17. COR. 2. Alfo if the firft has a given ratio to the fecond, and the excefs of the third above a given magnitude has alfo a given ratio to the fecond, the fame excefs fhall have a given ratio to the firft; as is evident from the 9th Dat. PROP. XXV. F there be three magnitudes, the excefs of the first whereof above a given magnitude has a given ratio to the fecond; and the excefs of the third above a given magnitude has a given ratio to the fame fecond. the firit fhall either have a given ratio to the third, or the exces 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 excefs of one of them above a given magnitude has a given ratio to the other. Ꭰ G 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 giA ven ratio to C; and becaufe FB, GE have each F of them a given ratio to C, they have a gia. 9. Dat. ven ratio to one another. but to FB, GE the given magnitudes AF, DG are added; thereb. 18. Dat. fore b the whole magnitudes AB, DE have either B CE a given ratio to one another, or the excefs of one of them above a given magnitude has a given ratio to the other. 18. IF PROP. XXVI. F 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 fhall have a given ratio to the other. Let 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 alfo let KD the excefs 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. 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 gi ven, wherefore HA is given. and because as GD to AB, fo is CG a. 2. Dat. to HA, and fo is b CD to HB; the ratio of CD to HB is given. b. 12. 5. alio becaufe KD has a given ratio to EF, as KD to EF, fo make CK to LE; therefore the ratio of CK to LE is given; and CK is given, wherefore* LE is given. and becaufe as KD to EF, fo is CK to LE, and fob is CD to LF; the ratio 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 BD T HA, LE being taken, the remainders AB, EF fhall cither have a given ratio to one another, or the excefs of one of them above a given magnitude has a given ratio to the other . "Another Demonftration. 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. A c. 9. Dat. d. 19. Dat. Because the excets of C above a given magnitude has a given ratio to AB, therefore AB together with a given magnitude has a a. 14. Dat. given ratio to C. let this given magnitude be AF, wherefore FB has a given ratio to C. alfo, F because the excefs of C above a given magnitude has a given ratio to DE, therefore1 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 B to C, therefore b 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 "." C E b. 9. Dat. PROP. c. 19. Dat. 19. PROP. XXVII. IF there be three magnitudes the excefs of the first of which above a given magnitude has a given ratio to the fecond; and the excefs of the fecond above a given mag. nitude 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 excefs of the firft 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. 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 a a. a. Dat. given; and CF is given, wherefore GH is gi ven; and AG is given, wherefore the whole G AH is given. and because as G B to CD, fo is b. 19. 5. GH to CF, and fo is b the remainder HB to the H remainder FB; the ratio of HB to FD is given. c. 9. Dat. and the ratio of FD to E is given, wherefore © the ratio of HB to E is given. and AH is given; B DE therefore HB the excefs of AB above the given magnitude AH has a given ratio to E. " Otherwise. Let AB, C, D be three magnitudes, the excefs EB of the first 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 excefs of AB above a given magnitude has a given ratio to D. Because EB has a given ratio to C, and the excefs of C above a given magnitude has a gid. 24. Dat. ven ratio to D; therefore the excess of EB above a given magnitude has a given ratio to D. let this given magnitude be EF, therefore FB the excefs of EB above EF has a given ratio to D. and AF is given, because AE, EF are given. A E F BCD there therefore FB the excefs of AB above the given magnitude AF has a given ratio to D." PROP. XXVIII. 25. I IF two lines given in pofition cut one another, the point See N. or points in which they cut one another are given. Let two lines AB, CD given in pofition cut one another in the point E; the point E is given. Because the lines AB, CD are given in pofition, they have always the fame fituation, and therefore the point, or points, in which they cut one another have C E B a. 4. Dcf. D C in which they cut one another, are likewife found; and therefore are given in position *. IF PROP. XXIX. F the extremities of a straight line be given in pofition; Because the extremities of the ftraight line are given, they can 26. be found1; let these be the points A, B, between which a straight a. 4. Def. but one straight line. and when the straight line AB is drawn, its magnitude is at the fame time exhibited, or given. therefore the ftraight line AB is given in pofition and magnitude. PROP. 27. IF PRO P. XXX. one of the extremities of a ftraight line given in pofftion and magnitude be given; the other extremity fhall alfo be given. Let the point A be given, to wit one of the extremities of a ftraight line given in magnitude, and which lies in the straight line AC given in pofition; the other extremity is alfo given. Because the ftraight line is given in magnitude, one equal to it a. 1. Def. can be found"; let this be the ftraight line D. from the greater ftraight line AC cut off AB equal to the leffer D. therefore the other extremity A B of the straight line AB is found, and tion, because any other point in AC, B_C upon the fame fide of A, cuts off between it and the point A a greater or lefs ftraight line than AB, that is than D. therefore the b. 4. Def. point B is given b. and it is plain another fuch point can be found in AC produced upon the other fide of the point A. 28. 2. 31. 1. IF PROP. XXXI. Fa ftraight lire be drawn thro' a given point parallel to a straight line given in pofition; that ftraight line is given in pofition. Let A be a given point, and BC a straight line given in pofition; has always the fame pofition, because thro' A parallel to BC. therefore the A E C b. 4. Def. straight line DAE which has been found is given b in position. PROP. |