AG to CF: The ratio of AG to CF is therefore given, and CF is given, wherefore AG A C E B DF G is given; and AE is given, and PROP. XXIII. IF, from two given magnitudes there be taken magni. tudes which have a given ratio to one another, the remainders fhall either have a given ratio to one another, or the excess of one of them above a given magnitude fhall 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 an other, 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 be the fame with the ratio of AE C to CF, then the remainder EB has the fame given ratio to the remainder FD. E B F D But if the ratio of AB to CD be not the fame with the ra tio of AE to CF, it is either greater than it, or, by inverfion, 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, fo make AG to CD; there fore the ratio of AG to CD is given, because the ratio of AE to CF is given; and CD is given, wherefore ↳ AG is given i A E GB C 10. 5. F D given; and because the ratio of AB to CD is greater than the C cause as AE to CF, fo is AG to CD, and fo is EG to FD; a 19. 5. the ratio of EG to FD is given: And GB is given; therefore EG, the excefs of EB above a given magnitude GB, has a given ratio to FD. The other cafe is fhown in the fame way. IF there be three magnitudes, the firft of which has a See N. given ratio to the fecond, and the excess of the fecond 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 excefs of AB above a given magnitude has a given ratio to E. A Let CF be the given magnitude, the excefs 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, fo make AG to CF; therefore the ratio of AG to CF is given; and CF is given, wherefore AG is given And becaufe as AB to CD, fo is AG to CF, and fo is b GB to FD; the ratio of GB G to FD is given. And the ratio of FD to E is given, wherefore the ratio of GB to E is given, and AG is given; therefore GB the excefs of AB above a given magnitude AG has a given ratio to E. a 2. dat. C b 19. 5. F c 9. dat. BDE COR. 1. And if the firft has a given ratio to the fecond, and the excefs of the firft above a given magnitude has a given ratio to the third; the excefs of the fecond above a given magnitude fhall have a given ratio to the third. For, if the fecond be called the firft, and the firft the fecond, this corollary will be the fame with the propofition. COR. 17. a 9. dat. COR. 2. Alfo, if the first has a given ratio to the fecond, and the excefs of the third above a given magnitude has also a given ratio to the fecond, the fame excefs fhall have a given ratio to the first; as is evident from the 9th dat. PROP. XXV. 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 fame fecond: The firft fhall either have a given ratio to the third, or the excess of one of them above a given magnitude fhall have a given ratio to the other. Let AB, C, DE be three magnitudes, and let the exceffes 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. D Let FB the excefs of AB above the given magnitude AF have a given ratio to C; and let GE the excefs of DE above the given magnitude DG have a given ratio to C; and because FB, GEF. have each of them a given ratio to C, they' have a given ratio to one another. But to FB, GE the given magnitudes AF, DG are addb 18. dat. ed; therefore 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 TF there be three magnitudes, the exceffes of one of which above given magnitudes have given ratios to the other two magnitudes; thefe two fhall either have a given ratio to one another, or the excefs of one of them above a given magnitude shall have a given ratio to the other. Let Let AB, CD, EF be three magnitudes, and let GD the excefs of one of them CD above the given magnitude CG have a given ratio to AB; and alfo let KD the excess of the fame CD above the given magnitude CK have a given ratio to EF, either AB has a given ratio to EF, or the excefs of one of them above a given magnitude has a given ratio to the other. a Because GD has a given ratio to AB, as GD to AB, fo make CG to HA; therefore the ratio of CG to HA is given; and CG is given, wherefore HA is given: And becaufe as a 2. dat. GD to AB, fo is CG to HA, and fo is b CD to HB; the ra- b 12. 5. tio of CD to HB is given: Alfo because KD has a given ratio C E CS. dat. D DF to EF, as KD to EF, fo make CK to LE; H therefore the ratio of CK to LE is given; and CK is given, wherefore LE is given: And because as KD to EF, fo is CK to LE, and A fob is CD to LF; the raito of CD to LF is given: But the ratio of CD to HB is given, wherefore the ratio of HB to LF is given: and from HB, LF the given magnitudes HA, LE being taken, the remainders AB, EF fhall 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. d 19. dit. "Another Demonftration. B Let AB, C, DE be three magnitudes, and let the exceffes 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. גי given mag. a 14, đạt G Because the excels of C above a given magnitude has a given ratio to AB; therefore AB together with nitude has a given ratio to C: Let this given magnitude be AF, wherefore FB has a given ratio to C: Allo, because the excels of C above A a given magnitude has a given ratio to D; therefore DE together with a given magni tude 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 the ratio b 9. dat, 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 excefs of one of them above a given magnitude has a given ratio to the others." Bb BC E c 19, das PROP. 19. a 2. dat. 19. 5. € 9. dat. PROP. XXVII. IF there be three magnitudes: The excess of the first of which above a given magnitude has a given ratio to the fecond; and the excefs of the fecond above a gi ven magnitude has also a given ratio to the third: The excess of the first above a given magnitude fhall 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 excefs 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 Because the ratio of GB to CD is given, as GB to CD, fo make GH to CF; therefore the ratio of GHA to CF is given; and CF is given, wherefore * GH is given; and AG is given, wherefore G the whole AH is given: And becaufe as GB to CD, fo is GH to CF, and fo is the re- ·H+ mainder HB to the remainder FD; the ratio of HB to FD is given: And the ratio of FD to E is given, wherefore the ratio of HB to B DE E is given: And AH is given; therefore HB the excess of AB above a given magnitude AH has a given ra. tio to E. "Otherwise, Let AB, C, D be three magnitudes, the excefs EB of the firft of which AB above the given magnitude AE has a given ratio to C, and the excefs of C above a given magnitude has a given ratio to D: The excess of AB above a given magnitude has a given ra tio to D. Because EB has a given ratio to C, and the excefs of C above a given magnitude has a gi d 24. dat. ven ratio to D; therefore the excess of EB E above a given magnitude has a given ratio to are |
