AG is given; and because the ratio of AB to CD is greater than the ratio of (AE to CF, to CD; AB is greater than AG: And AB, AG, are given; C therefore the remainder BG EG FD B is given And because as AE to CF, so is AG to CD, and so is a EG to FD; the ratio of EG to FD is given: And 19. 5. 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 See N. 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. C 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, Al so make AG to CF; therefore the ratio of AG to CF, is given: And CF is given, wherefore a AG is given: And because as G AB to CD, so is AG to CF, and so is b GB to FD; the ratio of GB 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 excess of AB B D above a given magnitude AG has a given ratio to E. a 2 Dat. b 19. 5. F E COR. 1. And if the first has a given ratio to the second, and the excess of the first above a given magnitude has 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. c 9 Dat. Cor 2. Also, if the first has a given ratio to the second, and the excess of a third above a given magnitude has 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. b 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 a 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 the given magnitude AF have a given ratio to C; and let GE, the A excess of DE above the given magnitude DI G CE DG have a given ratio to C; and because F FB, GE have each of them a given ratio to Dat. C, they have a givena ratio to one another. But to FB, GE the given magnitudes AF, 18 Dat. DG are added; 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 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 a HA is given: And 2 Dat. because as GD to AB, so is CG to HA, and so isb CD to b 12. 5. 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; H and CK is given, wherefore LEa is given: And because as KD to EF, so is CK to LE, and sob 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 HA, LE being taken, B 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 otherd. Another Demonstration. L K E9 Dat. F D 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 G D d 19 Dat. Because the excess of C above a given magnitude has a given ratio to AB: thereforea AB together with a given a 14 Dat. magnitude has a given ratio to C: Let this given magnitude be AF, wherefore FB has a given ratio to C: Also because the ex- A cess of C above a given magnitude has a given ratio to DE; therefore 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 the ratio of FB to GE is given: And b 9 Dat. 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. B C E © 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 GH to CF; therefore the ratio of GH to CF is given; and CF is given, ⚫ 2 Dat. wherefore a GH is given; and AG is gi- G ven, wherefore the whole AH is given: And because as GB to CD, so is GH to 19. 5. CF, and so isb the remainder HB to the remainder FD; the ratio of HB to FD is given And the ratio of FD to E is given, • 9 Dat. wherefore the ratio of HB to E is given : H F Bl DE And AH is given; therefore HB the excess of AB above a given magnitude AH has a given ratio to E. " Otherwise, "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 of AB above a F "given magnitude has a given ratio to D. "Because EB has a given ratio to C, " and the excess of C above a given mag"nitude has a given ratio to D; there* 24 Dat. " fored the excess of EB above a given B "magnitude has a given ratio to D: Let F D "this given magnitude be EF; therefore FB the excess " of EB above EF has a given ratio to D: and AF is "given, because AE, EF, are given: Therefore FB the "excess of AB above a given magnitude AF has a given ❝ ratio to D." IF two lines given in position cut one another, the see N. point or points in which they cut one another are given. Let two lines AB, CD, given in position, cut one another in the point E; the point E is given. Because the lines AB, CD, are given in position; they have always the same situation, and therefore the point, or points, in A which they cut one another have always the same situation: And because the lines AB, CD, can be found", the point, or points, in which they cut one another, are likewise found; and therefore are given in positiona. C. A a 4 Def. -B D E -B C D PROP. XXIX. Ir the extremities of a straight line be given in position; the straight line is given in position and magnitude. 26. Because the extremities of the straight line are given, they can be found a: Let these be the points A, B, be- a 4 Def. tween which a straight line AB can be drawn; this has an invariable position, because be A -B b 1 Postu tween two given points there can be drawn but one straight line: And when the straight line AB is drawn, its magnitude is at the same time exhibited, or given: Therefore the straight line AB is given in position and magnitude. late. |